Mathematical Statistics and Data Analysis (with CD Data Sets)

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ISBN: 9780534399429 | 0534399428
Cover: Hardcover
Copyright: 4/28/2006

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This is the first text in a generation to re-examine the purpose of the mathematical statistics course. The book's approach interweaves traditional topics with data analysis and reflects the use of the computer with close ties to the practice of statistics. The author stresses analysis of data, examines real problems with real data, and motivates the theory. The book's descriptive statistics, graphical displays, and realistic applications stand in strong contrast to traditional texts that are set in abstract settings.

Preface

Probability

(34)

Introduction

(1)

Sample Spaces

(2)

Probability Measures

(2)

Computing Probabilities: Counting Methods

(10)

The Multiplication Principle

(2)

Permutations and Combinations

(7)

Conditional Probability

(7)

Independence

(3)

Concluding Remarks

(1)

Problems

(9)

Random Variables

(36)

Discrete Random Variables

(12)

Bernoulli Random Variables

(1)

The Binomial Distribution

(2)

The Geometric and Negative Binomial Distributions

(2)

The Hypergeometric Distribution

(1)

The Poisson Distribution

(5)

Continuous Random Variables

(11)

The Exponential Density

(3)

The Gamma Density

(1)

The Normal Distribution

(4)

The Beta Density

(1)

Functions of a Random Variable

(6)

Concluding Remarks

(1)

Problems

(7)

Joint Distributions

(45)

Introduction

(1)

Discrete Random Variables

(3)

Continuous Random Variables

(9)

Independent Random Variables

(3)

Conditional Distributions

(9)

The Discrete Case

(1)

The Continuous Case

(8)

Functions of Jointly Distributed Random Variables

(8)

Sums and Quotients

(3)

The General Case

(5)

Extrema and Order Statistics

104

(3)

Problems

107

(9)

Expected Values

116

(61)

The Expected Value of a Random Variable

116

(14)

Expectations of Functions of Random Variables

121

(3)

Expectations of Linear Combinations of Random Variables

124

(6)

Variance and Standard Deviation

130

(8)

A Model for Measurement Error

135

(3)

Covariance and Correlation

138

(9)

Conditional Expectation and Prediction

147

(8)

Definitions and Examples

147

(5)

Prediction

152

(3)

The Moment-Generating Function

155

(6)

Approximate Methods

161

(5)

Problems

166

(11)

Limit Theorems

177

(15)

Introduction

177

(1)

The Law of Large Numbers

177

(4)

Convergence in Distribution and the Central Limit Theorem

181

(7)

Problems

188

(4)

Distributions Derived from the Normal Distribution

192

(7)

Introduction

192

(1)

Χ2, t, and F Distributions

192

(3)

The Sample Mean and the Sample Variance

195

(3)

Problems

198

(1)

Survey Sampling

199

(56)

Introduction

199

(1)

Population Parameters

200

(2)

Simple Random Sampling

202

(18)

The Expectation and Variance of the Sample Mean

203

(7)

Estimation of the Population Variance

210

(4)

The Normal Approximation to the Sampling Distribution of X

214

(6)

Estimation of a Ratio

220

(7)

Stratified Random Sampling

227

(11)

Introduction and Notation

227

(1)

Properties of Stratified Estimates

228

(4)

Methods of Allocation

232

(6)

Concluding Remarks

238

(1)

Problems

239

(16)

Estimation of Parameters and Fitting of Probability Distributions

255

(74)

Introduction

255

(1)

Fitting the Poisson Distribution to Emissions of Alpha Particles

255

(2)

Parameter Estimation

257

(3)

The Method of Moments

260

(7)

The Method of Maximum Likelihood

267

(18)

Maximum Likelihood Estimates of Multinomial Cell Probabilities

272

(2)

Large Sample Theory for Maximum Likelihood Estimates

274

(5)

Confidence Intervals from Maximum Likelihood Estimates

279

(6)

The Bayesian Approach to Parameter Estimation

285

(13)

Further Remarks on Priors

294

(2)

Large Sample Normal Approximation to the Posterior

296

(1)

Computational Aspects

297

(1)

Efficiency and the Cramer-Rao Lower Bound

298

(7)

An Example: The Negative Binomial Distribution

302

(3)

Sufficiency

305

(6)

A Factorization Theorem

306

(4)

The Rao-Blackwell Theorem

310

(1)

Concluding Remarks

311

(1)

Problems

312

(17)

Testing Hypotheses and Assessing Goodness of Fit

329

(48)

Introduction

329

(2)

