Document 6689

l
•
/\ Study of I-Iougaard Distributions, Hougaard
Processes and their Applications
by
St('phallil' i\1.C. Fook Chollg
•
A t hpt.b
1>11
hrnitted to the
Faclllty of Gf,lIlllal,c Studics and Hescarch
ill parti.1l flllfilllll(·111. of the r<'<iuireTllcllts for thc dcgrec of
M.lstPr of Science
DqlartllH'lIt of MatlH'lIIatics and Statistics
~lrGill
Universit.y
~I()ntr{-al,
Qlléb('c
Oct olH'f 1!)92
•
©
S
~I.c.
l'ooh Chong 1!Hl2
•
Abstract
JdH'~
'1111.1111' l', d,""
1h"11 "ppll(
III
plflYldl'
dl,,,,'llIp
dlllllYI·"tlg,lIioll of lIoup;aard dif>triblltioru>, lIougaard prowsscs and
.!lIOIl'>.
'IIJ('
fllrl.lll'r
11Ii'>igh1.
~,I)I'I(' III'W
ib tu
.JIIII
1111.0
ab~(~lIIblp
'111'11
fI\V1I
IlllililllWII,
'111'( I.Ji
•
•
Il
(.1'-.(",
IIIWb!
~yJ\thCbizc
kllown fCbults about the subject,
lb t.hporptic,ll foundations, to extl'nd cAisting methodb and
1lI!'1 hodb, 10 disnlbs alld illustratr applications, and finally to motivatc
lIbl~
of lIoup;aard dh,trihlltions and Ilougaald processcs
i\1t.h{)ll~11
thp family of lIougaard distributions is rclativcly
111111'1 :-,1.ill:-.1 Il 1.Jllb to 1II,lkl' grp,ttN
III
,Incl
Ig,lIlollb.
11I111Idl':-' th\' illVPlbP
.lJId t !JI'''P
illP
weil
CallsbJaIl,
kIlOWIl •
p;arnrna a.nd positive stable distributions as
RéslllUé
•
Il
,\[11 licéllion~. Son hut
des th00flt'S de
ha~t',
1',,\
dl
.1,' \'('unir H'~ rl'"ultats Cllnl\U~ "lIr h' slI\I'l. dt'
dlllillPI 1111
.\P['I\,II
1111'11 qllP 1.1
1.1111111.,
d{,
exist.antl'b, ct finalf'I'
dic,tl ibutiolls ct
d('s distributiull~
tard
(\,IIIS 1(,llr~ proprl'~ IP[ hl'II
pas {'lIfOr(' 1>1<'11 1011111\1' pal
cas particuliers tels CI li l' ;('l' ri
largcn~ént
ét.ldiés.
•
•
II
III'"
11'''
dll'lI!lI·I\I:-'. «(·II.IIW,
Acknowledgelnents
1
d"
.1111
l'"
l,l' ri
!llrI,
111.111.1111' ·.II!,:",·.II')II',
\,dll,'!"'·
i
l'
III' ',1'.1('
\1)
,/)('1 l''II!
dlld ill'>l).',lrl" !!,lItfllJulpc! Lo makI' t.he Ivrit.ing of tlte present thcsis a
dllli III'> )'/,IJI'r()lJ~ (orll rrblltlon of
111<111'/
1li"",,", al"o )!,o 10
1I/l1
1(II JtI.lllnll dl'
Illy
advi(p 1H'IJ)rd glc,ttly to iIlljlro\'c bot.1t
1"<1,,1, 1 dlll ).',rdldlll
('II!'I(
fdllllly, (,i'lf)('( ially lIly lllo!!rpr. Th('ir moral support
fol' tlll' fin:tncial ~llpport
hf'llIi'l ('ll'.\I(J(':t l,\,
I!P(hNChp .
•
•
Profp<'f,ur (; A \VhitlTlorl' for /tif> guidance.
<lJtrl (onlp,,1 of!.l1(' 1.I1f'i'l1'>
Loi'.! bll!
Id
P,
~IIP/'IVI'>()I,
Il) 111\
iii
pw"ided by the Fond3
pour
•
Contents
•
1
Introd uction
2
The Hougaard Distribution
3
2.\
Dp[ ivatioll and Pro!>Prti('f>
2.2
Graphiea 1 IIlllhtratioli of 1,11(' lIoul-!,a.1f(1 J)pll'>lly
l'.'
2.t!
Simlll.lIloll . . . . .
~)' )
Inference and Applications for the I10ugaard Distribution
~{.1
Parametcr Estimation hy thl' Mpthod of
:L2
p(lrameter E'itirnation by the' Mpthorl of M.lxirlllJlli Likpldlo(HI
:l 2.1
:l.:~
4
•
MOIlII'IIt.S
:\0
t>.I axilllll III QU:lf>i-lihl'lihood Eht.illl,tt.ioli
Appliratiollf> (Jf t hl' J1ol\gdald
J)1~tliIJIIlioll
.
The Hougaard Process and its Applications
4.1
The Hougaard Procpss . . . . . . . . . . . . .
,1.2
Stochai>tic ProceshPS Direc!.pd hy a
4.3
Appli( atiollh of th!' )[I)ugaard Pro( l'~h
4..1
Case Study . . . . .
J1oll~<laI'd
iv
Il
l'ro( P":-'
•
" r
1 )
•
List of Figures
') 1
1I(j'I~',,'''1
Il
d(,II~lt Il
fil III LlOI, witl, n
OJiO and fJ
OAS,"
0.20, ,md its
17
'.~'!
1I()1I1',,,.lld
Ikll~ity
fllllcl.loll wltll
ft
= 0.'10,1>
= O.GOandO
= 0.:30, and
ils
18
•
:~:\
lIoll)!;.I.lld
d('JI~lt.v
fil Il c ti()l1~ with
'!',
IlolI)!,oI.lId d(,JI~it.v fllllr1loll~
11
SlIllld,i!I'd hollllpj{, p,1f h
or .\,
\Vit.h
0
= 0..10 and
Il = 1.00 for thrcc valucs of X2. 19
ft
= 0.20 and
À2
= 0.80 for threc valucs
of Jl. 20
Il(O S, 1,1) PiOC('f,f,.
48
68
1'\
lilllfllllli Ploh.dlllit.y Plot. wh(,11 the outlier is presPIlt (Mcthod of Moments)
69
11
lfnifollil l'Ioh.dlility Plot. wll<'l1 the outlier is deletcd (Mcthod of .Momcnts)
69
,rI
n.IIIc/ClIIl Sc .1111'1 Plo! wh('1l t.h!' ollt.lier is deletcd (Mct.hod of Moments)
70
1 li
l'II i1'01111 l' 1 oh" bdi t.v Plot. wl\(,11 t.he 011 t1ier is present (M aximuTll Quasi-likelihood
71
1 ï'
P lIilulllI P ('o/Jahilt t) Plot WltPlI thp outlicr is dcletcd (Maximum Quasi-likclihood
:\ Il'1 hotl)
LS
•
. . . ,
71
ILIlIc!Olll S(.11 tPI Plot whplI 1h(, ouI lier il' dC'!etC'd (Ma.ximum Quasi-likclihood
72
\'
•
LD
...\ li t Ol o ITt'! a t iOIl
FUll{ t iOIl
1.10 Parti,)1 Alltllcnlldatioll FlIllctloll flll tilt'
•
•
:.1
for t hl' Sl~ '~
VI
~l;\
71
•
(~Jlaptel'
1
Introduction
III"
III
plllpll',"
Ihl'> 1 hl"I" 1'>
111"11 .lpplll.tlIIIII"
•
'1
IIIIII!!,.I.lId di"l"hlllillll~, lIollh.lald
lu ,llId,
Il'' \\1111- .11111' dl d"IlIIlIl,ll," III/.!, 1Ir ,II
1'1''''',.1'', Il.1\1' 1 1111'111"1 dhl .. pl""IIII.l1 1111 dpplll,111111i
j'dl.IIl"''''1 1.111111\ III IIIOh.rbrllt\ .11'11,11\
1'1I11Itlllll ....
.1 1'0 t 11Il 11111111 1.1 1111 il Il,1', t III' 111\ • '1 ..... ' ( ;" Il:-' '.1 d li, 1111'
,l',
,p"(I,iI ("1
hollr tir.
PICH'(,":-'I'') .11111
dbl"hlltIOll~ oIlid Ilrl'
lIollt!,ddld (11)~(j)11I11()dll!!'d d tltlf'I'·
'1'11(' l"lndv i~ d('II\('d IIOIll tlll'llI)!">lt", ..
!!.. 1l1l1ll.1 .r Il cl t Ir l' pO'. i t 1VI' ~ t a h II' Iii., 1l "Ill t IC III"
1 l'l' .lIld \\ 1,,11111111' (I!l!!:!) ddlllpd a llew 1'10«':-,,_ ('.IIII'd IIi!'
1111111) I.lhl"
1I01lg.l.lIli plll(l"~ "" .... (.<\ 011 Iht' 1/0llg.ldlll dl~llihlltl()II,
:! "'h 0111 11111)(11 1.1111 11''.1111, lOI tlt.· l'.lIlIil,\' 01 flollg Idlcl dl"lllhlllillll"
('''''pll'I
.1"111.111011 III
,11,111'.1
1 Ir .. 1I011~!.,".Ild dl'.lllhllll!lll
l'III' 111.1111 hjl4'( l,II
[',.1111111,\ .!1I1i
olllr.' Ilollg.l.lld 1'.lIl1il,\', Il'\111('1,\',
b
lit'
d<'IIHIII·
tll(' illl'PI:-'(' C.III:-."idll.
III<'
tl,,' l'"hltl\l' I.IId,- dl,tl\lllltl()II~, .111' pll'~t'IIII't!. :'\('\1. "OIlH' I!a:--Il propPI!"" 01
111t· 1.\1111" .IH· ,t.lll'd
•
I.I""C,
110111 tltp p{)~ili\f' ~tab'" di~lrihllti()11
ï
FI)IIII,t.\I\!\·, lOI IÏ\!'d Il, Il il'> ,,!r()\\1\ tirai thp
\"IH11i\'lIllori dl).jll'hlOIi IlI\HH
lIo\lg.\oIld
ralllrlv 1" ,III
\, .III 1I1I1I\(·dl.tll' l'(1Il~!'qll('I\(t' ()ftlri!"> Lllt..\ ).(\(ldl\··pullIl
•
,1PpIO\illlatioll ln Iltl' ffnll~,l.lId dl'Ibll\ 1'> (1111.III\('d
(('1'
l'hl" ,IJll'll'\III',III"1l l', Ii',t'd III th.!!,
:310 implt'llIl'llI ,1 Ill,l\i III 11 III qllol"i Itkt'ltllll\Hl 11It'11Hld 01 P.lI.IIIII'tt'l .'-.tllll.iI"'1l
propertips of Ihl' llOIli!.ol,lItl lli"lrrl>lItloll f,lIJ1tl~ likt' Il,, Illlllll,"l.,ltl\ ,'lId 1111'
it" p.dJ in i hl' IlPig,hholll ho()t! lIf /('ro .111' ,d-.() ('\.II11ll1I'd
l'lit' -.101'1'
.!
Dtlil'I
l",h.l\ltl\JI
,,{
III 1 1 III' Itll'.III·1 IIld.·!
r'lllltly lIIi!!,ht IlIcllldp d 1111I1Ihl'l lIf lil .... 1 p.l~~.tl-!.l' 111111' dhl ,,1>,,11011'0 Illtll'l'd, Il 1'> \\,,11 kll"'\ Il
Ihat thl' IIlV('lbl' C;a\l~~i"ll di"lllhlllilln which i" .1'>Pl'II.II,.I·'(' or Ih" 1It"!t'.,.\.lId dl·.lllhlllloli
b
thl'nl~t passagl' t.illlf'of.l \\,i,'II('1 pl'()«'~~.
étI'f!
•
SPI't'r.1I t1111~lldtlllll,>(l1 th,> 1111111'..\.\1<1 pd l '.
pr('wntpd to/!,ivl'.\ visllal picllllP oflhf' dhtllhutlC:n f.lIlllh. 'l'III' 11''IIl'.ol,lld Il dl
,tIld '1'.lqqU (HW1). l'hl' illtp).!,lal i"
li.l"
al~() ddficlllt ln tl ... d wlth (OlllpllLI! "HI.dl\
dlepoint approximatioJl ln thp lIoll~aafd il d.f
,\lI~ "" d ~ltI, ... titllt"
th"~ "dlllll., ·,'·1, ....
f."
wlridl compale tlll' eXMl p.d.l. (llJlll\lu\('d 111l1l1l'II(.lil,V) willt , III' "1'1'11I>.IIII.dll'"
': iii' "1'
proximatioll is g('II<'rally vpry 1!;()od. 'l'hl' .l[lproxIIII.ltlllll I~ "~I·d tu l'II': 01111'1 ~,I" pll'o Whll "
confi l'Ill th at tlll' )[ ou ga.lI'<l cl i 1> , ri!J 11 t in Il
highly skcwcd 1.0
('0111 Pri ~('il .1
wid (' l'I,,~~ () r d 1,,1 ri Il 111 IOJl'o,
Ilparly ~yllllll('t.rir di~lribllt.i()II~. 1·'10111 ~(Jlll!' Ilr t.liI'
that thl' Hougaal'd p.dJ. b fiat ill tltp IIl'il-\h !Jo Il 1h(Jod
lIougaal'd family, I.P., Iltl'
are
s(udied in
'l'Il!'
111""1
fl1-'.llIf~ 110111
pit', Il 1.... 'pp.II"1I1
',pl'! I.d
1
,J',!", "III",
i"v('I~(' C;.t!J"'~lall, thp !!,itl/llll<l ,llld ',III' p!J~IIIVI' ',1 "l,II' dl',1 JtI,IIIII1I1',
mOle dclailllext. 'l'hl! chaptl'r PJld~ witll
mg randarn variables from
•
or I.I'I!'
/-',f "
fol
t1IP
lIolJg,aard dl~t rr !Jill iOIl
adaptation of stand art! Illet bUlb.
cl hlllIlIl:III!JII
Ir'! 1IIIHIllI' fur ~',I'JI"I"t
TitI' '.tllllll;t!lr,p 1/11'1111111 1J1",,,lv""
of
fi
•
Ild, "
III' tll •• rI
Il''' ,1I1'!
"pplll.lI I(lll~ lor lIoIJi;.I1tn!
{JI polI.lllll'I.'[ P\I!lfl.I1!OIl ;!fl'
dbtributioll~
(oli~idpr('d,
aIl.' dealt with in chapter 3. Two
narnely, the method of moments and
1111' III.'IIIIHI of IllolXIIllllll1 IIh.,idiOOd. 'l'hl' rn,lxilllUIrl likclillood method is difficult to carry
1)f'ldll~"
qlll
(jf ,,, .. cOlnplc'xily of Ihl' I/ougaard dpm,ilY fllnction, a point which was rnen-
Il,1/11''' "oIIII"f :\ Il.11 ""llp .lIlprrlillp IIIdhod ofp,LfamctC[ c::;timation which is based on the
.lIldl,· p"llil "PP'()\llll.tlloll 10 Ihl' dplI»ity fUfiltion is devclopcd. The method is a slight
Ill.)(ldil.tIIOIi of tlJ(' (oIlVI'nlloll.1I Illethod of maximum Iikclihood
l
".iI
t hl'
1111111111/1
•
"II!!,!' 110111 1'1 flllOlIll1
i~
kllowlI
,t~
maximum quasi-likelihood cstimat.ion. Next,
()r 1II(' lIouga.\rc! di»tributiollb arc desrribed
c,
ft)
in the sense
n'pla('('d hy il s saddkpoint approximation in the likelihood
'l'hl' Hoc.lIlllllg 1II1'Ihod
pl dl 111,11 "pplll ,11101l~
l
i~
]] c)/11-'..1.11 cl JI Il.f
e~timation
The fi('lds of application
IIll'dlC/IlI'. t\IŒt of the applications to 1)(' jlfes('nted are applica-
11111/',01 Il)(' 111\1'1,>1' (;au»'>lcili di:-,Irihution (a hIH'cial
Cab('
of the Hougaard distribution) .
Il,,· ... lppII(.ttlllll~ f'''plelll IIJ(' f.Ht that tlH' inVPThC Gallsbian distribution is a first passage
t 1111" dl'" 1l/tul iOIl SllI('C' t11l' IllVPf»1' G,lUf,Slall distribution is a special case of the 1I0ugaard
1.111111\,
~(Illl('
(If Ih.'"" .q)pIH.tflolh cali Ill' gl'lINalized to t.11(' 1I0llgaard family.
('h''III('1 1 <11',,1» wllh Ihp lloll)!"IMd proresh and ils characleristics. The main special
("".'~
01 1III'
1I1(t!IOIi
PIOC('»~.
1h~, ill\"(,l"»(,
.1Il' pfl'~(,II\(·d
C;all~si,\n
proress, the gamma proce'iS and positive stable
'l'hl' jlllllp hphaviour of the lIollgaard process is described using a
(,("\ \ 1l'p 11'~I'llt a t iOIl. A ~.\1I1 pIf' p.lth of a 1I0ugaard process is simulated Ilsing the si Illulation
•li)!,1l 1 1t li III
01 « h.l()1 l'r 1 10 illtl»t ra 1(' 1hl' fMI t hat th(' process doC's exhibit a random num ber of
111111(1'" III I,tllt/Oltl
"I/l'~
Il''\' ul lIolli!.,I.II't/
P(()('l'~"l'~
ill ally )!,i\'l'II tllll<' illtl'f\"al. The chaptl'f thC'1l procccds to consider the
in
~lIhordinat('(1
proCl'loSClo. Subordination of a process involves
\ (111\\'1 t ill).!; ph\'~i( .• 1 1illll' in 1lit' Pl"O(,('SS to a ralldom operational time. The properties of the
•
Il Il ll).!"t ,\1 li pro(!'~~ th •• 1 ).!;i H' il pol l'lit ial to aet as an operational time or directing process
3
•
arc discussC't1. "'Ilh the lloUr."l.l!'t1 prou':,:, UM'li a:. ,l dit "tI in!!, proCt':.s, 1hl' lIIodl'ls 1h,ll
emcrge for SOIllC' standard par{'nt pron'Sf>l'h <ln' ilnaIYhl'd. For ('xampk. whl'II 1hl' jI.m' Il 1
proccss is Poisson tll(' outCOIllP b an intl'rPhtinl!: and Iracl,t1l\I' clllf>INillg POlhMl1I
The reslllting su bord inatl'd proc('ss is c,llIPd a
Poif>snll-II()II~.I.\f(1
the Hougaard procc"s a" such arc diSCIlSf>l'li ah
WI'\1.
pn)( l'hh
of the Hougaard
\HOCP"S,
it is slIggpf>l('d t h,lI, tll(',,1' h,III\(' .Ippli< al III Il '> 1ll.IY hl'
The case study involves the f1tting of wind pnprgy data tn
Goodness-of-fit diagnostics show that t1H' lI\od('1
•
18
rpé\f>llIl,lh\l'
,1
1111'1'.111111'
PI()('('~~I'~, ~Pl'I
to the Houga:trd proccss. Finally, a r(',tl applicat.ion of t IH' \Iollf.!,aard
•
A pplle.1I illllf> III
SII\("(' lhl'rl' l''\I:-.I:-..\ hiœ.lbh'
on applications of the in\'Prf,C' Gallsf>ian, p,all\lllil and pOf>il1VP "Llhll'
Plll(
pIOCI'~'>
l.tI I.I:-'I'S
,~1'1I1'1 alll,l'd
p"" is (Illlhidl'i l'I\.
\Iollf.!,.I.trd PIO('('hh luod.'l
•
Cllapter 2
The Hougaard Distribution
2.1
Derivation and Properties
III 1 hb ~,I>I 11011 Il
•
will 1)('
how the
:-.IIOWII
~taLle
HOllgd.!ld dbtlihlltioll f.lll1ily. Npxl, Home u&pful
distributions are used to generate the
propertie~
of the distribution family are
()('finit.ion 2.1.1 I,d .\1,.\2, ... br indrpcndcnt identically distributcd random variables.
l'II( tl/ ... I/'IbullOlI of .\
/8 ... lalJ/e
if fol' carh
thcre cxists constants (\' (0
Tl
<
Q
::;
2) and bn
.\1+.\2+···+ X n_b
nl/Ct
1111' Iht'
.-1/1111
(!t.~/1 dmtwlI as .\.
