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The Empirical FT .

The Empirical FT. What is the large sample distribution of the EFT?. The complex normal. Theorem. Suppose X is stationary mixing, then. Proof. Write. Evaluate first and second-order cumulants Bound higher cumulants Normal is determined by its moments. Consider. Comments .

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The Empirical FT .

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  1. The Empirical FT. What is the large sample distribution of the EFT?

  2. The complex normal.

  3. Theorem. Suppose X is stationary mixing, then

  4. Proof. Write Evaluate first and second-order cumulants Bound higher cumulants Normal is determined by its moments

  5. Consider

  6. Comments. Already used to study mean estimate Tapering, h(t)X(t). makes Get asymp independence for different frequencies The frequencies 2r/T are special, e.g. T(2r/T)=0, r 0 Also get asymp independence if consider separate stretches p-vector version involves p by p spectral density matrix fXX( )

  7. Estimation of the (power) spectrum. An estimate whose limit is a random variable

  8. Some moments. The estimate is asymptotically unbiased Final term drops out if  = 2r/T  0 Best to correct for mean, work with

  9. Periodogram values are asymptotically independent since dT values are - independent exponentials Use to form estimates

  10. Approximate marginal confidence intervals

  11. More on choice of L

  12. Approximation to bias

  13. Indirect estimate

  14. Estimation of finite dimensional . approximate likelihood (assuming IT values independent exponentials)

  15. Bivariate case.

  16. Crossperiodogram. Smoothed periodogram.

  17. Complex Wishart

  18. Predicting Y via X

  19. Plug in estimates.

  20. Large sample distributions. var log|AT|  [|R|-2 -1]/L var argAT [|R|-2 -1]/L

  21. Advantages of frequency domain approach. techniques for many stationary processes look the same approximate i.i.d sample values assessing models (character of departure) time varying variant...

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