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By D.F. Findley

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Additional resources for Applied Time Series Analysis I. Proceedings of the First Applied Time Series Symposium Held in Tulsa, Oklahoma, May 14–15, 1976

Sample text

If the index set is continuous and ~ H. L. Gray et al. 50 (21) if the index set is discrete. ;;;~ Although it is not necessary for our results to hold, it is convenient in the future to assume real processes. , T-ITI f o (x (t) -x) (x (t+ I T I) -x) dt f T o (22) [x(t)- xj 2dt for continuous data and integrals replaced by sums for discrete data. with these assumptions, we obtain G±(w;k,TO,h) = 2G k [ f TO o PT(T) cos 2WWT dT; h ] (23) and GT(w;k,NO,h) = eke 1 + 2 NO ~ m=l A PT(m) cos 2wwm; h j, (24) which we will take hereafter as the definition of the G-spectral estimator.

We will later show how this information is utilised in properly selecting k. A similar result holds for the continuous case (Theorem 5 below). It was first shown in [211 and comments corresponding to the above apply equally well to it. THEOREM S. p(t) inteqer m ~ £ ~ &et {X(t), - c < t < ... } ~! ess ~ n) s~ch ~ lim GT(w/k,TO,h) • 2 or- f to o p(t) oos 2wwtdt , (26) for every k > m. Proof. IV. s•• [261. Calculation of the G-Spectral E~ti~ator. Efficient methods for calculating the Gn - and en-transformations have, as mentioned before, been developed.

Initially, the so-called Gn - and en-transformations are defined and some simple examples are given to demonstrate their ability to increase the rate of convergence of certain types of sequences. Following this, theorems are given which establish sufficient conditions for a sequence to be extrapolated from a finite number of terms to its limit via these transformations. The above-described results are kept short so as not to be a deterrent, but are included since an understanding of them will arm the reader with excellent insight regarding the behavior of the G-spectral estimator given later.

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