Continuous Bivariate Distributions (eBook)

Preface 6
Contents 8
Univariate Distributions 36
Introduction 36
Notation and Definitions 37
Notation 37
Explanations 38
Characteristic Function 38
Cumulant Generating Function 39
Some Measures of Shape Characteristics 40
Location and Scale 40
Skewness and Kurtosis 40
Tail Behavior 41
Some Multiparameter Systems ofUnivariate Distributions 41
Reliability Classes 42
Normal Distribution and Its Transformations 42
Normal Distribution 42
Lognormal Distribution 43
Truncated Normal 43
Johnson's System 43
Box--Cox Power Transformations to Normality 44
g and h Families of Distributions 44
Efron's Transformation 45
Distribution of a Ratio 45
Compound Normal Distributions 45
Beta Distribution 46
The First Kind 46
Uniform Distribution 47
Symmetric Beta Distribution 47
Inverted Beta Distribution 47
Exponential, Gamma, Weibull, and Stacy Distributions 48
Exponential Distribution 48
Gamma Distribution 49
Chi-Squared and Chi Distributions 49
Weibull Distribution 50
Stacy Distribution 50
Comments on Skew Distributions 51
Compound Exponential Distributions 51
Aging Distributions 52
Marshall and Olkin's Family of Distributions 52
Families of Generalized Weibull Distributions 53
Logistic, Laplace, and Cauchy Distributions 54
Logistic Distribution 54
Laplace Distribution 54
The Generalized Error Distribution 55
Cauchy Distribution 55
Extreme-Value Distributions 55
Type 1 55
Type 2 56
Type 3 56
Pareto Distribution 56
Pearson System 57
Burr System 58
t- and F-Distributions 58
t-Distribution 58
F-Distribution 59
The Wrapped t Family of Circular Distributions 59
Noncentral Distributions 60
Skew Distributions 60
Skew-Normal Distribution 60
Skew t-Distributions 61
Skew-Cauchy Distribution 62
Jones' Family of Distributions 63
Some Lesser-Known Distributions 63
Inverse Gaussian Distribution 63
Meixner Hypergeometric Distribution 64
Hyperbolic Distributions 64
Stable Distributions 64
References 65
Bivariate Copulas 68
Introduction 68
Basic Properties 69
Further Properties of Copulas 70
Survival Copula 71
Archimedean Copula 72
Extreme-Value Copulas 73
Archimax Copulas 74
Gaussian, t-, and Other Copulas of theElliptical Distributions 75
Order Statistics Copula 76
Polynomial Copulas 76
Approximation of a Copula by a Polynomial Copula 78
Measures of Dependence Between Two Variables with a Given Copula 79
Kendall's Tau 79
Spearman's Rho 80
Geometry of Correlation Under a Copula 80
Measure Based on Gini's Coefficient 81
Tail Dependence Coefficients 81
A Local Dependence Measure 83
Tests of Dependence and Inferences 83
``Concepts of Dependence" of Copulas 83
Distribution Function of Z=C(U,V) 83
Simulation of Copulas 84
The General Case 85
Archimedean Copulas 85
Construction of a Copula 85
Rüschendorf's Method 85
Generation of Copulas by Mixture 87
Convex Sums 88
Univariate Function Method 88
Some Other Methods 89
Applications of Copulas 90
Insurance, Finance, Economics, andRisk Management 90
Hydrology and Environment 91
Management Science and Operations Research 92
Reliability and Survival Analysis 92
Engineering and Medical Sciences 92
Miscellaneous 93
Criticisms about Copulas 93
Conclusions 94
References 95
Distributions Expressed as Copulas 101
Introduction 101
Farlie--Gumbel--Morgenstern (F-G-M) Copula and Its Generalization 102
Applications 104
Univariate Transformations 104
A Switch-Source Model 105
Ordinal Contingency Tables 105
Iterated F-G-M Distributions 105
Extensions of the F-G-M Distribution 106
Other Related Distributions 109
Ali--Mikhail--Haq Distribution 110
Bivariate Logistic Distributions 111
Bivariate Exponential Distribution 112
Frank's Distribution 112
Distribution of Cuadras and Augé and Its Generalization 113
Generalized Cuadras and Aug Family (Marshall and Olkin's Family) 113
Gumbel--Hougaard Copula 114
Plackett's Distribution 116
Bivariate Lomax Distribution 118
The Special Case of c=1 121
Bivariate Pareto Distribution 122
Lomax Copula 123
Pareto Copula (Clayton Copula) 124
Summary of the Relationship BetweenVarious Copulas 126
Gumbel's Type I Bivariate Exponential Distribution 126
Gumbel--Barnett Copula 128
Kimeldorf and Sampson's Distribution 129
Rodríguez-Lallena and Úbeda-Flores' Family of Bivariate Copulas 130
Other Copulas 130
References to Illustrations 131
References 132
Concepts of Stochastic Dependence 138
Introduction 138
Concept of Positive Dependence and Its Conditions 139
Positive Dependence Concepts at a Glance 140
Concepts of Positive Dependence Stronger than PQD 141
Positive Quadrant Dependence 141
