Contents Preface Univariate Distributions

Transcription

1 Contents Preface... vii 0 Univariate Distributions Introduction Notation and Definitions Notation Explanations Characteristic Function Cumulant Generating Function Some Measures of Shape Characteristics Location and Scale Skewness and Kurtosis Tail Behavior Some Multiparameter Systems of Univariate Distributions Reliability Classes Normal Distribution and Its Transformations Normal Distribution Lognormal Distribution Truncated Normal Johnson s System Box Cox Power Transformations to Normality g and h Families of Distributions Efron s Transformation Distribution of a Ratio Compound Normal Distributions Beta Distribution The First Kind Uniform Distribution Symmetric Beta Distribution Inverted Beta Distribution ix

2 x Contents 0.6 Exponential, Gamma, Weibull, and Stacy Distributions Exponential Distribution Gamma Distribution Chi-Squared and Chi Distributions Weibull Distribution Stacy Distribution Comments on Skew Distributions Compound Exponential Distributions Aging Distributions Marshall and Olkin s Family of Distributions Families of Generalized Weibull Distributions Logistic, Laplace, and Cauchy Distributions Logistic Distribution Laplace Distribution The Generalized Error Distribution Cauchy Distribution Extreme-Value Distributions Type Type Type Pareto Distribution Pearson System Burr System t- andf -Distributions t-distribution F -Distribution The Wrapped t Family of Circular Distributions Noncentral Distributions Skew Distributions Skew-Normal Distribution Skew t-distributions Skew-Cauchy Distribution Jones Family of Distributions Some Lesser-Known Distributions Inverse Gaussian Distribution Meixner Hypergeometric Distribution Hyperbolic Distributions Stable Distributions References Bivariate Copulas Introduction BasicProperties FurtherPropertiesofCopulas SurvivalCopula... 36

3 Contents xi 1.5 Archimedean Copula Extreme-ValueCopulas Archimax Copulas Gaussian, t-, and Other Copulas of the Elliptical Distributions Order Statistics Copula Polynomial Copulas Approximation of a Copula by a Polynomial Copula Measures of Dependence Between Two Variables with a GivenCopula Kendall s Tau Spearman s Rho Geometry of Correlation Under a Copula Measure Based on Gini s Coefficient Tail Dependence Coefficients A Local Dependence Measure Tests of Dependence and Inferences Concepts of Dependence of Copulas Distribution Function of Z = C(U, V ) Simulation of Copulas The General Case Archimedean Copulas Construction of a Copula Rüschendorf s Method Generation of Copulas by Mixture Convex Sums Univariate Function Method Some Other Methods Applications of Copulas Insurance, Finance, Economics, and RiskManagement Hydrology and Environment Management Science and Operations Research Reliability and Survival Analysis Engineering and Medical Sciences Miscellaneous Criticisms about Copulas Conclusions References Distributions Expressed as Copulas Introduction Farlie Gumbel Morgenstern (F-G-M) Copula and Its Generalization

4 xii Contents Applications Univariate Transformations A Switch-Source Model Ordinal Contingency Tables Iterated F-G-M Distributions Extensions of the F-G-M Distribution Other Related Distributions Ali Mikhail Haq Distribution Bivariate Logistic Distributions Bivariate Exponential Distribution Frank s Distribution Distribution of Cuadras and Augé and Its Generalization Generalized Cuadras and Augé Family (Marshall and Olkin s Family) Gumbel Hougaard Copula Plackett s Distribution Bivariate Lomax Distribution The Special Case of c = Bivariate Pareto Distribution Lomax Copula Pareto Copula (Clayton Copula) Summary of the Relationship Between VariousCopulas Gumbel s Type I Bivariate Exponential Distribution Gumbel Barnett Copula Kimeldorf and Sampson s Distribution Rodríguez-Lallena and Úbeda-Flores Family of Bivariate Copulas Other Copulas References to Illustrations References Concepts of Stochastic Dependence Introduction Concept of Positive Dependence and Its Conditions Positive Dependence Concepts at a Glance Concepts of Positive Dependence Stronger than PQD Positive Quadrant Dependence Association of Random Variables Left-Tail Decreasing (LTD) and Right-Tail Increasing(RTI) Positive Regression Dependent (Stochastically Increasing) Left Corner Set Decreasing and Right Corner Set Increasing

