Nonparametric Estimation for Processes
This final chapter is concerned with questions of (asymptotic) unbiasedness, and consistent nonparametric estimation of some functions, such as bispectral densities of a class of second order processes. After some necessary preliminaries, we discuss spectral properties of separable, especially harmonizable, processes of second order in Section 1. Then we consider an asymptotically unbiased estimator, and a related function, of the spectral distribution of the process in the next section. These are usually not consistent estimators when the process is not stationary. So we need to use another procedure, the so-called resampling method, wherein the covariance between samples falls off at a reasonable rate (made precise later). This is described in Section 3 and using such a procedure it is possible to obtain a consistent estimator of the bispectral density of strongly harmonizable processes, and this is given for such a class. Then Section 4 contains a slightly more general second order family for which some related results are discussed. In Section 5 a limit distribution of the (nonparametric) estimator defined above for a strongly harmonizable class is presented. Thus the conditions imposed are progressively more stringent, but then we get more refined results. Several new avenues and possible improvements of the results are pointed out, with related exercises in the last section, usually with sketches of proofs, as complements to the preceding work.
KeywordsCovariance Function Spectral Function Limit Distribution Unbiased Estimator Nonparametric Estimation
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