Association of Random Variables 142
Left-Tail Decreasing (LTD) and Right-Tail Increasing (RTI) 143
Positive Regression Dependent(Stochastically Increasing) 145
Left Corner Set Decreasing and Right Corner Set Increasing 147
Total Positivity of Order 2 148
DTP2(m,n) and Positive Dependence by Mixture 150
Concepts of Positive Dependence Weaker than PQD 150
Positive Quadrant Dependence in Expectation 150
Positively Correlated Distributions 151
Monotonic Quadrant Dependence Function 151
Summary of Interrelationships 153
Families of Bivariate PQD Distributions 154
Bivariate PQD Distributions withSimple Structures 155
Construction of Bivariate PQD Distributions 158
Tests of Independence AgainstPositive Dependence 159
Geometric Interpretations of PQD and Other Positive Dependence Concepts 160
Additional Concepts of Dependence 161
Negative Dependence 162
Neutrality 163
Examples of NQD 163
Positive Dependence Orderings 164
Some Other Positive Dependence Orderings 167
Positive Dependent Ordering withDifferent Marginals 168
Bayesian Concepts of Dependence 169
References 169
Measures of Dependence 174
Introduction 174
Total Dependence 175
Functions 175
Mutual Complete Dependence 175
Monotone Dependence 176
Functional and Implicit Dependence 177
Overview 177
Global Measures of Dependence 177
Pearson's Product-Moment Correlation Coefficient 179
Robustness of Sample Correlation 180
Interpretation of Correlation 181
Correlation Ratio 184
Chebyshev's Inequality 184
and Concepts of Dependence 184
Maximal Correlation (Sup Correlation) 185
Monotone Correlations 186
Definitions and Properties 186
Concordant and DiscordantMonotone Correlations 187
Rank Correlations 188
Kendall's Tau 188
Spearman's Rho 189
The Relationship Between Kendall's Tau and Spearman's Rho 190
Other Concordance Measures 195
Measures of Schweizer and Wolff and Related Measures 196
Matrix of Correlation 197
Tetrachoric and Polychoric Correlations 198
Compatibility with Perfect Rank Ordering 199
Conclusions on Measures of Dependence 200
Local Measures of Dependence 200
Definition of Local Dependence 201
Local Dependence Function ofHolland and Wang 201
Local S and 202
Local Measure of LRD 202
Properties of (x,y) 203
Local Correlation Coefficient 203
Several Local Indices Applicable inSurvival Analysis 204
Regional Dependence 204
Preliminaries 204
Quasi-Independence and Quasi-Independent Projection 205
A Measure of Regional Dependence 206
References 206
Construction of Bivariate Distributions 211
Introduction 211
Fréchet Bounds 212
Transformations 213
The Marginal Transformation Method 213
General Description 213
Johnson's Translation Method 214
Uniform Representation: Copulas 215
Some Properties Unaffected by Transformation 216
Methods of Constructing Copulas 217
The Inversion Method 217
Geometric Methods 217
Algebraic Methods 218
Rüschendorf's Method 218
Models Defined from a Distortion Function 219
Marshall and Olkin's Mixture Method 219
Archimedean Copulas 220
Archimax Copulas 221
Mixing and Compounding 221
Mixing 221
Compounding 222
Variables in Common and Trivariate Reduction Techniques 225
Summary of the Method 225
Denominator-in-Common and Compounding 226
Mathai and Moschopoulos' Methods 226
Modified Structure Mixture Model 227
Khintchine Mixture 227
Conditionally Specified Distributions 228
A Conditional Distribution with aMarginal Given 228
Specification of Both Sets ofConditional Distributions 228
Conditionals in Exponential Families 229
Conditions Implying Bivariate Normality 231
Summary of ConditionallySpecified Distributions 231
Marginal Replacement 233
Example: Bivariate Non-normal Distribution 234
Marginal Replacement of a Spherically Symmetric Bivariate Distribution 234
Introducing Skewness 234
Density Generators 234
Geometric Approach 235
Some Other Simple Methods 236
Weighted Linear Combination 237
Data-Guided Methods 238
Conditional Distributions 238
Radii and Angles 239
The Dependence Function in theExtreme-Value Sense 240
Special Methods Used in Applied Fields 240
Shock Models 240
Queueing Theory 242
Compositional Data 243
Extreme-Value Models 243
Time Series: Autoregressive Models 245
Limits of Discrete Distributions 247
A Bivariate Exponential Distribution 247
A Bivariate Gamma Distribution 248
Potentially Useful Methods But Not in Vogue 248
Differential Equation Methods 249
Diagonal Expansion 251
Bivariate Edgeworth Expansion 252
An Application to Wind Velocity at the Ocean Surface 253
Another Application to Statistical Spectroscopy 253
Concluding Remarks 254