The Neyman-Pearson Paradigm

331

(6)

Specification of the Significance Level and the Concept of a p-value

334

(1)

The Null Hypothesis

335

(1)

Uniformly Most Powerful Tests

336

(1)

The Duality of Confidence Intervals and Hypothesis Tests

337

(2)

Generalized Likelihood Ratio Tests

339

(2)

Likelihood Ratio Tests for the Multinomial Distribution

341

(6)

The Poisson Dispersion Test

347

(2)

Hanging Rootograms

349

(3)

Probability Plots

352

(6)

Tests for Normality

358

(3)

Concluding Remarks

361

(1)

Problems

362

(15)

Summarizing Data

377

(43)

Introduction

377

(1)

Methods Based on the Cumulative Distribution Function

378

(11)

The Empirical Cumulative Distribution Function

378

(2)

The Survival Function

380

(5)

Quantile-Quantile Plots

385

(4)

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

389

(3)

Measures of Location

392

(9)

The Arithmetic Mean

393

(2)

The Median

395

(2)

The Trimmed Mean

397

(1)

M Estimates

397

(1)

Comparison of Location Estimates

398

(1)

Estimating Variability of Location Estimates by the Bootstrap

399

(2)

Measures of Dispersion

401

(1)

Boxplots

402

(2)

Exploring Relationships with Scatterplots

404

(3)

Concluding Remarks

407

(1)

Problems

408

(12)

Comparing Two Samples

420

(57)

Introduction

420

(1)

Comparing Two Independent Samples

421

(23)

Methods Based on the Normal Distribution

421

(12)

Power

433

(2)

A Nonparametric Method--The Mann-Whitney Test

435

(8)

Bayesian Approach

443

(1)

Comparing Paired Samples

444

(8)

Methods Based on the Normal Distribution

446

(2)

A Nonparametric Method--The Signed Rank Test

448

(2)

An Example--Measuring Mercury Levels in Fish

450

(2)

Experimental Design

452

(7)

Mammary Artery Ligation

452

(1)

The Placebo Effect

453

(1)

The Lanarkshire Milk Experiment

453

(1)

The Portacaval Shunt

454

(1)

FD&C Red No. 40

455

(1)

Further Remarks on Randomization

456

(1)

Observational Studies, Confounding, and Bias in Graduate Admissions

457

(1)

Fishing Expeditions

458

(1)

Concluding Remarks

459

(1)

Problems

459

(18)

The Analysis of Variance

477

(37)

Introduction

477

(1)

The One-Way Layout

477

(12)

Normal Theory; the F Test

478

(7)

The Problem of Multiple Comparisons

485

(3)

A Nonparametric Method--The Kruskal-Wallis Test

488

(1)

The Two-Way Layout

489

(15)

Additive Parametrization

489

(3)

Normal Theory for the Two-Way Layout

492

(8)

Randomized Block Designs

500

(3)

A Nonparametric Method--Friedman's Test

503

(1)

Concluding Remarks

504

(1)

Problems

505

(9)

The Analysis of Categorical Data

514

(28)

Introduction

514

(1)

Fisher's Exact Test

514

(2)

The Chi-Square Test of Homogeneity

516

(4)

The Chi-Square Test of Independence

520

(3)

Matched-Pairs Designs

523

(3)

Odds Ratios

526

(4)

Concluding Remarks

530

(1)

Problems

530

(12)

Linear Least Squares

542

Introduction

542

(5)

Simple Linear Regression

547

(17)

Statistical Properties of the Estimated Slope and Intercept

547

(3)

Assessing the Fit

550

(10)

Correlation and Regression

560

(4)

The Matrix Approach to Linear Least Squares

564

(3)

Statistical Properties of Least Squares Estimates

567

(13)

Vector-Valued Random Variables

567

(6)

Mean and Covariance of Least Squares Estimates

573

(2)

Estimation of σ2

575

(1)

Residuals and Standardized Residuals

576

(1)

Inference about β

577

(3)

Multiple Linear Regression---An Example

580

(5)

Conditional Inference, Unconditional Inference, and the Bootstrap

585

(2)

Local Linear Smoothing

587

(4)

Concluding Remarks

591

(1)

Problems

591

Appendix A Common Distributions

(3)

Appendix B Tables

(21)

Bibliography

(7)

Answers to Selected Problems

(16)

Author Index

(3)

Applications Index

(3)

Subject Index

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Mathematical Statistics and Data Analysis (with CD Data Sets)

Summary

Table of Contents

Supplemental Materials