5 Contents xiii Total Positivity of Order DTP 2 (m, n) and Positive Dependence by Mixture Concepts of Positive Dependence Weaker than PQD Positive Quadrant Dependence in Expectation Positively Correlated Distributions Monotonic Quadrant Dependence Function Summary of Interrelationships Families of Bivariate PQD Distributions Bivariate PQD Distributions with SimpleStructures Construction of Bivariate PQD Distributions Tests of Independence Against Positive Dependence Geometric Interpretations of PQD and Other Positive Dependence Concepts Additional Concepts of Dependence Negative Dependence Neutrality Examples of NQD Positive Dependence Orderings Some Other Positive Dependence Orderings Positive Dependent Ordering with DifferentMarginals Bayesian Concepts of Dependence References Measures of Dependence Introduction Total Dependence Functions Mutual Complete Dependence Monotone Dependence Functional and Implicit Dependence Overview Global Measures of Dependence Pearson s Product-Moment Correlation Coefficient Robustness of Sample Correlation Interpretation of Correlation Correlation Ratio Chebyshev s Inequality ρ and Concepts of Dependence Maximal Correlation (Sup Correlation) Monotone Correlations Definitions and Properties

6 xiv Contents Concordant and Discordant Monotone Correlations Rank Correlations Kendall s Tau Spearman s Rho The Relationship Between Kendall s Tau and Spearman srho Other Concordance Measures Measures of Schweizer and Wolff and Related Measures Matrix of Correlation Tetrachoric and Polychoric Correlations Compatibility with Perfect Rank Ordering Conclusions on Measures of Dependence Local Measures of Dependence Definition of Local Dependence Local Dependence Function of Holland and Wang Local ρ S and τ Local Measure of LRD Properties of γ(x, y) Local Correlation Coefficient Several Local Indices Applicable in SurvivalAnalysis Regional Dependence Preliminaries Quasi-Independence and Quasi-Independent Projection A Measure of Regional Dependence References Construction of Bivariate Distributions Introduction Fréchet Bounds Transformations The Marginal Transformation Method General Description Johnson s Translation Method Uniform Representation: Copulas Some Properties Unaffected by Transformation MethodsofConstructingCopulas The Inversion Method Geometric Methods Algebraic Methods Rüschendorf s Method Models Defined from a Distortion Function

7 Contents xv Marshall and Olkin s Mixture Method Archimedean Copulas Archimax Copulas Mixing and Compounding Mixing Compounding Variables in Common and Trivariate Reduction Techniques Summary of the Method Denominator-in-Common and Compounding Mathai and Moschopoulos Methods Modified Structure Mixture Model Khintchine Mixture Conditionally Specified Distributions A Conditional Distribution with a MarginalGiven Specification of Both Sets of Conditional Distributions Conditionals in Exponential Families Conditions Implying Bivariate Normality Summary of Conditionally Specified Distributions MarginalReplacement Example: Bivariate Non-normal Distribution Marginal Replacement of a Spherically Symmetric Bivariate Distribution Introducing Skewness Density Generators Geometric Approach Some Other Simple Methods Weighted Linear Combination Data-Guided Methods Conditional Distributions Radii and Angles The Dependence Function in the Extreme-Value Sense Special Methods Used in Applied Fields Shock Models Queueing Theory Compositional Data Extreme-Value Models Time Series: Autoregressive Models Limits of Discrete Distributions A Bivariate Exponential Distribution A Bivariate Gamma Distribution Potentially Useful Methods But Not in Vogue

