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 Introduction to Mathematical Statistics Hogg (Robert V.) & Craig (Allen T.) This Page provides (where held) the Abstract of the above Book and those of all the Papers contained in it. Colour-Conventions Disclaimer Books / Papers Citing this Book Notes Citing this Book

BOOK ABSTRACT:

Preface

1. We are much indebted to our colleagues throughout the country who have so generously provided us with suggestions on both the order of presentation and the kind of material to be included in this edition of our Introduction to Mathematical Statistics. Thus you will find, among other things, that
1. there is a greatly expanded discussion of random variables, that
2. the distribution of the order statistics has been incorporated with the work on transformations which are not one-to-one, that
3. an early proof of the independence of X and S2 (when sampling from a normal distribution) has been introduced, and that
4. limiting distributions are now found in Chapter 5.
Thus essentially all of the distribution theory which is needed for the remainder of the book is found in the first five chapters.
2. At the request of a number of teachers we have added some new exercises and a chapter on nonparametric methods. Although a number of other worthwhile suggestions were made, we have had to forego their inclusion in order to keep the book in reasonable bounds.
3. We hope, and believe, that this edition is written at the same mathematical level as were its predecessors, so that a good grounding in the calculus should be adequate mathematical preparation. This remark does not apply to the last chapter, which does require an elementary knowledge of matrix algebra.

Contents
1. Distributions of Random Variables – 1
• 1.1 Introduction 1
• 1.2 Algebra of Sets 3
• 1.3 Set Functions 7
• 1.4 The Probability Set Function 11
• 1.5 Random Variables 14
• 1.6 The Probability Density Function 21
• 1.7 The Distribution Function 29
• 1.8 Certain Probability Models 36
• 1.9 Mathematical Expectation 41
• 1.10 Some Special Mathematical Expectations 46
• 1.11 Chebyshev’s Inequality 54
2. Conditional Probability and Stochastic Independence
• 2.1 Conditional Probability 57
• 2.2 Marginal and Conditional Distributions 61
• 2.3 The Correlation Coefficient 69
• 2.4 Stochastic Independence 76
3. Some Special Distributions
• 3.1 The Binomial Distribution 86
• 3.2 The Poisson Distribution 94
• 3.3 The Gamma and Chi-Square Distributions 99
• 3.4 The Normal Distribution 104
• 3.5 The Bivariate Normal Distribution 111
4. Distributions of Functions of Random Variables
• 4.1 Sampling Theory 116
• 4.2 Transformations of Variables of the Discrete Type 121
• 4.3 Transformations of Variables of the Continuous Type 125
• 4.4 The t and F Distributions 135
• 4.5 Extensions of the Change of Variable Technique 139
• 4.6 Distributions of Order Statistics 145
• 4.7 The Moment-Generating Function Technique 154
• 4.8 The Distributions of X and nS22 163
• 4.9 Expectations of Functions of Random Variables 166
5. Limiting Distributions 171
• 5.1 Limiting Distributions 171
• 5.2 Stochastic Convergence 175
• 5.3 Limiting Moment-Generating Functions 178
• 5.4 The Central Limit Theorem 182
• 5.5 Some Theorems on Limiting Distributions 186
6. Interval Estimation
• 6.1 Random Intervals 190
• 6.2 Confidence Intervals for Means 193
• 6.3 Confidence Intervals for Differences of Means
• 6.4 Confidence Intervals for Variances 202
• 6.5 Bayesian Interval Estimates 207
7. Point Estimation and Sufficient Statistics
• 7.1 The Problem of Point Estimation 212
• 7.2 A Sufficient Statistic for a Parameter 215
• 7.3 The Rao-Blackwell Theorem 223
• 7.4 Completeness 226
• 7.5 Uniqueness 229
• 7.6 The Exponential Class of Probability Density Functions 231
• 7.7 Functions of a Parameter 234
• 7.8 The Case of Several Parameters 237
• 7.9 Sufficiency, Completeness, and Stochastic Independence 243
8. Further Topics in Point Estimation
• 8.1 The Rao-Cramer Inequality 248
• 8.2 Maximum Likelihood Estimation of Parameters
• 8.3 Decision Functions 258
• 8.4 Bayesian Procedures 261
9. Statistical Hypotheses
• 9.1 Some Examples and Definitions 265
• 9.2 Certain Best Tests 272
• 9.3 Uniformly Most Powerful Tests 280
• 9.4 The Sequential Probability Ratio Test 285
• 9.5 Minimax and Bayesian Tests 290
10. Other Statistical Tests
• 10.1 Likelihood Ratio Tests 297
• 10.2 Chi-Square Tests 308
• 10.3 The Distributions of Certain Quadratic Forms
• 10.4 A Test of the Equality of Several Means 321
• 10.5 Noncentral χ2 and Noncentral F 325
• 10.6 The Analysis of Variance 327
• 10.7 A Regression Problem 335
• 10.8 A Test of Stochastic Independence 339
• 10.9 Multiple Comparisons 343
11. Nonparametric Methods
• 11.1 Confidence Intervals for Distribution Quantiles
• 11.2 Tolerance Limits for Distributions 353
• 11.3 The Sign Test 357
• 11.4 A Test of Wilcoxon 359
• 11.5 The Equality of Two Distributions 365
• 11.6 The Mann-Whitney-Wilcoxon Test 371
• 11.7 Distributions under Alternative Hypotheses
12. Further Normal Distribution Theory
• 12.1 The Multivariate Normal Distribution 379
• 12.2 The Distributions of Certain Quadratic Forms 384
• 12.3 The Independence of Certain Quadratic Forms 388

BOOK COMMENT:

Collier Macmillan, 3rd Edition, 1997.

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