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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.

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Estimation of the spectral density function

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  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

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