8 xvi Contents Differential Equation Methods Diagonal Expansion Bivariate Edgeworth Expansion An Application to Wind Velocity at the Ocean Surface Another Application to Statistical Spectroscopy Concluding Remarks References Bivariate Distributions Constructed by the Conditional Approach Introduction Contents Pertinent Univariate Distributions Compatibility and Uniqueness Early Work on Conditionally Specified Distributions Approximating Distribution Functions Using the Conditional Approach Normal Conditionals Conditional Distributions Expression of the Joint Density Univariate Properties Further Properties Centered Normal Conditionals Conditionals in Exponential Families Dependence in Conditional Exponential Families Exponential Conditionals Normal Conditionals Gamma Conditionals Model II for Gamma Conditionals Gamma-Normal Conditionals Beta Conditionals Inverse Gaussian Conditionals Other Conditionally Specified Families Pareto Conditionals Beta of the Second Kind (Pearson Type VI) Conditionals Generalized Pareto Conditionals Cauchy Conditionals Student t-conditionals Uniform Conditionals Translated Exponential Conditionals Scaled Beta Conditionals

9 Contents xvii 6.5 Conditionally Specified Bivariate Skewed Distributions Bivariate Distributions with Skewed Normal Conditionals Linearly Skewed and Quadratically Skewed Normal Conditionals Improper Bivariate Distributions from Conditionals Conditionals in Location-Scale Families with Specified Moments Conditional Distributions and the Regression Function Assumptions and Specifications Wesolowski s Theorem Estimation in Conditionally Specified Models McKay s Bivariate Gamma Distribution and Its Generalization Conditional Properties Expression of the Joint Density Dussauchoy and Berland s Bivariate Gamma Distribution One Conditional and One Marginal Specified Dubey s Distribution Blumen and Ypelaar s Distribution Exponential Dispersion Models Four Densities of Barndorff-Nielsen andblæsild Continuous Bivariate Densities with a Discontinuous Marginal Density Tiku and Kambo s Bivariate Non-normal Distribution Marginal and Conditional Distributions of the Same Variate Example Vardi and Lee s Iteration Scheme Conditional Survival Models Exponential Conditional Survival Function Weibull Conditional Survival Function Generalized Pareto Conditional Survival Function Conditional Approach in Modeling Beta-Stacy Distribution Sample Skewness and Kurtosis Business Risk Analysis Intercropping Winds and Waves, Rain and Floods References

10 xviii Contents 7 Variables-in-Common Method Introduction General Description Additive Models Background Meixner Classes Cherian s Bivariate Gamma Distribution Symmetric Stable Distribution Bivariate Triangular Distribution Summing Several I.I.D. Variables Generalized Additive Models Trivariate Reduction of Johnson and Tenenbein Mathai and Moschopoulos Bivariate Gamma Lai s Structure Mixture Model Latent Variables-in-Common Model Bivariate Skew-Normal Distribution Ordered Statistics Weighted Linear Combination Derivation Expression of the Joint Density Correlation Coefficients Remarks Bivariate Distributions Having a Common Denominator Explanation Applications Correlation Between Ratios with a Common Divisor Compounding Examples of Two Ratios with a Common Divisor Bivariate t-distribution with Marginals Having Different Degrees of Freedom Bivariate Distributions Having a Common Numerator Multiplicative Trivariate Reduction Bryson and Johnson (1982) Gokhale s Model Ulrich s Model KhintchineMixture Derivation Exponential Marginals Normal Marginals References to Generation of Random Variates Transformations Involving the Minimum Other Forms of the Variables-in-Common Technique Bivariate Chi-Squared Distribution

