Probability and Statistical Inference

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ISBN: 9780824703790 | 0824703790
Cover: Hardcover
Copyright: 3/22/2000

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This gracefully organized text presents the rigorous theory of probability and statistical inference in the style of a tutorial, using worked examples, exercises, numerous figures and tables, and computer simulations to develop and illustrate concepts. Beginning with the basic ideas and techniques of probability theory and progressing to more rigorous topics, this treatment covers all of the topics typically addressed in a two-semester course in probability and statistical inference for graduate and upper-level undergraduate courses, including hypothesis testing, Bayesian analysis, and sample-size determination. The author reinforces important ideas and special techniques with drills and boxed summaries.

Preface

Acknowledgments

Notions of Probability

(64)

Introduction

(2)

About Sets

(3)

Axiomatic Development of Probability

(3)

The Conditional Probability and Independent Events

(9)

Calculus of Probability

(2)

Bayes's Theorem

(2)

Selected Counting Rules

(2)

Discrete Random Variables

(5)

Probability Mass and Distribution Functions

(4)

Continuous Random Variables

(9)

Probability Density and Distribution Functions

(5)

The Median of a Distribution

(1)

Selected Reviews from Mathematics

(4)

Some Standard Probability Distributions

(18)

Discrete Distributions

(4)

Continuous Distributions

(13)

Exercises and Complements

(15)

Expectations of Functions of Random Variables

(34)

Introduction

(1)

Expectation and Variance

(12)

The Bernoulli Distribution

(1)

The Binomial Distribution

(1)

The Poisson Distribution

(1)

The Uniform Distribution

(1)

The Normal Distribution

(3)

The Laplace Distribution

(1)

The Gamma Distribution

(1)

The Moments and Moment Generating Function

(9)

The Binomial Distribution

(1)

The Poisson Distribution

(1)

The Normal Distribution

(2)

The Gamma Distribution

(2)

Determination of a Distribution via MGF

(2)

The Probability Generating Function

(1)

Exercises and Complements

(10)

Multivariate Random Variables

(78)

Introduction

(1)

Discrete Distributions

100

(7)

The Joint, Marginal and Conditional Distributions

101

(2)

The Multinomial Distribution

103

(4)

Continuous Distributions

107

(12)

The Joint, Marginal and Conditional Distributions

107

(8)

Three and Higher Dimensions

115

(4)

Covariances and Correlation Coefficients

119

(6)

The Multinomial Case

124

(1)

Independence of Random Variables

125

(6)

The Bivariate Normal Distribution

131

(8)

Correlation Coefficient and Independence

139

(2)

The Exponential Family of Distributions

141

(4)

One-parameter Situation

141

(3)

Multi-parameter Situation

144

(1)

Some Standard Probability Inequalities

145

(14)

Markov and Bernstein-Chernoff Inequalities

145

(3)

Tchebysheff's Inequality

148

(1)

Cauchy-Schwarz and Covariance Inequalities

149

(3)

Jensen's and Lyapunov's Inequalities

152

(4)

Holder's Inequality

156

(1)

Bonferroni Inequality

157

(1)

Central Absolute Moment Inequality

158

(1)

Exercises and Complements

159

(18)

Functions of Random Variables and Sampling Distribution

177

(64)

Introduction

177

(2)

Using Distribution Functions

179

(11)

Discrete Cases

179

(2)

Continuous Cases

181

(1)

The Order Statistics

182

(3)

The Convolution

185

(2)

The Sampling Distribution

187

(3)

Using the Moment Generating Function

190

(2)

A General Approach with Transformations

192

(14)

Several Variable Situations

195

(11)

Special Sampling Distributions

206

(6)

The Student's t Distribution

207

(2)

The F Distribution

209

(2)

The Beta Distribution

211

(1)

Special Continuous Multivariate Distributions

212

(8)

The Normal Distribution

212

(6)

The t Distribution

218

(1)

The F Distribution

219

(1)

Importance of Independence in Sampling Distributions

220

(4)

Reproductivity of Normal Distributions

220

(1)

Reproductivity of Chi-square Distributions

221

(2)

The Student's t Distribution

223

(1)

The F Distribution

223

(1)

Selected Review in Matrices and Vectors

224

(3)

Exercises and Complements

227

(14)

Concepts of Stochastic Convergence

241

(40)

Introduction

241

(1)

Convergence in Probability

242

(11)

Convergence in Distribution

253

(11)

Combination of the Modes of Convergence

256

(1)

The Central Limit Theorems

257

(7)

Convergence of Chi-square, t, and F Distributions

264

(6)

The Chi-square Distribution

264

(1)

The Student's t Distribution

264

(1)

The F Distribution

265

(1)

Convergence of the PDF and Percentage Points

265

(5)

Exercises and Complements

270

(11)

Sufficiency, Completeness, and Ancillarity

281

(60)

Introduction

281

(1)

Sufficiency

282

(12)

The Conditional Distribution Approach

284

(4)

The Neyman Factorization Theorem

288

(6)

Minimal Sufficiency

294

(6)

The Lehmann-Scheffe Approach

295

(5)

Information

300

(9)

One-parameter Situation

301

(3)

Multi-parameter Situation

304

(5)

Ancillarity

309

(9)

The Location, Scale, and Location-Scale Families

314

(2)

Its Role in the Recovery of Information

316

(2)

