Continuous univariate distributions. Vol.2

Norman L.J.,Kotz S.,,Balakrishnan N. Continuous univariate distributions, vol. 2 ......Page 4
Copyright  ......Page 5
Contents ......Page 7
Preface  xiii ......Page 13
List of Tables xv  ......Page 15
TABLE 22.2 Standardized percentiles for Type 1 extreme value distribution 14......Page 34
6 Record Values, 17 ......Page 37
TABLE 22.4 Covariances of order statistics from extreme value distribution 18-19 ......Page 38
TABLE 22.6 Efficiencies of linear unbiased estimators of 0 for the extreme value distribution 30......Page 50
TABLE 22.7 Coefficients for the BLUEs of ? and 0 for complete samples 34......Page 54
TABLE 22.8 Values of Vl9 V2, and V3 for the BLUEs of ? and 0 for complete samples 35......Page 55
TABLE 22.9 Coefficients for the BLIEs of ? and 6 for complete samples 36......Page 56
9.4 Asymptotic Best Linear Unbiased Estimation, 37 ......Page 57
9.5 Linear Estimation with Polynomial Coefficients, 39 ......Page 59
TABLE 22.12 Efficiencies of linear unbiased estimators of ? with linear coefficients and quadratic coefficients (%) 40......Page 60
9.6 Maximum Likelihood Estimation, 41 ......Page 61
TABLE 22.14 Comparison of bias and mean square error of various estimators of ? and 0 for n = 10 and 20 and right censoring (r = 0) 48......Page 68
9.9 “Block-Type” Estimation, 51 ......Page 71
TABLE 22.16 Comparison of exact and F-approximation tolerance bounds 58......Page 78
TABLE 22.17 Percentage points tx y of the distribution of Tx in (22.144) 60......Page 80
TABLE 22.18 Distribution percentiles of Sx = (?** - Z\'t)/0** for Type-II right-censored samples of size n — s from a sample of size n 63......Page 83
TABLE 22.19 Percentage points for modified statistics W2, U2,and A2 67......Page 87
TABLE 22.20 Percentage points of the statistics\\tyfnD+,\\t4nD~,4nD, and yfn V when both ? and 6 are unknown 68......Page 88
TABLE 22.21 Critical values for the statistic DSP 69......Page 89
14 Applications, 72 ......Page 92
TABLE 22.23 Elements of the asymptotic variance-covariance matrix of the PWM estimators of the parameters of the generalized extreme value distribution 81......Page 101
TABLE 23.1 A comparison of four approximations for the cdf of the standardized mean T3 of samples of size 3 from a logistic population 122......Page 142
TABLE 23.2 Coefficients of the (n — i 4- l)th-order statistic X\'n_i+x in the linear estimator of a (by Jung’s Method) modified to make it unbiased 133......Page 153
TABLE 24.1 Coefficients and variances of best linear estimators of location (0) and scale () parameters 173......Page 193
TABLE 24.2 Efficiency of various estimators of 0, relative to best linear unbiased estimator 177......Page 197
TABLE 24.3 Coefficients of best linear unbiased estimators of expected value (0) and standard deviation (TABLE 24.5 Simulated percentage points /+ of the statistic L„ 180......Page 200
4.3 Simplified Linear Estimation, 181 ......Page 201
TABLE 24.7 Estimator  and its efficiency 182......Page 202
TABLE 24.8 The optimal spacing {A,}, the coefficients {?,}, and the ARE(**) of the ABLUE TABLE 24.9 Comparison of expected lengths of 100(1 - a)% conditional and unconditional confidence intervals for 0 (with  = 1) 186......Page 206
TABLE 24.10 Ratio of mean deviation to standard deviation, and j82 for Subbotin distributions 197......Page 217
TABLE 25.1 Actual and nominal values of /32 214......Page 234
TABLE 25.2 Comparison of estimates obtained from ML and moment estimates for a univariate beta distribution. Each row is for 100 replications 229......Page 249
TABLE 25.3 Best combinations of k and c 234......Page 254
TABLE 26.1 Comparison of N\' and N\"’ 288......Page 308
TABLE 27.1 Johnson’s empirical formulas for FVi V2 a for a = 0.95,0.975, and their accuracy 341......Page 361
TABLE 27.2 Quantiles 1-0 of the limit distribution F(a) for various a 352......Page 372
TABLE 28.1 Ratio of mean deviation to standard deviation for tvdistribution 366......Page 386
TABLE 28.2 Maximum absolute error of (28.15), c(d) = c X 10^ 376......Page 396
TABLE 28.3 Values of coefficients Br j in Appell polynomials, Ar(x) = xrL^0BrJxJ 377......Page 397
TABLE 28.4 Comparison of Hendricks’s approximation in (28.18) with the exact value 378......Page 398
