1 / 147

Estimation of the spectral density function

Estimation of the spectral density function. The spectral density function, f ( l ) The spectral density function, f ( x ), is a symmetric function defined on the interval [- p , p ] satisfying. and.

gisellez
Download Presentation

Estimation of the spectral density function

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Estimation of the spectral density function

  2. The spectral density function, f(l) The spectral density function, f(x), is a symmetric function defined on the interval [-p,p] satisfying and The spectral density function, f(x), can be calculated from the autocovariance function and vice versa.

  3. Some complex number results: Use

  4. Expectations of Linear and Quadratic forms of a weakly stationary Time Series

  5. Expectations, Variances and Covariances of Linear forms

  6. Theorem Let {xt:tT} be a weakly stationary time series. Let Then and where and Sr = {1,2, ..., T-r}, if r ≥ 0, Sr = {1- r, 2 - r, ..., T} if r ≤ 0.

  7. Proof

  8. Also since Q.E.D.

  9. Theorem Let {xt:tT} be a weakly stationary time series. Let and

  10. Expectations, Variances and Covariances of Linear formsSummary

  11. Theorem Let {xt:tT} be a weakly stationary time series. Let Then and where and Sr = {1,2, ..., T-r}, if r ≥ 0, Sr = {1- r, 2 - r, ..., T} if r ≤ 0.

  12. Theorem Let {xt:tT} be a weakly stationary time series. Let and Then where and

  13. Then where and Also Sr= {1,2, ..., T-r}, if r ≥ 0, Sr = {1- r, 2 - r, ..., T} if r ≤ 0.

  14. Expectations, Variances and Covariances of Quadratic forms

  15. Theorem Let {xt:t T} be a weakly stationary time series. Let Then

  16. and

  17. and Sr = {1,2, ..., T-r}, if r ≥ 0, Sr = {1- r, 2 - r, ..., T} if r ≤ 0, k(h,r,s) = the fourth order cumulant = E[(xt - m)(xt+h - m)(xt+r - m)(xt+s - m)] - [s(h)s(r-s)+s(r)s(h-s)+s(s)s(h-r)] Note k(h,r,s) = 0 if {xt:t T}is Normal.

  18. Theorem Let {xt:t T} be a weakly stationary time series. Let Then

  19. where and

  20. Examples The sample mean

  21. Thus and

  22. Also

  23. and where

  24. Thus Compare with

  25. If g(•) is a continuous function then: Basic Property of the Fejer kernel: Thus

  26. The sample autocovariancefunction The sample autocovariance function is defined by:

  27. or if m is known where

  28. or if m is known where

  29. Theorem Assume m is known and the time series is normal, then: E(Cx(h))= s(h),

  30. and

  31. Proof Assume m is known and the the time series is normal, then: and

  32. and

  33. where

  34. since

  35. hence

  36. Thus

  37. and Finally

  38. Where

  39. Thus

  40. Expectations, Variances and Covariances of Linear formsSummary

  41. Theorem Let {xt:tT} be a weakly stationary time series. Let Then and where and Sr = {1,2, ..., T-r}, if r ≥ 0, Sr = {1- r, 2 - r, ..., T} if r ≤ 0.

  42. Theorem Let {xt:tT} be a weakly stationary time series. Let and Then where and

More Related