Now we show 29 It is easy to see that \u03b8 B W t \u03b8 B \u03b8 1 B \u03c6 B X t \u03c6 B X t It

Now we show 29 it is easy to see that θ b w t θ b

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Now, we show ( 2.9 ). It is easy to see that ˜ θ ( B ) W * t = ˜ θ ( B ) ˜ θ - 1 ( B ) ˜ φ ( B ) X t = ˜ φ ( B ) X t . It suffices to show { W * t } is a white noise. This is left as a HW problem. 37
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2.6 Properties of X n , b γ X ( h ) and b ρ X ( h ) 2.6.1 For X n Recall that, for observations x 1 , . . . , x n of a time series, the sample mean is x = 1 n n X t =1 x t . The sample auto covariance function is b γ X ( h ) = 1 n n -| h | X t =1 ( x t + | h | - x )( x t - x ) , for - n < h < n. The sample autocorrelation function (sample ACF) is b ρ X ( h ) = b γ X ( h ) b γ X (0) . Estimation of μ X : The moment estimator of the mean μ X of a stationary process { X t } is the sample mean X n = n - 1 n X t =1 X t . (2.10) Obviously, it is unbiased; i.e., E( X n ) = μ X . Its mean squared error is Var( X n ) =E( X n - μ X ) 2 = n - 2 n X i =1 n X j =1 Cov( X i , X j ) = n - 2 n X i =1 n X j =1 γ X ( i - j ) = n - 2 n X i - j = - n ( n - | i - j | ) γ X ( i - j ) = n - 1 n X h = - n 1 - | h | n γ X ( h ) = γ X (0) n | {z } is Var( X n ) when { X t } are iid + 2 n n - 1 X h =1 1 - | h | n γ X ( h ) . Depending on the nature of the correlation structure, the standard error of X n may be smaller or larger than the white noise case. Consider X t = μ + W t - 0 . 8 W t - 1 , where { W t } ∼ WN(0 , σ 2 ), then Var( X n ) = γ X (0) n + 2 n n - 1 X h =1 1 - | h | n γ X ( h ) = 1 . 64 σ 2 n - 1 . 6( n - 1) σ 2 n 2 < 1 . 64 σ 2 n . 38
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And if X t = μ + W t + 0 . 8 W t - 1 , where { W t } ∼ WN(0 , σ 2 ), then Var( X n ) = γ X (0) n + 2 n n - 1 X h =1 1 - | h | n γ X ( h ) = 1 . 64 σ 2 n + 1 . 6( n - 1) σ 2 n 2 > 1 . 64 σ 2 n . If γ X ( h ) 0 as h → ∞ , we have | Var( X n ) | ≤ γ X (0) n + 2 n h =1 | γ X ( h ) | n 0 as n → ∞ . Thus, X n converges in mean square to μ . If h = -∞ | γ X ( h ) | < , then n Var( X n ) = n X h = - n 1 - | h | n γ X ( h ) = γ X (0) + 2 n h =1 ( n - h ) γ X ( h ) n = γ X (0) + 2 n - 1 h =1 h i =1 γ X ( i ) n γ X (0) + 2 X i =1 γ X ( i ) = X h = -∞ γ X ( h ) = γ X (0) X h = -∞ ρ X ( h ) . One interpretation could be that, instead of Var( X n ) γ X (0) /n , we have Var( X n ) γ X (0) / ( n/τ ) with τ = h = -∞ ρ X ( h ). The effect of the correlation is a reduction of sample size from n to n/τ . Example 2.10. For linear processes, i.e., if X t = μ + j = -∞ ψ j W t - j with j = -∞ | ψ j | < , then X h = -∞ | γ X ( h ) | = X h = -∞ | σ 2 X j = -∞ ψ j ψ j + h | X h = -∞ σ 2 X j = -∞ | ψ j | · | ψ j + h | = σ 2 X j = -∞ | ψ j | X h = -∞ | ψ j + h | = σ 2 X j = -∞ | ψ j | 2 < To make inference about μ X (e.g., is μ X = 0?), using the sample mean X n , it is necessary to know the asymptotic distribution of X n : If { X t } is Gaussian stationary time series, then, for any n , n ( X n - μ X ) N 0 , n X h = - n 1 - | h | n γ X ( h ) ! . 39
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Then one can obtain exact confidence intervals of estimating μ X , or approximated confidence intervals if it is necessary to estimate γ X ( · ). For the linear process, X t = μ + j = -∞ ψ j W t - j with { W t } ∼ IID(0 , σ 2 ), j = -∞ | ψ j | < and j = -∞ ψ j 6 = 0, then n ( X n - μ X ) AN(0 , ν ) , (2.11) where ν = h = -∞ γ X ( h ) = σ 2 ( j = -∞ ψ j ) 2 .
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  • Spring '15
  • Dewei Wang
  • Statistics, Stationary process, ACF, Xt, γx

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