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. . ) (under the assumption that the spectral density exists). For this reason, a vast amount of periodical and monographic literature is devoted to the nonparametric statistical problem of estimating the function tJ( T) and especially that of leA) (see, for example, the books [4,21,22,26,56,77,137,139,140,]). However, the empirical value t;; of the spectral density I obtained by applying a certain statistical procedure to the observed values of the variables Xl' . . . , X , usually depends in n a complicated manner on the cyclic frequency). . This fact often presents difficulties in applying the obtained estimate t;; of the function I to the solution of specific problems rela ted to the process X . Theref ore, in practice, the t obtained values of the estimator t;; (or an estimator of the covariance function tJ~( T" are almost always "smoothed," i. e. , are approximated by values of a certain sufficiently simple function 1 = 1
List of contents
I Properties of Maximum Likelihood Function for a Gaussian Time Series.- 1. General Expression for the log Likelihood.- 2. Asymptotic Expression for the "Principal Part" of the log Likelihood.- 3. The Asymptotic Differentiability of Gaussian Distributions with Spectral Densities Separated from Zero.- 4. The Asymptotic Differentiability of Gaussian Distributions with Spectral Densities Possessing Fixed Zeros.- Appendix 1.- Appendix 2.- Appendix 3. Remarks and Bibliography.- II Estimation of Parameters by Means of P. Whittle's Method.- 1. Asymptotic Maximum Likelihood Estimators.- 2. Properties of Asymptotic Maximum Likelihood Estimators in the Case of Strictly Positive Spectral Density.- 3. Consistency, Asymptotic Normality, and Asymptotic Efficiency of the Estimator $$mathop theta limits^ sim $$ in the Case of Spectral Density Possessing Fixed Zeros.- 4. Examples of Determination of Asymptotic Maximum Likelihood Estimators.- 5. Asymptotic Maximum Likelihood Estimator of the Spectrum of Processes Distorted by "White Noise".- 6. Least-Squares Estimation of Parameters of a Spectrum of a Linear Process.- 7. Estimation by Means of the Whittle Method of Spectrum Parameters of General Processes Satisfying the Strong Mixing Condition.- Appendix 1.- Appendix 2.- Appendix 3. Remarks and Bibliography.- III Simplified Estimators Possessing "Nice" Asymptotic Properties.- 1. Asymptotic Properties of Simplified Estimators.- 2. Examples of Preliminary Consistent Estimators.- 3. Examples of Constructing Simplified Estimators.- Appendix 1. Remarks and Bibliography.- IV Testing Hypotheses on Spectrum Parameters of a Gaussian Time Series.- 1. Testing Simple Hypotheses.- 2. Testing Composite Hypotheses (The Case of a Sequence of General "Asymptotically DifferentiableExperiments").- 3. Testing of Composite Hypothesis about a Parameter of a Spectrum of a Gaussian Time Series.- Appendix 1. Remarks and Bibliography.- V Goodness-of-Fit Tests for Testing the Hypothesis about the Spectrum of Linear Processes.- 1. A Class of Goodness-of-Fit Tests for Testing a Simple Hypothesis about the Spectrum of Linear Processes.- 2. X2 Test for Testing a Simple Hypothesis about the Spectrum of a Linear Process.- 3. Goodness-of-Fit Test for Testing Composite Hypotheses about the Spectrum of a Linear Process.- Appendix 1. Remarks and Bibliography.
About the author
Samuel Kotz, PhD, honorary Doctor of Science, is professor and research scholar at the Department of Engineering Management and Systems Engineering at George Washington University in Washington, D.C.§