TILl> :-.1.\1>11' di:-.t tihllt iOIlH
h.\\ l' L.lpl.\\ P
011
li
/s
cal/cd the chamcleristic exponent.
tll(' positiw llu1I1bers have (\'(0,1] and apart from scale factors
Il,\11:-.1'01111
I,(.~):::: I,-'{(\xp(-,~.\)}
•
n
= cxp(-SO)
5
(s 2: 0).
(2.1)
•
This distribution is dl'Ilotl'd hy P(o,n,O). TI\{'
distributions. For (} < 1, using 1
=
-0
(',lS(' (\
=
1 ('oln':-'IHlIlds to tht' dt')!,\'III'I,ItI'
in tlll' r(':-.lIlt pron'd Ily I-'t'Il('r (1 !171 ), Il
;I~U, \\('
obtain the density fun,tion
\Veshallnow d<'lllOllstrate how J1ougaard(l!lI'(i)dPl"lvpd th(' dbttihlltioll f.lllltI" "(o,b,O)
from the htabk distribution!> 011 the !>oslt.i\,(' IIlI1IIbl'l"!>
Lpt.\ h.I\'1' tlH'
~l.lIlddld ~,t.lhh·
di&tribution P(O',O', 0) as givpn by (2.1), whpf(' n«O, 1]. Npxt, dpftlH' 1'(o,h,O) wl\('f(' h '·0
to be the distribution of (8/n)I/"X. The forrpspollding d('IIi->It.y fllllrtiol\ IS obLIÎIlf'd Il,,
.1
change of v,'Lriable techniqll(' and is giv{'11 hy
(:l :\)
•
for a < 1, where fOt is the function in (2.2). Th!' Lclplcl('(' t.r<llli->forlll of (h/n)l/of.\ ih /!,IVf'1I
by
1
00
exp(-.,>x)h(x)dx
= ('xp(-h.~rr/n)
(h " 0)
Thu&,
[00 exp(
-.'lX
vO
+ 0.'/' / n )h( x )d.r. =) (.~ ~ 0).
il follows therefore that for () 2: 0 and 0((0,1),
h(x)
= exp( -Ox + h()"/n)/-;,(f)
i& a prohability dCIIhity functioll (Jr.fiIlPc! on tlll' (H)!'>!t ivp I1lll1dH·rh. TIJI' (li:-,tribIJI iOll i. . dl'l1ot.l'd
•
by P(cx,IJ,O).
(j
•
')01/11'
(;1) '/
plopPl1 il", of 1)( (x, 0, ()) ..,bown hy Hougaard arc as follows:
h,. pd f for
n < l,Ii> 0,0 2 0 b
(j <"
J(.r,H,Ii,O).::
('xp(-().l
r
r(k(): + 1)
+ Ml " /0')-,1 ~
~ --1.;-,-(-bx7rX k=1
'/ h.. ahflvl'
JI'~III1. follow~
Cl
k.
(2.7)
/a) sm(ak7r).
•
din'c tly from (2.:2), (2.3) and (2.6).
( Il) 'l'Iii' La pla( (' 1r.lll~forllI
li>
(2.8)
(1)
J'èJl fixc'd
() Fil 1 flxl'd
ft
,lIld h, tht' falllily if, pxponcntial with natural observation
0, II.
(cl) FOI (J .. () ,llId
ft
Ih
.Ill
<
('XIHJlIPlitial dispersion mode!.
l, t.h(' 11th
1lI0lllPlit
(h/o)"I"I'( J - II/n)/I'( 1 - 1/). For ()
•
Tht'
"1 h (1iIIIIIIall1., k > 1,
= Ii( 1 -
(l') JI' \ Jllllowh J>(o,h,O) ,111\1 ('
0
exiFts ;r and only if v
> 0 or
ct
= 1, aIl
< a, in which case it equals
moments exist and E(X)
= 6(}o-l.
IS
\.1.
( 1) Th (' JI. d.r
x and parameter
fi) ( Cl , li, ()) i S
n)(1- et) .. . (1.: - 1 - a)(}o-k.
(2.9)
> 0, t1u' disttibution ofcX is P(a, cOb, O/c).
1111 i III 0
cl al.
(g) 'J'hl' haddll'Jloillt 'IJlJlloxilllat.ion fol' th<, density of P(a,6,O) is
\\ hl'Il' Il :.. 6g~::))· j"(J')
COI\VNgf'S
to the exact density under the limit
<1/(0.1). 'l'hl' ,lppro"illlalioll is ('Xold only for
ft
6()O --> 00,
a.,tixed,
= 1/2.
(h) TI\l' dislllhutioll of 1)(n,b,B) is infinitt'ly divisible. That is, for any n, P(a,6,B) can be
•
l\'I)(('M'II!t,d
as Ih(' di~lrihlllioll oftll('
SUlll
of n indcpcndent P(a,o/n,(}) random variables .
7
•
N'ext, some p('pliminaries ne('(kd in chaptN :l will tH' (kriwd.
The second and thinl moments abolit tht'
IIH'.1Il
hcl\'t' tht'
sponding cUIllulants [Kendall and SLlart(1977), p, n].
~,11l1l' \.lllll'~ .1:'
FOI" :-.
O. t hl'
tht'ir
1'01'\'1'
M'(,(llid .\1111 1hill!
moments about the mean are therefore /?,iven by ('2.9) as
l'2.II)
and
\:1 = 6(1 - n)(::! - n)Ij"-,I,
(:!,11)
The I10ugaard dis tri bu tian family P( Ct, D, 0) is ri!!;ht hkl'w('d. Posit i\'l' ht..lhll'
(the case where () = 0) arc knowlI
1.0
he right Hk('w{'d. Frolll (2.12), il.
IH'
(',UI
<11:,1..1 hll 11011:'
:'('('11
1Il.11 \
1 •
Il
whenever () > 0
•
lIougaard does not elaboratp
011
prop<'rty (') hu t I)('ca u:.('
1.111'
model provides an alternative parallletp.izal.ioll I.h .. t will Il!' u:,('d
('x p01l1'1\1 I.ti dl:' 1)f'1 hllll\
1,11.(,1 W('
:-.h.dll'xpdnd on
propcrty (c).
Jprgenscn (1988) provided the following dpfillitioll for an
PXPOII('lIti,t!
dislH'fSIOII IIllldPl.
A randoIn variable X follows an cxponcllLial dis[lPfsion rnod('1 if il.:-. p.d.r. is J.'.iv('11 hy
for suitahlc functions a and
K"
wherc 11 > 0 and "1 varÏPh il1
,UI 1111.1'1
v,lI
(J"
t.Il1~
'l',d
hill'
When X has the density function (2.1:l), iL i& d{'notpd by X "" /0)/)(/",(1'1.) Wlll'IP Il, - f-'(r)
is the expectation of X and
(J'?
= l/r,
is a disJH'rsioll parallll'I.N, TIIf' lIIom!',,!. p,I'I\I'I,.tillJ.!;
function of an ED(p"0'1.) random variabll! b hltOW/l hy .Jl1rgl'lIh('1I t,o IJI'
•
It was also shawn by .J0rgcnscn that tlte paramcters ri and "( ill ttl(' I~ O(JI, a:l) an' ortho~(jfl,t1.
•
Il 1'> ll'dddy Vl'lifll'd 1li ,JI 1 \if'
1I(J1I~1"I,lld':-, pdldrlll't!'f<'
followillg idflutitich chtablish a corrcspondence betwcen
and thOhf' for thp exponcntial dispersion mode! in (2.1J).
(2.1.5)
(2.16)
III .. ddrIÎoll,
(2.17)
Il(.r,
1J}
1 ~),(kn+l)(
(k)
=- - ~
k'
- (/)10'
'Tl 0' - X- Ct/)k'
SIIl 0' '7r •
7rJ:J.=1
lI:-.rllg
•
/.( ,l
Ihl'
III
(()ll('!'IHllld{'II(('
of
par,ulletPr~
(2.18)
•
givcll in (2.15) and (2.16), and the expression for
n, ~lIb~f.it.lIt.iOll illto th{' llIoJ\H'nt generating function in (2.1-1) gives
(:! 1
(2.19)
=
{'xp{ _(6/n)1/(I-")( -Q'Y -
=
pxp{ -(6/0-)[(0 -
QS/TJY" + bO" /a}
st - OO]}
(2.20)
\\' Il 1dl .\('( ()rd IIII!. t () propl'fty (h) is the moment gencrating fllTIction of a random variable,
\ '" l'i Il. b. 0).
1t I~
\\ 01
[1(n.I\O) .I~
pll\\('1
•
III Ihl'
111'1 l'. Cl
is Ih:pd.
t h 11\('11 t iOllill!!, t hal. .J{1fgl'I1~Cll
0111'
of Ihl'
11\('.111.
c1.IS~{,S
rOllsidercd the Ilougaard distribution family
of llXpOIH'lItial dispcrsion modeI whcre the variance is a
Ik !'lhO\\'{'d that [or the l',lInily, P(u,6,O), the variance is of the [orm
(2.21)
•
where
Jl
is the 11I<.'an orthe distribution eUHI p = (2-0)/(1-0). III p.IIII<III.Ir. il
=L
that p = 3 ror the in\"('rsp Galls~i.ln 1.1l1lily (n == l/~), .lIId p
(n: --. 0). From !HO!Wrty (2.D) dpri\'('d by lIoug.l.lnl, \\(' cali
proportion al to
Another key
Jl(2-L.)/(I-ct),
pro pert y
~('('
\\'.I~ ~h,)\\'11
for Ihl' ~.II1lIl\.1 r.llmly
Ih.1I th(' valÎ.ll\(l' \!
i~
a f(':-'ult. which agll't's with .JMgl'II~('II'S Il'stllt
dl'rived by .Jorg<'llhell whi('h follows ah
a (OIl~('qll"Il(,('
of 1hl'
r.1I 1
that a lIougaard distribution is an l'xpoIH'nti.ll displ'l'sion lIlolkl I,;J)U(,(T~) i~ .t~ foll()\\'~
For small values of the dispersioll par.t!l\('IPf
(Tl, .111 .H·(
IIr.lIl' fl.Hldh'puilll .lpPIU'{IIII.tlIOIl
1:-,
given by
where v(·) is the varialle<.' function and
D(X,Jl)
b th" d('victn('(' fllliclioll
JJ(.r,/d I~ dl'fil\('d
as
•
VCr,/l)
= 2[t(J',M -
t(:r,II)]
wltere t(X,ll) = "(X - ,\:(,), Îl is thp lIl,lxiHllI1II JiI..plihood 1'!1lillla!.or of Il -- '.'(1) .1\111 P{Jl) 1\:" ( "( ).
In the following steps a. saddlepoillt approximation in t('ftll!1
he derivcd rrom (2.22). Restating (2.17)
h(,)
w('
or IMrrlIllP!.p!!1 "( ami
11 will
have
= -(-(q)"/o.
'l'hm"
('2 '20)
Conscq \lcn tly,
•
Next,
('l '1.7)
10
•
Il. =
f(x,{l)
= -Y(X -
t. '( Î, ) =
(2.28)
x.
2,.-1
I(
x/n.) = (1 - a)a----r=;;- /xo 1-0
• (1 - 0')0'
I)(:r,fl)=~[
)
(2.29)
br-I
)-0
_"
-Îx+Kb)].
(2.30)
:rI-a
N('\ 1"
(2.31 )
•
n(x)
= (1
-
0')0'-0
l'III,dly, lJ~ill/!, (2 :W), (2.:!2) ,lIId the fart t.hat. (12
2-u
I( 1-0' )Xl-o.
= 1/17 in
(2.32)
(2.22) the following is obtained
(2.33)
\\'II('J('
lf
= (t/( 1 -
{J ::-::
(2 - 0')/(1 - 0'). Substituting (2.15), (2.16) and (2.17) in
(h,ll. t h('
lIou/-',aard p.d f. has the infinite series representation (2.7)
n) ,Illd
Sil LI!, il, I~ kllown
.Intl t hl' (()II('~p()llding ~,\(Idl('p()illt n\prc~('ntation
Il d.1.
(,III
1'1I11rtll>ll
Il(' ('xpr('s~('d in
,\II
(2.10). It can he shown that the Hougaard
il\tq~ral fonn as weIl.
For hrevity, the Hougaard density
!(J';n,I\IJ) will bl' (lPlIotpd by !(x). Inverting the Laplace transform (2.8), the
f"lh l\\ III/!, Ih obt.lIl1l'd
•
(2.34)
1l
•
whcrc a is ehost'll sueh 1hat a 2: -H. After cMrying ouI t h(' inll'rlll('di,lk
~t('p~. 1 hl' dl'Il~II.\
fUllction f(:r) can be srcIl to have tll(' follo"villg illh'gr<ll fl'Il['t'sl'lItallOll
f(:r) = -1 exp( -O,r
7r
whcrc C
2.2
= 8:1' -
b~ 1~
~
+ -)
l'X p[ - ( (}
II
fi
),0;(\
ros( Il Ir /~) 1('lJ~U
')
t!,~
(~)s" sin(mr/2),
l~
Graphical Illustration of the I-Iougaard Dellsit.y
II~
Now that wc have studied SO!llC' of thl' prop<'rl.ips of th(' lIollga,lld dh-tllhllllol\. Id
examine the shape of the probability (IPllsity fUllrtioll, f(J';o,b,(J).
The fact that f(x; a, 8, 0) is dt'fitH'd as ail i"fi"itl' SIIIII in
only be calculatcd lluI11cric,t1ly
•
\1
(~.7) 1111',111:-' 1 h.11
f>illlp"~ .t:-.
il. ,1PIH'<l.fl>. COtllpll(,lllol\l>
computation of f(.r;O',b,O) I1ear thp origin will''' 1.11(, l",r.l.IIll'l,('r
representation of f(x; a, 8, 0) in (2.7), It
Fortran language
wa~
x-nI.
CéllI
hl' S!'('II I.lIat
ft
il-.
1.1 l'pp.
for larJ.',(~ k b('colI\(' lItllll,llI.l)!;('ahll'. 1\
(.111
111111)('1
il .11
,111:-'1'
with 1.111'
Hdl'nilll!: t.o
is IclfJ.',f' Wltl'II
J'-"
b, 0)
0,
Il to a :-.ppcilipd dpgrl'(' of lU'(' Il 1 My. II IIWPVI'I, t III'
computation of f(3.:;n,8,O) is not as
x is smal1. IIence, tlle tel'ms
f(,/,
ft
1:-'
I.IIJ.',I' .LIId
plOJ.',loIm Wllt.t(·11
designed to nlllTleIical1y cornput(' thl' lIou)!;a;l/d dPIIl>ity fllll(
a limiteù range of parametcr valups and this prograrn
1111'
wa~ IIM'd 1,0101111>111.1'
I.III!I
f(I) fil l'
III
for
:onlliP
of the graphs that appear later.
1'0 ~ct a bet ter
U Il dl'rstand i fig
of t he Il at Il rc of t 1)(' Il ifIi clIl tipf>
of the dClIsity fUllctioll, the illleg/"t! /'('Pl('f>('ntation
The aitn of the discuf>f>ion to follow is
1,0
(~.:Jr,)
point Ollt thl'
III VIII v(·d
for 011'
(Ol1I1P(
"('11..,11,
i Il 1, hl' ( 11/11 JI Il ta t.ioll
Y will hl'
(0/1:01111'1('(1.
tioll IJI'twl'PII int.I'J.!,I.d
(~
:1:)) alld
an analogous integral that ernerges in th!' work of Hardin, SaTrlorodllibky il/III T'HI'III( I!J!)}).
These authors have dpvploped an algorithrn for
•
(ornplltiTl~
tlJ(' rl'J!,/I'f>:Olon of
0/11'
ht,lbl,.
random variable lIpon another. Af> a by-procI1Ht, thpir progralfl (O/Ilplltl':-' thl' pro),'lbiIJl,y
IL
•
dl'II',JI Y
"f
Iwo
fil II< IIIJI! of a ~ t.tbl!' r<lllrlom va riabIP, The rpgression corn putation in volves a quotient
1I1!l'gr.d" Ih.JI
fi,,, Irtfp)!,r.dc,
lllll~t
!JI' {>vaillatpd 1I1JrflPrically. They found that the computation of
l'lIl.ld.., ('(JrJlpll<;t1loll~,
i
,;rJ
One> of thf! integrals is of the same form as the integral
"
exp[-(-)sÜ cos (0'7l'/2)]COfi(C) ds
n
Il
wlil! li .IPIH>.tr:-.
III
(2.36)
(2.:1!». From tlH'ir Irtve:-.tigation and a study of (2.35), it is found lhat the
Ifll l'g/'' /ICI i Il (1.:l;)) dPI .ly:-. ('XpOllPlI tially while o~~ilIati ng, in many cases increasingly rapidly,
.dJOIlI. /'(>/0. Fil/I.hl'IIIIO/i', t.ltPy di:-.(ov('rpd tltat use of standard int<'gration techniques like
SlIlip:-'OIl\
l'III!'
J!,:tVP.L vPI'y poor approximation. The authors dcveloped lIurnerical methods
Il.I:-.(·d OH v.l/iahll' :-.1,<'p:-.il:p and (lIt.ofr poillts t.hat work<'d sllcressflllly and these can be
.Ippllf'd \\'11"
11i11IO/'
lIIodificat.ions tn l'valUaling (2.35).
Np;..t., it will br' :-.I\owll f hal .dl
•
flll).!,1 fi
0111 d f
h.iI thl' 1(' i:-. Ii 1111'
dl'rivativ('~
d<'"~i1,y
of a I10ugaard d<'lIsity fllnction are zero at the
iH the Ileigh bOIl rhood of th<.' origm. This propcrty of
tlll> IlolJJ!,.I:lld d('II:-.ity :-'lI1!).!,<':-.ts t.ha1. the dIstribution has similar propcrties to those ofafirst
p.I:-~agl' 1111\1'
di:-t.ribllt.ioll of il diffusion prucess. For any difrusion process, a certain lapse of
1illl!' is f('«(llil('d \w[or<' it hit.s a sp<,cified barrier. Bence, in order for any density function
1111' di:-t.rihut.ioll of t.lt(' first. pas:-.age time of snch a proccss, the dcnsity fllnction must
101)('
hol \'f'
Il
\'\'I,V
li t.t 1<' dPII:-.i 1y
(,()J\(t'1I1.r
will h\' M'Pli t h.lt. 1hl'
llollf.!,.I.lld
lkll~ily rail
a I.<'d in t II<' [I<'igh bourhood of zero. In the forthcoming steps
lIoll~.\(lf(1
dist.ribut.ion satisfies this reqllircment. From (2.35), the
hl' w/'il('11 as
1(:1')
MJO
= exp( -Ox + -)g(x)
(2.37)
Q
wh!'/'('
•
1
g(J') = 11'
lL'-) exp[-(_)sacos(mr/2)]cos(C)
b
ds,
0
Q
13
(2.38)
•
with
C
= 8.1" -
/1
(-),.;"
n
~1II(\'Ir/1).
The funclioll g(J:) I~ ,1 positivp ~t.1hll' p.d.f..
1
ordl'r
]11
prm'p th.ll, /('\0) -- 0 lm
10
= 0,1,2, ... , it is sulficit'Ilt Lo pwVP that f/('\O) == 0 for 1
=
n.I,:! ....
HpIH'.dt'dly
differentiating g(:r), w(' have
(:! :I!l)
wher('
Ifl
= I,:I,!), .. ,
if
== n, :l,-1, . ,
',,(C) = {
H>!.( C)
1
and
/(, = {
•
if i ::-: t1k or 'Ik
1
if z = t1k
-1
Lelting y
+ a for
HOl1\P
+ 1 or -II,; + 2
for
kc {O, l,:l, :1,
~()IIH'
h{O, l, :l,:I,
,J,
= (0/0)8°,
(2,10)
wh cre
l';
= -y sil1( mr /2).
