{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

21 - Introduction to Time Series Analysis Lecture 21 1...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Introduction to Time Series Analysis. Lecture 21. 1. Review: The periodogram, the smoothed periodogram. 2. Other smoothed spectral estimators. 3. Consistency. 4. Asymptotic distribution. 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Review: Periodogram The periodogram is defined as I ( ν )= 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 . Under general conditions, X c ( ν j ) , X s ( ν j ) are asymptotically independent and N (0 , f ( ν j ) / 2) . Thus, E I ν ( n ) ) f ( ν ) , but Var ( I ν ( n ) )) f ( ν ) 2 . 2
Background image of page 2
Review: smoothed periodogram If f ( ν ) is approximately constant in a band of frequencies [ ν k L/ (2 n ) , ν k + L/ (2 n )] , we can average the periodogram over this band: ˆ f ( ν k )= 1 L ( L - 1) / 2 summationdisplay l = - ( L - 1) / 2 I ( ν k l/n ) = 1 L ( L - 1) / 2 summationdisplay l = - ( L - 1) / 2 ( X 2 c ( ν k l/n )+ X 2 s ( ν k l/n ) ) . 3
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Review: smoothed periodogram Under general conditions, the X c ( ν k l/n ) and X s ( ν k l/n ) are asymptotically independent and N (0 , f ( ν k l/n ) / 2) . Thus, E ˆ f ( ν ( n ) ) f ( ν ) and Var ˆ f ( ν ( n ) ) f 2 ( ν ) /L . Notice the bias-variance trade off : 1. Our assumption that f is approximately constant on [ ν L/ (2 n ) , ν + L/ (2 n )] becomes worse as L increases, so the difference between ˆ f ν ( n ) ) and f ( ν ) (the bias) will increase with L . 2. The variance of our estimate, Var ˆ f ν ( n ) ) decreases with L .
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}