Completeness

318

(9)

Complete Sufficient Statistics

320

(4)

Basu's Theorem

324

(3)

Exercises and Complements

327

(14)

Point Estimation

341

(54)

Introduction

341

(1)

Finding Estimators

342

(9)

The Method of Moments

342

(2)

The Method of Maximum Likelihood

344

(7)

Criteria to Compare Estimators

351

(7)

Unbiasedness, Variance and Mean Squared Error

351

(3)

Best Unbiased and Linear Unbiased Estimators

354

(4)

Improved Unbiased Estimator via Sufficiency

358

(7)

The Rao-Blackwell Theorem

358

(7)

Uniformly Minimum Variance Unbiased Estimator

365

(12)

The Cramer-Rao Inequality and UMVUE

366

(5)

The Lehmann-Scheffe Theorems and UMVUE

371

(3)

A Generalization of the Cramer-Rao Inequality

374

(1)

Evaluation of Conditional Expectations

375

(2)

Unbiased Estimation Under Incompleteness

377

(3)

Does the Rao-Blackwell Theorem Lead to UMVUE?

377

(3)

Consistent Estimators

380

(2)

Exercises and Complements

382

(13)

Tests of Hypotheses

395

(46)

Introduction

395

(1)

Error Probabilities and the Power Function

396

(5)

The Concept of a Best Test

399

(2)

Simple Null Versus Simple Alternative Hypotheses

401

(16)

Most Powerful Test via the Neyman-Pearson Lemma

401

(12)

Applications: No Parameters Are Involved

413

(3)

Applications: Observations Are Non-IID

416

(1)

One-Sided Composite Alternative Hypothesis

417

(8)

UMP Test via the Neyman-Pearson Lemma

417

(3)

Monotone Likelihood Ratio Property

420

(2)

UMP Test via MLR Property

422

(3)

Simple Null Versus Two-Sided Alternative Hypotheses

425

(4)

An Example Where UMP Test Does Not Exist

425

(1)

An Example Where UMP Test Exists

426

(2)

Unbiased and UMP Unbiased Tests

428

(1)

Exercises and Complements

429

(12)

Confidence Interval Estimation

441

(36)

Introduction

441

(2)

One-Sample Problems

443

(13)

Inversion of a Test Procedure

444

(2)

The Pivotal Approach

446

(5)

The Interpretation of a Confidence Coefficient

451

(1)

Ideas of Accuracy Measures

452

(3)

Using Confidence Intervals in the Tests of Hypothesis

455

(1)

Two-Sample Problems

456

(7)

Comparing the Location Parameters

456

(4)

Comparing the Scale Parameters

460

(3)

Multiple Comparisons

463

(6)

Estimating a Multivariate Normal Mean Vector

463

(2)

Comparing the Means

465

(2)

Comparing the Variances

467

(2)

Exercises and Complements

469

(8)

Bayesian Methods

477

(30)

Introduction

477

(2)

Prior and Posterior Distributions

479

(2)

The Conjugate Priors

481

(4)

Point Estimation

485

(3)

Credible Intervals

488

(5)

Highest Posterior Density

489

(3)

Contrasting with the Confidence Intervals

492

(1)

Tests of Hypotheses

493

(1)

Examples with Non-Conjugate Priors

494

(3)

Exercises and Complements

497

(10)

Likelihood Ratio and Other Tests

507

(32)

Introduction

507

(1)

One-Sample Problems

508

(7)

LR Test for the Mean

509

(3)

LR Test for the Variance

512

(3)

Two-Sample Problems

515

(7)

Comparing the Means

515

(4)

Comparing the Variances

519

(3)

Bivariate Normal Observations

522

(7)

Comparing the Means: The Paired Difference t Method

522

(3)

LR Test for the Correlation Coefficient

525

(3)

Tests for the Variances

528

(1)

Exercises and Complements

529

(10)

Large-Sample Inference

539

(30)

Introduction

539

(1)

The Maximum Likelihood Estimation

539

(3)

Confidence Intervals and Tests of Hypothesis

542

(13)

The Distribution-Free Population Mean

543

(5)

The Binomial Proportion

548

(5)

The Poisson Mean

553

(2)

The Variance Stabilizing Transformations

555

(8)

The Binomial Proportion

556

(3)

The Poisson Mean

559

(1)

The Correlation Coefficient

560

(3)

Exercises and Complements

563

(6)

Sample Size Determination: Two-Stage Procedures

569

(22)

Introduction

569

(4)

The Fixed-Width Confidence Interval

573

(6)

Stein's Sampling Methodology

573

(1)

Some Interesting Properties

574

(5)

The Bounded Risk Point Estimation

579

(5)

The Sampling Methodology

581

(1)

Some Interesting Properties

582

(2)

Exercises and Complements

584

(7)

Appendix

591

(42)

Abbreviations and Notation

591

(2)

A Celebration of Statistics: Selected Biographical Notes

593

(28)

Selected Statistical Tables

621

(12)

The Standard Normal Distribution Function

621

(5)

Percentage Points of the Chi-Square Distribution

626

(2)

Percentage Points of the Student's t Distribution

628

(2)

Percentage Points of the F Distribution

630

(3)

References

633

(16)

Index

649

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Probability and Statistical Inference

Summary

Table of Contents

Supplemental Materials