Carrying out the int('gratioIl giVC'h,
9
.!.il l , ~ + l
,
(I l l .
(0)= -l\.(n/h) " -1(---)".(1'),
'Ir
f.l'
lX
where
j,
•
l
+
l
1 O'lr
= -(-)- = fi
2
'
(
z+
'Ir
1)2
Now, since
bin[-(l
+ I)~l = 0,
Itl
l
= I,:\,!}"
..
(~ ~ 1)
•
Ir
(Oé,[--(l+ I)~I=O,
JI
11I111Jw:-, thal, !t,( F) : () lor ail
l=0,2,-1, ... ,
l.
So,
'1(')(0)
= 0 for allz =
jI'I(O) == 0 for allt
III
ot hl'I Wllld:-" II,
IIIII(
tloll
IS
VI'I,V
(,III
n.IL
1)(,
:tlld that élt the origin th{' density fuuction takes on value O.
•
Il
1,0
(lppCM1>
~h()\v tlll~ «()IIc1I1~ivpjy.
tll
]('qlllll'd
JllodllCed
1111' lIollg.l.trd p.d.r.
III
= 0,1,2, ....
IlId('d th,tt ill tlU' intcrval [0, ç], E. srnall, the Hougaard density
('OII(
NI'\I, ~,1I1111' dl.lgl.llIl~ ,11('
1111.1111111
0,1,2, ...
illilstratc the faet that the saddlepoint approxto be quitc good, although further rescarclt is
Figure 2.1 shows the IIougaard density function for
0 ·Ir" h =: 0 (iO alld () == 0.20 and itH saddkpoint approximation, obtaincd using (2.10).
1\ 1101,1\1'1 (OIl1parbllll or thl' lIollga,ml dl'Ilsity function with its saddlcpoint approximation
I~ ~',IVI'II
III
ligll!'!' 1.1 fur thp
pl(l\illl.ilioll b <I"lt('
.lIId
(.1 Il ('.10ily
dt'I\'dty
IIIII(
tl~lllg
IH'
clo~p
(0111
~d
=
(UjO and () = O.:W. The ap-
for !JoLh s('Ls of paraIllPLPff,. Sin(e t}\(, saddk'point scems precise
pli t l'd for (Illy ad III issi bic spI, or pdfallletprs, ail su bscq \lent Hougaard
tioll ).!,raphs will !JI'
th!'
of paranl<'Lers n = 0..10,b
~.Iddh'poil\t.
r('JlI'('~ent.ed
by their saddlepoint approxima.tion.
approximatioll, a ('ouplp or graphs arc plotted in an attempt to
dlll"t 1.111' hn\\' t h(' shape' of t Il(' lIoll~aard p.<i.f. changes whcn two characterÎstics of the
1It1l1g.t.1I d di~lllbllt iOll
MI'
h<'pt !ixpd wlllic anotlH'r charactNistic is varied. The graph in
IÎ!!,IIIl' :?:~ ~11O\\'~ tlllt'(' lIoll~.\ard dpnsiLy rUII('tions with
•
dlill'I\'llt
(\ -
\"\II.\Ill'l'0
0'
== 0.4, the same mean Ji.
= 1, and
\2. Thp gr.tph in fi~ure 2..t shows three Hougaard dcnsity functions with
(}:.!, t hl' ~a 1I1t' \"Iri.\ 1\('('
\
'2
= O.H, ,Hld difrNent means Il.
15
•
A look at fig,urt's
ail the
~.l-
2.1
graph~ .HC IIlliIllOd.l!
('onfillll~ 1Il.IIIY (lftl\(' prllpf'llil'~ ill\l'~tl!!.,lll'd III
:-'1011'0\'('1, Ihl' plolll'd dl~tlihIlIHIII~ 1.llIgl'
t'IOIII
li'! 1 IlIdl,,'d,
Itlgltl., ~kf'\\"d
ta nearl:: sylllll1elric dbtrihlltioll~, r('II(,ctlll~ Ill(' f.HI IIt.11 tIlt' 1I011!~.I.lld dhtldllltlllil 1.1111111
cantains a '''l'ge
c!.lSS
of dl~tl'ibllt iom. And. frolll li!!,lIn' 2.:1, il I~ .Ipp.trl'III t 1t.1I 1lit, Hou).!,.ldl\l
dellsity fUl1ction is \'('l'y fiat IIpar tht' origill. This is iil .1).!,n'I'IIIl'1I1 Wli il 1111' 1hl'(1I1'1 Il.t! \\'01 k
donc carlin
•
•
-
•
--- ----------------------------------------------------------------,
-
(
1
1
\
\
1
1
\
\
\
\
1
\
1
J
\
1
1
1
1
1
J
•
-
-------1'-------------------r----------------~~----------------_.~
2
x
1 l"
1111' .}
1.
I!Pl::!d,lrd
l
•
4
3
'.ol(
t
dl'Ili>ily fllnct.ion \\ith
fUlIClioll
fi
== 0..1.5, éJ
0.60 and B
..... , Approximating fUIlclion
17
0.20, and its
..------------------------- ------------- ----
•
-
-----
o
Cl)
o
C\J
o
--------...
-!~
-
•
.. -4... . . . . . . . .
..... ..
.. ...........
o
o
-,------
o
-------
2
x
l'igure 2.2: HougaarJ ùcnsity fUllctioll \\Ith n
-- 0 10,0 _ (J.(iO
dllrl ()
saddlr:point approximation.
Exact fUllCtioll
•
. . .... AppJ'()\illl.iI ing
rlllilllOl1
-- ------
-----------------------------------,
•
.
/.
. /
'.J
\.
\
\
!.
\
.' ....
.".
\
\
1
.
\
\
.
1
\
,.
\
1
\
1
\
\
\.
•
,
)
- -o ()
---------r----------,r----------r-------------r-------------.-~
0.6
0.8
x
Il:',111 1 ' '2:! !lollg.lard
____ \ 2 ::- ()
•
04
02
(lPll<,ity fUllctlOllS \\oith () == OAO and Il = 1.00 for three values of X2.
I!)
...... \2 = 0.10
1 ~)
- . - . - X2 = 0.06
1a
/\
•
i \
i \
ii
i
,
"i
'\,
,1
\,
,1
\,
\
\
\
\
\
\
\
1....)
o
\
\
... \
•
o
ci
.- ..
L..-,.-_ _ _ _ _ _ _ _, - -_ _ _ _ _ _ _ _, -_ _ _ _ _ _ _ _ -
o
-
-
2
1
3
x
FigUlc 2A: lIoug,aard dCllsity fllll('tioIl~ \\ilh
_ I l = 1.00
•
-
0'
= 0.'2;) dlld
....... /1=2.00
\~;--
():-,O 1"11111""
-
-/1
·IO()
\,.111",
111' /
•
2.:~
'J J",
Sp(~cial
Cases of the Hougaard Distribution
(,(J1l1pri<;(·"
]]r)lW,d.ll d dl'J1',lf y fllll( t.lrHI
r.I'I", .JII'
tlJ{'
ill\r'I,>I' (;,lll;,;,idll,
,1,
gal/lllld. and
wide c1ails of distributions. Sorne of il.s special
~tahle
distributions; they are presented in this
, ,.( fIl J 11
f)"fi Il il ion 2 .3.1 A mwlmll 1J(JT'wblc X luI.'; (m in verse Gaussian dislribulwn, denoted by
IL'Ut, À), II il.\]/ rl J.
i.~ fjW('/t
Ily
(2.42)
• () fl/lt! À
Il
1/'//1 Il
'/'lit
.? ()
(·()""(.~JlO1tdl1tg
Laplace transfon/! of X zs
(2.43)
FIJI n
•
:
]
(:0111','01.111
/'2
illld
lixpd () > 0, t.hp Ilouga,ard distribution P( 0',6,0) corresponds to the inverse
di:-o! libll! iOIl !G(jl, /\) with
(2.44)
'l'hl' Il
fl).\
.-
I]ohl'd
LI'!
1/'21'.1;'('
fOI III.
\1:-, ('" :\IIIÎ Ill'
dt
th!' ollly
yl't ,\ lIut lier
•
,\ '-. ()
for which f(x;o,o,O) as defined in (2.7) corresponds
UIIU!O/ll
::'11 bdass of
/I.~/f!l flllH'!wl!
will,; > II Th(,
0'
= 1/2
is the only case for which the
the lIougaard famlly of distribu tions.
l'(l/'iablr X lUIs a gamma distribution, lienoted by G(>..,{3), if
18 y/l'CH
f( JO; ,\ ./l)
/l'/It Il
OHI'
"I:-on, as poiHtl'd out in §2.1,
l){'finition 2.a.2 :\
I/~ Il/rI/JUIllllfy
1;'
by
1
.\-1
-x
= l'( >")13
\ x cxp( If) for x > 0,
COITC8]10ndmg
(2.45)
Laplace iral1sform of X is
(2.46)
'21
•
It can be shawn that for (}
G(,\,,]) with ,\ = <1 and
-4
0 and fix('d {}
J = 1/0.
;\pplyin~
> 0,
P(o.~,O) ll('roll\('~ IIH' g.\llIlll.1 dl"llIhlltlllll
L'llôpital's mIt' 10 t Il\' 10goHil hl (If 1hl' LI pl.\( ('
transforlll of P( n, <1,0) yidds
0+.\
lim -(b/n)[(O +8)'"' - (J"]
-(Iln(-f}-)
1"\'-0
Cl
17)
(~
IX)
III [( 1 + .~ / 11)-l']
The correspondillg, Laplac('
tl'all~forlll
is t hus
(2. 1~l)
A nothcr special case of the lIoug.tard d ist ri bu t. iOIl is t 11(' p{)~it ivp ~I
which the IIougaard distribution is ;t('tually dprivp(1. Ah
•
distributions correspond to 0 = 0,
For
ally
1.('"
2.4
hI.· di~t ri bill iOIl fi 11111
I\olpl! (',ldil'r,
thl'
]I():-.lIlvl' ~1.lbll'
P(o,b,O).
B, the IIolIga.ard distribution
to 1. Thus, the I1ougaa.rd distributioll
<1
!H'COIlIPS
I>C('OIll('S
(,oll('('lIl,r,1I1'1I al, t 1", poillt 1> .. "
dpgPIIt'r:lI,('
III
this
lirllltill).\
Il
t"lItho
t .1:-'1'.
Simulation
This s('ctioll d('scribes
iL
~illllllatioJl tpchniqll(, for g('JlI'Jatilli!, 1.lIIlIorll
lIaugaal d dcnsity fu Hction J( x;
0',
Oll!.(
()IJI"~ 'rolll
h, ()) Ilsitrg a l'IjpctioJl fd~()r Ith rll [:'('(' ()"vr or (1 'JXf)
The rejcctioll algorithlll requin'fi a dcnsity f\ludioll fl( T), aud
ft
«(JlI~Llld, (' /
.1
1.
l, ~IJ("
that
J(x)
Recall that the
•
d(,Jl~ity fllllctioll of
~
cg(x)
for ail x.
('2
SO)
P(ex, h, 0) is
J( x) = exp( -()x + 00" / n )y( x),
('2 !)I)
•
'/ ;1 king;
fJ( J') tu IH' th!' following po~itive stable density,
q(:r)
1 ~
= -L...7r.J: k-=l
1'(I.:O't1) -6 k •
( - ) sm(ak1r)
k!
Ox"
(2.52)
.Inti
c = exp(Myi la),
Il (.III
IJ('
""1'11
(2.53)
I.bat thl! followillg illPClllality
f(x)
- - = cxp(-Ox) $ l,
('g( x )
l', ·"dl'.fll'cl lo( .dl
1.'(0111 t.h{'()(('/I1
(2.54)
,1'
2 O.
(j
fi of \)cvroyp (198(i), if (J is uniformly distributed on
Il''1I11ally dl"t.rihut('d with paramctN 1 and independcnt of
•
[0,1] and E is expo-
then X = (p(TrU)1 E)(1-a)/a
(J,
fllllllw!> t.h.lf, }' =-= (hjo:)I/"X h,l~ distribution P(O',b,O) and the density function
H.llldolll
St.,']! 1.
V.II
iatl's with d('n~ity
(:l'I\('(,lIP LWI)
f
01\
R+ can be obtained by carrying through the follow-
illdPIH'lldcnt numbcrs,
Ul
from auniform dcnsity on the interval [0,1]
.\Ild " h()m ail (''\]lollplltial d('nf>it.y with paramcter 1. Let
l' W
g(x) in
=
TrU}.
Then, caleulate y and
- a)) .!.j(l-a)!O'
v
(2.55)
W
h"1 ('
y
= [ i\il~(nlJ))I!(1_o)sill(~lJ(1
(
.~m( QW)
SlII li'
(2.56)
Sfllp_~
•
)!.(llll'l'.Itl'd
(;('11('1,,11('
a
~PlOl\(1
ilul(']lelldent uniform Humber,
in ~t('P l, ralculate t = cg(x)/f(x)
1.1I1d1l1ll 1I1l1ll ber
=exp(Ox).
lt oIII t h(' dpllsity fUllrtioll f(.r;
23
0:,
If
1l2.
U2t
Then, using the value x
$ 1, then x is the required
b, 0). Otherwise, repeat the two steps.
•
According ta Devroye, the constant (' l'quaIs thl' pxp<,c('(llll1ll\lH'r of ilt'ratlOlI:-. rt'qllin'd
to gcneratc one random Humber x. TIH' pr(,Cl'dinp; pnl("ItIrP Îs not pr,\( tll,ll \\ IH'II
is large. A simulation method for the' Houg"lard dbt ribution whirh \:-. /!,('nt'r,lily ,ll'pli, "hk
rcmains to he deveioped.
•
•
M}" /n
•
Cllapter 3
Inference and Applications for the
Hougaard Distribution
•
III .l(lplJ( .d,iollh (Jf tlH' lIollgaard distribution, infcrences about its pararnctcrs arc often
Iw,·d,·d. TIH' h<llllplc· daLt 1111<1('1 htudy arc
("Il~.(I,,·d
~oltlctim('s
complete' but freqllently thcy are
SIII VIV.I! 01 lifp d.1I d, for l'xalllplc, are oft.cn in( ompletc lH'cftllsC of ccnsoring.
'l'III' d.1I .1 an' «'I\:-,ol('d wh('11 thp ('xact ~urvival tirnc:-. of sOllle IIl11t~ withlll the sample are
Il Il klluwIl.
ul
Thi~
(ft,lpt.('r airm, at. dC'vdoping paramet('r c&tirnation mcthods for applications
1I01l)!;.I,lId di~trihut.iol\s.
Two ('onvelltional mcthods of parameter estimation, namely,
11\1' 11\1'1 h()(lof 1II01ll(,1It.:-. and the It\ethod of maximum iikelihood are presented. The thesis
«(111('('1111;11(':-. (lll illfl'l(,l\c(, 1\I('lhods for samples with complete data. Inferences based on
ll'Il!->Ol('d
d.lla
ail'
Idl
'l'hl' 11\1'1 hoc! of
IlIl'!
•
hod
lrholld
1'01111
j:.,
IIH'I
Lo
allother :-.llldy.
1lI0llwtlis
is historically olle the oldcst mcthods of estimation. The
:-.11'.lif.!.h 1forwa rd as will be
S{'{'II
in §:l.1. On the other hand, the maximum like-
hml for lIollf.!.aMd dbtribut.iolls is difficult t.o implemcnt bccausc of the complex
or ! hl' 11(l1lf.!..I,ll'd (l.d. f..
'1'0 offspt t Il(' di (ficull ics with the maximum likclihood method,
25
•
another com'sponding Illet hod deriVl'd from t lH' s.uldll'point approxim.tI ion 1II 1hl' lIollg.I.1I'I1
p.dJ. is developcd; the I11<'thod is ('alled maXlllIum qll:lhi-likt'hhood Phlil!laIIOll. l\1.IXillllllll
likclihood mcthods art' important
!W(",lUt.C
thcy han' gond HI.llihl Il.tI propl·rlips. FOI (.,t'stllll.llol~ .trI' IOIlSISII'lIt
ample, under ((uite gl'lIl'r;tl comlltions, maximlllll Iikl'lihood
.Illd
asymptotically normal \Vith the parame ter to \)(' eHt.imal('d as nW,ln and wil h a variall("(' llO
greater than that of any other estimator. For buth
('~till\atloll
IIwthods, lonlidpllCl' inll'I v.tls
for the parameters are constructcd.
Sorne of the special cases of the lIougaard p.dJ., sllclt
(0' = 1/2) and the gamma p.dJ.
((~
= 0),
.I~
hav(' the p,lfal\lptl'f
1lU' in\,('lw (;,llIs:--ian p.d.r.
H
knowlI hlll III 01111'1 C.ISp:--
0' must be inferrcd [rom the data. lIowl'ver, in fe rl' 1\ CI' rl'sttlt.f> fOl 1111' llIVl'If>P (;.IIl:--si.\l1
distribution <l1ld the gamma distribution an' wpll known for hl'th thl' llll'l.hod of IIIOIIlt'Ilt.S
•
and the maxillluJll Itkpltllood lIIethod. '} hw" thl' cha ptN will fonll,
011 IIIC"II'III"1'
111('1 h(uls
far the general IIollgaald p.dJ. wlH'f(' n is IInknown.
After diseuf>sing estimation Illpthods i" gplINal,
tions in suell
(liver~c
SOIll('
al'eas as econolllÏ<'s, ml'dicinp and
applicat.ions of J(()IIf.!;.léml dihtrihu
f1H't('orolo~y
will IH' dPhl fI'lI'd
1vl.lIly
of these applications involve the lIougaard distriuution as a Itlodpl for Ilf.. dat" HOllg.lilld
(1986) initially praposed the Hougaard distribution as
Cl
fra.ilty 1110(11·1. lu, tH.I· ill t.11lf> «(JIII.l'Xt.
is discussed at sorne lcngth at the end of the chaptPr.
3.1
Let
Parameter Estimation by the Method of MOlnents
{Xl,X2,"'X n }
he a raudom sarnple of hize n from a lIollgaard poplll,JI,joll, l'(o,n,r))
When ail parameters are to b(' estimated and 0 is assurned to lu! pOhÎl,ivI',
•
Wf' ('all
l'qllat.!'
the first sam pie moment (the sa.rnple rnean) with the corr<'hpondiIlg poplliatioll 1Il01lll'IIL
Likewise, wc can equate the r,econd and thirrl
2fl
barn pif'
rnOlnl'lIth abOlit tl/l' 1I11'an witJl tllI'lr
•
III,tflhillg pfJp'll,tI,lon lllOlllerrb about li\(' mettn. In principle, any three moments may be
Il',(·<1 1,111 III thi., 'Ippli( ,ttioll, the !lIf';W and the next two moments about the mean are
hll~, 1I11d/'f 1.111'
(fJlI.,idf·fl'd. '1
1!,IV/'11 ill
!l~
J,
WI'
obtai/l
t1lP
hypotlrf':-,b that 0 > 0 and using the population moments as
follo\'llng
:-,('t
of cqllatiolls.
(3.1 )
(3.2)
b(l- 0)(2 - n)8Ct-3,
wh!'l/'
1111
~
l',
11/1.
\ ~11
n1 L,,=I
1
_
:t, - -)1..
X
.trI< 1n3 -
(,
fil
"n
(
L..1=1
XI
(3.3)
-x-)3 .
Not 1('(' f Ir ,rt,
ml
()
1n2-(1-a)'
•
17/2
()
1It:l
(2 - a)
-=
,
(3.5)
(3.6)
ft
iJ
1- 2R
1- R'
~(1 - cr),
7n2
g
mdiii-l.
Tht' ~l'I of (':-.1 illlalt's will hl' 'ldIllÏh:-,ible only if ()
Il
.111
(3.7)
(3.8)
(3.9)
< R < ~, bccause only then will 0 < Q < 1.
IlIadllllhSihll' l'stilllatl' for n arisl's for ally givclI sample, it is an indication that either
t hl' d.11 a .1)('
•
(3.4)
11I.ld('(l'la t t'.
