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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
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.