References 255
Bivariate Distributions Constructed by the Conditional Approach 261
Introduction 261
Contents 261
Pertinent Univariate Distributions 262
Compatibility and Uniqueness 263
Early Work on ConditionallySpecified Distributions 264
Approximating Distribution Functions Using the Conditional Approach 264
Normal Conditionals 265
Conditional Distributions 265
Expression of the Joint Density 265
Univariate Properties 266
Further Properties 266
Centered Normal Conditionals 266
Conditionals in Exponential Families 268
Dependence in Conditional Exponential Families 269
Exponential Conditionals 269
Normal Conditionals 272
Gamma Conditionals 272
Model II for Gamma Conditionals 273
Gamma-Normal Conditionals 274
Beta Conditionals 275
Inverse Gaussian Conditionals 276
Other Conditionally Specified Families 277
Pareto Conditionals 277
Beta of the Second Kind (Pearson Type VI) Conditionals 278
Generalized Pareto Conditionals 280
Cauchy Conditionals 281
Student t-Conditionals 282
Uniform Conditionals 283
Translated Exponential Conditionals 284
Scaled Beta Conditionals 285
Conditionally Specified BivariateSkewed Distributions 286
Bivariate Distributions with SkewedNormal Conditionals 286
Linearly Skewed and Quadratically Skewed Normal Conditionals 288
Improper Bivariate Distributions from Conditionals 288
Conditionals in Location-Scale Families with Specified Moments 288
Conditional Distributions and the Regression Function 289
Assumptions and Specifications 289
Wesolowski's Theorem 290
Estimation in Conditionally Specified Models 290
McKay's Bivariate Gamma Distribution andIts Generalization 292
Conditional Properties 292
Expression of the Joint Density 292
Dussauchoy and Berland's Bivariate Gamma Distribution 292
One Conditional and One Marginal Specified 293
Dubey's Distribution 293
Blumen and Ypelaar's Distribution 294
Exponential Dispersion Models 294
Four Densities of Barndorff-Nielsenand Blæsild 295
Continuous Bivariate Densities with a Discontinuous Marginal Density 295
Tiku and Kambo's BivariateNon-normal Distribution 296
Marginal and Conditional Distributions of the Same Variate 297
Example 298
Vardi and Lee's Iteration Scheme 298
Conditional Survival Models 299
Exponential Conditional Survival Function 299
Weibull Conditional Survival Function 300
Generalized Pareto ConditionalSurvival Function 301
Conditional Approach in Modeling 301
Beta-Stacy Distribution 301
Sample Skewness and Kurtosis 302
Business Risk Analysis 303
Intercropping 303
Winds and Waves, Rain and Floods 304
References 307
Variables-in-Common Method 311
Introduction 311
General Description 312
Additive Models 313
Background 313
Meixner Classes 314
Cherian's Bivariate Gamma Distribution 315
Symmetric Stable Distribution 315
Bivariate Triangular Distribution 315
Summing Several I.I.D. Variables 316
Generalized Additive Models 317
Trivariate Reduction of Johnson and Tenenbein 317
Mathai and Moschopoulos' Bivariate Gamma 318
Lai's Structure Mixture Model 318
Latent Variables-in-Common Model 319
Bivariate Skew-Normal Distribution 320
Ordered Statistics 321
Weighted Linear Combination 322
Derivation 322
Expression of the Joint Density 322
Correlation Coefficients 322
Remarks 323
Bivariate Distributions Having a Common Denominator 323
Explanation 323
Applications 324
Correlation Between Ratios with aCommon Divisor 324
Compounding 325
Examples of Two Ratios with a Common Divisor 325
Bivariate t-Distribution with Marginals Having Different Degrees of Freedom 327
Bivariate Distributions Having a Common Numerator 327
Multiplicative Trivariate Reduction 327
Bryson and Johnson (1982) 328
Gokhale's Model 328
Ulrich's Model 329
Khintchine Mixture 329
Derivation 329
Exponential Marginals 329
Normal Marginals 330
References to Generation of Random Variates 330
Transformations Involving the Minimum 331
Other Forms of the Variables-in-Common Technique 331
Bivariate Chi-Squared Distribution 331
Bivariate Beta Distribution 332
Bivariate Z-Distribution 332
References 333
Bivariate Gamma and Related Distributions 337
Introduction 337
Kibble's Bivariate Gamma Distribution 338
Formula of the Joint Density 338
Formula of the Cumulative Distribution Function 339
Univariate Properties 339
Correlation Coefficient 339
Moment Generating Function 339
Conditional Properties 340
Derivation 340
Relations to Other Distributions 341
Generalizations 341
Illustrations 341
Remarks 342
Fields of Applications 342
Tables and Algorithms 343
Transformations of the Marginals 343
Royen's Bivariate Gamma Distribution 343
