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Unformatted text preview: 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 Review: Periodogram The periodogram is defined as 2 2 I() = Xc () + Xs (). 1 Xc () = n 1 Xs () = n n cos(2t)xt , t=1 n sin(2tj )xt . t=1 Under general conditions, Xc (j ), Xs (j ) are asymptotically independent and N (0, f (j )/2). Thus, EI(^(n) ) f (), but Var(I(^(n) )) f ()2 . 2 Review: smoothed periodogram If f () is approximately constant in a band of frequencies [k - L/(2n), k + L/(2n)], we can average the periodogram over this band: 1 ^ f (k ) = L 1 = L (L-1)/2 l=-(L-1)/2 (L-1)/2 I(k - l/n) 2 2 Xc (k - l/n) + Xs (k - l/n) . l=-(L-1)/2 3 Rev...
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