IIlroll~i:-. t ('lit wi th tlll' lIollgaard modcl or, possibly, that the sam pIe size is
Fm ill~t.\I\(,(,. if Il'ft -skl'wl'd h,lIl1ple data are liai vely fitted tü a Hougaard model,
27
•
the ratio R as dcfinrd in (:J.6) wi1l1w \l('gatin' brcausl'
the condition 0
< R<
The variances and
~ whirh
111 Il st
hold
IIII<lN
williw lIeg,lllv!';
t Ill' 1l0tl~,I,lrd
('ovariaIH'('S of t!te 1ll01lU'llt
the jackJ..nife lIlethod [Efron (19R1)].
/11.\
vinl.\tillg
111\1(11'1
l'si illl,llorH C.II\ b"
For il 1ll01llpnt ('~tl\lIatOl
IU'lIn"
e~1 11ll.lkd (,.I~ily llsillg
i of .\
~illl!,l(' p.lI,llIlI'll'f "
the estimated variance is given by
p.IO)
where n is the sam pie size,
observation and
f(:)
"Y(:) is the TllOtlH'l1l ('sti 1II,ttl' of " oht.\Î 1I(·d hy dt·kt Ill).!; t hl'
= '* I::~I fW.
Il Il
For a parallll'tcr V('ctor,,' :::- (n ;, ()), t.hl' l'l'hlllt. Ill' 1':rIUII
(1982) givcs the following cst.imat('d covarian('c lllatrix VI for t.hl' 1I1011l1'Ilt l'htllll,1I01 )-.
(:l.Il)
•
where
The square foots of the diagonal entri(\foj of VI ar(' P:-,1,IlIl,Lt.pd :-,f.,llld.lJd
estimators. The correlation matl'ix may abo 1)(' C<JllIplltf'd frolll VI t.u
of linear depclIdclIce of Hl('
vidual confidence intervals for
a~pect
t!H~
pararnetNi:> can
t)(~
(\x;lIIlplp,
tilt'
or
t.hl'
l'X!,t·l1t.
]J
S"hf'f and Wild (1 !)).;!))J, IIldl
(·OllstfUct.I'd, hit:-,(·d on tltl'/ di:-,trlbllt.ioll
of simultaIlPOllS infen'nrp). Ll'ttllll!;
of the l dibtribution with n -
•
l'X.IIIIIII!'
1'2)
e~timators.
Also, according to the jackknife theory [1>ep, for
(ignoring the
('rl(ll~
n
l"-TJ.fJ/~
def!;rl!l'f, of fr(,l'dolll, Wlll'fl'
1t
Îh
dl'lIo11' titI' 1
tlll' ... 1IIIIpll' hizf' .. /Id Tl 11JI'
number of pararnctcrs e:-,timat(!d in th!' lllOd('l, (1 - (J) 1OOo/r, (ollfid"l1( f' IiI/iiI,<, fOf
'2H
(i/'!. fr.1f 1.11,·
par,IIIIPl"f<.,
•
'(3,13 )
(3.14)
iJ ± tn wh"11'
/J",,, '/J6b dllel
(3.15 )
3 ,{l/2 VOO,
/Joa a/(' thp {'orrpsponding estirnated standard crrors for &,
6 and Ô.
'/'0 i/llp/ov(, t.hp qualit.y of infprPHrcs dcriwd by jackknifing and to produce confidence
/ilINV.dl> for
I./If' paralllPl,prs thiLt cncornp..tf>S ollly admissible values, the jackknifing is ap-
plil'd II('fI' hy (.trryilll!; out. su/I,able tr,l/IsfOrrnatlollH
1/ d Il:·..fO/ 11I.ll,ion 1()
!. l~;'_1 f'.!(o0l)
•
IOI'I/H,d
(t)
,tIld
parcllllP!.prs
alld
<l
l;{) =
1,0 tlll'
lOI!; I,r<lll~forrna lion to 6 and
~
L::::l h(0(0),
n.
pararncters; specifically, a logit
50, letting t 1 (
i50j) = 1n[6(0/( 1 -
the covariance matrix for the vcctor of trans-
(In[al( 1 - a)L In(6), In(8») is
(3.16)
\\' IIt'I ('
il (ltvi) -
t-;O
t2(b0)) - t-;(0
t3(8(0) -
t-;O
11(n(2l)-t~ t2(b(;))-t;ô l3(8(;)-G()
t 1 «(l(fI) -
t-;(:)
t2( 6(:)) -
fa(:)
3 ,{3/2 tv Ot ,
{3.18)
t--;O ± t n-3,{3f2 'W8,
(3.19)
t-;{:j ± t n - 3 ,{3/2 'WB,
(3.20)
11(,)
•
± tn -
t3(é(,:) -
0')], In(6) and In(O) arc respcctivcly
(1 - rl)IOO<j{l ('ollfid('IIC1' int.('rvaIH for In[O'/(l ...-
t;O
(3.17)
29
•
standard errors fol' In[&/(I- ci )], In(b) .tIld In(").
3.2
Parameter Estiulation by the Method of Maxiullllll Likelihood
The maximum likdihood IIH'thod of pa.rallH't<'r ('stilllatioll I~ l'OlI:-.idNPd lir~t fOi thl' ~itll
ation wherc the sitmple rOIll('S from ,1 lloll~aard popillat iOIl /'(n, Il, 0), for wliu li .dl (If t lu'
parameters arc Ilnknown. The situation w\i('f('
ll'
i:-. hllown i~
I.lk('1I IIp ,I~ .1 ~(,lllIld C.I~(,.
P( Q, 0,8). The salllple log-Iik('li hood fil nrtion i!>
Il
•
l(n,8, 0)
= -1I(1.E +lIM)" /0 1- L
h(.r,;n, Il)
(:1.'21 )
,=1
where
f
h(x,;n, 6) -= log[ _(1l'X 1 )-1
r(h~,+ 1)( -hx;"/n)k ~ill(nbr )].
k=1
(:1.'2'2)
•
Diffcrcntiating 1((1,0,0) with rC'!>IH'ct to 0 and (lquat.ing t.hp dprivatiVl' 10 I.l'rll)!,ivI':-'
iJ/(o,b,O)jiJO
It thclI follow& tlmt, for lixed vaillps of
0'
-
(J
= -/Ir + nM)"-1 = o.
alld
n,
titI'
lIIaximlllll likplihood
(J if>
1
of the Iik<'lihood flllletion berauM', for a.1I b > 0, - 1 <
•
of
= (l/n);;-=-j.
Undcr the ab ove conditiolls, il. cali be !o.cell th,LI. (:J.:H) ('orf('~pollrlh
second derivative ib
P .... t 1111,1101
alway~ po~itiV<'.
:$0
H
-
1.0
l '" 0 alld
tlll' JSlo"al
(J /
maXlllIlJl1i
0, t Ilf' f()lIllwill~
•
Il
(JIN'/vf>d Ih,JI Ihl' /rt,LXiHJlJTn likplJtlood ('btimator for
h
t II" 111"I}II)(1
or IlIOI/JI'''' h lIhlllJ!;
Hf'lll/llllI~ 1()
rd li
.Il1d
t}lf'
1/1.1\11/III.f'd, tlr,LI 1'>
1-'.1.1I11f'1I1 tPI hlll'lll('
f 11111 1iO!1
(:L 1) arp l'qllivalent in thb cab('.
1II!'1 !Jod of Ill,lXirnuITI likclihood we notice in (3.24) that
Sllb,>llllItJlI~
ft
il
fUII(
(ail }H'
iJ
is a function
(:J 21) ill (:3.21) giVPh a log-likelihood function, now partially
tlOIl
of pararnetc'rh a and fJ only. As a final stcp, a numerical
Ilhl'd to ohtain th(' pair
Thl'Il, () «1/1 }H' (akulah'u by rl'placing
",llIlIall''>
e in (3.24) and that obtained by
(Q,b) which maximizcs the log-likelihood
Ct
and /j in (3.24) by their corresponding
'l'hl' 1IIlIlIl'ri(,d bl'ilr<h for the pair is not casy howcver hccause as mentioned
III '!~ '2 1}1f' 11()11).',;J,lld plOl>dbrllty dl'Ilhlty fUllction does not hehave weil computationally.
,\I..,IJ, Ihl' pll'lIll'lIth of thl' illfollll,Llioll lIlatrix are difficllit to compute so that the
(')\,/11.11111' III,LI!IX
•
h 110"
esti~nated
e;lbily av.tilclble
NI'\I, tilt' (.lh!' whe!!' ct ih krrOWIl is considercd becallse of its Iink with exponential
dl~llI'lhlOIi t hm!'.v.
LI/lllly i~
.111
From §2.1, it Il> krlowlI that when a is fixed the lIougaard distribution
!'\I)()II<'lIlial di~pNhiol\ lIIodcl, say RD(ll,a 2 ). So, the sample log-likelihood is
71
Ih,1!l
= L {lrt[a(x,; 1/)] + 1][x'Î -
K(Î)]}
(3.26)
.=1
wlrl'I'P
(I(J';
1/) I~ giVl'1l
by (2,IX) ,wd
1]
by (2.16).
'l'hl' [11.1\111111111 Irk('li!tood Phtl/llators of 'Y and 1] are obtaincd
1)\ ('qll.1I ill)!. hol h iU{),
,/)/ih
In
the usual way, that is,
alld iJ/(-y, 1/)/D,] to zcro. Proceeding to do 50,
Il
iJl(Î,I/)/ÜÎ
= L1J[x,- K'({)] = O.
,=1
:'\ ()\\'.
~1I1(
(' '/ '> 0,
K'('7)
•
Bul,
.(3.27)
1[0111
= K'(i) = x.
(3.28)
('2.lï), il followh that
(3.29)
31
•
so that
p.;\() )
Then, (3.28) and (3.30) yie\d
,= __ .
1(X)_1
"-1.
n
Observe that
i
fi
ran be ca\cu\atcd without r('fr-(,lIcP to tll<' vaIn!' of '1.
The iuC'qnality that follows confirms thal.
i
in (:1.:n) m<lximil.('!. t hl' lo).';-likpliho(ld p.1(i)
(:I.;I~ )
since
11,
> 0, 17 > 0 and V(IL} > O.
The next step is ta sparch for the maximulII likPlihood p!.t.illlator of '/
(3.26) with resp(lct
•
1,0
77 and tlwn ('qnatillg ta
ZN()
Dd["'I'PIII i.lI.ill).';
giws
I3y inspection of a(x; 77) as exprl'ssed in (2.18) and its dPflV,tt.iV(· wilb rI'SIII'( t
\'0 ,/, il
I!.
clearly difficult ta solve for the maximulll likdihood pstimator of 1/ frolll p.:I:\).
A point worth noting is that estimators
i
and
r,
in thlh casl' arl' .1!.ymplol.J< ally illdl'-
p<,ndent. To demonstrate th is ract, wc ohtain I.h(· !.l'cond
log-likelihood (:L26) with respect
1,0
part.Ial
dPri va t.1 VI' or
1.\11' S.IIII l'II'
'1 and 1 ah
11
2
Û /(f,TI)/ÛTIÛ, =
I)x. - ,,'h)] =
u(x --II) .
• =1
Since E(X)
= Il, it follows
tltat the pxpcctation of thib
51'(
ond dNivativl'
E[ rp/(f, 77) / ür/rh] == I~[n( X - IL)]
•
Ih ZI'/O, 1.1'.,
= O.
The orthogonality of 77 and, is indicated by (3.:i5). By btall<lard ahylrlptotic t.Il1'ory, il. is
known tIrat
(il, i') is
asym ptotically normally distributf'd
witll
a covdfi;UII ('
III al. rix P;I vr'n
by
•
III"
111','('1
0111'
1111'
(Jf tll" 1 orrc':-'!HJllcllng Illforlflrttlon lJl,ltrix. The entrics of the information matrix
"
C't:l)C'(
l,tliOlI:-'
of 1II(! lIf'gatlvl' :-,ec olld derivativps of the log-likelihood ftlnction. From
(:~ :VJ), II, follow,> 11t,1I titI' Illformatlon matrix ha,,'; zero off-diagonal entries. COllscqucntly,
lit,.
10\,,111<1111 l' III,ltnx
wlti! li fl'hUltf> from tlH' inver&ion of the information matrix
If'l!) I/fl dld/!,I)!I.d l'Iltlll':-'. Tltl~ illlpliph thrtt ~ and
aJso
ha~
l' arc asymptotically indcpendent.
Maxilllum Quasi-likclihood Estimation
:L2.1
rvt"Xilllllll1 lilwldlood c:-,tim,Ltion in t.hp g<'ncral ('as(' is not practical, as pointed out alrcady.
Il OWI'VI'I,
(~
III
:l:q,
111,\ XIIIIU III Cf lIahi-li kplJ
\':,11111,11 1011.
a pr,lcI iraI
hl' dpvplolH'd ,Ih
1.111
tlw, :-'('( t.iol\,
III
hood Cf> ti ma tion, baf>cd on the saddl<'point approximation
il Tllpl
rtild effe('tive alternative to m,lximum likelihood
hud for oht,tining maxim1lm qllélsi-likclihood cstirnates is
p/I· ... f·II!.('d. III lIt(' (·xp('(·:-,~i(J1I for thl' h,Ullpl<' log-likelihood function, the saddlcpoint approx-
•
Illl.IIHII!
hy (:l.;~;l) b
1-\1\'1'11
1/
1(11, Il. Î) '-
\\'1.1'(('
1/
1111' 1101
~
in pla('!' of the exact density function, as follows.
H'I
1/
--Ill 'I - --III[
2
11
aq n
-1]- ""'
(,7 L-.
1=1
= (1- 0)/(1- a).
1=1
x-
q
1
+ 1J,1ix- n1J li (ï),
(3.36)
Observe that no account is taken of
III.di 1.1 Il).!, 1011~tal1l th,tt would be lequired ta make the saddlcpoint approximation
FOI .1
a pmper p.t!
!ixl'd valul'
t'qll.lt Ill)!;
(If
H,
r.
th(' maximum likelihood C'btirnators of, and 17 may be obtained
hol Il U/(n, 1, Il)/;71 alld ()[(n", '1)/0'1 ta 0 and solving the resulting equations.
i)
il[
•
P
)1- -2 LIn XI
17r(1-o
- 0/(1 -- 0) ,lIld Il
!I11l\ lioll r(.r)
11\
Il!'.l·d
1(0, 'I, Î)
= 1111Ï5 -
7l11K.'b)
= n1J(x -
JL)
(3.37)
(3.38)
33
•
For Ully fixed
0', s~tting
C'quation (3.aï) to lero yi('hlH th(' fnllo",ing.
it = JO.
Thus, the last teflll in (3.38)
(.1 :I!I)
bCCOIlH'H
P
Now, setting (3.:J8) to
Z('ro
10)
yieldf>
or
(:\ Il)
Consequently,
•
"
[(0,1/,Î)
Il
,n
n'l
]J ,~
11
-III[ (
1- - L..,III.,..2
21fl-n)
2.,,:1
'2
= -ln(I/) 2
SI11CC
=
ft may he observed from (3.11) that
r,
is a
n
'2
f\lndion of parallll'\,pr n ,.1011('
likelihood in (3.12) is a]f>o a fundion of pararnetpr n aJo/l(·.
ou t a one dimensional search for the vah)(' of n that
replacillg a by â.
31
f,(J
t Il,11
1.11/'
Jol-',
/11'110', t1H' III'XI. :>11'1' 1:> III (,l''Y
!li ,Lxi III iZl'h
obtained from (:.L11) by rpplacir1g n hy h. SlIllilarly,;y
•
(:1 1'2)
('fUI
(:L,12). Thl'/I, i,
"1' «(JlIqJ\Jt.l'd
f'IJIII (:~
(.JlI
1)/·
'10) !ly
•
Il l.trl Il!' dl mOIl:-.1 r,t1,pd t hat tll(' critical point (il, 6, Ô) is indeed the maximum likelihood
l'',lllll,doi IIf tir" paldTllPlpr
Vpct.Of
(n',Il,f)), For any fixed
Q,
(3.43)
'dlll l'
Il./
D,
1/,
,>
(J
,1IId li)!· varitlll(,f' function v(-) is always positive. Similarly,
(3.44)
hlllC l'
11
... (J
,1I1e1
1/ > 0,
FillaJly,
(3.45)
'l'ltll~, l(o,
.1
~I()h.d
i"i)
iu (;L,I~) f('pr(,~(,lIl~ a. globalllla.ximum for any fixed
hl',l/('h, tir!' optulI,dity of (à, 1), i) is
•
'1"""1
:-'II('1r
that. ('
Since â is found by
as~ured,
NI'xl, (o,tfic!('I1( (' inlC'fv,!Is for th(' parameters
1)(·.\ l'''ldlllI'lPI' v('(lol'
Q.
= (a,l/,,)
Q,
band () will be constructed.
alld let
(=
Let (
(6,r,,7) df'note its maximum
Irh .. liltoocl p~t.ill\,l1ol', Trc'.\lllig ( as a maximum likclihood estimator, its asymptotic
(O\'.\llolU«·
lllat.I'IX
b gÎwu by thp illver~e of the Fisher information matrix, namely,
(3.46)
\\ III'/('
I( Il, Il, Î)
Ih
.I~ iu (;~,:H». Th(, subscript zero on E[
Ili'dd,' I~ ('Vdillall'd .rI Ih('
1'.\1.\1111'1('(:-.,
tllll'
palametl'l'
]0
indicates that the quantity
vedor (= (o. To obtain confidence limits for ,he
Il i~ (1l1I\PUÎ('ut lu (',dCIlI,llp li\(' local eslimatc of Var((), which is
(3.47)
•
[(/1/(0, '/. Î )/()(iJ(1~=,
=
(Jl'l/Do:2
[Pl/Dnô", D21/ ô o: u,
(}21/Dl/Dn
u2[/ D",2
iJ2[ /
ih Un ()2[/D,Dl/
:~5
(3.48)
D2[ / UllUi
8 21/Ôi 2
(=(
•
where I( Q, 77, 1) is dCl\oLcd by 1 for ~i III pli ri Iy. N ow.
of the log-lihelihood fllnrtion, evalllatl'd .It (
Wl'
= ~.
will giH' 1h<' lil'cond p.lrl 1.11 dt'rÏ\.\ 1ln':,
Ihal Ml' rt'quin'd for (;1 I~). \\'(' h.l\t'
already noted that
P
I!I)
Next,
so that
(.1 fd)
Similarly,
•
The expressions (3.51)-(:3.55) may be eva.:llatpd nllllll'ri('ally for f>rJl.dl "
1.0
ol!Lll1i
tllI' If'
quired clements of matrÎx (a.1~),
It is weil kl\oWII thal the pstirnator ( is <ti,y III ptotic <Lily Ilorrllal wlI,h
IIIl'd!1 (0 .JIld «()
variance matrix (3An). Using the local estima!'(' (:$.'17) of th!' trtu' <lllylllpt.oli<
(OV,lIldlllf'
matrix (3.'16), (1 - ,8) 100% confidence lirnits for thp par<llll!'t.l'fs ,1 rI'
•
(:1.:;1 )
:w
•
(3.58)
whl'I"
Il,,,,,0'1'1 <llld 11-1'1 .!fI'
rJI'!.) fr,j( til/' of tliP "tandard normal distribution.
:3.3
'J'III'
(JI
i,
and
v")''')'
are
thl' (orrpl>ponding cstirnat.cd standard crrors for li, fJ and
V Qcn V'f/'f/
and
Applications of the Hougaard Distribution
1-',.11111/1,1
Wlill
',(JIIIf'
all(1 1.111'
li wrll 1)1'
01
111(',>1'
IIlv('r~('
IJl'(,~(,lIt(,d
(;,llll>l>ian
!J('f'f'
a~
(li~tribtJt.ions
have fOllnd extensive applications, a fcw
illuHt.rat.joIlH. As th('!'(, arc !'pecial I10llgaard distributions,
applil.11 i{)l1~ IIl1glrt logir,t1ly 1)(' gPllCralized to the Ifougaard family.