Formula of the Cumulative Distribution Function 343
Univariate Properties 344
Derivation 344
Relation to Kibble's BivariateGamma Distribution 344
Izawa's Bivariate Gamma Distribution 344
Formula of the Joint Density 344
Correlation Coefficient 345
Relation to Kibble's BivariateGamma Distribution 345
Fields of Application 345
Jensen's Bivariate Gamma Distribution 345
Formula of the Joint Density 345
Univariate Properties 346
Correlation Coefficient 346
Characteristic Function 346
Derivation 347
Illustrations 347
Remarks 347
Fields of Application 348
Tables and Algorithms 348
Gunst and Webster's Model and Related Distributions 348
Case 3 of Gunst and Webster 349
Case 2 of Gunst and Webster 350
Smith, Aldelfang, and Tubbs' BivariateGamma Distribution 350
Sarmanov's Bivariate Gamma Distribution 351
Formula of the Joint Density 351
Univariate Properties 351
Correlation Coefficient 351
Derivation 352
Interrelationships 352
Bivariate Gamma of Loáiciga and Leipnik 352
Formula of the Joint Density 353
Univariate Properties 353
Joint Characteristic Function 353
Correlation Coefficient 353
Moments and Joint Moments 353
Application to Water-Quality Data 354
Cheriyan's Bivariate Gamma Distribution 354
Formula of the Joint Density 355
Univariate Properties 355
Correlation Coefficient 355
Moment Generating Function 355
Conditional Properties 355
Derivation 356
Generation of Random Variates 356
Remarks 356
Prékopa and Szántai's Bivariate Gamma Distribution 357
Formula of the CumulativeDistribution Function 357
Formula of the Joint Density 357
Univariate Properties 358
Relation to Other Distributions 358
Schmeiser and Lal's Bivariate Gamma Distribution 358
Method of Construction 358
Correlation Coefficient 359
Remarks 359
Farlie--Gumbel--Morgenstern Bivariate Gamma Distribution 359
Formula of the Joint Density 359
Univariate Properties 360
Moment Generating Function 360
Correlation Coefficient 360
Conditional Properties 360
Remarks 360
Moran's Bivariate Gamma Distribution 361
Derivation 361
Formula of the Joint Density 361
Computation of Bivariate Distribution Function 361
Remarks 361
Fields of Application 362
Crovelli's Bivariate Gamma Distribution 362
Fields of Application 362
Suitability of Bivariate Gammas forHydrological Applications 362
McKay's Bivariate Gamma Distribution 363
Formula of the Joint Density 363
Formula of the CumulativeDistribution Function 363
Univariate Properties 363
Conditional Properties 363
Methods of Derivation 364
Remarks 364
Dussauchoy and Berland's Bivariate Gamma Distribution 364
Formula of the Joint Density 364
Mathai and Moschopoulos' Bivariate Gamma Distributions 366
Model 1 366
Model 2 367
Becker and Roux's Bivariate Gamma Distribution 368
Formula of the Joint Density 368
Derivation 368
Remarks 369
Bivariate Chi-Squared Distribution 369
Formula of the CumulativeDistribution Function 369
Univariate Properties 369
Correlation Coefficient 370
Conditional Properties 370
Derivation 370
Remarks 370
Bivariate Noncentral Chi-Squared Distribution 371
Gaver's Bivariate Gamma Distribution 371
Moment Generating Function 371
Derivation 372
Correlation Coefficients 372
Bivariate Gamma of Nadarajah and Gupta 372
Model 1 372
Model 2 373
Arnold and Strauss' Bivariate Gamma Distribution 374
Remarks 375
Bivariate Gamma Mixture Distribution 375
Model Specification 375
Formula of the Joint Density 375
Formula of the CumulativeDistribution Function 376
Univariate Properties 376
Moments and Moment Generating Function 376
Correlation Coefficient 377
Fields of Application 377
Mixtures of Bivariate Gammas ofIwasaki and Tsubaki 377
Bivariate Bessel Distributions 377
References 378
Simple Forms of the Bivariate Density Function 383
Introduction 383
Bivariate t-Distribution 384
Formula of the Joint Density 384
Univariate Properties 384
Correlation Coefficients 385
Moments 385
Conditional Properties 385
Derivation 386
Illustrations 386
Generation of Random Variates 386
Remarks 386
Fields of Application 387
Tables and Algorithms 387
Spherically Symmetric Bivariate t-Distribution 388
Generalizations 388
Bivariate Noncentral t-Distributions 388
Bivariate Noncentral t-Distribution with =1 389
Bivariate t-Distribution Having Marginals with Different Degrees of Freedom 389
Jones' Bivariate Skew t-Distribution 391
Univariate Skew t-Distribution 391
Formula of the Joint Density 391
Correlation and Local Dependence for the Symmetric Case 392
Derivation 392
Bivariate Skew t-Distribution 393
Formula of the Joint Density 393
Moment Properties 393
Derivation 393
