Statistical Analysis of Stationary Time Series

Stationary Stochastic Processes and Their Representations: 1.0 Introduction; 1.1 What is a stochastic process?; 1.2 Continuity in the mean; 1.3 Stochastic set functions of orthogonal increments; 1.4 Orthogonal representations of stochastic processes; 1.5 Stationary processes; 1.6 Representations of stationary processes; 1.7 Time and ensemble averages; 1.8 Vector processes; 1.9 Operations on stationary processes; 1.10 Harmonizable stochastic processes Statistical Questions when the Spectrum is Known (Least Squares Theory): 2.0 Introduction; 2.1 Preliminaries; 2.2 Prediction; 2.3 Interpolation; 2.4 Filtering of stationary processes; 2.5 Treatment of linear hypotheses with specified spectrum Statistical Analysis of Parametric Models: 3.0 Introduction; 3.1 Periodogram analysis; 3.2 The variate difference method; 3.3 Effect of smoothing of time series (Slutzky's theorem); 3.4 Serial correlation coefficients for normal white noise; 3.5 Approximate distributions of quadratic forms; 3.6 Testing autoregressive schemes and moving averages; 3.7 Estimation and the asymptotic distribution of the coefficients of an autoregressive scheme; 3.8 Discussion of the methods described in this chapter Estimation of the Spectrum: 4.0 Introduction; 4.1 A general class of estimates; 4.2 An optimum property of spectrograph estimates; 4.3 A remark on the bias of spectrograph estimates; 4.4 The asymptotic variance of spectrograph estimates; 4.5 Another class of estimates; 4.6 Special estimates of the spectral density; 4.7 The mean square error of estimates; 4.8 An example from statistical optics Applications: 5.0 Introduction; 5.1 Derivations of spectra of random noise; 5.2 Measuring noise spectra; 5.3 Turbulence; 5.4 Measuring turbulence spectra; 5.5 Basic ideas in a statistical theory of ocean waves; 5.6 Other applications Distribution of Spectral Estimates: 6.0 Introduction; 6.1 Preliminary remarks; 6.2 A heuristic derivation of a limit theorem; 6.3 Preliminary considerations; 6.4 Treatment of pure white noise; 6.5 The general theorem; 6.6 The normal case; 6.7 Remarks on the nonnormal case; 6.8 Spectral analysis with a regression present; 6.9 Alternative estimates of the spectral distribution function; 6.10 Alternative statistics and the corresponding limit theorems; 6.11 Confidence band for the spectral density; 6.12 Spectral analysis of some artificially generated time series Problems in Linear Estimation: 7.0 Preliminary discussion; 7.1 Estimating regression coefficients; 7.2 The regression spectrum; 7.3 Asymptotic expression for the covariance matrices; 7.4 Elements of the spectrum; 7.5 Polynomial and trigonometric regression; 7.6 More general trigonometric and polynomial regression; 7.7 Some other types of regression; 7.8 Detection of signals in noise; 7.9 Confidence intervals and tests Assorted Problems: 8.0 Introduction; 8.1 Prediction when the conjectured spectrum is not the true one; 8.2 Uniform convergence of the estimated spectral density to the true spectral density; 8.3 The asymptotic distribution of an integral of a spectrograph estimate; 8.4 The mean square error of prediction when the spectrum is estimated; 8.5 Other types of estimates of the spectrum; 8.6 The zeros and maxima of stationary stochastic processes; 8.7 Prefiltering of a time series; 8.8 Comments on tests of normality Problems Appendix on complex variable theory Bibliography Index.