Silil (' t li!' illVl'lhl' (;;Wl>hl,11l dil>t.l'iblltioll has iLs origin in Brownian motion, it is not surpll'dll/-',
•
'>tlIIiI'
tll h('(' ils applicatlollh in t.he lIatllral and physical sciences. \Ve shaH also present
ot 1i!'1 .lpplll aliollh olltwlp the natural sciences. Many of the applications to be pre-
hl·lIll·d !'xplolt. t hl'
\Vith dldt. Ih
<III
r,l( 1.
t.h.lI. th(' firHt passage time to a fix<>d barrier in a Brownian motion
illVl'lhP (;alll>hiall variable [Chhikara and Folks (1989)].
1)ill('J('lIt 1 all'p;ori(·s of applicat.ion of the inverse Gaussian distribution found in the
Il II'I .tllI H'
.!rI'
rl'portl'd h('('('. A llH'diral a pplication of the gamma distribution is presented
ol~ \\ l,II
(.1) t \1 /111/ Mod( /. A t hl'olpt.icaJ applk,ttioll of the inverse Gaussian distribution in metet
Il
III",!.!, \
\\'.h
WlJhlllPll'd hy Ba l'lh;J{'y (1980). Ile used thC' distribution as a model for wind
"PI'('Ii" (1I1It .1I11illg lo\\' fl'('(!1H'llcÎPh of low SIH'eds, at a given site.
(Il) \ 1/ /lItll'l'urdHl81' '1'11111' Mm/cl. Banerjee and Bhattacharyya (1976) applied the inverse
(:.Il1:-hi,1I1 dlhtrihlltion
•
(,0
an
aJlaJy~is
of data on family purchases of toothpaste.
They
pll:-.llllatt'd t hat for l'aeh f,lInily tl\(' time betwcen consecutive purchascs is modelled by an
11I\('lh('
(:.\lI:-~i.lll
dbtrrhulÏolI. 'l'II(' <tl>slIlI1ptioll is reasonable sincc the tube capacity may
:3ï
•
be considered as a barrier. Thel!, if tool h p.1~tl'
drift, the time until thl' next
purcha~l'
iL
fo\low:> Brownia Il llIot iOIl
Wl'r<'
fitll'd ln d,lt.\ fr011l \'2
fiVl'-Yl'dr pl'riod. 'l'hl'
to estimate the parameters of c(leh ltoUbl'llllld.
1I\,lXIIIIIIII\
h po:-.it i \'l'
hOlll>l'h(lld~
liJ..l'lthllod
(,hi-~qU,Ul' V;O(ldlll'~~
computed for the l2 households and nOlle W,l:> signilirant dt t hl' i)
significance levcls ranged from 0.053 to O.n;H with tlll'
W Il
di~tlÎhlltl\lll.
has ail illvl'r:-'{, (iallssl.llI
Individual inVNbl' Gallssian dibtribllticlIs
al lcast 100 purchascb ovcr
IlSa~l'
d('~r(,l's
11I.IJ..illf,
hud W.I"
Il:-'1'11
of-fit :-.1.\1 i:-.1i< ~
W('I ('
!l('f(('111
IIwl
1('\,('1 - 1 hl' ohM'( \'1'<1
of f(,I'doll\ r.lIIglll)!.
\rO(1l :\
to 6.
(c) A Strike Duralzon Modrl. Lancast<'r (1972) IISl'c1 th(' invl'fsl' (;alls:-'l.tII
describe strike duration da\.a" L('ttillg l d('IIO\.(' thp
strikc, it was postulat<'d that therl' pxisb
•
two negotiating partiC's at time 1. lt
entircly
011
,1
s(',llar
tÎlIlP
X(t). The origin and !..calp of .\"(/)
to
('I,\p:-'I'<I sille!' 1hl' lu'gillllill!!, (If 1hl'
11\1'.1:-'11[('
wa~ ~lIppOl;(,d
dl~t.rihllt()(1
1)[ a~J('I'\I\(,IIt.
.\ (1) 1)('1 WI'I'\I 1hl'
th,tf. Ihl' dllJal,(OIl of Ih(' sI J(J..(' dl'pl'lI«1:-.
WN!'
dlOf-1'1I f-O t.h,1l \'(0) - () .11. th" ht..Irl
of the strike, and the strik<, is settlpd, that. is, agr<'('!lH'nt is r<'dch"d, .lI. I.ltl' fir:-.t pllllil. 1
for which X(t) = 1. The time until beltlclIH'1I1. is the dllrat.1011 of titI'
~Irikl'
Asslllllin~ ,1
I3rownian motion for the process {X(l)} it thl'n follows that thl' fin. !. p.ISS.I/!;1' t.illll' of tlll'
proccss to the absorbing barrier X
= 1 has an inverse (:allhhi.U1 distrihut.ion.
The data consisted of the duration of strikes rpporll'f! hy thl' Britiflh Minbl.ry of L.dJllllr
betwcen 1965 and 1972, in the United KingdoUl. The dat.a W('f(' dividl'd Jllt. . ('i/!;ltt. illdllht.rlf':-'.
(1) metal manufacturing, (2) non-('I(,ctrical engilleering, (:l)
(4) vehicles, (,5) construction, (G)
~hip buildiJ,,~,
(7)
di~trrbllt.(/)1I
tl;UI~port. ;11111
tradl's <Llld hl'rvÎ! l'S,
(X) 1'11'rt.l'Îc.d
/II,l(
hllll'I y
The data weIe truncated - the Humber of htnk('!.. la...,ting khh tllan a day W('JI' flOI. Il'('(Hd('d
by the Ministry - and, the data were only availabl(, in a grollJH'd form
•
compriscd the strikcs lasting more than a clay but
:m
h!~~
titan 2
ddy~,
t}ll!
Th" lin. t grolJp
~I'(f)rtd
WOllp t}JI'
•
IIIL(·"
Â','>lllllllli!, <Ill
1/110
~ r1dy~
LI .lllll!, 1I10ll' tlt,1l1
but I"s~ than :J days, et~ .
trlVl'r~1' (;;LII~f>i,jn
modf'1 witll distribution function F(t), Lancaster took
ac C(JlJlI1. tlll' f,l.( t 1.ha1. thl' datrl. wprp- trllncated by working with an assumcd population
l/locl!'1
ltavllI~
thl'
fIlodilic~d di~1
riblltion fllllction,
"'r(l)
= [F(I) -
}\ 111111)('1 /( al itC'I,t1 IVP Plo(,I'<llIr(' baf>pd
011
F(l)]/[l - F(l)].
(3.59)
the distribution l'Tel) was used to obtain 'maxi-
11111111 Ilkl'lilhlod C'f>tllll.ÜPio; of IL and), frorn t.he truIlcated, groupcd data.
\VII h t11f' ,!id of t.hc' nllllH'rical proccdurp an illv('f~e Gaussian distribution was fittcd to
1III' ~I 1d,l'
cl III ,11 iOIl
d,tf,j
fOI c'ac h illd ustry. Ove rail, the in verse Gaussian fi ts were good. Only
lOI 1hc' plc'< t.ri< ,LI 11I<LrhiIlP/ y illdllf>try was the chi-square goodncss-of-fit statistic significant
.11,
•
t 1)(' 10 IH'I'(,PIit. kw!. Th<' rdativc silllilarity of the estimates of Il and), in industries 1
10 li f.1I/-!'/-!'<'i;!,,,d t.h.ll. t.h(' ohf>f'rvations for thcsc 6 industries could be dcscribcd by a single
dlf>lliblltioll. Af> ,1 llIatt.!'r of f,lCt, the fit of the inverse Gaussian to the poolcd data for
IlIdll~1 1J('~ I·(j
(d)
W,If> quit!' !?;ood.
'/'/'W'I/' J)YIWIIW· ..,
Wp
1I0W
('OllSldN an application of the gamma distribution in cardi-
Illo~y Wi~1' d 1/1 (I!HiX) asslIlIl<'d a !?;éltnllla distribution for tlH' time T from injcctio'n of a
p.1I
1i< 1(' 111101 he' Illood st/'(',11I1 until it is pxcrpl,('d or dcpositcd into the
IlIil'llioll of
,1
OOllCS.
Following the
~lIb~I,III<'(' int.o th!' I>lood, the cOllcentration of that suostance can oe mOlli-
t III l'd .11 ~ Il b~l'q 111'11 t 1111\(' poi n1.&. From such data the probability distrib utlon of T rnay be
('~IIII1.t1I'd
.\n illj('clioll of t.hp nllclidc Calciulll was considcred in the application by Wise
cl Ill. Tht' .tlll hOls fittt'd .t galllllla distribution to the data obtained from monitoring the
10111 l'III r,ll iOIl of t.1lt' III1c1idc ovcr a IClIgth of time.
•
((') :1
["ft Ill/II' Model.
Il ha& alrC'éldy opcn
!loUg.I.1I d li hot 1 i hu 1ion i~
.t& éI
Ilot cd
that a potential area of application of the
IlllHipl for survival data. The inverse Gaussian distribution
39
•
has bcen ext,ellsiv<,ly u8<,d as a Iift'tinH' mod<,1,
a~
ha:, Ihl'
g.lIl1lll:t
di:,trihllfioll .
Suppose that F(t) c1l'notes the distribulion fUlldion of tlll' f.lilult' linlt' l'
nism. The reliabilit y R( t) of the mech.l.IlislIl i:-. ddilled
fail before time t. Hellce, R(t)
=1-
.lS
lIH'dl.l
1'01'.1
1ht' prob.1 bi\ily t h.1I i t dm':,
1101
F(t). And, 1ht' faihm' r.11(' 1'(1) al 11111(' , IS (It'fillpd
.1:'
r(t) = f(I)/ R(t), where f(l) is the probahility dpllsity fllllfllon for T.
Chhikara and Folks (1977)
~howpd
that t11l'
illvl'r~l'
Gaussiall dbl rihlll 1011 h.I:'
rate r(t) that is nOIl- monotonie, ini tially illcrPélsing and t.IH'1I
dp("\
f.1Î\" ....
Pilsi n!!;. Tht'Y .dsn ::.how<,d
that the asymptotic (long rUII) failllf(' ratp is cOllstant and ('qu.d to À/'2I,l. TIII'Y
lIol. .. d
this failnre rate pattern is very prevalPllt in appli(',ttiolls alld frt'<!II(,lllly
1h.. iIlVl'r:,<,
Gaussian an appropriat.e lifpti!ll(' 1110<1('1
Allot.h<'I' ('«(lI.lIly 1III pur \..III\.
the inverse Gauhsian a attractive lifptillll' lIlodp\
i~
\.h('
phY~Î<.d .I~I)('(
111.1 hPS
(1'.11111'<'
1Il.11
wltH h 11l.1I\1'!">
1 of" Blownl.11l
which gives rise to the inversp Ga,ussiall af. the (in,\' pass.I/!,1' \.illlP disl 1ibnl Ion
•
.1
11101 jOli
(:ldllkar.1
and Folks illustrated the llhe of the Înverf.(, Gausf.ian as a IifPl.illlP 111011('\ by .lpplyillJ!. Il
to data reported by Von Alvell (1!>64)
011
the dnration of f(·p.tir I.IIll('h for .111 ,Ii,horlte'
communication transceiver. They round tiraI, th(' inVNh(' Gauf>hi,L11 lIIodp\ plOvitl,·d
il
J!;ood
fit. The calculated value of 0.0526 for th!' Koirnogorov-Sllliruov ht.üihtir is Ilot hil!;lIdi( .1111,
at the 5 percent significancc lewl.
(f) A Frazlty Mode 1. A key application of thp J[oug.I,trCl dU'Jtrihutioll ih flh
.L
fradt.y l!loch·l,
in hcterogeneous populations. The g('IINal Il oug.Lard d Ih trI hu tion W,Lh ("'ri vpd hy 1/ OIlI!;.la rd
(1986) to serve ah a frailty IlIodpl.
COflllIIOIl
lifp tahlp ",('I.II(J(b d,hhlllllP
undcr investigation is homogelleollh. But it Ih 1II0rl' ,Lppropriat.(·
•
1,0
tll,L1,
t.hl' poplll.lI,ioll
(OlIhldl'l'
1,".11, 1,0
individual in a population thpre rorn'Il[)(J/ldll a rl'!at.iv(' rillk wllich ih rrll',Lhllrpd hy
il
called frailty, say Z. The hazard rate at ag<' t for any JH'rSOll wil,h fr;lIlt.y h il>
t1WII
1'.1<"
qll.lllfify
givl'lI
as ZIL(t), where Il.(t) is an age effect ahhurned to he l.omJnoll to ail irHlivldll.L1h. JlOllg.lflfd
•
dl',1 1 dIIJlI(jIJ.-,
hdVI'
fI.ltl'y IIIOdl'l, II. Ii>
pr(JJJI'rtJ('~
d(·~ir.lbl('
tltat llIakp thf'fT\ attractive as frailty distributions.
that
f.lIrttly of c1il-,l,f1blltiom; if. (Io:,('d
.11ll(J/l~
hllfVIVCHf. of ally
~ive/l
th(~
UJI(Jpf
For a
di1->tribution for Z be f>\lch that the corresponding
mortality &elcction, that is, the distribution of Z
app /llust b(' from the same family. It is also debirahle that
t.h" dH,I/lbutlOIJ of h 1)(' infinitely divbihle and that the distribution of log Z he infinitely
TIIf' nl'I'd lor t1\(. illlillit(· divisibility of Z is justified hy 1I0ugaard (1986) based on
titI' fl/II()WIII~
( 01 1
IlIod,,1 for rOlllpeting risks. Suppose thcre are k causes of death with the
('i>pOIHIIlIP; fI .lil tipi> hl, . .. , hk and bd.f-,diJl(' hazards III (t), ... , 11k( t). The hazard of death
d III' 1() (.lll1->P 1 {()lId itinn al on t.hp fr.lIlty is th(,11 h,ll t (t) and the total hazard is
Ai>nllllllll~
•
Ih.l!, tlll'r!' eXIi>ts 11(1) <llld pof-,itivp
('{)n~tallts
Ch"
.,C Ie
2::: 1 Z,JLt (t).
such that JLt(t)
tlll' t"t.1! h.lz.lJ'Il b Il(/»=~=1 (',Z" givillp; ribe to a model with fra.ilty Z
= Ctll(t),
= 2:7::1 ctZ,.
If
t J". dlf.tl ihl/t.ioll of h if. illfillit('ly illdivibible, buch a. brcakdown into cauSeb would then be
)lJ.~1
dÎl'd. tu. ilJdlc.tI,p<l by propprty (h),
PC a, fi, ())
is infinitcly divisible.
II ii> .d:-;() of pl adÏral hpll(·[it if the frailty distribution satisfies produet results. For
1I1~t.11J«·. 1'10111 J(()ll~.lclfd
Il .1I1t Y Z,~
.1:-;
1'( Î', 111,0),
([!HW) it follows that if frailty Z. is distrihuted as P(Q,O,O) and
W h('1 P
ZI and Z'2 arc ind<'pcndent, then prod uet Z ::: Z 1 Z~/o follows
/}(OÎ,h"II'OI--y,O). Prd(·rahly, the frailty difltribution should be a product of independent
1<\c>lllir.dly di:-.tribut.pd variablps, that is, log Z should be infinit<,ly divisible. This property
b
~,It.I~fi(·d
~1'1l1'1
.t!
SOlllt'
,1~~llIlll'd
•
hy pOfliliV<' stalll!' and gamma distributions but not by the IIüugaard family in
1('i>llIls whirh follow when Z is distributcd as P(a,o,O) will be presented. It is
Ihal Ih(· h<lz,1I'l1 for
.t
pprson with frailty Z is ZI'.(t). The integra,tcd individual
·Il
•
hazard is defined as
= fol p(r)d.r.
M{t)
p.(iO)
Supposing that the conditional slIrvivor dist ribllt ion gl\'<,n fra li ty i!> 1'( '/' >
il n
=--
- /, .\1 (1) 1.
1'\ p[
~1I1\'I\'1I1
it then follows from the fOflll of the Laplac<' I.ransforlll in (2.l'Q th.lt tht' poplll.IIIOII
fUllction is
b
P(T> i) = (lXp{ --[(8 + M(l)r' - O"]}.
(:Uil)
n
Moreover, from Ilollgaard (1981\), it is
kllOWII
11I.tt h is di!>t.rihlll<,d
.IIllOIII~
:"lIrVI\'IlI:"
.1:"
P(a,o,O+ M(t)).
The usefulness of the I10ugaard distribution for .Ipplir.\l.ioll a:..
.1
fl.ully 11I11t1 1'1
trated in IIolIgaard (1986) wl\('re h<' 1Il0<1<,1s th(' fr.Lilty of (l.lIipllts :-'lIlrPrIIlf,!;
infarctioll by a lIolIgaard dbtriblltion, P(n,b,lJ). 11('
•
W.IS
1'10111
I~
11111'"
1I1\,()(;lIdl.d
hU(lplll'd wil.h 1.\1(. d.lla of 11·10
patients admitted dllring the years 1977-l!)79 at Glostrup lIospil..ll, \)"11111.11 J..
III t hl' fir:..1.
few days after infarction the mortality rat<, is hif,!;h and t.\H'n d('('I!IH'S. II. IS po:..llIl.lfl'd 1h.11
this decrease in mortality is dlle to h<,t.('rog<'lIl'il.y hill<P t.h ..
(>.11.11'111:-.
ar .. 1'1((11'( I(·d III dillf'1
grcatly in the way Uwir hearts )lPrforrn.
Choosing the individual hazard r~'.;c Jt(t) 1.0 h<, 1, thl' prohahilit.y of lll'.It." ill d.lv t., t.h."
is, from time t - 1 until l is
Pt = P(7'
> t-l)-P(7' > l)
=
[1
{
+ (l-
ex p [-
1)/8]-6 - (1
t5 { ( (} + t - 1Y'
+ t/0)-6
- (}" } / lX 1 -
('x P[- Il { (() 1 t)" - 0" } / (f 1
(:UI'2)
Applying the model to the sUfvival data for the patipnt.h, t.h!' Iikl'llIlOod fil II! tlflll If. IlIilXI
mized to get the estirnates for
•
n,a and (}.
Sill(,(' IU'art failllre Îh <ln import.ant. dihlillJ!;lIif.hing
characteri&tic in a patient's prognohi&, the rnùdcl is alfio
patient& with and without failure.
42
fitt(~d
fieparatl'Iy to tlu'
~IOllpH
of
(H
Il),
(().
ft
1
1
•
IIl'fp/IIg/'lJpjty
fil
tlll'
1101/1
rI ,1
mpa<,urpc] by 1hl'
f'~tiTllatpd
TII1I~, for
01 /1I'f ~rolll'.
1III' f.!,fOIlI'
10 :\
<."!lWI.f!
7
By
for patients
~plittifl~
tlJ('
pati(~lIt.~,
Hougaard &cparated a group of zero mortality
of p,tf,il'lIt.:-. wit.h }}('arl. fa.lluf(·, explaining the decrea&e of hetcrogeneity from
1I1111~,.t"ld\
qllJfI'
00
jI,IIWlltH wlt.houl. hparl failllTe the IllortnJity is practically nil cornpared
i1tudy
1:-' ail
(·frl'ct.iv(· applica.tlon of the heterogeneity concept and
(1l'dlly t!l(' lIi1pfulm':-'s of tll(· Ilougaard distribution <1& a model for frailty.
•
•
coeffiripnt of variation. A heterogeneity
1 lé> obI dllll'd for ,dl patil'f1t~, :\.7 for patients with h('art failure and
!)
WII "oill.
1.0
I~
·13
•
Chapter 4
The Hougaard Process and its
Applications
•
4.1
The Hougaard Process
Lee and Whitmorc (1!)!)2) defincd a falllily
ofsto(ha~ti(" prO(!'h"'('h
Il.1:-.('(1
011
tlll' IIoli/.\" .. rd
distribution family introduced in chapter 2. Thifl !IPW typP of pwn% Ih (.III"d .1 1/011/.\.1.11 d
proccss Il (ft, b, 0). A lIougaard IHOCPflfl /1 (n, b, 0) ih dpli III·d
.IS .t
hlo<lI.lsl,l(
IlIl JI
Phh ."J'( 1),
with S(O) = 0 and LaplaCl' traIlsforlll
l~[('xp{ -uS(/)}] =
wherc 0 < u: < l, li > 0, () 2
('xp{ -l~[(f)
ft
+ 1/)"
- (J"]},
(·1 I)
o.
