《STATISTICAL ANALYSIS OF STATIONARY TIME SERIES》

CHAPTER1.Stationry Stoehastic Processes and Their Reprssentafions21

1.0Introduction21

1.1What Is a StochasticProcess?21

1.2Continuity in the Mean24

1.3Stochastic Set Functions of Orthogonal Incerments25

1.4Ortheogonal RePresen tations of Stochastic Processes27

1.5Stationary Processes29

1.6Represen tations of Station ary Prcesses33

1.7Time and Ensemble Averages42

1.8Vector Processes44

1.9Operatons on Stationary Processes49

1.10Harmonizable Stochastic Processes59

CHAPTER 2.STATUSTUCAK Questions uhen the Specrum is Known61

2.0Introduotion61

2.1Prelininares61

2.2Prediction65

2.3InterPolation82

2.4Filtering of Stationary Processes85

2.5Treatmant of Linear Hypotheses with Specified Spectrum86

Chapter3.Staistical Analysis of Parametric Models91

3.0Introduction91

3.1Periodogeam Analy sis91

3.2The Variate Differaence Method94

3.3Effect of Smoo thing of Time Series (Slutzky 's Theorem)95

3.4Serial Correlation Coefficients for Normal White Noise97

3.5Approximate Distributions of Quadratic Forms101

3.6Testing Autoregerssive Schemes and Moving Averages106

3.7Estimation and the Asymptotic Distribution of the Coef.ficients of an Autoregressive Schems111

3.8Duscussion of the Methods Desctibed in theisChapter115

CHApter 4.Etimaion of the Spectrum117

4.0Introduction117

4.1A General Class of Estim ates117

4.2An Optimum Property of S pectrograph Estomtes123

4.3A Remark on the Bias of Spectrograph Estimates129

4.4The Asymptotic Variance of Spectrograph Estimates131

4.5Another Class of Estim ates137

4.6Special Estimates of the Spectral Density145

4.7The Mean Square Error of Estimates153

4.8An Example from Statistical Optics155

CHAPTER 5.Applications155

5.0Introduction158

5.1Derivations of Spectra of Random Noise158

5.2Measuring Noise Spectra160

5.3Turbulence163

5.4Measuting Turbulence Spectra171

5.5Basic Ideas in a Statistical Theory of Ocean Waves174

5.6Other AppLications177

CHAPTRER6.Distibution of Spectral Estimates179

6.0Introduction179

6.1Preliminary Remarks179

6.2A Heuristic Derivation of a Limit Theorem182

6.3Preliminary Considerations185

6.4The atment of Pure White Nois188

6.5The General Theorem190

6.6The N ormal Case195

6.7Remarks on the Nonno rmal Case198

6.8Spectral Analy sis With a Regression Present202

6.9Alterintive statistics and the Corresponding Limit Theorems207

6.10AlternativeStatistics and the Vorresponding Limit Theorems207

6.11CONFIDENCE Bad for the Spectral Density208

6.12Spectral Analysis of Some ArtificiallY GENERATED Time SERIES214

CHAPTER7.Problems in Linear Estimaion226

7.0Preliminary Discussion226

7.1Estimating Regression Coefficients321

7.2The Regression Spectrum233

7.3Asymptotie Expression for the Covariance Motrices235

7.4Elements of the Spectrum241

7.5Polynmial and Trigonomertic Regression245

7.6More General Trigonom etric anf Poly nomial Regression248

7.7Some Other Types of Regression254

7.8Detection of Sigeals in Noise255

7.9Confidence Intervals and Tests257

CHAPTER8.Assorted Problems360

8.0Introduction260

8.1Prediction when the Con jectured Spectunmis not the Trus One260

8.2UnifirmConvergence of the Estimsted Spectal Density to the The Trus Spoctal Density262

8.3The Asymptotic Distribution of an Integral of a Spectro-geanph Estionate263

8.4The Mean Square Error of Prediction When the Spectrum isEstim ated267

8.5Other Types of Estim ates of the Spectrum270

8.6The Zeros-and Maxima of Stationary Stochastic Processes271

8.7Prefiltering of a Time Series273

8.8Comments on Tests of Normality275

Problrms278

APPENDIX ON COMPLRX V ARIABLE THEORY288

BIBLIOGRAPHY294

INDKX299