Possible Application due to Flexibility 394
Ordered Statistics 394
Bivariate t-/Skew t-Distribution 394
Formula of the Joint Density 394
Univariate Properties 395
Conditional Properties 395
Other Properties 395
Derivation 395
Bivariate Heavy-Tailed Distributions 396
Formula of the Joint Density 396
Univariate Properties 396
Remarks 396
Fields of Application 396
Bivariate Cauchy Distribution 397
Formula of the Joint Density 397
Formula of the Cumulative Distribution Function 397
Univariate Properties 397
Conditional Properties 397
Illustrations 398
Remarks 398
Generation of Random Variates 398
Generalization 398
Bivariate Skew-Cauchy Distribution 399
Bivariate F-Distribution 399
Formula of the Joint Density 400
Formula of the CumulativeDistribution Function 400
Univariate Properties 400
Correlation Coefficients 400
Product Moments 400
Conditional Properties 401
Methods of Derivation 401
Relationships to Other Distributions 401
Fields of Application 402
Tables and Algorithms 402
Bivariate Pearson Type II Distribution 403
Formula of the Joint Density 403
Univariate Properties 403
Correlation Coefficient 403
Conditional Properties 403
Relationships to Other Distributions 403
Illustrations 404
Generation of Random Variates 404
Remarks 404
Tables and Algorithms 404
Jones' Bivariate Beta/Skew Beta Distribution 404
Bivariate Finite Range Distribution 405
Formula of the Survival Function 405
Characterizations 406
Remarks 406
Bivariate Beta Distribution 406
Formula of the Joint Density 406
Univariate Properties 407
Correlation Coefficient 407
Product Moments 407
Conditional Properties 407
Methods of Derivation 407
Relationships to Other Distributions 408
Illustrations 408
Generation of Random Variates 408
Remarks 408
Fields of Application 409
Tables and Algorithms 410
Generalizations 410
Jones' Bivariate Beta Distribution 411
Formula of the Joint Density 411
Univariate Properties 412
Product Moments 412
Correlation and Local Dependence 412
Other Dependence Properties 412
Illustrations 413
Bivariate Inverted Beta Distribution 413
Formula of the Joint Density 413
Formula of the CumulativeDistribution Function 413
Derivation 413
Tables and Algorithms 414
Application 414
Generalization 414
Remarks 414
Bivariate Liouville Distribution 414
Definitions 415
Moments and Correlation Coefficient 416
Remarks 417
Generation of Random Variates 417
Generalizations 418
Bivariate pth-Order Liouville Distribution 418
Remarks 418
Bivariate Logistic Distributions 419
Standard Bivariate Logistic Distribution 419
Archimedean Copula 421
F-G-M Distribution with Logistic Marginals 421
Generalizations 421
Remarks 421
Bivariate Burr Distribution 422
Rhodes' Distribution 422
Support 422
Formula of the Joint Density 422
Derivation 423
Remarks 423
Bivariate Distributions with Support Above the Diagonal 423
Formula of the Joint Density 423
Formula of the CumulativeDistribution Function 424
Univariate Properties 424
Other Properties 424
Rotated Bivariate Distribution 424
Some Special Cases 425
Applications 426
References 426
Bivariate Exponential and Related Distributions 433
Introduction 433
Gumbel's Bivariate Exponential Distributions 434
Gumbel's Type I BivariateExponential Distribution 435
Characterizations 435
Estimation Method 435
Other Properties 435
Gumbel's Type II BivariateExponential Distribution 436
Gumbel's Type III BivariateExponential Distribution 437
Freund's Bivariate Distribution 438
Formula of the Joint Density 438
Formula of the CumulativeDistribution Function 438
Univariate Properties 438
Correlation Coefficient 439
Conditional Properties 439
Joint Moment Generating Function 439
Derivation 439
Illustrations 440
Other Properties 440
Remarks 440
Fields of Application 441
Transformation of the Marginals 441
Compounding 441
Bhattacharya and Holla's Generalizations 442
Proschan and Sullo's Extension ofFreund's Model 442
Becker and Roux's Generalization 443
Hashino and Sugi's Distribution 443
Formula of the Joint Density 443
Remarks 443
An Application 444
Marshall and Olkin's Bivariate Exponential Distribution 444
Formula of the CumulativeDistribution Function 444
Formula of the Joint Density Function 445
Univariate Properties 445
Conditional Distribution 445
Correlation Coefficients 445
Derivations 446
Fisher Information 446
Estimation of Parameters 446
Characterizations 447
Other Properties 447
Remarks 448
Fields of Application 450
Transformation to Uniform Marginals 450
Transformation to Weibull Marginals 451
Transformation to Extreme-Value Marginals 451
