Estimation problems connected with stochastic processes

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Virginia Polytechnic Institute


A brief introduction to the concepts and terminology of spectral analysis and a review of the standard methods for cross-spectral estimation, based on discrete time history data, are incorporated in Chapter 1.

Co-spectral and quadrature-spectral estimators which are characterized by non-negative spectral windows are developed in Chapter 2. While the spectral windows for the co-spectral estimators are non-negative for all relevant values of the assignable constants, certain restrictions on these constants are necessary to assure the non-negativity of the quadrature-spectral window. The properties of these estimators are considered in detail.

In Chapter 3 randomized co-spectral and quadrature spectral estimators are presented. These estimators depend on the random selection of sets of time differences, as opposed to the systematic evaluation of all possible time differences for the standard estimators. By suitable choices of probability distributions for the time differences and of weight functions, the expectations of the randomized estimators can be made equivalent to the expectations of the standard estimators or the estimators of Chapter 2. Since the randomized estimator is much simpler to use than the standard estimator, these estimators are compared in terms of their variances, given that they have equal expectations. The choice of probability distributions to yield minimum variance, given that the expectation is specified, is considered.

Extremely simple co-spectral and quadrature-spectral estimators, for the case where the coefficients of the Fourier series expansions of realizations of the processes over a finite time interval can be obtained by means of suitable analog equipment, are developed in Chapter 4. The expectations, variances and covariances of these estimators are derived.