Continuous Univariate Distributions Volume 2 Second Edition NORMAN L. JOHNSON University of North Carolina Chapel Hill, North Carolina SAMUEL KOTZ University of Maryland College Park, Maryland N. BALAKRISHNAN McMaster University Hamilton, Ontario, Canada A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Brisbane Toronto Singapore
Contents Preface List of Tables xiii xv 22 Extreme Value Distributions 1 1 Genesis, 1 2 Introduction, 2 3 Limiting Distributions of Extremes, 4 4 Distribution Function and Moments, 11 5 Order Statistics, 15 6 Record Values, 17 7 Generation, Tables, and Probability Paper, 23 8 Characterizations, 25 9 Methods of Inference, 26 9.1 Moment Estimation, 27 9.2 Simple Linear Estimation, 28 9.3 Best Linear Unbiased (Invariant) Estimation, 32 9.4 Asymptotic Best Linear Unbiased Estimation, 37 9.5 Linear Estimation with Polynomial Coefficients, 39 9.6 Maximum Likelihood Estimation, 41 9.7 Conditional Method, 49 9.8 Method of Probability-Weighted Moments, 50 9.9 "Block-Type" Estimation, 51 9.10 A Survey of Other Developments, 53 10 Tolerance Limits and Intervals, 55 11 Prediction Limits and Intervals, 59 12 Outliers and Robustness, 64 13 Probability Plots, Modifications, and Model Validity, 66 vii
iii CONTENTS 14 Applications, 72 15 Generalized Extreme Value Distributions, 75 16 Other Related Distributions, 86 References, 93 23 Logistic Distribution 113 1 Historical Remarks and Genesis, 113 2 Definition, 115 3 Generating Functions and Moments, 116 4 Properties, 118 5 Order Statistics, 123 6 Methods of Inference, 127 7 Record Values, 135 8 Tables, 137 9 Applications, 138 10 Generalizations, 140 11 Related Distributions, 147 References, 154 24 Laplace (Double Exponential) Distributions 164 1 Definition, Genesis, and Historical Remarks, 164 2 Moments, Generating Functions, and Properties, 165 3 Order Statistics, 168 4 Methods of Inference, 172 4.1 Maximum Likelihood Estimation, 172 4.2 Best Linear Unbiased Estimation, 175 4.3 Simplified Linear Estimation, 181 4.4 Asymptotic Best Linear Unbiased Estimation, 183 4.5 Conditional Inference, 185 4.6 Other Developments, 187 5 Tolerance Limits and Prediction Intervals, 187 6 Related Distributions, 190 7 Applications, 201 References, 201 25 Beta Distributions 210 1 Definition, 210 2 Genesis and Random Number Generation, 212 3 Properties, 217
CONTENTS IX 4 Estimation, 221 5 Applications, 235 6 Approximations and Tables, 238 6.1 Approximations, 238 6.2 Tables, 244 7 Related Distributions, 247 8 Products, Quotients, and Differences of Independent Beta Variables, 256 References, 264 26 Uniform (Rectangular) Distributions 276 1 Definition, 276 2 Genesis, 277 3 Historical Remarks, 278 4 Generating Functions, Moments, and Order Statistics, 279 5 Characterizations, 281 6 Estimation of Parameters, 286 7 Estimation Using Order Statistics Censored Samples, 293 8 Tables of Random Numbers, 295 9 Related Distributions, 296 9.1 Mixtures of Two Uniform Distributions, 302 9.2 Other Related Distributions, 304 10 Applications, 307 10.1 Corrections for Grouping, 308 10.2 Life Testing, 308 10.3 Traffic Flow Applications, 310 10.4 Applications in Statistical Tests and Models, 312 11 Random Number Generation, 312 References, 314 27 F-Distributions 322 1 Introduction, 322 2 Properties, 325 3 Order Statistics, 332 4 Tables, 332 5 Approximations and Nomograms, 334 6 Applications, 343
* CONTENTS 7 Pearson Type VI Distributions, 344 8 Other Related Distributions, 346 8.1 "Generalized" /-"-Distributions, 348 8.2 Other Related Distributions, 350 References, 355 28 (-Distributions 362 1 Genesis and Historical Remarks, 362 2 Properties, 363 3 Tables and Nomograms, 368 3.1 Tables, 368 3.2 Nomograms, 372 4 Approximations, 374 5 Applications, 395 6 Pearson Type VII Distributions and Their Modifications, 396 7 Other Related Distributions, 403 References, 422 29 Noncentral x ^Distributions 433 1 Definition and Genesis, 433 2 Historical Notes, 435 3 Distribution, 435 4 Moments, 447 5 Properties of the Distribution, 450 6 Estimation, 451 7 Tables and Computer Algorithms, 458 8 Approximations, 461 9 Applications, 467 10 Related Distributions, 470, References, 473 30 Noncentral ^-Distributions 480 1 Definition and Genesis, 480 2 Historical Remarks, 481 3 Properties, 481 4 Tables and Computer Programs, 488 4.1 Tables, 488 4.2 Computer Programs, 489
CONTENTS xi 5 Approximations, 491 6 Estimation of the Noncentrality Parameter A,, 495 7 Related Distributions, 499 7.1 Doubly Noncentral /-"-Distributions, 499 7.2 Noncentral Beta Distributions, 502 References, 504 31 Noncentral (-Distributions 508 1 Definition, 508 2 Historical Remarks, 508 3 Applications and Estimation, 509 4 Moments, 512 5 Distribution Function, 514 6 Approximations, 519 7 Tables, Charts, and Computer Algorithms, 527 8 Related Distributions, 533 8.1 Noncentral Beta Distribution, 533 8.2 Doubly Noncentral /-Distribution, 533 8.3 Modified Noncentral (-Distribution, 537 8.4 Distribution of Noncentral (-Statistic When the Population is Nonnormal, 538 References, 539 32 Distributions of Correlation Coefficients 545 1 Introduction and Genesis, 545 2 Derivation of Distribution of R, 547 3 Historical Remarks, 556 4 Distribution of R in Non-normal Populations and Robustness, 559 5 Tables and Approximations (Asymptotic Expansions), 569 5.1 Tables, 569 5.2 Approximations Using Transforms, 571 5.3 Asymptotic Expansions of the Distribution of R, 578 6 Estimation of p: Additional Robustness Considerations, 580 6.1 General Remarks, 580 6.2 Point Estimation, 583 6.3 Maximum Likelihood Estimation, 590
XII CONTENTS 6.4 Estimation of Common p Based on Several Samples, 592 6.5 Miscellaneous Estimation Problems, 597 7 Sample Covariance, 599 8 Circular Serial Correlation, 601 9 Noncircular Serial Correlation, 605 10 Leipnik Distribution, 612 11 Multiple Correlation Coefficient, 617 References, 627 33 Lifetime Distributions and Miscellaneous Orderings 639 1 Introduction, 639 2 Life Distributions, 640 3 Birnbaum-Saunders Distributions and Transformations, 651 4 Ordering and Classification of Distributions, 663 4.1 Basic Definitions and Bounds, 663 4.2 Reliability Classification of Orderings, 668 4.3 Alternative Stochastic Classification of Orderings, 672 References, 681 Abbreviations 691 Author Index 693 Subject Index 711