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Mathematical Statistics and Data Analysis

by RICE

ISBN13: 9780534209346
ISBN10: 0534209343
Additional ISBN(s): 9780534399429
Edition: 2nd
Format: Paperback
Copyright: 1994-06-01
Publisher: Duxbury Press
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1. Mathematical Statistics and Data Analysis > ISBN13: 9780534209346

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Summary

1. PROBABILITY Introduction / Sample Spaces / Probability Measures / Computer Probabilities: Counting Methods / Conditional Probability / Independence / Concluding Remarks / Problems 2. RANDOM VARIABLES Discrete Random Variables / Continuous Random Variables / Functions of a Random Variable / Concluding Remarks / Problems 3. JOINT DISTRIBUTIONS Introduction / Discrete Random Variables / Continuous Random Variables / Independent Random Variables / Conditional Distributions / Functions of Jointly Distributed Random Variables / Extreme and Order Statistics / Problems 4. EXPECTED VALUES The Expected Value of a Random Variable / Variance and Standard Deviation / Covariance and Correlation / Conditional Expectation and Prediction / The Moment-Generating Function / Approximate Methods / Problems 5. LIMIT THEOREMS Introduction / The Law of Large Number / Convergence in Distribution and the Central Limit Theorem / Problems 6. DISTRIBUTIONS DERIVED FROM THE NORMAL DISTRIBUTION Introduction / Chi-Squared, t, and F Distributions / The Sample Mean and Sample Variance / Problems 7. SURVEY SAMPLING Introduction / Production Parameters / Simple Random Sampling / Estimation of a Ratio / Stratified Random Sampling / Concluding Remarks / Problems 8. ESTIMATION OF PARAMETERS AND FITTING OF PROBABILITY DISTRIBUTIONS Introduction / Fitting the Poisson Distribution to Emissions of Alpha Particles / Parameter Estimation / The Method of Moments / The Method of Maximum Likelihood / Efficiency and the Cramer-Rao Lower Bound / Sufficiency / Concluding Remarks / Problems 9. TESTING HYPOTHESES AND ASSESSING GOODNESS OF FIT Introduction / The Neyman-Pearson Paradigm / Optimal Tests: The Neyman-Pearson Lemma / The Duality of Confidence Intervals and Hypothesis Tests / Generalized Likelihood Ratio Tests / Likelihood Ratio Tests for the Multinomial Distribution / The Poisson Dispersion Test / Hanging Rootograms / Probability Plots / Tests for Normality / Concluding Remarks / Problems 10. SUMMARIZING DATA Introduction / Methods Based on the Cumulative Distribution Function / Histograms, Density Curves, and Stem-and-Leaf Plots / Measures of Location / Measures of Dispersion / Boxplots / Concluding Remarks / Problems 11. COMPARING TWO SAMPLES Introduction / Comparing Two Independent Samples / Comparing Paired Samples / Experimental Design / Concluding Remarks / Problems 12. THE ANALYSIS OF VARIANCE Introduction / The One-Way Layout / The Two-Way Layout / Concluding Remarks / Problems 13. THE ANALYSIS OF CATEGORICAL DATA Introduction / Fisher's Exact Test / The Chi-Square Test of Homogeneity / The Chi-Square Test of Independence / Matched-Pair Designs / Odds Ratios / Concluding Remarks / Problems 14. LINEAR LEAST SQUARES Introduction / Simple Linear Regression / The Matrix Approach to Linear Least Square / Statistical Properties of Least Squares Estimates / Multiple Linear Regression--An Example / Conditional Inference, Unconditional Inference, and the Bootstrap / Concluding Remarks / Problems 15. DECISION THEORY AND BAYESIAN INFERENCE Introduction / Decision Theory / The Subjectivist Point of View / Concluding Remarks / Problems / APPENDIXES / A. COMMON DISTRIBUTIONS / B. TABLES / BIBLIOGRAPHY / ANSWERS TO SELECTED PROBLEMS / AUTHOR INDEX / INDEX TO DATA SETS / SUBJECT INDEX

Probability

(32)

Introduction

(1)

Sample Spaces

(2)

Probability Measures

(3)

Computing Probabilities: Counting Methods

(8)

The Multiplication Principle

(1)

Permutations and Combinations

(6)