The farnily of Hougaard :-.tochastic prO(PflhP~ i!lrllldl'~ illVl'fhl' (;.III~hl.111 l'roI Ph~f''', p,,·,t
live slablC' motions and gamma prO(PflM':-' ,th :-')II·d.1I ('.tM'~, a.'> w" :-.1I.dl h('"
•
distribution when {) is relaced by M. Thu:" it cali he dpdu('('d tltal. tltt.
has the IIougaard distribution P( ft, hl, 0).
/lOW.
riill<10/ll
VMI.lbll' SU)
•
.\,
III
,j
Il ...
of 1Iii'
«,fl"I'!jljl'fJ(1'
1'1"11
,c,
ob'>PfY,ttion, it follows from (2.44) that when
Ct
= 1/2
1/((t,h.fI). S(/) ha:, 1111' iIIY(~r:,(! (;allf>Slan dif>trihution !C(/Ll,>.t 2 ), wherc
'1.h~
!1
pr (jl
,dHJ\'1'
'11111'>, thl' l/ullg;,tard process 1I0/2,h,8) is an inverse Gaussian
l'."
~lllld,tlly, wllf'lI
"dll«l,fi
À
\\'111'11
Il
J III.dl\. [(JI
1/0.
--,0 rt.
Ir
i~ kllOWII
from !i2.:J that 8(t) is distributed as G(>.t,f) where
I/(O,Ii,f}) is a gamma prOCCf>b.
111'11(1',
l, III 1,f1,O) l~ thl' dl'tf'fllllHibtic proceSb {hl}.
fi
{S(/)} 1'> ,\ p(J~itl\,(, f>tabIP motion with in.icx
(J.
Ct,
as indicated by the
11)1 III ,,1 IliI' I..q>Jdl l' 1 r" Il,,fur III al (J = 0,
( 4.2)
J
•
J.I'I'
10111 (1 1),
\1,(' (.III
dlld \\ïllllllll)l'
{,'(I J} 11\
~('P 111,d
(I()!)~)
dl'II\llIg Ife,
{S(f)} hale> positive, stationary and indcpcndcnt incrcments.
cI",>< IdH'd tlJ(' :',lIl1plc path behaviour of a Hougaard process
U,\'y IpPIp!'>pIl1atioll, which is
III [';[pxp{ -llS(l)} 1
(4.3)
\\111'11'
1/(/1') ---: - . -li- - 1 (1 - 0)
( 4.4)
111
Il''111 1h.'
"ho\'(' 1('pl\''''('Iltalioll, 11. 1~ hnowll that the sam pie path of a lIougaard process
.'\lllhll-..\
1.11111()111
111I1l\1H'r of j11111p~ of random sizes in any given time interval. The number
III 1I1111P" 1'\( ('('d 1Il)!;
a
).',1\1'11
le>il.p
11\ Il') \\ hl< Ir dppPllds 011 ~il.('
•
j'X' .r-1-CI'exp(-(}x) dx.
l'lIt' l(lllm\ III~ i"
III 11111'1\.11
.l
li'
IL'
in a unit time intNval is Poisson distributed with mean,
'lr('ording 10 (.tA).
hl'1l1 i"t i( dp!i \'ùt ion of (,1.3). The number of jumps of size (w, w + dw)
(tU), dl'llotl'd by S",dll'
IS
Poisson distributed with mean -tdJl(w). Notice that
·15
•
the mean of the above Poisson variablP has a lH'g,ltivp
~igll
,lltarllt'd ln Il
dccreasing functioll dnd 1H'l1CC has a lll'gatiVl' gr"dl('llt. For ('Mh
by jumps of size (lL',
11'
+ dw)
is the produrt of the jUlllp
of that size occurring in intNval (0,1); II('IIC(', tht' contribution
aggregate of th(' cOIlt.ributlollb wNwdll', lakpll o\'('r a.1I
S(I) =
l
tht' r(lllt lihut iOIl to S( t)
Il',
~i/,(' Il'
1'(/1') I~ a
\!t'I .. IU:-'l'
alld t Il\' lIulIlht'r (lf jlllllp:-'
i~ II'NII.dlt'
pO~~lbh' V,IItH'~
or
S(t)
II'.
'l'h.t!
1hl'
Ih\'11
IS
1:-',
e'-,
o
wN",d/l'.
The Laplace transfarrn of Nwdw, which is a Pobsoll v.triablp \Vit h
1111',111
--Idl/{ II'),
I~
,..,ivl'lI
by
( \ li)
and heIlcc that of
?J)
N wdw by
<'xp[-ldJl(1I')(r-"'" - 1)]
•
(.1. 7)
Hel1ce, the log-Laplace
tran~form of S'(l)
=
J;'; wN",dw h,
The algorithm fol' generating randorn HurnlH'rs flOIII
scribed in §2.4 can be
Ubpd
,t
to hirnulate a lIollgaard prn«(,h~.
u p succehsively to prod lice the
proc('~:-..
The
t.hl' pic! III"
haviaur suggcsted by the Lévy reprebentatÎon. FOi l'x,l.Inpll', tlrl'
•
increrncnts that are of magnitude greater than 0.10 alld,
tatian in (4.3), the Humber of jllmps excpedillg
IOIl:-.r.,h
of I.wo ""11
,tri' gl'lI l 'r<ll!·d illdl'pl'Jldl'lIt.1y ,lIld
j'HIl pi> III
thi~
~"!'''I
,.<ld,,"
1() ( o,d, r III t III' 1,,·
n.lIllpl!' (),llh
a«ordill~
rnagnitlld!'
dl'
Fi~III(' ,\ 1 rl'pf(,~,l'lIt~ tlll'
simulated path of a /l(0.5, 1, 1) procehS. 'l'III' illlli>t.r,II.1'd hrllllpll' (Ml.h
dled increments per unit intprval. The ÎlIen'llll'lIb
dl~t.llbllt.i(J1I ,t~
lIollgrl;lld
~how~
1.0 th" JJ·vy
III il 1/1111.
Iwo
rl'(lI"~I'1I
liulf' ill1.l·rv,11 if.
•
l '!JI"
,011
dl '-.1/ JlIII!
pd
w/l
li
IfIf"LlI
~.OIJ(j.
1 r:
-1'((
l.,,)
Il\I"UI
l'fJ
:r -:~/2 cxp( -x) dx.
010
•
•
v,iluc is calculatcd [rom (-1 A) by cval uation
'l'lib
·1 ï
(4.9)
•
_. -
a
.--~
-
--_........--
,--_..--C\J
a
•
0
0
-
--.
L-~--------------r-------------~------------.--r----
00
0.2
04
06
Figure 4.1: SimulaLcd sam pIc path of a 1/ (O"rJ, 1,1)
•
1 Cl
(llllll'
,
•
Stochastic Proccsses Directed by a Hougaard Process
1.2
ï III',
Il'JlI I)t'glll:. WII h
',l"
M'lrkov
1" d
111"11"',
thplI {X(S(/»} J!->
{."·(l)} i:, a proccs& with nonnegative, ind<'pendent incre-
~ubordill(Ü<'d
il
procebb.
{X(S(t»)} is said to hc subordillated
tll" p.lll·IiI IHIH'I'''-'' {.y(t)} u!->irrg {SU)} af, operation al tÎm('. {Set)} is known as the
dIIP(tllll~
'.IIlllf'
PII/II"-.'. {X(I)} .IIHI {S(I)} arp uf,ually
dll/;,II.i1irHI;' of t 1If'
~J.llldplI)I()t
1
pxpl.watioll of what JS rneant hy a bubordinated process. If
pro(I'!->~ illld
(,\ (I)}
1"
Il,,('
tak(~n
to he illd('J)(llld(lllt processes. Next,
of huhordillated pro(css(',<, are providpd.
,1IId Taylor (I!Hi7) lI\()(I!·lIpc! stock price fluctuations by a subordinated pro-
r';';', {.\ (S( I)}. S( 1) rpprpf,('lIlpd t.!tp cumulative volume of transactions up to calendar
.Jlld X(.... ) tlll' f,t()(k lllicp c11<lng(' ohb('rv('d during a volume of transactions s.
1111((' f
III,iI\,,,. '.f'llhr·
•
.III
111'1
dl)'/'
liJoili
III
If)
,1111"
(!ir)();,f'.t nlll)(Jldlllated prO«'bl-> X(S(t» as a mode! for l->ecurity priee changes
pl 1«(';'
11'11<1
to
(h.tIl1!,('
IIIOIP in rUcordance with the level of economic activity
wlth 1111' ~illlpl(' pa!->!->.I/!'p of timp.
1)(·
.1
It
III
Mandplbrot's mode!, {X(s),s ~ O} was taken
Wil'Iu'r plO('Pf,f, and {S(i), t ~ O} a positive 1>tablc proce1>s.
Other authors like
n,II!t!'\' .11111 HII!->rhl'llClorf(I!l!H), Hachev and Samorodnitsky (1991)and Westerfield (1977)
h.\ \ (' .IInll
I)(II(,~.
111111:-,
Il;,(·11
!-> \1 h()rd i 1\.11 pd
Lo modpl f,PC Il rit y price1>, stock retuflls and option
FOI illn!.1III (', followill)!. !tIn inv('~II!!,atioll of a largp sample of daily comrnon stock re-
\\"1' . . 11'1 fll'ld rOllnd ('vidPII('(l 1hat tpnd('d to support Mandelhrot's subordinated model
{ \ ( 'i li))} pl 1'''1'11 Il'd
IlIr 1(,,1'>\' \\'Ilh 1h,'
.lIlO\'p.
\'oltllllP of
III III \, ~lIh(lrdlll.\tl'd IlIodpl
\.1110111\ l' ~lIllllldill,lIl'd
•
JlI'U(,('~1'o('f,
T!tp absolu tp magnitude of stock price change was foùnd to
trallnclctiollSj
110\\,('\'(,[,
modd or
a
finding which is indecd consistent with Mandel-
Wpstcrfipld's results did not agree with the infinite
~land('lhrot.
.\lIothl·1 plltl'Illial \l1'oP or l'oubordinalion is in equipmcnt usage. X(.s) may reflect the
dt'lt'llor,llioll of
,1 1II.H'!til\(,
.Ift('r ... hOllr5 of opPfatioIl whilc Set) may reprcsellt the total
·19
•
hours of opl'ratioll of the lllaehÏlH.' durin~ tinll' int('r,\"t1 [O.q. IIN('. {S(i)} 1l\(',\~IlIl':- t ht'
randOIll opcl'ationalusp of the Ill.lchin(' o\,('r t Ïlm'.
Usually, the choiCl.' of t hl' opPrat jonal t j IIW nH',ISII n' H, dPI {'rlllill('d hy t Ill' Il.1111 1(' of t 1\1'
application. For cXilmplc, at
il.
tl'l('phoIlP exchdngp t 11(' I('\,pl of MII\'lty dlltÎlI/!, ,III h01l1' of
thc night would not be cOlllparable with that
is busicr. It is dcsirable to lIItrotlllC(,
,l
durin~
rhall~('
pN
th.1I t Ill' '''111'111
pl(l«("'~
dir{'ctlll~ Pl()('('~~ i~ ~l'll'dl'd sn
1h.11 1 hl'
in tht' 1 ill\l'
is recluced to a statiouéll'y proCPSb; in otlll'r words,
expectcd Humber ofcalls
an hour of thl' d.lv will'Il thl' (';..(h.IIl~(,
ft
~('ah, HO
unit of opl'rat iOIl al titll(, f('lIlains conslant.
FOI
processes, thc dirceting proecss Illllht hav(' I\oll-n('f.!,ativ(' ill('I('IlIl'nls. For
applications the dirccting prorCf>S also lias hl a,tionary alld
cabC of the stock priee chang('
•
pIOCC~S
11I<!l'\H'lIdPIII.
.1
,11-\11,.11 Il III 11h1'1
11\('11'1111'111:--,
of Mandl'Ihrot .lIld Taylor. TitI'
posscsses thcse proprrti('s .1IId thus lIlay bl' hllitahll' a.h
.dl IIII\(' 1I'I.III'd
.1:-'
1I001/~",,,d
of
ill 11\1'
PIO( "'-0',
din'( t illl-\ pro( p""
Before colllIncnting Oll thr 1!'lationship hptw('(,1\ a slIhordinal"tllll(l( ('~:-.
timc proccss, let us lirst illtroducc the notion of ,1 nat.ural
I.illl(,
alld Il'-0 dllt'ct III~'.
pnJ((':-':-' .Ih d"fill«'d Ity Lc'"
and Whitmorc (1992).
Defi nit ion 4.2.1 A rULl uml lim(' pr'ocrss {S (t)}
(J)
TloT!T/('[jalivc, slationar'y Will mdcpcTu/f'1tl
(2) E[S(l)]
=t
t.'l
a slt}('Iw:; 1U' ln'()(', ,'l," IJIlllt
17U'1'e7T1f' nis
aud
fol' ail t.
The expected incrcIncnt of prO<:l'!-.h {S(t)} thal ()(CUfh durillg
by E[S(t
+ 1) -
ally
unit t.int" 1111,('1 v,t! 1<,
I;IV«'II
S(t)] = 1. III oLller words, a natural ti,,\(' I)f()('''~h {S'(/)} 111",1',\111'<, 11If'
passage of Lime a.t the fla.n\(' !'x!)('cted ratf' ah ib tilllP IMr;ul\('I.,'r 1.
When {S(t)} is a IIougaard procl'!>s lI(n,IJ,O), 1,'[8(1)]
•
natufal time proceHS the condition
/)0,,-1
= 1>10,,-1.
So, for {S(I)} 1.0111'
== 1 iflll"(,(,h~ary, ln pafl.illll,lf, {S(I)}
tillle inverse Gausbian IHoePfls IIO/2,h,O)
Will'Il b()-1/2
GO
= ] ,wei
<1,
/Iatur.l1
01
Ih il
1i.",UfilJ
fwlf'
~,lllltrl.1.
•
I/(U),fJ) Wh!'11 hO- 1 =-: 1.
(d'"''''
011 Il!I' {JIIJf'r Ir .1 IId, "LJ!JII'
Il',
.J
.1",1".(
.111'.('
prOf f'''''(,:-'
(Jf t.hp fortn Il (0:, h, 0) cannat be natural time pro-
1';[,','( i)] dol';' nof (';"1.,1,. Thl'fl'fon~, whcn a pOf>ltive stable proress is used as
dill'( 1III).!, PIO( (..... ~
TIJI' dl,/)/'lJdI'1I1
WI' l'XJ)!'(
(>
t. II. fo J1ltroc!IHI) ;,orne erratir behaviour into its suhordinated
IH't.wl'l'l1 t.hl' hlrllOrrlillrlt."c! proc('hs {Y(t)} = {X(S'(t))} and its dir<,cting
For imtancc, when Y(t) rcpresents the
{S(I)} (0111 1", (Jf pl.I(fi(,t! imporl,t1lc(,.
pl'H'"''
1'1111' IIIIJ\I'III('IJI III
,1 h('(
1I11ty ,l/Id 8(/) t.h!' volulO(' of trading in the security then this
01 (Olfl,l,llion lH'lw('('[\ priee rnovernents anù transaction volume.
d"(lI'lld"1I1 (' .IPPP,llh ,Ih
1\111I.dly, WI'"ll'llipld (I!J77) fOlllld t.hat the priee rnovement is c10sely connected ta the
1 1.llIh,I<lII)(1 volulII<'.
•
Whl'II t 11<' 1I0111-'>.I,lId pro( l'hh Ih
Illllll('b, dn
for
('111<'1/1,<'
\Ihcd ah
a dirccting proeess, intcr<,sting and manageable
ht.llldard pan'nt. prOCl'sses. The Poisson-Hougaarù process,
;'0111('
dl'lilll'd by Lpp alld Wlllt II\O!'(' (1!)!)2), 18
Olll'
('x ample.
I)Plillitioll '1.2.2 [,Il {.Y(/)} 1)(' a slandard IlOntogencous Poisson process with X(O) = 0
{S(/)} !J,
11I1t1
/'lit
Il.
Ilu
l.i'l·
Il l/olJ!lll(JI'(I,)/,()('('ss ll(n,~,B) as
,"I//WUlllllllld
]J/(J('I ,"8 {r(t)}
.Il1d \\ hif IIIUIt' (l!)!):!)
l'I<HI'''''
'1'11.111:-',
())('l'I<l((':-':-'
= {X(.'i(t))}
~hllW th,t! ,1
P'(/)}
defincd in §1.1, independent
28
cal/cd
Cl
of
{x(tn.
Poisson-lIougaard process.
POiS8oll-Hougaard process is a clustering Poisson
= {X(S(l))} is idcntical to an intcger-valued compound
l\lI',~,ilJl Pl()(('~:-' {2=~~:) (J)}, wh('n' N(t) is a Poisson process with intensity parameter
.11111 {(J)} ~. . . ~ 1
\h.\ :-'('<1 IJ('JI('('
of i.i.d. IlIh'!!pr-valucd random variables. {Q J} ~1 is independent
Id {St!)}. Tht' 1I1llllhl'r of r1\1~(,IS arriviJlg in till\c intcrval
•
"'I.'I'~
,Ill' () 1. (J~, (}
\ 1) l'Ill'
11tl~I('J'~
1,.· ,
III
À
q \ (1)
(O,t] is N(t) and the cluster
Thp following rcsllits were obtaincd.
Pli)} .1Irivl' .l('(OIdlng to
51
(t
homogeneous Poisson process at rate À
•
where
(LlO)
(2) The probability gPIlt'r<l,ting fUllctioll of IIH' dusl!'1
l~(rQ)
(3) The pl'Ohability
lIIaf,~
= [(H + 1)"
fundion
of
q
l'tl/lUI 1
- (0 -1- 1 -
q, t!Pllot('d
~i/,(,
i~
1)"
( 1. 11)
(1"1
by P'I' fi -- 1,2, .
,I~
If 1/:= 1
( II:!)
il fi
(4) Q has finite mOlllPllts of ail
ordpr~
= '2, :1, ..
if (1 > 0, and only of ()rd('I~
~III,dll'l 1b,,"
n If (}
()
(5) The probability gPIlt'fatillg fllllctioll of }l'(t) b
E(rl/(t l ) = exp{-t,\[(O-l- 1 - 1')" - (J'"l/[(O-j- 1)" - O"I}.
•
The authors devclopcd infNPnce !Iwtho<h, for t.hp
( 1 1:1)
P()i~:'OIl-II()ll/!;,I,\f(1 pnH l'~,~, fOl hol
complete and ccllsorcd samplcs, and iIlustrated tJlPir appli< ,ltlOI\ t.u illt.I'I('~1. 1',111' tille
A second bllbordillated prO('('bS disCllb:,pd by L('(' alld Whi11l1011' (1!1!12)
I~
il
111 .. 111111,
1.11('
~t.d,l{,-
IIougaard proccss.
Definition 4.2.3 Let {X(l)} be
(l
stable
7/tollOn
1IJzth X(O)::: () (Lud {S(I)} Iw
process ll(o:,8,O), zndcpendenl of {X(t)}. 'l'hm, {y(t)}
:=
IL Ifmlllllll1d
{)f(S(t))} z.~ ('(dIu/il .~/llblf-
llougaard proccss.
Unlike the Pois:,on- I10ugaal Il proc(':-,:" in 1.'/'III'f,t! t hf'
IIougaard prOlebS is complex and
•
illustrate the point
WP
~I"lbll'
<>nCOJnr>a:'~(>~ d WHh: r<lll).';1'
Il o IIJ.!"I ,If d
pl III l',,,,
of :,tO( h,,:-,I,J(
shaH pre:-,pnt the pxarnple provid"t! by LI'I' ,llId
X(t) '" N(O,a 2 t) i1> a Browniall motion that b dirpcted hy an
IIIVI'I:'''
I~ 1I0!.
