20 - Introduction to Time Series Analysis Lecture 20 1...

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Introduction to Time Series Analysis. Lecture 20. 1. Review: The periodogram 2. Asymptotics of the periodogram. 3. Nonparametric spectral estimation. 1
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Review: Periodogram The periodogram is defined as I ( ν ) = | X ( ν ) | 2 = 1 n vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle n summationdisplay t =1 e - 2 πitν x t vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle 2 = X 2 c ( ν ) + X 2 s ( ν ) . X c ( ν ) = 1 n n summationdisplay t =1 cos(2 πtν ) x t , X s ( ν ) = 1 n n summationdisplay t =1 sin(2 πtν j ) x t . The same as computing f ( ν ) from the sample autocovariance (for ¯ x = 0 ). 2
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Asymptotic properties of the periodogram We want to understand the asymptotic behavior of the periodogram I ( ν ) at a particular frequency ν , as n increases. We’ll see that its expectation converges to f ( ν ) . We’ll start with a simple example: Suppose that X 1 , . . . , X n are i.i.d. N (0 , σ 2 ) (Gaussian white noise). From the definitions, X c ( ν j ) = 1 n n summationdisplay t =1 cos(2 πtν j ) x t , X s ( ν j ) = 1 n n summationdisplay t =1 sin(2 πtν j ) x t , we have that X c ( ν j ) and X s ( ν j ) are normal, with E X c ( ν j ) = E X s ( ν j ) = 0 . 3
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Asymptotic properties of the periodogram Also, Var ( X c ( ν j )) = σ 2 n n summationdisplay t =1 cos 2 (2 πtν j ) = σ 2 2 n n summationdisplay t =1 (cos(4 πtν j ) + 1) = σ 2 2 . Similarly, Var ( X s ( ν j )) = σ 2 / 2 . 4
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Asymptotic properties of the periodogram Also, Cov ( X c ( ν j ) , X s ( ν j )) = σ 2 n n summationdisplay t =1 cos(2 πtν j ) sin(2 πtν j ) = σ 2 2 n n summationdisplay t =1 sin(4 πtν j ) = 0 , Cov ( X c ( ν j ) , X c ( ν k )) = 0 Cov ( X s ( ν j ) , X s ( ν k )) = 0 Cov ( X c ( ν j ) , X s ( ν k )) = 0 . for any j negationslash = k . 5
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Asymptotic properties of the periodogram That is, if X 1 , . . . , X n are i.i.d. N (0 , σ 2 ) (Gaussian white noise; f ( ν ) = σ 2 ), then the X c ( ν j ) and X s ( ν j ) are all i.i.d. N (0 , σ 2 / 2) . Thus, 2 σ 2 I ( ν j ) = 2 σ 2 ( X 2 c ( ν j ) + X 2 s ( ν j ) ) χ 2 2 .
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