TABLE 28.5 Approximations—values of Ua using expansion (28.20) 379......Page 399
TABLE 28.6 Approximation to tv a using (28.21) 380......Page 400
TABLE 28.7 Comparative table of approximate and exact values of the percentage points of the ^-distribution 381......Page 401
TABLE 28.8 Bounds on the equivalent normal deviate u(t) for tv 384......Page 404
TABLE 28.9 Values of a, b, c, d, and e for the approximation in (28.39) 387......Page 407
TABLE 28.10 Comparison of approximations (28.42), (28.43), and (28.45) with exact values of Ua 388......Page 408
TABLE 28.11 Exact and approximate tail areas for the ^-distribution with v degrees of freedom 390......Page 410
TABLE 28.12 Coefficients for the approximation in (28.52) 392......Page 412
TABLE 28.13 Coefficients for polynomial approximation (28.54b) of student’s t percentage points UVy(X) 394......Page 414
5 Applications, 395 ......Page 415
6 Pearson Type VII Distributions and Their Modifications, 396 ......Page 416
TABLE 28.16 Corrective factors for distribution of tv in nonnormal (Edgeworth) populations 408-410......Page 430
TABLE 28.17 Comparison of approximations for Da (vx = v2) 415......Page 435
References, 422 ......Page 442
TABLE 29.1 Mean-square-error ratios, MSE^/MSE^A\' - v)+) 457......Page 477
TABLE 29.2 Errors of Johnson (29.68), Patnaik (29.59), Pearson (29.60), Abdel-Aty (29.61a), and Sankaran (29.61b-d) approximations for v = 2, 4 and 7 464......Page 484
5 Approximations, 491 ......Page 511
7.2 Noncentral Beta Distributions, 502 ......Page 522
TABLE 32.1 Formulas for the cdf of R 551......Page 571
TABLE 32.2 Leading terms in expansions 562......Page 582
TABLE 32.3 Moment values for the distribution of R{ 608......Page 628
TABLE 32.4 Distributions of first serial correlation coefficient 611......Page 631
TABLE 33.1 Least-squares efficiency for values of a 662......Page 682
TABLE 33.2 Bounds on distributions classified by hazard rates 665......Page 685
TABLE 33.3 Bounds on distribution functions 666......Page 686
1 Genesis, 1 ......Page 21
2 Introduction, 2 ......Page 22
3 Limiting Distributions of Extremes, 4 ......Page 24
4 Distribution Function and Moments, 11 ......Page 31
5 Order Statistics, 15 ......Page 35
7 Generation, Tables, and Probability Paper, 23 ......Page 43
8 Characterizations, 25 ......Page 45
9 Methods of Inference, 26 ......Page 46
9.1 Moment Estimation, 27 ......Page 47
9.2 Simple Linear Estimation, 28 ......Page 48
9.3 Best Linear Unbiased (Invariant) Estimation, 32 ......Page 52
9.7 Conditional Method, 49 ......Page 69
9.8 Method of Probability-Weighted Moments, 50 ......Page 70
9.10 A Survey of Other Developments, 53 ......Page 73
10 Tolerance Limits and Intervals, 55 ......Page 75
11 Prediction Limits and Intervals, 59 ......Page 79
12 Outliers and Robustness, 64 ......Page 84
13 Probability Plots, Modifications, and Model Validity, 66 ......Page 86
15 Generalized Extreme Value Distributions, 75 ......Page 95
16 Other Related Distributions, 86 ......Page 106
References, 93 ......Page 113
1 Historical Remarks and Genesis, 113 ......Page 133
2 Definition, 115 ......Page 135
3 Generating Functions and Moments, 116 ......Page 136
4 Properties, 118 ......Page 138
5 Order Statistics, 123 ......Page 143
6 Methods of Inference, 127 ......Page 147
7 Record Values, 135 ......Page 155
8 Tables, 137 ......Page 157
9 Applications, 138 ......Page 158
10 Generalizations, 140 ......Page 160
11 Related Distributions, 147 ......Page 167
References, 154 ......Page 174
1 Definition, Genesis, and Historical Remarks, 164 ......Page 184
2 Moments, Generating Functions, and Properties, 165 ......Page 185
3 Order Statistics, 168 ......Page 188
4.1 Maximum Likelihood Estimation, 172 ......Page 192
4.2 Best Linear Unbiased Estimation, 175 ......Page 195
4.4 Asymptotic Best Linear Unbiased Estimation, 183 ......Page 203
4.5 Conditional Inference, 185 ......Page 205
5 Tolerance Limits and Prediction Intervals, 187 ......Page 207
6 Related Distributions, 190 ......Page 210
References, 201 ......Page 221
1 Definition, 210 ......Page 230
2 Genesis and Random Number Generation, 212 ......Page 232
3 Properties, 217 ......Page 237
4 Estimation, 221 ......Page 241
5 Applications, 235 ......Page 255