1)/'1t.IVII)11
WhIIIlIOI",
n,
.1
'1'0
whl'rl'
(;dll~>"I,lJI IJJIIlI'~')
•
{SU)}, Whl'fl' ,<,'(/),-.." f(,'(f,(U'l). Th" invc~r~e Gauosian process was defined as a natural
1JIIJI'
PIIH 1'\'>
~,() 1!J,Lt
tlJI' p(rpc t of
Illit ".IVIlI)!; 1,0 Llkp illio :1C(OIlIJ!,
III
/,1'1'
<lfld Whil./lloff'
(l!)!)~),
~lJb()rdination
on
t1H'
parent process can be studied with-
the hC'alinp; eff(lct due to the directing proccss. According
{X(S(l)} Itah tlte Laplace transforrn
(4.14)
.Il/ri
Il,,, prob.t1l1lil.y dl'Ilhity fl/l1(
tioll
(4.15)
wIw/('
/Il --
../(1 2/--;.1. /linT)
,1IIe1 1\",\ (
Il) t!Pllol('s a llIodified Des&el function of the third kind.
SJIICP 1, 1If' IIIVI'lll(' (;allslliall prOC(1Sh is a natural time process and thc Drownian motion is
:--t.allo'J.try III Il)('all, il, follows !'hat bath the parent process {X(t)} and the subordinatcd
•
PIlII('hh {X(S(t))} h,IV(, lIu'all 0 and variancc
,111<1
1lit'
(J'lt
at time t. Thus, the subordinatcd process
Iltl' P.tI('lIt (ll()({'hh bo1.ll h.lv(' tht' &,ttllC fir&t and second momcnts. Moreovcr, since
:--l1bOlcil/l,t!,('d
(>I0(
Ph&
tht' parent prOC('bS arc bath symmetrical it is difficult ta
<1IIe1
di:--I JlIgUI~II t ht' I,wo pro('('&&ps. Th!' efr(lct of subordination may he apparcnt only when
loohill/!,
,II,
\ ()1I~1I1(,1
t'cl. Th(' sl.l/lda rd I!'pci
Ih.1I Ill' th\'
t.lt!' hurt.ohis of (',\,( h prOf('HS, and J)(lrhaps Ilot Hlltil ('ven higlH'r momcnts are
c111\'(till~ PIO«'""
lIH'aS\l r(l
is
a.
of k Il ri osis for the su bordinatpd process is 3/ f3 while
11('11<'(', tht' two proc('f>&es have the same kurtosis when
,J _ 1
A ,,1.1 hl(' pl (l( ('l'oh di Il'flet! by <iTluther Htable proccss, referred to as a stable-stable proccss,
I~ ,1 t '.lrI abk
"11H11'!. A htahlp-stabl(' proCPSh lies in the the stable-Hougaard class. Dy
1,\,11('1 (Ini'!), wlll'Il tll<' p,m'nt pW('(lSS is a stable process with charactcristic exponent
•
oIl1c1 Ih(' dill'I'IIII/!,
PIO«'~"
b ,\ stahlt' procpss \Vith characteristic exponent
~lIhllldJII.lt\·d PIIH'('h" 1" .t ~t.tblt' pro('('hS
\Vith characteristic exponent
V2,
Vl1/2.
VI
the resulting
•
Next,
a
a
subordinated prOCl'Sll ha\'ing
lIougaard
S(/)
f'V
proce~s
pron'I>s
,t g,lI11llla
dS t ht' !l,\H'1I1 proll'~~ { \'
as the' dir(,cting pron'~s {St!)} i~ n>llsllh'f('d.
(In ,I\ld
\ (1) "" (,'(.\1. d)
Ll'I
,I\ld
P{o:,bt,O),
The cha.racteristic fuuction of the suhordinal<'d pn)('('!-.s {.\'{Slt))}
l,III 1I',I\Idy lit' dl'
rived from the following proposition pH'st'nLl'd by Ln' and Wllltllllllt' (1!l~1:!). LI'! {\(t)}
and {S(t)} be two : 'leJlendent procpsws with statiollary .11111
IIItll'Ilt'IHII'1I1
IIIC H'lIIl'lIt~
Moreover, suppose that {S(t)} h,ts lIonlH'g.tt.iv(' in('f('1ll1'Il1.S. ASSUllll' t !t.1f { \ (!)} !t.I.., 1hoir
actcristic function E[exp(wX(t))] = PXp(lIi'{U)) itlld t!t,lt {St!)} !t .. .., L.lpl.1l1' Il.111..,101111
E[exp( - u8( t))]
= exp( -t'}'( IL)).
The characteristic fundion
Theil, {.\ (8( l))} !tas dl.trad,(,1 istÏt' fil Il et Itlll l'X P {
or {X (f)}
is
~iv('11
hy PX p(t ~,( It)) W111'1 (' I/'{ 1/)
!hu) and the Laplace tran8form of {S(l)} by ('xp(-/)'(ll))
00
•
].
W!tNI'
)(11)
1) (
r/'( I/))}
,\
1
III (
(h/oll(ll f /1)"
Applying the abov(' proposition t.1Il'n yi('lds t.hp foll()win~ ('!t.lr.\( t"ll~t)( fil 1)( I.IOIJ fO)
{X(S(l))}.
E[l'xp(wX(S(I)))] = exp[( -lb/n)[(O +,x III( 1 - {Jill))" As in thc previous example, {S(t)} is
~ct 1,0
of subordination can be studied, fn'e from th('
Equating E[5'(I)]
= 00"-1
iL
M
(J"lI
( 1 111)
natura.1 tilllP pro(!':,:.;:,o I.h,II. t.h,·
"fI(,ft
aHllg "rrp('t. dll!' t.u t.h" dlfl'( t ill/-', prOf (':-.:-..
t.o l, il. then follow!-. frolll (;J,Hi) th,lI t.1I1·
(11111111.1111,
~('II"I,II,IIIJ.',
functioll of the subordinat(·d prOI:I':-'5 {X(S(t»} 18
l\(u) = In{E[uX(5'(t»)]} = exp[( -W)[(l
n
+ ~O In(1
- flll))" -- Il]
(;J 17)
The mean of the subordinatcd procp:-.s {X(5'(l»} i:-. giVl'II by
(tJ 1H}
•
The parent !HOCCSll {X(l)} al:,o hü:-.
rncall
t'x(J. So, in orrkr
1.0 1IIVI':-.I.Ig"lIf'
ordination, higllcr moment!> of both thc I,an'nt and the !-.ubordill,LI.f'd
I.h,'
l'flf'f 1.
of hub
plrHf'1·>'>f'~' ,III' dl'rlvl'd
•
.tIId
t/iI'lI
(ollip.trl'd. Th,. '>l'(ond dl'Iivative of I\(u) evaluated at u
(4.19)
).',IVf'l'. 1.1)(· van,lIJ( l' for
t Iii' ()t/wc
.III III(
pl (){
t.1J(~
l'.llhordinal,(·d
proces~.
Th(' variance for the parent proccss, on
Ii.LIld, il'. O.(j'l.. Sin('(' ,,\(0.- 1)/() < 0, subordination has the effect of introducing
l""",,,
V,llldll(,('
tlH' subordillal.ed
ill
jlJOCCSS
as compared to the associated parent
("oe,
1.3
•
= 0,
Applications of the Hougaard Process
Of'( Il rfl Il Jo!;
in 1'(·,11 li fp have nonnegative, stationary and il1dcpendcnt incre-
1\I.IIIY
JI l!l( PhhflS
111('111 h
Ir k(· 1 hl' Il o Il).;.l il rd prOf l'l'.:-'. 11Idl'pd, t h(' applications lhat will be discussed hcre exploit
thl' Illllllotollir
!l.II,III'(·
alld I.hl' j1llll!> hl'haviour of the I1ollg.Lard prorcss .
Th(· llolJ/!,.I<lld dist.lihllt.ioll alld tllf' lIolJp;.tard pro('('ss are relativ<,ly new, but the inverse
(;.lll:-':-'I<l1l .uld ga III 111,\ l'''()('(,:-'hl':-' have uecn
011
tll(' stene for a long timp. Wasan (1968)
IlIt lod1l( (·tI 1111' illvPrsP (;allf,hiilll prO(,(·8h. lIaf,ofer (196/1) and Whitrnore (1990) have made
1I~.4' 01
thl'
\\'.11('1
flo\\'
illVl'I~1' (;,IIl~l'.idll plll('('~l'.
.tnt! wlIld
('Il ('fgy.
as a stOtJl.tiltic mode! for natllral phenomenon such as
W(' hhall now hridly descrihc thcs(1 two applications of the
lIlV(,I~1' <:.I1I~loi.lll IIIO('(·Sh.
(a) tI'lllt'/' ,.,'omYI'. A waler rewrvoir lias infinite storage capacity. Let X(t) be the
((llltilll\(}l1~
input of wat<'f which Pllters th(' r('servoir during time interval (0, t]. Denote the
lOlItt'llt III' th(' Il'M'I\'oir dt linl(' 1 hy thp continuous random variable Z(t), with Z(O) = c.
~IOI('II\'t'I,
I{·t r Ill' tIt(·
rcltp
of watpr relp<ts(' from the reservoir, Ihe rate bcing constant
pl()\'llkd thl' !t''<{'l\oil' is Ilot PllIpt,\'. TIIl'Il, for illhtallce, Ihe wat('r content of the reservoir
•
'I~
1-',i\('11 hy 1.'(1)
= (' + .\ (1) -
1'1
dS
IOIl~
as the resprvoir contains water in the iriterval
55
•
(O,t]. Hasofrr (U)6I) was cOllcerned with th(' probability dbtributioll of tht' tillll' tllitii tl\('
rcscrvoir was first elllpty. IIr ~olv('d titis prohklll Ily postlliatin~ th.lt t lu' tn!.ll W.\tl'I IlIpllt
X(t) fo11ows an invcfsr Gamsidll pro('('S~. lIa~()fl'r\, dl~l'msilln of tht' I1\odpl i~ t 111'01'('\11"011
and no attrmpt is lIladr to apply the III{Hkl 10 a Il'al r,tsp.
(b) lVlTul enCl'!lY. A practical a ppli(',lI iOIl of t.hl' i Il Vl'rM'
(;,111 ~~I.III
for wind cnNgy. Whitmorc (HJDU) postlllat.{'d t.hat th{' total \Vint!
pr(}l'l'~:-'
1'1\('1 gy
1'>
.\~
.\
lIIodd
T( ... ) 1!,l'lIl'lat pt!
at a location in an illtcrval (0, ... ] follows an inVl'fSt' Gall~~i,tIl pro('Pss. 'l'III' lit of 1111' 1II0dl'i
to data for this particnlar applicatioll showpd t.hat Hw IlI{HII'I
W,lh
Ilot I1l1r('.\~ol\.\hll' 'l'hl'
case study in the IIcxl section g('t\pralizps Whilmore's Illodd by ('01I:-.idPI ill~ 1111' 11(11).\<1,11 d
process as a model
fOf
wind
PII('f~y.
The pol<'l1tial lise of a. Ilongd.trd pr()('('~~ ,,~ a dil'I'ctillg
plon'!,h
ill
was poinled out ill §,1.2. 'l'hl' fart tltal a Hong.lard PI'O('('HS pOhS(,~h(,h
•
h1\
hordin.tI.(·" IlIodl'l:-.
1I011lH'I-'"d.ivl',
ary and indcpendpllt. incretnrllts makes it ail aUr.tet ivp candida 1.(' [or a. dir('( t 1Il~
Ilougaard proce&ses may find
iL
widl' range ofapplicatiollh ar-. difPIl.illf!:
applications of subOl'dillatcd plOcessp~ ma.y 1)(' lIlodifi('d
1.11 !t'LYP
sl.lIioll
pllH P:--:--.
PIO((,~'>f':-', PXI~,tlll,!.
t.ltp 1I1111~a.trd IlI()(('~~
.1'> .1
directing pl'O('('ss. For cxamplp, M andPlbrot and Ta.ylor (1 !)(H» pOht.ul,II,('d t.h.1I :-'('( 11111 y
If'
turn series can he modelled by Wicner proceHSl'S dir('ct(~d by po:-.itivl' htahlp prof l'h:-'I'h. Th"11
proposaI can be gencralizcd by as:-,ullling a JloU!!;'Ldrd pro(p!'>:-. for th!' dif('IIIII~ prof p:,h. III
faet, the gCllcralized proposai wa,s used by Lee and VVhit,llofP ( 1!)!J~)
III
t l\I'ir
.11'1'111 .1I.ioli of
Poisson-Houga,ard processes to int<!re~t ra.te spries.
4.4
•
Case Study
Wind enefgy can he a userul wurce of efH'rgy. Il ('an abo be f'xt.rpHlf'ly dl'ht./III t.JVp
hJl
both rcason!'> il
t.Ill~
iH
important ta have
doll
undprht,tndIllg o[ itf> IIl1df'rlying
IIIIHI .......
III
•
,1,,11011 1h,.
lIoll~.'l<lrd
pro(I'''''' will bp
Ile, 1I',I'flJlllf''-,'J lOI !hi ...
l'lIl'I)~Y
'1 ~II'
d,LI <1
IJ'.JlIg t.h,.
11IVI'I~1'
(<1,,('
~t.lJdy
Callhhiall
IJ~l'd
'J'hl' .1.1 t.l
plllpO~(>
('omidpf{~d a~
a rnod('1 fOf a total wind energy process .
will hp pxarnined by rneamofa
foJJow1> a
prO(,~H a~
~iJTIilar (,rt~('
cas('~tlldy
ofactualwind
study ronducted by vVhitmore (1990)
a model.
ill 1h" a/laly:,i~ wprp Plovided hy the Canadian Clirnate Centre, Atmo-
',pIU'11I EIIVIIOIIIlJ('lIt Sl'rVICe, Jo;lIvirOllfllPllt Canada to the Crmadul7l Journal of Statistics
1() 1 .t ( 01,,(' "t.\ldy 0/1
11I.ld(,
lIc,l'd
('X
tll'lIl('
W
i /Id ~I)('('(b (Cl'Il Uprnan and Whitrnore (1987». The data were
.1v'lIlabl<' for thih rphC'arch hy Whitmore. Gilly a portion of the original data set was
~llJdy
lor thp
n'I){Jrtpc! h('rc. ThiH Iirnited data set contained hourly wind speeds (in
kill/hr) 11I1',1:'IIIPd al. tll!' VaJlcollvPf w('ath{'r station, Canada, for the calendar year 1947,
.. loLt! of
S7(jO
o!J:-"·lvatioll". 'l'II!' OhSf'fvations WCf(' recorded hOllrly, eithcf fronl a chart
1('«11<11'1 III hy Oh:-'PIVIlIf,\ tlll' 11IOVPIIH'lIts of a dial over <tonr-minllte period.
•
Il H
Il
'1111 HI
Il "pd
From the
101 t h(' d,I.1,(\ (,011<,( t.iOIl, t h{' data can hc descrihed as su bjecti wly detcrmined
[11'-11111111 h' Illl'dia Il
WIII(I
SIH'P<!s. Mor<'ovpr, the technicians fecording the
0
bS(,fvations were
lold tll lIl:-'J('f,\,lld :-.hort ~lIsts and lull8. The instructions to ignore gusts and lu11s, the one-
lIIillUl" 1IIl':t"IIII'IIII'II!. illtI'I'V.ll, alld t.h<> ('!l'lIIent ofsubj{'ctivity in measurement can aIl cause
t li \' 11\1'01:-'1111'<1
pl o ('('Mi
to di rrpr fl'olll th l' tfll<' process. 'l'Il{' firbt effect (ignoring gusts and
Itlll'i) IlIlght p:'pl'rially hav(' th<, ('rrpct of IIl1dpfstating the inherent volatility ()~ the actual
I)n)(('~:,
L('t F( ... ) d('lIol.<, t II<' tot al wind Pll('rgy gl'll('ratNI at the station over time interval [0, s].
n l'(s)/iI ...
i" th{,11 1 h(' wint! ('Il<,rgy flux ,tt t.illl(' ~. Morpovpr, (trcording to llardslcy (1980),
l
k(I'(.,n whl'I(' k j" ,\ pl'Oportioll.lhty const.ant and t'Cs) is the wind velocity at
1\1I\\'~. Thll~,
•
th(' lotoll winll PlIl'lgy pr()r(,~h il:> givca by
(1.20)
57
•
(·I.~O) \\'a~ ('ho~t'n
For the sake of conveniC'llct" II\(' const.lIIt IL III
readings
~pt'I'd
in hl' 1 HOllrl\' "inti
tir
\Vere cubed ,lIHi tlll' (ubl'd \'.tlups WNl' ~ullllll('d 0\'('1 1I01l0\,\'d,IPPIlI).', illtt'I\"t1~
five days cadi in
order
to cn'ate
ï:l Pl\('rgy
a S('qll('I\('(' of
1Il(
n'tlU'lItt-.
fOI t 1)('
for choosing {lve-day intervals is ('xpl.tilWd 1.ltl'r 'l'Ill' tot.lI wind 1'1l1'lgV
be approximated by tht>
\'('.11.
pHI(
l'ht'
n',It-.OIl
l'~:-' t ht'Il
t .111
SUrCPf,SlV(' hllm~
7~ =L~'I~,I= 1,2, ... ,ï:~,
( 1 ~ 1)
)=1
where b.'I J denotes the total wind
CI\Np,y
il\(rplI\Pllt p:q)('rit'lIlpd dlJlill~ thl' Jtll
fI\'1'
d,Iv
interval.
A time series plot. of the T, 'H is glVPII ill ligurp
process hcl'c. As was lIot<'d ill §,t.l,
hypothebis is that the wind
•
.I.~.
J1()l1~;l<lId pIlHl'ht-.('i'l
(,I\<'rgy Jllon'bl>
in §4.1, the wind clIcrgy illrr('nH\lIt~ b.'/~,l
'1'(8) i:, .1
Il .'pp",lr., th.lt WI' h.lv\' .1 jllllip
l'xhihlt
thl~
hllid
'l'hl'II, .I~,
1I01l/!;.I.trd PIIII('I>I>
= 1,2, ... ,7:1,
l)I'h,IVIOIII
of
will (Ollfltitlltl'.1
0111
.,110\\'11
1.llIdOIlI I>.tlllpll'
from a lIolIgaard distributioll, say l'(n, h, 0).
Next, the parameters of the
po~t.lllatcd ([oll!!.aard
prt)((,flh.trl' l'htllll.llf'd, 1t11>1,
ITIcthod of moments and thCll ti\(' lIlethod of 1Il.1ximlllll
qll,\fli
ll~lllg
titI'
likf'liltood
(a) Method of Moments.
The parametcrs D,b a.nd 0 are (\slÎlIlated frolll t h(, 7:1 ohhf'rv,II,lollfl
mornclI t~ as dcscriut'd i fi §:I.l. Th" v.1111eR of
O.7GGO and
dcrived
•
7n3
from
= :1.5:Ja.lJ.
The ratio li
i~
t11(~
salllpl" 1Il01ll"lltfl
0.12:11. Thf'
(3.7), (a.i5) alld (:3.!J),arl' ah
folloWR.
(.,t
O.712H.
0
!1.27HO .
b
O..1!!;H.
:)K
IIhl1l/-';
al P 1/1)
1Il0lllf'rd. ('I>I,Îm.d.Ph
t
-
11f'
Illl't
hot!
Il. 7,1-1 '2,
III
111;,
(Jf t 1If' pU.lIl1l'tl'lf>,
•
'J 'J'
k .~, h, t h"1 t hl' IIfJ llJ.';.j.tJd pfOI ('~~ i,
fJf'(
,':1 flll.\
II., \I\.1IIJ'- r'JI Ih,.