11 Contents xix Bivariate Beta Distribution Bivariate Z-Distribution References Bivariate Gamma and Related Distributions Introduction Kibble s Bivariate Gamma Distribution Formula of the Joint Density Formula of the Cumulative Distribution Function Univariate Properties Correlation Coefficient Moment Generating Function Conditional Properties Derivation Relations to Other Distributions Generalizations Illustrations Remarks Fields of Applications Tables and Algorithms Transformations of the Marginals Royen s Bivariate Gamma Distribution Formula of the Cumulative Distribution Function Univariate Properties Derivation Relation to Kibble s Bivariate Gamma Distribution Izawa s Bivariate Gamma Distribution Formula of the Joint Density Correlation Coefficient Relation to Kibble s Bivariate Gamma Distribution Fields of Application Jensen s Bivariate Gamma Distribution Formula of the Joint Density Univariate Properties Correlation Coefficient Characteristic Function Derivation Illustrations Remarks Fields of Application Tables and Algorithms Gunst and Webster s Model and Related Distributions Case 3 of Gunst and Webster

12 xx Contents Case 2 of Gunst and Webster Smith, Aldelfang, and Tubbs Bivariate Gamma Distribution Sarmanov s Bivariate Gamma Distribution Formula of the Joint Density Univariate Properties Correlation Coefficient Derivation Interrelationships Bivariate Gamma of Loáiciga andleipnik Formula of the Joint Density Univariate Properties Joint Characteristic Function Correlation Coefficient Moments and Joint Moments Application to Water-Quality Data Cheriyan s Bivariate Gamma Distribution Formula of the Joint Density Univariate Properties Correlation Coefficient Moment Generating Function Conditional Properties Derivation Generation of Random Variates Remarks Prékopa and Szántai s Bivariate Gamma Distribution Formula of the Cumulative Distribution Function Formula of the Joint Density Univariate Properties Relation to Other Distributions Schmeiser and Lal s Bivariate Gamma Distribution Method of Construction Correlation Coefficient Remarks Farlie Gumbel Morgenstern Bivariate Gamma Distribution Formula of the Joint Density Univariate Properties Moment Generating Function Correlation Coefficient Conditional Properties Remarks Moran s Bivariate Gamma Distribution Derivation Formula of the Joint Density

13 Contents xxi Computation of Bivariate Distribution Function Remarks Fields of Application Crovelli s Bivariate Gamma Distribution Fields of Application Suitability of Bivariate Gammas for Hydrological Applications McKay s Bivariate Gamma Distribution Formula of the Joint Density Formula of the Cumulative Distribution Function Univariate Properties Conditional Properties Methods of Derivation Remarks Dussauchoy and Berland s Bivariate Gamma Distribution Formula of the Joint Density Mathai and Moschopoulos Bivariate Gamma Distributions Model Model Becker and Roux s Bivariate Gamma Distribution Formula of the Joint Density Derivation Remarks Bivariate Chi-Squared Distribution Formula of the Cumulative Distribution Function Univariate Properties Correlation Coefficient Conditional Properties Derivation Remarks Bivariate Noncentral Chi-Squared Distribution Gaver s Bivariate Gamma Distribution Moment Generating Function Derivation Correlation Coefficients Bivariate Gamma of Nadarajah and Gupta Model Model Arnold and Strauss Bivariate Gamma Distribution Remarks Bivariate Gamma Mixture Distribution Model Specification Formula of the Joint Density

14 xxii Contents Formula of the Cumulative Distribution Function Univariate Properties Moments and Moment Generating Function Correlation Coefficient Fields of Application Mixtures of Bivariate Gammas of IwasakiandTsubaki Bivariate Bessel Distributions References Simple Forms of the Bivariate Density Function Introduction Bivariate t-distribution Formula of the Joint Density Univariate Properties Correlation Coefficients Moments Conditional Properties Derivation Illustrations Generation of Random Variates Remarks Fields of Application Tables and Algorithms Spherically Symmetric Bivariate t-distribution Generalizations Bivariate Noncentral t-distributions Bivariate Noncentral t-distribution with ρ = Bivariate t-distribution Having Marginals with Different Degrees of Freedom Jones Bivariate Skew t-distribution Univariate Skew t-distribution Formula of the Joint Density Correlation and Local Dependence for the SymmetricCase Derivation Bivariate Skew t-distribution Formula of the Joint Density Moment Properties Derivation Possible Application due to Flexibility Ordered Statistics Bivariate t-/skew t-distribution Formula of the Joint Density