6.1 Approximations, 238 ......Page 258
6.2 Tables, 244 ......Page 264
7 Related Distributions, 247 ......Page 267
8 Products, Quotients, and Differences of Independent Beta Variables, 256 ......Page 276
References, 264 ......Page 284
1 Definition, 276 ......Page 296
2 Genesis, 277 ......Page 297
3 Historical Remarks, 278 ......Page 298
4 Generating Functions, Moments, and Order Statistics, 279 ......Page 299
5 Characterizations, 281 ......Page 301
6 Estimation of Parameters, 286 ......Page 306
7 Estimation Using Order Statistics— Censored Samples, 293 ......Page 313
8 Tables of Random Numbers, 295 ......Page 315
9 Related Distributions, 296 ......Page 316
9.1 Mixtures of Two Uniform Distributions, 302 ......Page 322
9.2 Other Related Distributions, 304 ......Page 324
10 Applications, 307 ......Page 327
10.2 Life Testing, 308 ......Page 328
10.3 Traffic Flow Applications, 310 ......Page 330
11 Random Number Generation, 312 ......Page 332
References, 314 ......Page 334
1 Introduction, 322 ......Page 342
2 Properties, 325 ......Page 345
4 Tables, 332 ......Page 352
5 Approximations and Nomograms, 334 ......Page 354
6 Applications, 343 ......Page 363
7 Pearson Type VI Distributions, 344 ......Page 364
8 Other Related Distributions, 346 ......Page 366
8.1 “Generalized” F-Distributions, 348 ......Page 368
8.2 Other Related Distributions, 350 ......Page 370
References, 355 ......Page 375
1 Genesis and Historical Remarks, 362 ......Page 382
2 Properties, 363 ......Page 383
3.1 Tables, 368 ......Page 388
3.2 Nomograms, 372 ......Page 392
4 Approximations, 374 ......Page 394
7 Other Related Distributions, 403 ......Page 423
1 Definition and Genesis, 433 ......Page 453
3 Distribution, 435 ......Page 455
4 Moments, 447 ......Page 467
5 Properties of the Distribution, 450 ......Page 470
6 Estimation, 451 ......Page 471
7 Tables and Computer Algorithms, 458 ......Page 478
8 Approximations, 461 ......Page 481
9 Applications, 467 ......Page 487
10 Related Distributions, 470 ......Page 490
References, 473 ......Page 493
1 Definition and Genesis, 480 ......Page 500
3 Properties, 481 ......Page 501
4.1 Tables, 488 ......Page 508
4.2 Computer Programs, 489 ......Page 509
6 Estimation of the Noncentrality Parameter A1? 495 ......Page 515
7.1 Doubly Noncentral F-Distributions, 499 ......Page 519
References, 504 ......Page 524
2 Historical Remarks, 508 ......Page 528
3 Applications and Estimation, 509 ......Page 529
4 Moments, 512 ......Page 532
5 Distribution Function, 514 ......Page 534
6 Approximations, 519 ......Page 539
7 Tables, Charts, and Computer Algorithms, 527 ......Page 547
8.2 Doubly Noncentral ;?Distribution, 533 ......Page 553
8.3 Modified Noncentral ;-Distribution, 537 ......Page 557
8.4 Distribution of Noncentral ;-Statistic When the Population is Nonnormal, 538 ......Page 558
References, 539 ......Page 559
1 Introduction and Genesis, 545 ......Page 565
2 Derivation of Distribution of R, 547 ......Page 567
3 Historical Remarks, 556 ......Page 576
4 Distribution of R in Non-normal Populations and Robustness, 559 ......Page 579
5.1 Tables, 569 ......Page 589
5.2 Approximations Using Transforms, 571 ......Page 591
5.3 Asymptotic Expansions of the Distribution of R, 578 ......Page 598
6.1 General Remarks, 580 ......Page 600
6.2 Point Estimation, 583 ......Page 603
6.3 Maximum Likelihood Estimation, 590 ......Page 610
6.4 Estimation of Common p Based on Several Samples, 592 ......Page 612
6.5 Miscellaneous Estimation Problems, 597 ......Page 617
7 Sample Covariance, 599 ......Page 619
8 Circular Serial Correlation, 601 ......Page 621
9 Noncircular Serial Correlation, 605 ......Page 625
10 Leipnik Distribution, 612 ......Page 632
11 Multiple Correlation Coefficient, 617 ......Page 637
References, 627 ......Page 647
1 Introduction, 639 ......Page 659
2 Life Distributions, 640 ......Page 660
3 Birnbaum-Saunders Distributions and Transformations, 651 ......Page 671
4.1 Basic Definitions and Bounds, 663 ......Page 683
4.2 Reliability Classification of Orderings, 668 ......Page 688
4.3 Alternative Stochastic Classification of Orderings, 672 ......Page 692
References, 681 ......Page 701
Abbreviations, 691......Page 711
Author Index, 693 ......Page 713
Subject Index, 711 ......Page 731
cover......Page 1