Conditional Probability

(6)

Independence

(3)

Concluding Remarks

(1)

Problems

(9)

Random Variables

(36)

Discrete Random Variables

(13)

Bernoulli Random Variables

(1)

The Binomial Distribution

(2)

The Geometric and Negative Binomial Distributions

(1)

The Hypergeometric Distribution

(2)

The Poisson Distribution

(5)

Continuous Random Variables

(11)

The Exponential Density

(2)

The Gamma Density

(3)

The Normal Distribution

(4)

Functions of a Random Variable

(4)

Concluding Remarks

(1)

Problems

(7)

Joint Distributions

(42)

Introduction

(2)

Discrete Random Variables

(2)

Continuous Random Variables

(10)

Independent Random Variables

(2)

Conditional Distributions

(7)

The Discrete Case

(1)

The Continuous Case

(6)

Functions of Jointly Distributed Random Variables

(8)

Sums and Quotients

(3)

The General Case

(5)

Extrema and Order Statistics

100

(3)

Problems

103

(8)

Expected Values

111

(52)

The Expected Value of a Random Variable

111

(11)

Expectations of Functions of Random Variables

116

(3)

Expectations of Linear Combinations of Random Variables

119

(3)

Variance and Standard Deviation

122

(7)

A Model for Measurement Error

126

(3)

Covariance and Correlation

129

(6)

Conditional Expectation and Prediction

135

(7)

Definitions and Examples

135

(5)

Prediction

140

(2)

The Moment-Generating Function

142

(7)

Approximate Methods

149

(5)

Problems

154

(9)

Limit Theorems

163

(14)

Introduction

163

(1)

The Law of Large Numbers

163

(3)

Convergence in Distribution and the Central Limit Theorem

166

(7)

Problems

173

(4)

Distributions Derived from the Normal Distribution

177

(8)

Introduction

177

(1)

X2,t, and F Distributions

177

(2)

The Sample Mean and the Sample Variance

179

(3)

Problems

182

(3)

Survey Sampling

185

(54)

Introduction

185

(1)

Population Parameters

186

(2)

Simple Random Sampling

188

(18)

The Expectation and Variance of the Sample Mean

189

(7)

Estimation of the Population Variance

196

(3)

The Normal Approximation to the Sampling Distribution of X

199

(7)

Estimation of a Ratio

206

(7)

Stratified Random Sampling

213

(10)

Introduction and Notation

213

(1)

Properties of Stratified Estimates

214

(4)

Methods of Allocation

218

(5)

Concluding Remarks

223

(2)

Problems

225

(14)

Estimation of Parameters and Fitting of Probability Distributions

239

(60)

Introduction

239

(1)

Fitting the Poisson Distribution to Emissions of Alpha Particles

239

(4)

Parameter Estimation

243

(3)

The Method of Moments

246

(7)

The Method of Maximum Likelihood

253

(20)

Maximum Likelihood Estimates of Multinomial Cell Probabilities

259

(2)

Large Sample Theory for Maximum Likelihood Estimates

261

(5)

Confidence Intervals for Maximum Likelihood Estimates

266

(7)

Efficiency and the Cramer-Rao Lower Bound

273

(7)

An Example: The Negative Binomial Distribution

277

(3)

Sufficiency

280

(5)

A Factorization theorem

281

(3)

The Rao-Blackwell Theorem

284

(1)

Concluding Remarks

285

(1)

Problems

286

(13)

Testing Hypotheses and Assessing Goodness of Fit

299

(46)

Introduction

299

(1)

The Neyman-Pearson Paradigm

300

(3)

Optimal Tests: The Neyman-Pearson Lemma

303

(3)

The Duality of Confidence Intervals and Hypothesis Tests

306

(2)

Generalized Likelihood Ratio Tests

308

(2)

Likelihood Ratio Tests for the Multinomial Distribution

310

(6)

The Poisson Dispersion Test

316

(2)

Hanging Rootograms

318

(3)

Probability Plots

321

(6)

Tests for Normality

327

(3)

Concluding Remarks

330

(1)

Problems

331

(14)

Summarizing Data

345

(42)

Introduction

345

(1)

Methods Based on the Cumulative Distribution Function

346

(11)

The Empirical Cumulative Distribution Function

346

(2)

