Introduction to Mathematical Statistics

An exceptionally clear and impeccably accurate presentation of statistical applications and more advanced theory. Included is a chapter on the distribution of functions of random variables as well as an excellent chapter on sufficient statistics. More modern technology is used in considering limiting distributions, making the presentations more clear and uniform. Title Page......Page 1 CONTENTS......Page 5 Preface......Page 9 1.1 Introduction......Page 13 1.2 Set Theory......Page 15 1.3 The Probability Set Function......Page 24 1.4 Coditional Probability and Independence......Page 31 1.5 Random Variables of the Discrete Type......Page 40 1.6 Random Variables of the Continuous Type......Page 49 1.7 Properties of the Distributiion Function......Page 56 1.8 Expectation of a Random Variable......Page 64 1.9 Some Special Expectations......Page 70 1.10 Chebychev's Inequality......Page 80 2.2 Conditional Distributions and Expectations......Page 86 2.3 The Correlation Coefficient......Page 103 2.4 Independent Random Variables......Page 112 2.5 Extension to Several Random Variables......Page 119 3.2 The Poisson Distribution......Page 128 3.3 The Gamma and Chi-Square Distributions......Page 143 3.4 The Normal Distribution......Page 150 3.5 The Bivariate Normal Distribution......Page 158 4.1 Sampling Theory......Page 167 4.3 Transformations of Variables of the Continuous Type......Page 175 4.4 The Beta, t, and F Distributions......Page 191 4.5 Extensions of the Change-of-Variable Technique......Page 198 4.6 Distributions of Order Statistics......Page 205 4.7 The Moment-Generating-Function Technique......Page 215 4.8 The Distributions of X_bar and nS^2/sig^2......Page 226 4.9 Expectations of Functions of Random Variables......Page 230 *4.10 The Multivariate Normal Distribution......Page 235 5.1 Convergence in Distribution......Page 245 5.2 Convergence in Probability......Page 250 5.3 Limiting Moment-Generating Functions......Page 251 5.4 The Central Limit Theorem......Page 258 5.5 Some Theorems on Limiting Distributions......Page 265 6.1 Point Estimation......Page 271 6.2 Confidence......Page 280 6.3 Confidence Intervals for Difference of Means......Page 288 6.4 Tests of Statistical Hypotheses......Page 292 6.5 Additional Comments About Statistical Tests......Page 300 6.6 Chi-Square Tests......Page 305 7.1 Measures of Quality of Estimators......Page 319 7.2 A Sufficient Statistic for a Parameter......Page 326 7.3 Properties of a Sufficient Statistic......Page 334 7.4 Completeness and Uniqueness......Page 341 7.5 The Exponential Class of Probability Density Functions......Page 345 7.6 Functions of a Parameter......Page 350 7.7 The Case of Several Parameters......Page 353 7.8 Minimal Sufficient and Ancillary Statistics......Page 360 7.9 Sufficiency, Completeness, and Independence......Page 365 8.2 Fisher Information and the Rao-Cramer Inequality......Page 375 8.3 Limiting Distributions of Maximum Likelihood Estimators......Page 384 8.4 Robust M-Estimation......Page 399 9.1 Certain Best Tests......Page 407 8.2 Uniformly Most Powerful Tests......Page 417 9.3 Likelihood Ratio Tests......Page 425 9.4 The Sequential Probability Ratio Test......Page 437 9.5 Minimax, Bayesian, and Classification Procedures......Page 445 10.1 The Distributions of Certain Quadratic Forms......Page 458 10.2 A Test of the Equality of Several Means......Page 464 10.3 Noncentral X^2 and Noncentral F......Page 470 10.4 Multiple Comparisons......Page 473 10.5 The Analysis of Variance......Page 478 10.6 A Regression Problem......Page 483 10.8 The Distributions of Certain Quadratic Forms......Page 490 10.9 The Independence of Certain Quadratic Forms......Page 498 11.1 Confidence Intervals for Distribution Quantiles......Page 509 11.2 Tolerance Limits for Distributions......Page 512 11.3 The Sign Test......Page 518 11.4 A Test of Wilcoxon......Page 520 11.6 The Mann-Whitney-Wilcoxon Test......Page 526 11.8 Linear Rank Statistics......Page 539 11.9 Adaptive Nonparametric Methods......Page 542 B Tables......Page 555 C Answers to Selected Exercises......Page 564 INDEX......Page 571 The Fifth Edition Of This Text Offers A Careful Presentation Of The Probability Needed For Mathematical Statistics And The Mathematics Of Statistical Inference. Robert V. Hogg, Allen T. Craig. Includes Bibliographical References And Index.