《Mathematical Statistics》

1PRELIMNARIES1

1．1 Sample spaces and events1

1．2 Definitions and rules for combining and decomposing events2

1．3 Fields of sets8

1．4 Probability measure10

1．5 Extension of a probability measure15

1．6 Statistical independence16

1．7 Random variables19

1．8 Integration of random variables21

1．9 Conditional probability24

1．10 Conditional random variables25

Problems26

2DISTRIBUTION FUNCTIONS30

2．1 Preliminary remarks30

2．2 Distribution functions of one-dimensional random variables31

2．3 Common types of one-dimensional random variables34

2．4 Distribution functions of two-dimensional random variables39

2．5 Common types of two-dimensional random variables43

2．6 Distribution functions of k-dimensional random variables49

2．7 Common types of k-dimensional random variables51

2．8 Functions of random variables53

2．9 Conditional distribution functions59

2．10 Finite stochastic processes68

Problems69

3MEAN VALUES AND MOMENTS OF RANDOM VARIABLES72

3．1 Introduction72

3．2 Mean value of a random variable73

3．3 Moments of one-dimensional random variables74

3．4 Moments of two-dimensional random variables77

3．5 Moments of k-dimensional random variables79

3．6 Means,variances,and covariances of linear functions of random variables82

3．7 Mean values of conditional random variables83

3．8 Least squares linear regression87

Problems92

4SEQUENCES OF RANDOM VARIABLES96

4．1 Definition of a stochastic process96

4．2 Probability measure for a stochastic process96

4．3 Convergence in probability99

4．4 Almost certain convergence106

4．5 Kolmogorov＇s inequality107

4．6 The strong law of large numbers108

Problems110

5CHARACTERISTIC FUNCTIONS AND GENERATING FUNCTIONS113

5．1 Case of a one-dimensional random variable113

5．2 Case of a k-dimensional random variable119

5．3 Characteristic functions of independent random variables120

5．4 Characteristic functions of a sequence of random variables122

5．5 Determination of distribution functions from moments125

Problems129

6SOME SPECIAL DISCRETE DISTRIBUTIONS133

6．1 The hypergeometric distribution133

6．2 The binomial distribution136

6．3 The multinomial distribution138

6．4 The Poisson distribution140

6．5 Discrete waiting-time distributions141

6．6 Distributions in the theory of runs144

Problems150

7SOME SPECIAL CONTINUOUS DISTRIBUTIONS155

7．1 The rectangular distribution155

7．2 The normal distribution156

7．3 The bivariate normal distribution158

7．4 The k-variate normal distribution163

7．5 The gamma distribution170

7．6 The beta distribution173

7．7 The Dirichlet distribution177

7．8 Distributions involved in the analysis of variance183

Problems187

8SAMPLING THEORY195

8．1 Definition of a random sample195

8．2 Means and variances of mean,variance,and other symmetric functions of a sample198

8．3 Sampling theory of sample sums and means203

8．4 Sampling theory of certain quadratic forms in samples from a normal distribution208

8．5 Sampling from a finite population214

8．6 Matrix sampling222

8．7 Sampling theory of order statistics234

8．8 Order statistics in samples from finite populations243

Problems245

9ASYMPTOTIC SAMPLING THEORY FOR LARGE SAMPLES254

9．1 Convergence of sample mean in probability254

9．2 Limiting distribution of sample sums and means256

9．3 Asymptotic distribution of functions of sample means259

9．4 Asymptotic expansion of distribution of sample sum262

9．5 Limiting distributions of linear functions in large samples from large finite populations266

9．6 Asymptotic distributions concerning order statistics268

Problems274

10LINEAR STATISTICAL ESTIMATION277

10．1 Introductory comments277

10．2 Minimum variance estimators for the mean and variance of a population from random samples279

10．3 Estimators for parameters in linear regression analysis283

10．4 Interval and ellipsoidal estimators for the parameters in normal regression theory289

10．5 Simultaneous confidence intervals:multiple comparisons290

10．6 Normal linear regression analysis in experimental designs297

10．7 Estimation of variance components from linear combinations of random variables305

10．8 Estimators for variance components in experimental designs308

10．9 Linear estimators for means of stratified populations313

10．10 Linear estimator for mean of stratified populations in two-stage sampling318

Problems323

11NONPARAMETRIC STATISTICAL ESTIMATION329

11．1 Introductory remarks329

11．2 Confidence intervals for quantiles329

11．3 Confidence intervals for quantile intervals332

11．4 Confidence intervals for quantiles in finite populations333

11．5 Tolerance limits334

11．6 One-sided confidence contours for a continuous distribution function336

11．7 Confidence bands for a continuous distribution function339

Problems342

12PARAMETRIC STATISTICAL ESTIMATION344

12．1 Differentiation of parametric distribution functions345

12．2 Point estimation350

12．3 Point estimation from large samples358

12．4 Interval estimation365

12．5 Interval estimation from large samples371

12．6 Multidimensional point estimation376

12．7 Multidimensional point estimation from large samples379

12．8 Multidimensional confidence regions381

12．9 Asymptotically smallest confidence regions from large samples384

Problems389

13TESTING PARAMETRIC STATISTICAL HYPOTHESES394

13．1 Introductory remarks and definitions394

13．2 Test of a simple hypothesis398

13．3 The likelihood ratio test402

13．4 Asymptotic distribution of likelihood ratio in large samples408

13．5 Consistency of likelihood ratio test411

13．6 Asymptotic power of likelihood ratio test413

13．7 The likelihood ratio test of a simple hypothesis417

13．8 The likelihood ratio test of a composite hypothesis419

Problems422

14TESTING NONPARAMETRIC STATISTICAL HYPOTHESES428

14．1 The quantile test428

14．2 The nonparametric simple statistical hypothesis430

14．3 The problem of two samples from continuous distributions441

14．4 The method of randomization462

Problems468

15SEQUENTIAL STATISTICAL ANALYSIS472

15．1 Introductory remarks472

15．2 The basic structure of a sequential test474

15．3 Cartesian sequential tests479

15．4 The probability ratio sequential test482

15．5 Application of probability ratio sequential test to binomial distribution494

15．6 Sequential estimation496

Problems498

16STATISTICAL DECISION FUNCTIONS502

16．1 General remarks502

16．2 Definitions and terminology502

16．3 Minimax solution of the decision problem504

16．4 Bayes solutions of the statistical decjsion problem508

16．5 Remarks on extensions and generalizations511

Problems512

17TIME SERIES514

17．1 Introductory remarks514

17．2 Stationary time series515

17．3 The spectral function of a stationary time series517

17．4 Estimation of mean and covariance function of a stationary time series522

17．5 Estimation of spectral distribution523

17．6 Statistical tests for parametric time series526

17．7 Testing a normal noise for whiteness533

17．8 Linear prediction in time series535

Problems537

18MULTIVARIATE STATISTICAL THEORY540

18．1 Multidimensional statistical scatter540

18．2 The Wishart distribution547

18．3 Independence of means and internal scatter matrix in samples from k-dimensional normal distributions555

18．4 Hotelling＇s generalized Student distribution556

18．5 The multidimensional Model I analysis of variance test561

18．6 Principal components564

18．7 Discriminant analysis573

18．8 Distribution of eigenvalues in discriminant analysis581

18．9 Canonical correlation587

Problems592

REFERENCES AND AUTHOR INDEX603

SUBJECT INDEX623