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Author Dzhaparidze, K. author Parameter Estimation and Hypothesis Testing in Spectral Analysis of Stationary Time Series [electronic resource] / by K. Dzhaparidze New York, NY : Springer New York, 1986 http://dx.doi.org/10.1007/978-1-4612-4842-2 324 p. online resource

SUMMARY

. . ) (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 tJ̃( Tยป are almost always "smoothed," i. e. , are approximated by values of a certain sufficiently simple function 1 = 1

CONTENT

I Properties of Maximum Likelihood Function for a Gaussian Time Series -- 1. General Expression for the log Likelihood -- 2. Asymptotic Expression for the โ{128}{156}Principal Partโ{128}{157} 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. Whittleโ{128}{153}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 $$\mathop \theta \limitŝ \sim$$ 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 โ{128}{156}White Noiseโ{128}{157} -- 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 โ{128}{156}Niceโ{128}{157} 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 โ{128}{156}Asymptotically Differentiable Experimentsโ{128}{157}) -- 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

Mathematics Applied mathematics Engineering mathematics Probabilities Mathematics Probability Theory and Stochastic Processes Applications of Mathematics

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