Transformation of Marginals: Approach of Muliere and Scarsini 451
Generalization 452
ACBVE of Block and Basu 453
Formula of the Joint Density 453
Formula of the CumulativeDistribution Function 453
Univariate Properties 453
Correlation Coefficient 453
Moment Generating Function 454
Derivation 454
Remarks 454
Applications 455
Sarkar's Distribution 455
Formula of the Joint Density 455
Formula of the CumulativeDistribution Function 456
Univariate Properties 456
Correlation Coefficient 456
Derivation 456
Relation to Marshall and Olkin's Distribution 456
Comparison of Four Distributions 457
Friday and Patil's Generalization 457
Tosch and Holmes' Distribution 458
A Bivariate Exponential Model of Wang 459
Formula of the Joint Density 459
Univariate Properties 459
Remarks 459
Lawrance and Lewis' System of Exponential Mixture Distributions 460
General Form 460
Model EP1 460
Model EP3 461
Model EP5 461
Models with Negative Correlation 462
Models with Uniform Marginals 462
The Distribution of Sums, Products, and Ratios 462
Mixture Models 462
Models with Line Singularities 462
Raftery's Scheme 463
First Special Case 463
Second Special Case 463
Formula of the Joint Density 464
Formula of the CumulativeDistribution Function 464
Derivation 464
Illustrations 464
Remarks 465
Applications 465
Linear Structures of Iyer et al. 465
Positive Cross Correlation 466
Negative Cross Correlation 466
Fields of Application 467
Moran--Downton Bivariate Exponential Distribution 468
Formula of the Joint Density 468
Formula of the CumulativeDistribution Function 468
Univariate Properties 468
Correlation Coefficients 468
Conditional Properties 469
Moment Generating Function 469
Regression 469
Derivation 470
Fisher Information 470
Estimation of Parameters 471
Illustrations 471
Random Variate Generation 471
Remarks 472
Fields of Application 473
Tables or Algorithms 474
Weibull Marginals 474
A Bivariate Laplace Distribution 475
Sarmanov's Bivariate Exponential Distribution 475
Formula of the Joint Density 475
Other Properties 476
Cowan's Bivariate Exponential Distribution 476
Formula of the Cumulative Distribution Function 476
Formula of the Joint Density 477
Univariate Properties 477
Correlation Coefficients 477
Conditional Properties 477
Derivation 478
Illustrations 478
Remarks 478
Transformation of the Marginals 478
Singpurwalla and Youngren's Bivariate Exponential Distribution 478
Formula of the CumulativeDistribution Function 479
Formula of the Joint Density 479
Univariate Properties 479
Derivation 479
Remarks 479
Arnold and Strauss' BivariateExponential Distribution 480
Formula of the Joint Density 480
Formula of the CumulativeDistribution Function 480
Univariate Properties 480
Conditional Distribution 480
Correlation Coefficient 481
Derivation 481
Other Properties 481
Mixtures of Bivariate Exponential Distributions 481
Lindley and Singpurwalla's Bivariate Exponential Mixture 481
Sankaran and Nair's Mixture 482
Al-Mutairi's Inverse Gaussian Mixture of Bivariate Exponential Distribution 482
Hayakawa's Mixtures 483
Bivariate Exponentials and Geometric Compounding Schemes 483
Background 483
Probability Generating Function 483
Bivariate Geometric Distribution 484
Bivariate Geometric DistributionArising from a Shock Model 484
Bivariate Exponential Distribution Compounding Scheme 485
Wu's Characterization of Marshall and Olkin's Distribution via a Bivariate Random Summation Scheme 487
Lack of Memory Properties of Bivariate Exponential Distributions 487
Extended Bivariate Lack ofMemory Distributions 489
Effect of Parallel Redundancy with Dependent Exponential Components 489
Mean Lifetime under Gumbel's Type I Bivariate Exponential Distribution 490
Stress--Strength Model and Bivariate Exponential Distributions 491
Basic Idea 491
Marshall and Olkin's Model 492
Downton's Model 492
Two Dependent Components Subjected to a Common Stress 492
A Component Subjected to Two Stresses 493
Bivariate Weibull Distributions 493
Marshall and Olkin (1967) 494
Lee (1979) 494
Lu and Bhattacharyya (1990): I 495
Farlie--Gumbel--Morgenstern System 495
Lu and Bhattacharyya (1990): II 495
Lee (1979): II 496
Comments 496
Applications 496
Gamma Frailty Bivariate Weibull Models 497
Bivariate Mixture of Weibull Distributions 497
Bivariate Generalized Exponential Distribution 498
References 498
Bivariate Normal Distribution 508
Introduction 508
Basic Formulas and Properties 510
Notation 510
Support 510
Formula of the Joint Density 510
Formula of the CumulativeDistribution Function 511
Univariate Properties 512
