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Estimation of the spectral density function

This article discusses the estimation of the spectral density function for a weakly stationary time series using the autocovariance function. It also covers complex number results and expectations, variances, and covariances of linear and quadratic forms. The Discrete Fourier Transform is introduced for estimating the spectral density function.

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

  11. Expectations, Variances and Covariances of Quadratic forms

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

  13. and

  14. 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.

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

  16. where and

  17. Examples The sample mean

  18. Thus and

  19. Also

  20. and where

  21. Thus Compare with

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

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

  24. or if m is known where

  25. or if m is known where

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

  27. and

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

  29. and

  30. where

  31. since

  32. hence

  33. Thus

  34. and Finally

  35. Where

  36. Thus

  37. Estimation of the spectral density function

  38. The Discrete Fourier Transform

  39. Let x1,x2,x3, ...xT denote T observations on a univariate one-dimensional time series with zero mean (If the series has non-zero mean one uses in place of xt). Also assume that T = 2m +1 is odd. Then

  40. where with lk = 2pk/T and k = 0, 1, 2, ... , m.

  41. The Discrete Fourier transform: k = 0, 1,2, ... ,m.

  42. Note:

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