,'JIll ".Il "Ild () 1.lk,1l1!. th"ll
Cjlldllflll('~ D'f~
";,tIJII,tt(~d
él
~o()d
mod('1 for the total wind energy series,
,Hl'comput('Cl using a P(Ct,o,B) distribution
v,dul', Attually, the
11fi' '"ddll'poll!t "P/H(Jxllllatioll (2 :i:l), T10fllldli/.pc! to \lnity,
Il Illf', Il!I' Il,I.dIOII'> (2 /:,)-(2 lï). Ali
11JI' ,,1. "lv"IIIJlI
JI,
al /"'Iiod
1101,10111\ " IlIod"I.llf' dldllU' of
() Ill) 1) III 1)1'
The e"tirnates of
Î, TJ
and
"X<l1
"'1"''' l ,·tI \\ Il'' t III' p(Jll'llt I,d
0111
IIl1iforrn probahility plot in figure 4.3. The p-value
IIrlllg III a "dm pie of 73 obsC'rvations (a probability of
(J('(
T11I~ ~lIgg('st.~
t)
,L
arecomputpd using approximation
i" found to be O.n99g.517. A value this large or largcr
fj
thal thp ohSPfvatioll may be an outlier. 50 the analysis
lier remov<'d. The revised estimates are
•
(t
-= 0 1105,
o
il"
Il''1
Il'\
h
I"pd l'~t 1111.111'" dll' Ill<lfhl'dly dlfr{'f('llt [rorn the
l"I'~"1l1
1-'
.1
d,'1111111'
11'11111\.11
(JI
"1I1111I',101l
1"
,d th.,
1111,'
initially obtaincd whcn the out-
('Olllp,trif>Oll of figure ,I.:~ and figure ·104 shows that there
dl' .lldpd
IlIlPI(J\"11lf'1I1
ill tllp IiI of tlll' lJlode! whell the outlier is discdrded. \Vith the
th" IIl1tlH'1 dll' lit of tllp lIollg,\,\f(1 pro('('ss appears to be quite good. The same
,.111
h" dl d\\ Il fr (llJl t h(' r,llIdOlll ~catt('f plot of the p-valu<,s in figure 4.5, which
",\.'" 1111 IlIdltoillllll
11I,·.I'IlIt'"
OIWS
l'I!2,III1' 1 1 <I,..,pl.\)'-, 111(, \llllfoflll prohability plot based on lIIoment estimation
\\111, ,,1>''''1\.\111)11
•
~lIh~('q')('lIt (l-v,d,){'~
'/ h,' P \.11111'" ·111' dl:-,pl.l)l'd III
Il)
l'.
arc computed \Ising
v.lll'" "'" ll'qlllf,·d for 1h(' (omputatlon afP obtained from the estimates of Ct, 0 and B,
/,( -r)
J .. ,
p-value~
.iI
(JI
~t!OlI!!. 1I(l1l-~latl()lI.t!'it~. It th\l~ follow" that the wind energy series
li\ l' d.l\ 1IIIt'n.t1~
olltlll'l "bllh !-\'('111~
h \\\'\I
(0
Il'PfP~('lIt(ld by a lIougadrd proc('"s, with the exception
li\' ollbidl' the ~cop(' of thb mode!. The olltlier might not
59
•
secm so cxtremc if gusts and lull __ lI,lll b('l'Il rl'ford('d t hroup;hotlt t h('
.\ftl'r fl'IlIO\'al of tht' po~~ihk llutlIt'r, tl\(' ,In,d~:-'I~
(olllplt'!('d h~ l'~tllll.ltlllg tht' prt'I'l
sion of tlte e~tilllate~ tlsing the j.JCkkllift' plOn'dult, ,I~ d('~rIiIH'd III fi:1 1. ll:-,ing (:1 Il) ,111\\
(3.12), the C'stimated covarial\('p mat Il'>: for fi,
The corresJlondi ng correl,ttloll
•
1:-'
~('fI('~ •
The correlation
Loeffici('nt~
h and Bb
foulld to hl'
O.IO()()
-(U~Sl
-O.:IIIS
-O.3SS 1
1.G:J 1fi
1.:,:,1:\
-(U 1U~
1 :,:d:\
1 :):t:'!")
1.0000
-O.!):2!)S
-O/{·\:'Î
-O.!)2m~
1 0000
o !JS02
-() S,I:,7
O.!I~02
1 0000
III Olt
( 1 :.':..')
rix is
Iwill),'; IPI,ttivply
(10:-'1' 1o 0111'
ill
,t1)~I)IIII!'
that the parameter ('~tilll,ÜOI~ are IlIghl) (orrl'I.tll'd wlth
('(I(h
V,tllI!'
nt 1)('1
1" ,t (1".11 1I\f11<
\1"111).';
dtlOIl
(:11:1), (:11,1)
and (3.15),9.')% confidmcp int.Nvab fOI 0,0 and () ar"
~
(1
1)
< 0 < Il :20!H,
- 0 !)007
(rU!')
The confidence intervals for thp IMrampt('ff,
intf'rvals for
Q
0
.tlld () an' 1 (·I,If IVl'ly
Will .. ,
ill
f,1f
l,IIi!'
and b ('ncompa~~ i/ladlrli~flib", V,l"/I'~. 'l'hl' III<1dlfliflfll}lI"ty (dl) 1)1' ,l'v'Old .. d bv
imp\ctllellting the jackknifp IHO('('<!IIf1'
COllsidcring the
•
ft,
011
,tppropri;JI(' tr,lfI'-,ff)rrll,Jlloll~ of thl'
tl,U1~f()rlll<lll()lI~ ~llg).!/·:-,I('d III
admiosiblc cf>timatc of
0:, obtdill(~d
f!:l 1.
';(,7) I~ [<Jlllld
whell the !jïth o1J""I\rctllfJlI j.,
causcsln[Q(s7j/(l- (i(57j)] to be un(j('filwd.
'1'0
(J.'j:I~
If) IH' 0
11I·11'1(·d fr{J111
rf'llIl'dy to thl'-, proIJ!r'IlI, Il
1'.I1,1I11f'1"I',
'1 Iii',
1/1
t.11f' '-,dlllpll'
1'> ,1'>',IIIIJ"d
th,JI
•
lit,.
J/1.J{lrlli~~ibll' ('~t.illiatf'
II)
l,
VI'I
1(il
.1 Vdlul' (Io~('
of
i\ "d, 1lu'
1,0
11.1 I1~flJllIl('d
pOnnlbly an:,p:, hecau:,e
is near 0; henc(>, Cl(57) is then set to
t hl' b01JlIdary O. 'J'hen, lIsing (3.16), the covariance matrix for the
pa 1 a/lll·tl'r:, i:,
1'01 1 (,<'p()lIdill~
12.:30~n
-224lf.i
-1.26.59
-2.241G
0.4346
0.2:171
-1.21).19
0.2371
0.1422
( 4.27)
(,(Hf(·lation lIlatrix is
1.0000 -0.9692
-0.9568
-o.%n
1.0000
0.9536
-O.!).1(iR
0.9536
1.0000
Fil 1111 (:I.lX)-P 20), il. I.h(,/I follows that (1 -
•
Ct
( 4.28)
,13)100% confidence intervals for In[a/(I- a)],
!J,(h) <llIcl IIl((}) .\1(' IP:'P('( t.ivply
- !Ll:)X()
< In[n/(I - Cl)] < 4.8393,
- O.RI94 < In(b)
< 1.8108,
O.2i3{j < ln( 0) < 1. 778~.
Frll.dh. il
(.\11
( 4.29)
(4.30)
(4.31)
('(\sily hp showlI that
.0001
< Cl < .9921,
( 4.32)
(4.33)
1.314i
( 4.34)
,dIU\'!'
wilfidt'/ICC int<'fvals for a, li and D, it can be deduccd that even after
t 1.\ m.folllllllg t Ill'
para llH't t'rs, the Îlltervals remain wide; causcd in part by the high multi-
(o'WIII
•
< D < 5.9191.
t hl'
wlli 1\(',111 t \' of 1Ill' paralll<'tt'r ('Stlll\,\ tors which inflates the standard crrors of the parameter
61
•
f('Il.lr.llll('!ri.~a­
estimators. This condition sllgg('sts a n'parametriu!ion is in or<l('r. Surh a
tion is considcred in conncction wit.h the maximulII ql1.1,si-hJ..plihood 1lI('! hod whu'h
i~
L\J..I'II
up ncxt.
AfaxÎm1l11l Quasl-IlkclzllOOd !\fctllOd. 'l'Il(' paranwtl'fl> n," .llld (} .HI' .d~(} l'I>I lilial <'II frol\l 1hl'
sam pIe of 73 observations using the maximulIl qllasi-liJ..plillOod I\\('! hud lI('sl nhl'd in
The sam pIc mean
value of
0'
x is found to he 0.7142 flO t.ha!, by (:l.:m), JI = 0.7111
that maximizes tht, log-lik<'lihood fllnrtlOn
à
Then, 1Î is obtained from (3.'11) alld
'l'hl' I>I',\\( h
~i:1 ~.1
r'lI
Ihl'
p .. 12) yÎl'hb
= 0.23011
i from (3.'10), r<'pla(lng n hy
l\
III
1',lIh
(.I~1'
'l'III'
estimates are
•
i,
1 '{i·H~(i
(,1 :11;)
-O.!HG2.
The corresponding estima tes for li and (} art' obtaincd by flol vi Ill!; U((' ('q Il.11 1011 li (2. If;)
,llId
(2.15), yielding
(·U7 )
(·1 :IH)
The est' 'nates are
b = 1.01H,
(~
iJ = 1.500,
(~.~O)
IIere again the p-values arc romputed as a tPllt of fit of th" lIougaard
•
IIfI){Pflfl IIlOdl·1
The uniform probability plot is given i li figure 4,f), AIl bpfon', t}u' p- val UI' for
X6
is very dOlle to one, being O,9D9!)!J18 in thill C'ai:e. Thb vaIlle illdicat(·" that.
62
:I!J)
IIhfll'r v;t1 iOIl
f"
1:-'
,t
cll·ar
•
(JlIIII"1 .llld. IJI'fI((·f()lth. th('
,llIitly"b is (itrried out without this observation. The revised
u
7/
=
0.2000,
2.2116,
-l.ia07
FI)!.IIIt'
,1 7 rll:.,pl.1Yl-> a
(·,,111I1;1!,IOII
•
111011 1111' fil
tI IIi fO/l1l
( 4.41)
Il = 1.:J68,
( 4.42)
0=:2.501.
( 4.43)
prohability plot ba&cd on maximum quasi-likelihood parameter
with ()h~NVilti()1I .rn discardpd. It (ail be observed from figure tI.6 and figure 4.7
l',
~11~hlly b('UN wh('11 ob1>ervatioll
X6,
the outlier, is omittcd. Figure 4.7 shows
1 iI .. l 1 h(' postlllatf'd lIollgaard procpss is a reasonable mode!.
1hl' r .1IHlollI 1>( ,II 1.1'1 plot of t h(' p- valtl<'s w 1)('11 obsprvation
(()IIII1I1I~ 1 hl' ~t.tli()llarity
of thp
X6
Figure 4.8, which displays
is discardcd from the analysis,
1>('ri('~.
1'\1'\1, ('()lIfidl'lJ((' illl('rval~ for
li\(' paramcters 0,7J and 'Y will be constructed. 'From
(:~ 1'1), (:I;d). (:1 ri:!), (:1 ri:\). (:LGI) and
(:L.15), the following second partial derivatives of
(4.44)
(4.45)
( 4.46)
•
( 4.47)
( 4.48)
63
•
(1 I!l)
From thc aboyccll'lllC'nt5 ofthC' information matrix and
tlSl'
nfp.II), th\' followlII!!.
('~tllll,lIt'd
covariance matrix is obtainC'd.
Vîr(~) =
0.002~
-(LOOG5
0.001 !l
-00065
0.I[l,17
-000 Il
0.0019
-0.00 Il
o OI,IR
The corresponding correlation matl'ix 15
l.OOOO
- (l.:J 118
O.2!)7!)
-0.3118
1.0000
-O.OH:W
0.2!)7!)
-O.(H)2!)
1.0000
Finally, use of (:t5G), (3.57) ,uHI
•
(:Lr)~{)
giv(' thp following t'OIdidPIlt'1'
(l. O!)G 7
<
1A1 12
< 1/ <
-1.:J697
( 1 fd)
Illt('rv.d~
< O.:W:l:l.
ft
2
!Ht~X.
< 'Y < - O.H92:!.
It is noteworthy lhat th(' confidence intPrval for
0
('xdudpl'> 1/2,1.111' illV('11'>1'
<:.1II:-'~,1.11I
'<lhl'
considered by Whitmore (1990).
In arder to comparC' the accuracy of the two el'>tilllat,ion 11I1't!Jodl'> l'lIIploYl'd
st udy, the parameters 11 and 'Y of tll<, rnaxilll tllll q u<l:-.i-likp!i hood IIll'thod
in terms of the param<'lC'rs a,
4>'
= (a
{j
0) and
J'
= (à
6 {j).
{j and
0 wili( il an!
llf,('d
III
1,1111'> (.1:-'1'
Il'p.11 ,LI 111'1 rizl'd
cl fi'
in tllI' lllpthod Ilf /110/111'111.:-.
I,l't
The corn l'II tation of 1/11' ('sti lII,d,l'd (OV..ri.lll( l' 111,11 Il x of
~ is analogous to that of the el'>tirn;~ted covariance matrix Y.lr(() givl'lJ III (:117)
cver, lhe
•
~eculld
PdI'lieti derivetlivp in (:JA9) heing
D21( a, 0, 0)/ DoDO cvalllat('<1 .lt
cp = ~ i:-. ;l,PI'O. III
f,let,
Il
;l,l'ro <101'1'> 1101
(,111
IJI' :-.howlI
for both the exact Iikplihood alld thf' qUclhi-likplihood fUllctlOIl:-'
lIow
J11'(1':-.:-.,uily IIl1ply
t,}ldt i)ll /
SO, t/II'
1 h,JI,
l)hO() _ n()"
f'II'IrII'llh
1,
of t/JI'
•
"c,llllldlf'd
(()V<lfI<JIICP
ail( ft, h, (J)I (Jhi)()
IIlctlrix of ~ arc givPII by (3 ..5J )-(3 ..55), with replacement of 1],
1"",:<"
= }t~:P( Ir, b" + h, 0 + h) -le à, li" + h, 8) -
l( â, li, (J + h)
-
-
"
il, i
2
+l( â, 6,- 0)]/
h .
( 4.52)
J-c If
('X ,lIlJ
pl,',
flOIll
iJll(o,h,O)/iJni)h
(;t!j~)
t!J(' followi nI!; spcond partial derivative can be obtained.
•
-,
,-2
1.,._",:- 1~I~iP(ri+h,li+h,(J)-/(ô:,Ii+h,8)-I(â+h,Ii,O)+I(â,6,O)]/h.
(4.53)
TI .. · ",lIlIpl,. log-lih.plihood l(n,h,O) can he cOlllpllted from its equivalent farm l(a,1],,)
l'Xp/f'hhl'd in (:~.:H». For any ~iV('1I valucs of a, li and 0, there arc corresponding values for
Î
.wel 'I, ohLIÏII,lhl(' frolll (2.15) and (2.Hi).
11:-'11I1-', I!J" 1IH'l.hod dpscrilH,d abovc, the estimated covariance matrix of ~ is
•
0.0028 -0.0085
tll('
0111<1
'l'hl'
(O(((':-,polldlllg
.11111\(' 101'/('1.111011
hlglil\'
((II
1'1'1.llt'd
\VIth
-0.0117
-0.00~5
0.OG02
0.lH2
\ -0.0117
0.11~2
0.2962
(4.54)
('stim.l1.pd correlation matrix is
1.0000
-O,(>GO'l
-O.(;G()tI
1.0000
-OAOG5
0.8519
-0.4065
(4.55)
1.0000
lIIatrix indieal.ps tha.t the parameter estimators â,
(',\.eh othPf.
g and ê arc quite
IL llI.ty he recalled that these parameter estimators
('\hillilet! high 1lI1111llolliuparity Illldcr the method of moments as weIl.
(1 - d) IO()~:{, confidpl\cl' illtl'rvals for n, f, and (J arc analogous to those for â,
•
\\ltllit .11(' /!,I\l'Il
fJ .and i
b~ p.5(j)-(:3.5H). Thll:-', the rl'qllircd confidence limits are
( 4.56)
65
•
where
lIoe.,
llb.5
and
1'00
ar(' tlH' corrpf,ponding l'htilllatt'd :-.land,lId
obtainable from the estimated covariann' matrix of
;p,
::1 1/'1
('I\llI~ fOI
à, lt
i~ ,Ih dPlillt'd IlIt'\'i()II~I~'
and Ô are computed using (/1.37) and (-US). TIlt' !)5% fOnlid('nu' inh'l v.llh
.trt'
tJ,
,1IIt!
b
.1IIt!
t!\('n
O.O!){;7 < n < O.:W;l3,
0.8870 < li < 1.8 /190,
( 1 IjO)
< (} < 3.5G77.
( 1 li 1)
1 tl:Jtl3
From a direct cornparison of the illcqualitit's (tl.:tl)-(,1,;l.1) wilh th ..
(4.61), it is obvions lhat the rnethod of maximum quasi-likl'lihood provl<ll'f,
•
confidence intervals for the parametcrs n, fi and 0 than do\'s t.h<'
For both estimation IIIcthods, the fit of tl:e lIollgaard
1Jr()(
111 .. 1hod
!).'/~
sprips l,a,Y('
1.0
l'A(:F
gl'nphl->
[l'vl'.d
f,light,
present, which is Ilot blJrIHlf.illg f.irH·(' w('athN
several dayn al, a timc. Overall, il, cali be
:-'yht,('I11~
cOIlrllld(~d
SOBIt'
tl'llel t,O
p.lrtl.d
dr.lwlI
slight correlation at lag one. This I('adf.
oc
1II01l11'1I1.s
111'('11
<tilt!
to belicvp tltal.
of
It Il,II1oWI'r
fi~Il[(' 'L!)) .111.1
that the seriC's appears independent and stationary. '1'h<' ACF
Ul->
IlIlIt
('f,f, appt',11 h 1(1111.1' /!;ood Wlt.h
the outlier rellloved. The e:-.lirnated <llllocorrelalion fllll('tioll (ACF,
é1l1tocorrclation functioll (PACF, figure tl.IO) of the
(.I.:,!l)
1111'<\11.111111':-'
,Ill t 0101
11111111'111
l'
hl,ow
.1
n'I,ll.lolI III .ly
.1 1111 at.ifJlI
fil 1
that th\' ttllll' :-'l'rJ('h for '.'/illd ('III'IJ!;'y
incrernents, rne<l.sured al. five-day intervals, if> approxilllatply hlaliollary ,llld IlId"I)I'lId/'lIl..
It may he tempting t.o illterpolate the HOllgaard prO(Pllh lo
incrernents al. interval& shorter than fi day& hut this Îh Ilot
•
shOl tcr than .5 days, thcrc
cxi~tn
fI'(
('~ti"I.II.f'
dl/' wllid l'III'IJ-';1
ollllll/'Tld .. d
AI.
f>trongpr hprial (kp(,lldPIIC P hO I.h,ll. thl' l'III'IV.Y
(j(;
1111.1'1
v.d"
qll.11I1
:1.11''-.
•
Il'/JlI 10 dl'pMt f/olll titI' a~~llrnptloll of
1
IJIIé,ldp/lrlv, tli!' Ilo1Jv,a<ll'd
pro(P~h
lJIdPIwndcnt increments which is necpssary when
mode!,
•
•
6ï
•
o
LO
o
••
o
20
40
LO
Perlod
Fig,Ule '1.2. Tllne SPIIP'> Plot
•
fOI
Ill" 'l', ".
•
- - - ---
--
--- --
----------------------
-----------------------l
y=
t_rt'n',or.
- -
- -
:1'-,(:
.........
~
._~---------
~
....
... . . '
-- . -- -- - -------~-------~-------------.---I
06
04
02
08
10
ExpGcted probûbilty
l,
1"111"
,1 '1
1:lltf'}111l
Pro!J,dlllity Plut.
fOI
the p-vaJucs wh(,\1 the ontlicr is present .
•
----
----------------------------------------,
,." ,
\
;
,]
'J
]
...... " , .....
_o,
,
..
-----~------------~--------------~--------------~
0::'
04
06
08
E xpcc!cd probabilty
!'IlII(lIlll
•
P[(lh.tbilIty l'lot for the
p-\"tlue~
when the outlier is deleted .
10
•
o
co
o
q
.0
rJ
c.o
0
.0
•
0....
a..
'0
0
è
0
(f)
.0
0
~
0
o
o
--,-----
o
20
40
( ~1
PCfJod
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