Correlation Coefficients 512
Conditional Properties 512
Moments and Absolute Moments 512
Methods of Derivation 513
Differential Equation Method 513
Compounding Method 514
Trivariate Reduction Method 514
Bivariate Central Limit Theorem 514
Transformations of DiffuseProbability Distributions 514
Characterizations 515
Order Statistics 517
Linear Combination of the Minimum andthe Maximum 518
Concomitants of Order Statistics 518
Illustrations 520
Relationships to Other Distributions 520
Parameter Estimation 521
Estimate and Inference of 522
Estimation Under Censoring 523
Other Interesting Properties 523
Notes on Some More Specialized Fields 525
Applications 525
Computation of Bivariate Normal Integrals 526
The Short Answer 526
Algorithms---Rectangles 526
Algorithms: Owen's T Function 530
Algorithms: Triangles 533
Algorithms: Wedge-Shaped Domain 534
Algorithms: Arbitrary Polygons 535
Tables 535
Computer Programs 535
Literature Reviews 536
Testing for Bivariate Normality 536
How Might Bivariate Normality Fail? 537
Outliers 537
Graphical Checks 538
Formal Tests: Univariate Normality 542
Formal Tests: Bivariate Normality 545
Tests of Bivariate NormalityAfter Transformation 552
Some Comments and Suggestions 553
Distributions with Normal Conditionals 555
Bivariate Skew-Normal Distribution 555
Bivariate Skew-Normal Distribution of Azzalini and Dalla Valle 555
Bivariate Skew-Normal Distribution ofSahu et al. 555
Fundamental Bivariate Skew-Normal Distributions 557
Review of Bivariate Skew-Normal Distributions 557
Univariate Transformations 557
The Bivariate Lognormal Distribution 557
Johnson's System 559
The Uniform Representation 561
The g and h Transformations 561
Effect of Transformations on Correlation 561
Truncated Bivariate Normal Distributions 563
Properties 563
Application to Selection Procedures 564
Truncation Scheme of Arnold et al. (1993) 566
A Random Right-Truncation Model of Gürler 566
Bivariate Normal Mixtures 567
Construction 567
References to Illustrations 567
Generalization and Compounding 568
Properties of a Special Case 568
Estimation of Parameters 568
Estimation of Correlation Coefficient for Bivariate Normal Mixtures 569
Tests of Homogeneity in NormalMixture Models 570
Sharpening a Scatterplot 570
Digression Analysis 571
Applications 571
Bivariate Normal Mixing withBivariate Lognormal 572
Nonbivariate Normal Distributions with Normal Marginals 572
Simple Examples with Normal Marginals 572
Normal Marginals with Linear Regressions 573
Linear Combinations of Normal Marginals 573
Uncorrelated Nonbivariate Normal Distributions with Normal Marginals 573
Bivariate Edgeworth Series Distribution 574
Bivariate Inverse Gaussian Distribution 574
Formula of the Joint Density 574
Univariate Properties 575
Correlation Coefficients 575
Conditional Properties 575
Derivations 575
References to Illustrations 576
Remarks 576
References 577
Bivariate Extreme-Value Distributions 593
Preliminaries 593
Introduction to Bivariate Extreme-Value Distribution 594
Definition 594
General Properties 594
Bivariate Extreme-Value Distributions in General Forms 595
Classical Bivariate Extreme-Value Distributions with Gumbel Marginals 596
Type A Distributions 596
Type B Distributions 598
Type C Distributions 600
Representations of Bivariate Extreme-Value Distributions with Gumbel Marginals 601
Bivariate Extreme-Value Distributions with Exponential Marginals 602
Pickands' Dependence Function 602
Properties of Dependence Function A 603
Differentiable Models 603
Nondifferentiable Models 604
Tawn's Extension of Differentiable Models 604
Negative Logistic Model of Joe 605
Normal-Like Bivariate Extreme-Value Distributions 606
Correlations 606
Bivariate Extreme-Value Distributions with Fréchet Marginals 607
Bilogistic Distribution 607
Negative Bilogistic Distributions 608
Beta-Like Extreme-Value Distribution 608
Bivariate Extreme-Value Distributions with Weibull Marginals 609
Formula of the CumulativeDistribution Function 609
Univariate Properties 609
Formula of the Joint Density 609
Fisher Information Matrix 610
Remarks 610
Methods of Derivation 610
Estimation of Parameters 611
References to Illustrations 611
Generation of Random Variates 611
Shi et al.'s (1993) Method 611
Ghoudi et al.'s (1998) Method 612
Nadarajah's (1999) Method 612
Applications 612
Applications to Natural Environments 612
Financial Applications 614
Other Applications 614
Conditionally Specified Gumbel Distributions 614
Bivariate Model Without HavingGumbel Marginals 615
Nonbivariate Extreme-Value Distributions with Gumbel Marginals 616
