Continuous Univariate Distributions Volume 1 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 xv xvii 12 Continuous Distributions (General) 1 1 Introduction, 1 2 Order Statistics, 6 3 Calculus of Probability Density Functions, 14 4 Systems of Distributions, 15 4.1 Pearson System, 15 4.2 Expansions, 25 4.3 Transformed Distributions, 33 4.4 Bessel Function Distributions, 50 4.5 Miscellaneous, 53 5 Cornish-Fisher Expansions, 63 6 Note on Characterizations, 67 Bibliography, 68 13 Normal Distributions 80 1 Definition and Tables, 80 2 Historical Remarks, 85 3 Moments and Other Properties, 88 4 Order Statistics, 93 5 Record Values, 99 6 Characterizations, 100 7 Approximations and Algorithms, 111 vii
Viii CONTENTS 8 Estimation, 123 8.1 Estimation of f, 123 8.2 Estimation of a, 127 8.3 Estimation of Functions of and a, 139 8.4 Estimation from Censored Data, 146 9 Siriiulational Algorithms, 152 9.1 Box-Muller Method, 152 9.2 Marsaglia-Bray's Improvement, 153 9.3 Acceptance-Rejection Method, 153 9.4 Ahrens-Dieter Method, 155 10 Related Distributions, 156 10.1 Truncated Normal Distributions, 156 10.2 Mixtures, 163 10.3 Other Related Distributions, 168 Bibliography, 174 14 Lognormal Distributions 207 1 Introduction, 207 2 Historical Remarks, 209 3 Moments and Other Properties, 211 4 Estimation, 220 4.1 6 Known, 220 4.2 6 Unknown, 222 4.3 Graphical Estimation, 236 5 Tables and Graphs, 237 6 Applications, 238 7 Censoring, Truncated Lognormal and Related Distributions, 240 8 Convolution of Normal and Lognormal Distributions, 247 Bibliography, 249 15 Inverse Gaussian (Wald) Distributions 259 1 Introduction, 259 2 Genesis, 260 3 Definition, 261 4 Moments, 262 5 Properties, 266 6 Estimation of Parameters, 270
CONTENTS ix 7 Truncated Distributions Estimation of Parameters, 276 7.1 Doubly Truncated Distribution, 277 7.2 Truncation of the Lower Tail Only, 278 7.3 Truncation of the Upper Tail Only, 279 8 Conditional Expectations of the Estimators of the Cumulants, 279 9 Related Distributions, 281 9.1 Reciprocal of an Inverse Gaussian Variate, 281 9.2 Birnbaum-Saunders Distribution, 283 9.3 Generalized Inverse Gaussian Distributions, 284 9.4 Mixtures of IG(/i, A) with Its Complementary Reciprocal, 285 9.5 Other Related Distributions, 287 10 Tables, 289 11 Applications, 290 Bibliography, 292 16 Cauchy Distribution 298 1 Historical Remarks, 298 2 Definition and Properties, 299 3 Order Statistics, 303 4 Methods of Inference, 306 4.1 Methods Based on Order Statistics, 306 4.2 Maximum Likelihood Inference, 310 4.3 Conditional Inference, 314 4.4 Bayesian Inference, 315 4.5 Other Developments in Inference, 317 5 Genesis and Applications, 318 6 Characterizations, 321 7 Generation Algorithms, 323 7.1 Monahan's (1979) Algorithm, 323 7.2 Kronmal and Peterson's (1981) Acceptance-Complement Method, 324 7.3 Ahrens and Dieter's (1988) Algorithm, 326 8 Related Distributions, 327 Bibliography, 329 17 Gamma Distributions 337 1 Definition, 337 2 Moments and Other Properties, 338
