Asymptotic distributions of sample mean and ACVF Felix Dietrich and Gundelinde Wiegel University of Trento 29th of October 2013 Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 1 / 24 Overview 1 Repetition 2 ¯ Distribution of X 3 Distribution of γˆ Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 2 / 24 What we already know Let {Xt }t∈Z a stationary process with E[Xt ] = µ and ACVF γ(·). Definition (Estimator of µ) n X ¯n := 1 X Xi n i=1 . Definition (Estimator of ACVF) n−h γˆ (h) = 1X ¯n )(Xt+h − X ¯n ) (Xt − X n for 0≤h ≤n−1 i=1 Definition (Estimator of ACF) ρˆ(h) = Felix Dietrich and Gundelinde Wiegel (UTN) γˆ (h) γˆ (0) ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 3 / 24 Mean Estimator ¯n = For X 1 n n P Xi we have already shown i=1 ¯n ] = µ E[X and ¯n ] = Var [X X |h| 1− γ(h) . n h<|n| Hence ¯n ] → 0 Var [X ∞ X ¯n ] → nVar [X γ(h) h=−∞ if if γ(h) → 0, ∞ X |γ(h)| < ∞ . h=−∞ ¯n distributed? But how is X Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 4 / 24 ¯n Distribution of X Theorem ∞ P Let {Xt } a stationary process, defined as Xt = µ + ψj Zt−j with j=−∞ P ¯n is {Zt } ∼ IID(0, σ 2 ) for each t ∈ Z and ∞ 6 0 < ∞. Then X j=−∞ |ψj | = asymptotically normal i.e. ¯n ∼ AN µ, ν X n P∞ P 2 where ν = h=−∞ γ(h) = σ 2 ( ∞ j=−∞ ψj ) . Remark: {Xn } is asymptotically normal distributed with µn and σn > 0 for n large enough if for n → ∞ Xn ∼ AN(µn , σn2 ). Felix Dietrich and Gundelinde Wiegel (UTN) Xn −µn i.d. −→ σn Z where Z ∼ N(0, 1) . We write: ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 5 / 24 Tools Proposition Let {Xn }n∈N and {Ynj }j∈N,n∈N be random vectors such that (i) Ynj −→ Yj for each j ∈ N n→∞ (ii) Yj −→ Y j→∞ (iii) lim lim supP(|Xn − Ynj | > ε) = 0 for every ε > 0. Then j→∞ n→∞ i.d. Xn −→ Y . n→∞ Lemma (Chebyshevs inequality) Let X be a random variable such that E [X ] = µ and Var[X ] = σ 2 < ∞ then the following inequality holds for any real number k > 0 P[|X − µ| ≥ k] ≤ Felix Dietrich and Gundelinde Wiegel (UTN) σ2 k2 ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 6 / 24 Tools II Definition (m-Dependence) A stationary process {Xt } is called m-dependent with 0 < m ∈ N if for each t ∈ Z {Xj , j ≤ t} and {Xj , j ≥ t + m + 1} are independent. Theorem (Central Limit Th. for Stationary m-Dependent Sequences) Let {Xt } be a stationary and m-dependent sequence of r.v. with mean P zero and ACVF γ(·). If νm = m γ(j) = 6 0, then j=−m ¯n ] = νm lim nVar [X ¯n ∼ AN 0, νm . X n (1) n→∞ Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ (2) 29th of October 2013 7 / 24 Proof of the Theorem Xtm := µ + m X ψj Zt−j j=−m and n Ynm X ¯nm := 1 Xtm . := X n t=1 {Ynm }n∈N is an 2m-dependent sequence. With the central limit theorem for any m ∈ N: ⇒ Ynm −→ Xm as n → ∞ (in distribution) P where Xm ∼ N µ, n1 m j=−m γ(j) . Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 8 / 24 Proof of the Theorem II Hence we get √ 2 n(Ynm − µ) −→ Ym ∼ N 0, σ ( m X j=−m 2 ψj ) = m X γ(j) j=−m P P∞ 2 2 2 Since σ 2 ( m j=−m ψj ) → σ ( j=−∞ ψj ) > 0, we can show that Ym −→ Y as m → ∞ (in distribuition) P 2 where Y ∼ N 0, σ 2 ( ∞ j=−∞ ψj ) .Thus we showed property (i) and (ii) of the first proposition. For (iii) we consider Var √ n X X X ¯n − Ynm ) = nVar 1 n(X ψj Zt−j → σ 2 ( ψj )2 . n t=1 |j|>m Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ |j|>m 29th of October 2013 9 / 24 Proof of the Theorem III Hence n X X 1 X ψj Zt−j = lim σ 2 ( ψj )2 lim lim sup Var m→∞ m→∞ n∈N n i=1 |j|>m |j|>m =0 Chebyshevs inequalitiy ⇒ (iii) √ ¯n − µ) −→ Y ∼ N 0, σ 2 ( n(X ∞ X ψj )2 j=−∞ Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 10 / 24 Distribution of γˆ Now to the distribution of n−h γˆ (h) = 1X ¯n )(Xt+h − X ¯n ) (Xt − X n for 0≤h ≤n−1 . i=1 Theorem (Distribution of γˆ ) P Let {Xt } be a two-sided MA-process with Xt = ∞ j=−∞ ψj Zt−j , P∞ 2 {Zt } ∼ IID(0, σ ), where j=−∞ |ψj | < ∞ and E[Zt4 ] = ησ 4 < ∞. Then for any h ∈ N γˆ (0) γ(0) .. . V . ∼ AN .. , n γˆ (h) γ(h) with V = Cov (γ(p),P γ(q))p,q=0,...,h = (η − 3)γ(p)γ(q) + ∞ k=−∞ (γ(k)γ(k − p + q) + γ(k + q)γ(k − p)) p,q . Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 11 / 24 Tools First of all we show some properties of n γ ∗ (h) = 1X Xt Xt+h n with h∈N . t=1 Lemma (Lemma 1) P P∞ {Xt } MA-process Xt = ∞ j=−∞ ψj Zt−j , where j=−∞ |ψj | < ∞ and 4 4 E[Zt ] = ησ < ∞. For p, q ∈ N we obtain lim nCov [γ ∗ (p), γ ∗ (q)] n→∞ = (η − 3)γ(p)γ(q) + ∞ X (γ(k)γ(k − p + q) + γ(k + q)γ(k − p)) k=−∞ Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 12 / 24 Proof of Lemma 1 As {Zt }t≥0 is i.i.d. with mean 0 and variance σ 2 plus the property E [Zt4 ] = ησ 4 < ∞, we obtain 4 ησ if t = s = u = v E [Zt Zs Zu Zv ] = σ 4 if s = t 6= u = v 0 else. E [Xt Xt+p Xt+p+h Xt+p+h+q ] XXXX ψi ψj+p ψk+p+h ψl+h+p+q E [Zt−i Zt−j Zt−k Zt−l ] = i j = (η − 3)σ k l 4 X ψi ψi+p ψi+p+h ψi+p+h+q + γ(p)γ(q)+ i γ(h + p)γ(h + q) + γ(h + p + q)γ(h) Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 13 / 24 Proof of Lemma 1 II Now back to γ ∗ . We know that E [γ ∗ (h)] = γ(h) Cov[γ ∗ (p), γ ∗ (q)] = E [γ ∗ (p)γ ∗ (q)] − E [γ ∗ (p)]E [γ ∗ (q)] X = n−1 (1 − n−1 |k|)Tk , k<|n| where Tk : = γ(k)γ(k − p + q) + γ(k + q)γ(k − p) X + (η − 3)σ 4 ψi ψi+p ψi+k ψi+k+q i Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 14 / 24 Proof of Lemma 1 III Absolute summability of {ψj }j∈Z ⇒ {Tk }k∈Z is absolutely summable. ∗ ∗ lim n Cov[γ (p), γ (q)] = n→∞ ∞ X Tk k=−∞ = (η − 3)γ(p)γ(q) ∞ X + γ(k)γ(k − p + q) + γ(k + q)γ(k − p) . k=−∞ Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 15 / 24 With this Lemma we show the convergence of the distribution of γ ∗ for a truncated MA. Lemma (Lemma 2) {Xt } MA-process Xt = m X ψj Zt−j , j=−m where P∞ j=−∞ |ψj | < ∞ and E[Zt4 ] = ησ 4 < ∞. Then for any h ∈ N γ(0) γ ∗ (0) .. . V . ∼ AN .. , n γ ∗ (h) γ(h) with V = Cov (γ(p),P γ(q))p,q=0,...,h = (η − 3)γ(p)γ(q) + ∞ k=−∞ (γ(k)γ(k − p + q) + γ(k + q)γ(k − p)) p,q . Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 16 / 24 Proof of Lemma 2 Yt0 := (Xt Xt , Xt Xt+1 , . . . , Xt Xt+h ) . {Yt } is again stationary and 2m + h-dependent and n 1X Yt = n t=1 n n 1X 1X Xt+0 Xt+0 , . . . , Xt Xt+h n n t=1 !0 = (γ ∗ (0), . . . , γ ∗ (h))0 . t=1 Central Limit Theorem: Let λ ∈ Rh+1 . n 1X 0 λ Yt ] = lim nVar [λ0 (γ ∗ (0), . . . , γ ∗ (h))0 ] lim nVar [ n→∞ n→∞ n t=1 = lim nVar [λ0 γ ∗ (0) + . . . + λh γ ∗ (h)] = lim n n→∞ Felix Dietrich and Gundelinde Wiegel (UTN) n→∞ ¯ and γ Asymptotic distributions of X ˆ h X λi λj Cov [γ ∗ (i), γ ∗ (j)] j,i=1 29th of October 2013 17 / 24 Proof of Lemma 2 II = Lemma 1 λ0 V λ > 0 with V = P (η − 3)γ(p)γ(q) + ∞ k=−∞ (γ(k)γ(k − p + q) + γ(k + q)γ(k − p)) p,q Central Limit Theorem ⇒ γ(0) n 1X 0 1 λ Yt ∼ AN λ0 ... , λ0 V λ n n t=1 γ(h) for every λ ∈ Rh+1 . Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 18 / 24 Lemma 3 Now back to our original two-sided MA. Lemma (Lemma 3) {Xt } MA-process Xt = ∞ X ψj Zt−j , j=−∞ where P∞ j=−∞ |ψj | < ∞ and E[Zt4 ] = ησ 4 < ∞. Then for any h ∈ N γ(0) γ ∗ (0) .. . V . ∼ AN .. , n γ ∗ (h) γ(h) with V = Cov (γ(p),P γ(q))p,q=0,...,h = (η − 3)γ(p)γ(q) + ∞ k=−∞ (γ(k)γ(k − p + q) + γ(k + q)γ(k − p)) p,q . Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 19 / 24 Proof of Lemma 3 The idea is to apply Lemma 2 on the truncated sequence Xt,m := m X n ψj Zt−j and therefore ∗ γm (p) := 1X Xt,m Xt+p,m n t=1 j=−m Lemma 2 ⇒ ∗ (0) − γ (0) γm m √ i.d. .. n −→ Ym ∼ N(0, Vm ) . . ∗ γm (h) − γm (h) Besides Vm → V and Ym → Y ∼ N(0, V ). Also we can show, that √ ∗ lim lim supP[ n|γm (p) − γm (p) − (γ ∗ (p) − γ(p))| > ε] = 0 m→∞ n→∞ Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 20 / 24 Proof of the Theorem Left γˆ . Hence √ instead of n(γ ∗ (p) − γˆ (p)) = op (1) γ∗ for n→∞ γ(0) γˆ (0) .. . V . ∼ AN .. , n γˆ (h) γ(h) Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 21 / 24 Some interesting corollaries Corollary (Distribution of ρˆ) P Let {Xt } be the stationary process given by Xt = ∞ j=−∞ ψj Zt−j − µ, P∞ 4 4 where j=−∞ |ψj | < ∞ and E[Zt ] = ησ < ∞. Then for each h ∈ N W ρˆ(h) ∼ AN ρ(h), n where ρˆ(h)0 = [ˆ ρ(1), ..., ρˆ(h)]0 and ρ(h)0 = [ρ(1), ..., ρ(h)]0 and W the Covariance Matrix, whose elements are given by Bartlett’s formula: wij = ∞ X [ρ(k + i)ρ(k + j) + ρ(k − i)ρ(k + j) + 2ρ(i)ρ(j)ρ2 (k) k=−∞ − 2ρ(i)ρ(k)ρ(k + j) − 2ρ(j)ρ(k)ρ(k + i)] Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 22 / 24 Some interesting corollaries Corollary (Distribution of ρˆ II) P Let {Xt } be the stationary process given by Xt = ∞ j=−∞ ψj Zt−j − µ, P∞ P∞ 2 where j=−∞ |ψj | < ∞ and j=−∞ ψj |j| < ∞. Then for each h ∈ N W ρˆ(h) ∼ AN ρ(h), n where ρˆ(h)0 = [ˆ ρ(1), ..., ρˆ(h)]0 and ρ(h)0 = [ρ(1), ..., ρ(h)]0 and W the Covariance Matrix, whose elements are given by Bartlett’s formula: wij = ∞ X [ρ(k + i)ρ(k + j) + ρ(k − i)ρ(k + j) + 2ρ(i)ρ(j)ρ2 (k) k=−∞ − 2ρ(i)ρ(k)ρ(k + j) − 2ρ(j)ρ(k)ρ(k + i)] Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 23 / 24 The End Felix Dietrich and Gundelinde Wiegel (UTN) ¯ and γ Asymptotic distributions of X ˆ 29th of October 2013 24 / 24

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