Positive or Negative Correlation 617
Fields of Applications 617
References 618
Elliptically Symmetric Bivariate and Other Symmetric Distributions 621
Introduction 621
Elliptically Contoured Bivariate Distributions: Formulations 622
Formula of the Joint Density 622
Alternative Definition 623
Another Stochastic Representation 623
Formula of the Cumulative Distribution 624
Characteristic Function 625
Moments 625
Conditional Properties 626
Copulas of Bivariate Elliptical Distributions 626
Correlation Coefficients 626
Fisher Information 626
Local Dependence Functions 627
Other Properties 627
Elliptical Compound BivariateNormal Distributions 628
Examples of Elliptically and Spherically Symmetric Bivariate Distributions 629
Bivariate Normal Distribution 629
Bivariate t-Distribution 629
Kotz-Type Distribution 629
Bivariate Cauchy Distribution 629
Bivariate Pearson Type II Distribution 630
Symmetric Logistic Distribution 630
Bivariate Laplace Distribution 630
Bivariate Power Exponential Distributions 630
Extremal Type Elliptical Distributions 631
Kotz-Type Elliptical Distribution 632
Fréchet-Type Elliptical Distribution 634
Gumbel-Type Elliptical Distribution 635
Tests of Spherical and Elliptical Symmetry 637
Extreme Behavior of Bivariate Elliptical Distributions 637
Fields of Application 638
Bivariate Symmetric Stable Distributions 638
Explanations 638
Characteristic Function 638
Probability Densities 639
Association Parameter 639
Correlation Coefficients 639
Remarks 640
Application 640
Generalized Bivariate Symmetric Stable Distributions 641
Characteristic Functions 641
de Silva and Griffith's Class 641
A Subclass of de Silva's Stable Distribution 642
-Symmetric Distribution 642
Other Symmetric Distributions 643
lp-Norm Symmetric Distributions 643
Bivariate Liouville Family 643
Bivariate Linnik Distribution 643
Bivariate Hyperbolic Distribution 644
Formula of the Joint Density 644
Univariate Properties 644
Derivation 645
References to Illustrations 645
Remarks 645
Fields of Application 646
Skew-Elliptical Distributions 646
Bivariate Skew-Normal Distributions 647
Bivariate Skew t-Distributions 647
Bivariate Skew-Cauchy Distribution 648
Asymmetric Bivariate Laplace Distribution 648
Applications 648
References 649
Simulation of Bivariate Observations 653
Introduction 653
Common Approaches in the Univariate Case 654
Introduction 654
Inverse Probability Integral Transform 655
Composition 655
Acceptance/Rejection 656
Ratio of Uniform Variates 656
Transformations 657
Markov Chain Monte Carlo---MCMC 657
Simulation from Some Specific Univariate Distributions 658
Normal Distribution 658
Gamma Distribution 659
Beta Distribution 660
t-Distribution 660
Weibull Distribution 661
Some Other Distributions 661
Software for Random Number Generation 661
Random Number Generation in IMSL Libraries 662
Random Number Generation in S-Plus and R 662
General Approaches in the Bivariate Case 662
Setting 663
Conditional Distribution Method 663
Transformation Method 664
Gibbs' Method 664
Methods Reflecting theDistribution's Construction 665
Bivariate Normal Distribution 665
Simulation of Copulas 667
Simulating Bivariate Distributions withSimple Forms 668
Bivariate Beta Distribution 668
Bivariate Exponential Distributions 669
Marshall and Olkin's BivariateExponential Distribution 669
Gumbel's Type I BivariateExponential Distribution 669
Bivariate Gamma Distributions and Their Extensions 669
Cherian's Bivariate Gamma Distribution 669
Kibble's Bivariate Gamma Distribution 670
Becker and Roux's Bivariate Gamma 670
Bivariate Gamma Mixture of Jones et al. 670
Simulation from ConditionallySpecified Distributions 670
Simulation from Elliptically Contoured Bivariate Distributions 671
Simulation of Bivariate Extreme-Value Distributions 672
Method of Shi et al. 672
Method of Ghoudi et al. 672
Method of Nadarajah 673
Generation of Bivariate and MultivariateSkewed Distributions 673
Generation of Bivariate Distributions with Given Marginals 673
Background 673
Weighted Linear Combination andTrivariate Reduction 674
Schmeiser and Lal's Methods 675
Cubic Transformation of Normals 676
Parrish's Method 676
Simulating Bivariate Distributions with Specified Correlations 676
Li and Hammond's Method for Distributions with Specified Correlations 676
Generating Bivariate Uniform Distributions with Prescribed Correlation Coefficients 677
The Mixture Approach for Simulating Bivariate Distributions with Specified Correlations 677
References 678
Author Index 684
Subject Index 696