15 Contents xxiii Univariate Properties Conditional Properties Other Properties Derivation Bivariate Heavy-Tailed Distributions Formula of the Joint Density Univariate Properties Remarks Fields of Application Bivariate Cauchy Distribution Formula of the Joint Density Formula of the Cumulative Distribution Function Univariate Properties Conditional Properties Illustrations Remarks Generation of Random Variates Generalization Bivariate Skew-Cauchy Distribution Bivariate F -Distribution Formula of the Joint Density Formula of the Cumulative Distribution Function Univariate Properties Correlation Coefficients Product Moments Conditional Properties Methods of Derivation Relationships to Other Distributions Fields of Application Tables and Algorithms Bivariate Pearson Type II Distribution Formula of the Joint Density Univariate Properties Correlation Coefficient Conditional Properties Relationships to Other Distributions Illustrations Generation of Random Variates Remarks Tables and Algorithms Jones Bivariate Beta/Skew Beta Distribution Bivariate Finite Range Distribution Formula of the Survival Function Characterizations

16 xxiv Contents Remarks Bivariate Beta Distribution Formula of the Joint Density Univariate Properties Correlation Coefficient Product Moments Conditional Properties Methods of Derivation Relationships to Other Distributions Illustrations Generation of Random Variates Remarks Fields of Application Tables and Algorithms Generalizations Jones Bivariate Beta Distribution Formula of the Joint Density Univariate Properties Product Moments Correlation and Local Dependence Other Dependence Properties Illustrations Bivariate Inverted Beta Distribution Formula of the Joint Density Formula of the Cumulative Distribution Function Derivation Tables and Algorithms Application Generalization Remarks Bivariate Liouville Distribution Definitions Moments and Correlation Coefficient Remarks Generation of Random Variates Generalizations Bivariate pth-order Liouville Distribution Remarks Bivariate Logistic Distributions Standard Bivariate Logistic Distribution Archimedean Copula F-G-M Distribution with Logistic Marginals Generalizations Remarks

17 Contents xxv 9.18 Bivariate Burr Distribution Rhodes Distribution Support Formula of the Joint Density Derivation Remarks Bivariate Distributions with Support Above the Diagonal Formula of the Joint Density Formula of the Cumulative Distribution Function Univariate Properties Other Properties Rotated Bivariate Distribution Some Special Cases Applications References Bivariate Exponential and Related Distributions Introduction Gumbel s Bivariate Exponential Distributions Gumbel s Type I Bivariate Exponential Distribution Characterizations Estimation Method Other Properties Gumbel s Type II Bivariate Exponential Distribution Gumbel s Type III Bivariate Exponential Distribution Freund s Bivariate Distribution Formula of the Joint Density Formula of the Cumulative Distribution Function Univariate Properties Correlation Coefficient Conditional Properties Joint Moment Generating Function Derivation Illustrations Other Properties Remarks Fields of Application Transformation of the Marginals Compounding Bhattacharya and Holla s Generalizations