The Survival Function

348

(5)

Quantile-Quantile Plots

353

(4)

Histograms, Density Curves, and Stem-and-Leaf Plots

357

(4)

Measures of Location

361

(9)

The Arithmetic Mean

361

(3)

The Median

364

(1)

The Trimmed Mean

365

(1)

M Estimates

366

(1)

Comparison of Location Estimates

367

(1)

Estimating Variability of Location Estimates by the Bootstrap

367

(3)

Measures of Dispersion

370

(2)

Boxplots

372

(2)

Concluding remarks

374

(1)

Problems

375

(12)

Comparing Two samples

387

(56)

Introduction

387

(1)

Comparing Two Independent Samples

388

(22)

Methods Based on the Normal Distribution

388

(8)

An Example---A Study of Iron Retention

396

(4)

Power

400

(2)

A Nonparametric Method---the Mann-Whitney Test

402

(8)

Comparing Paired Samples

410

(7)

Methods Based on the Normal distribution

411

(2)

A Nonparametric method---the signed Rank Test

413

(2)

An Example---Measuring Mercury Levels in fish

415

(2)

Experimental Design

417

(7)

Mammary Artery Ligation

417

(1)

The Placebo Effect

418

(1)

The Lanarkshire Milk Experiment

419

(1)

The Portocaval shunt

419

(1)

FD&C Red No. 40

420

(1)

Further Remarks on Randomization

421

(1)

Observational Studies, Confounding, and Bias in Graduate Admissions

422

(1)

Fishing Expeditions

423

(1)

Concluding Remark

424

(1)

Problems

425

(18)

The Analysis of Variance

443

(40)

Introduction

443

(1)

The One-Way Layout

443

(12)

Normal Theory; the F Test

445

(6)

The Problem of Multiple Comparisons

451

(1)

Tukey's Method

451

(2)

The Bonferroni Method

453

(1)

A Nonparametric Method---The Kruskal-Wallis Test

453

(2)

The Two-Way Layout

455

(16)

Additive Parametrization

455

(3)

Normal theory for the Two-way Layout

458

(7)

Randomized Block Designs

465

(4)

A Nonparametric Method---Friedman's Test

469

(2)

Concluding Remarks

471

(1)

Problems

472

(11)

The Analysis of Categorical Data

483

(24)

Introduction

483

(1)

Fisher's Exact Test

483

(2)

The Chi-Square Test of Homogeneity

485

(4)

The Chi-Square Test of Independence

489

(3)

Matched-Pairs Designs

492

(2)

Odds Ratios

494

(4)

Concluding Remarks

498

(1)

Problems

498

(9)

Linear Least Squares

507

(64)

Introduction

507

(4)

Simple Linear Regression

511

(18)

Statistical Properties of the Estimated Slope and Intercept

511

(4)

Assessing the Fit

515

(11)

Correlation and Regression

526

(3)

The Matrix Approach to Linear Least Squares

529

(3)

Statistical Properties of Least Squares Estimates

532

(12)

Vector-Valued Random Variables

532

(5)

Mean and Covariance of Least Squares Estimates

537

(2)

Estimation of σ2

539

(2)

Residuals and Standardized Residuals

541

(1)

Inference about β

542

(2)

Multiple Linear Regressions---An Example

544

(4)

Conditional Inference, Unconditional Inference, and the Bootstrap

548

(3)

Concluding Remarks

551

(1)

Problems

552

(19)

Decision Theory and Bayesian Inference

571

(26)

Introduction

571

(1)

Decision theory

571

(16)

Bayes Rules and Minimax Rules

573

(5)

Posterior Analysis

578

(2)

Classification and Hypothesis Testing

580

(4)

Estimation

584

(3)

The Subjectivist Point of View

587

(10)

Bayesian Inference for the Normal Distribution

589

(3)

Bayesian Analysis for the Binomial Distribution

592

(5)

Concluding Remarks

597

(1)

Problems

597

Appendix A Common Distributions

Appendix B Tables

Bibliography

A25

Answers to Selected Problems

A31

Author Index

A43

Index to Data Sets

A45

Subject to Index

A46

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Mathematical Statistics and Data Analysis

Summary

Table of Contents

Supplemental Materials

Rewards Program