X CONTENTS 3 Genesis and Applications, 343 4 Tables and Computational Algorithms, 344 5 Approximation and Generation of Gamma Random Variables, 346 6 Characterizations, 349 7 Estimation, 355 7.1 Three Parameters Unknown, 356 7.2 v Some Parameters Unknown, 360 7.3 Estimation of Shape Parameter (j8 and y Known), 368 7.4 Order Statistics and Estimators Based on Order Statistics, 370 8 Related Distributions, 379 8.1 Truncated Gamma Distributions, 380 8.2 Compound Gamma Distributions, 381 8.3 Transformed Gamma Distributions, 382 8.4 Convolutions of Gamma Distributions, 384 8.5 Mixtures of Gamma Distributions, 386 8.6 Reflected Gamma Distributions, 386 8.7 Generalized Gamma Distributions, 388 Bibliography, 397 18 Chi-Square Distributions Including Chi and Rayleigh 415 1 Historical Remarks, 415 2 Definition, 416 3 Moments and Other Properties, 420 4 Tables and Nomograms, 422 5 Approximations and Computational Algorithms, 426 6 Characterizations, 441 7 Simulational Algorithms, 443 8 Distributions of Linear Combinations, 444 9 Related Distributions, 450 10 Specific Developments in the Rayleigh Distribution, 456 10.1 Historical Remarks, 456 10.2 Basic Properties, 456 10.3 Order Statistics and Properties, 459 10.4 Inference, 461 10.5 Prediction, 474 10.6 Record Values and Related Issues, 475
CONTENTS 10.7 Related Distributions, 479 Bibliography, 481 19 Exponential Distributions 494 1 Definition, 494 2 Genesis, 494 3 Some Remarks on History, 497 4 Moments and Generating Functions, 498 5 Applications, 499 6 Order Statistics, 499 7 Estimation, 506 7.1 Classical Estimation, 506 7.2 Grouped Data, 509 7.3 Estimators Using Selected Quantiles, 510 7.4 Estimation of Quantiles, 521 7.5 Bayesian Estimation, 522 7.6 Miscellaneous, 526 8 Characterizations, 534 8.1 Characterizations Based on Lack of Memory and on Distributions of Order Statistics, 536 8.2 Characterizations Based on Conditional Expectations (Regression), 540 8.3 Record Values, 544 8.4 Miscellaneous, 544 8.5 Stability, 545 9 Mixtures of Exponential Distributions, 546 10 Related Distributions, 551 Bibliography, 556 20 Pareto Distributions, 573 1 Introduction, 573 2 Genesis, 573 3 Definitions, 574 4 Moments and Other Indices, 577 4.1 Moments, 577 4.2 Alternative Measures of Location, 577 4.3 Measures of Inequality, 578
Xii CONTENTS 5 Estimation of Parameters, 579 5.1 Least-Squares Estimators, 580 5.2 Estimators from Moments, 580 5.3 Maximum Likelihood Estimation, 581 5.4 Estimation Based on Order Statistics, 584 5.5 Sequential Estimation, 588 5.6 Minimax Estimation, 588 5.7 Estimation of Pareto Densities, 589 5:8 Estimation of Pareto Quantiles, 590 5.9 Censored Data, 592 5.10 Bayesian Estimation, 594 6 Estimation of Lorenz Curve and Gini Index, 595 7 Miscellaneous, 596 8 Order Statistics and Record Values, 599 8.1 Order Statistics, 599 8.2 Record Values, 601 9 Characterizations, 603 10 Product and Ratios of Pareto Random Variables, 605 11 Applications and Related Distributions, 607 12 Generalized Pareto Distributions, 614 Bibliography, 620 21 Weibull Distributions 628 1 Historical Remarks, 628 2 Definition, 629 3 Order Statistics, 637 4 Methods of Inference, 641 4.1 Moment Estimation, 641 4.2 Best Linear Unbiased Estimation, 644 4.3 Asymptotic Best Linear Unbiased Estimation, 647 4.4 Minimum Quantile Distance Estimation, 651 4.5 Modified Moment Estimation, 652 4.6 Maximum Likelihood Estimation, 656 4.7 Modified Maximum Likelihood Estimation, 660 4.8 Bayesian Estimation and Shrinkage Estimation, 661 5 Tolerance Limits and Intervals, 663 6 Prediction Limits and Intervals, 667 7 Record Values, 671 8 Tables and Graphs, 675
CONTENTS Xiii 9 Characterizations, 679 10 Simulation Algorithms, 682 11 Applications, 684 12 Related Distributions, 686 Bibliography, 695 Abbreviations 723 Author Index 725 Subject Index 749