18 xxvi Contents Proschan and Sullo s Extension of Freund s Model Becker and Roux s Generalization Hashino and Sugi s Distribution Formula of the Joint Density Remarks An Application Marshall and Olkin s Bivariate Exponential Distribution Formula of the Cumulative Distribution Function Formula of the Joint Density Function Univariate Properties Conditional Distribution Correlation Coefficients Derivations Fisher Information Estimation of Parameters Characterizations Other Properties Remarks Fields of Application Transformation to Uniform Marginals Transformation to Weibull Marginals Transformation to Extreme-Value Marginals Transformation of Marginals: Approach of Muliere andscarsini Generalization ACBVE of Block and Basu Formula of the Joint Density Formula of the Cumulative Distribution Function Univariate Properties Correlation Coefficient Moment Generating Function Derivation Remarks Applications Sarkar s Distribution Formula of the Joint Density Formula of the Cumulative Distribution Function Univariate Properties Correlation Coefficient Derivation Relation to Marshall and Olkin s Distribution

19 Contents xxvii 10.8 Comparison of Four Distributions Friday and Patil s Generalization Tosch and Holmes Distribution A Bivariate Exponential Model of Wang Formula of the Joint Density Univariate Properties Remarks Lawrance and Lewis System of Exponential Mixture Distributions General Form Model EP Model EP Model EP Models with Negative Correlation Models with Uniform Marginals The Distribution of Sums, Products, and Ratios Mixture Models Models with Line Singularities Raftery s Scheme First Special Case Second Special Case Formula of the Joint Density Formula of the Cumulative Distribution Function Derivation Illustrations Remarks Applications Linear Structures of Iyer et al Positive Cross Correlation Negative Cross Correlation Fields of Application Moran Downton Bivariate Exponential Distribution Formula of the Joint Density Formula of the Cumulative Distribution Function Univariate Properties Correlation Coefficients Conditional Properties Moment Generating Function Regression Derivation Fisher Information Estimation of Parameters

20 xxviii Contents Illustrations Random Variate Generation Remarks Fields of Application Tables or Algorithms Weibull Marginals A Bivariate Laplace Distribution Sarmanov s Bivariate Exponential Distribution Formula of the Joint Density Other Properties Cowan s Bivariate Exponential Distribution Formula of the Cumulative Distribution Function Formula of the Joint Density Univariate Properties Correlation Coefficients Conditional Properties Derivation Illustrations Remarks Transformation of the Marginals Singpurwalla and Youngren s Bivariate Exponential Distribution Formula of the Cumulative Distribution Function Formula of the Joint Density Univariate Properties Derivation Remarks Arnold and Strauss Bivariate Exponential Distribution Formula of the Joint Density Formula of the Cumulative Distribution Function Univariate Properties Conditional Distribution Correlation Coefficient Derivation Other Properties Mixtures of Bivariate Exponential Distributions Lindley and Singpurwalla s Bivariate Exponential Mixture Sankaran and Nair s Mixture Al-Mutairi s Inverse Gaussian Mixture of Bivariate Exponential Distribution Hayakawa s Mixtures

21 Contents xxix Bivariate Exponentials and Geometric Compounding Schemes Background Probability Generating Function Bivariate Geometric Distribution Bivariate Geometric Distribution ArisingfromaShockModel Bivariate Exponential Distribution Compounding Scheme Wu s Characterization of Marshall and Olkin s Distribution via a Bivariate Random Summation Scheme Lack of Memory Properties of Bivariate Exponential Distributions Extended Bivariate Lack of Memory Distributions Effect of Parallel Redundancy with Dependent Exponential Components Mean Lifetime under Gumbel s Type I Bivariate Exponential Distribution Stress Strength Model and Bivariate Exponential Distributions Basic Idea Marshall and Olkin s Model Downton s Model Two Dependent Components Subjected to a Common Stress A Component Subjected to Two Stresses Bivariate Weibull Distributions Marshall and Olkin (1967) Lee (1979) Lu and Bhattacharyya (1990): I Farlie Gumbel Morgenstern System Lu and Bhattacharyya (1990): II Lee (1979): II Comments Applications Gamma Frailty Bivariate Weibull Models Bivariate Mixture of Weibull Distributions Bivariate Generalized Exponential Distribution References Bivariate Normal Distribution Introduction Basic Formulas and Properties

22 xxx Contents Notation Support Formula of the Joint Density Formula of the Cumulative Distribution Function Univariate Properties Correlation Coefficients Conditional Properties Moments and Absolute Moments Methods of Derivation Differential Equation Method Compounding Method Trivariate Reduction Method Bivariate Central Limit Theorem Transformations of Diffuse Probability Distributions Characterizations Order Statistics Linear Combination of the Minimum and themaximum Concomitants of Order Statistics Illustrations Relationships to Other Distributions Parameter Estimation Estimate and Inference of ρ Estimation Under Censoring Other Interesting Properties Notes on Some More Specialized Fields Applications Computation of Bivariate Normal Integrals The Short Answer Algorithms Rectangles Algorithms: Owen s T Function Algorithms: Triangles Algorithms: Wedge-Shaped Domain Algorithms: Arbitrary Polygons Tables Computer Programs Literature Reviews Testing for Bivariate Normality How Might Bivariate Normality Fail? Outliers Graphical Checks Formal Tests: Univariate Normality Formal Tests: Bivariate Normality

23 Contents xxxi Tests of Bivariate Normality After Transformation Some Comments and Suggestions Distributions with Normal Conditionals Bivariate Skew-Normal Distribution Bivariate Skew-Normal Distribution of Azzalini and Dalla Valle Bivariate Skew-Normal Distribution of Sahuetal Fundamental Bivariate Skew-Normal Distributions Review of Bivariate Skew-Normal Distributions Univariate Transformations The Bivariate Lognormal Distribution Johnson s System The Uniform Representation The g and h Transformations Effect of Transformations on Correlation Truncated Bivariate Normal Distributions Properties Application to Selection Procedures Truncation Scheme of Arnold et al. (1993) A Random Right-Truncation Model of Gürler Bivariate Normal Mixtures Construction References to Illustrations Generalization and Compounding Properties of a Special Case Estimation of Parameters Estimation of Correlation Coefficient for Bivariate NormalMixtures Tests of Homogeneity in Normal MixtureModels Sharpening a Scatterplot Digression Analysis Applications Bivariate Normal Mixing with Bivariate Lognormal Nonbivariate Normal Distributions with Normal Marginals Simple Examples with Normal Marginals Normal Marginals with Linear Regressions Linear Combinations of Normal Marginals Uncorrelated Nonbivariate Normal Distributions with Normal Marginals Bivariate Edgeworth Series Distribution Bivariate Inverse Gaussian Distribution

24 xxxii Contents Formula of the Joint Density Univariate Properties Correlation Coefficients Conditional Properties Derivations References to Illustrations Remarks References Bivariate Extreme-Value Distributions Preliminaries Introduction to Bivariate Extreme-Value Distribution Definition General Properties Bivariate Extreme-Value Distributions in General Forms Classical Bivariate Extreme-Value Distributions with GumbelMarginals Type A Distributions Type B Distributions Type C Distributions Representations of Bivariate Extreme-Value Distributions with Gumbel Marginals Bivariate Extreme-Value Distributions with Exponential Marginals Pickands Dependence Function Properties of Dependence Function A Differentiable Models Nondifferentiable Models Tawn s Extension of Differentiable Models Negative Logistic Model of Joe Normal-Like Bivariate Extreme-Value Distributions Correlations Bivariate Extreme-Value Distributions with Fréchet Marginals Bilogistic Distribution Negative Bilogistic Distributions Beta-Like Extreme-Value Distribution Bivariate Extreme-Value Distributions with Weibull Marginals Formula of the Cumulative Distribution Function Univariate Properties Formula of the Joint Density Fisher Information Matrix Remarks

25 Contents xxxiii 12.8 Methods of Derivation Estimation of Parameters References to Illustrations Generation of Random Variates Shi et al. s (1993) Method Ghoudi et al. s (1998) Method Nadarajah s (1999) Method Applications Applications to Natural Environments Financial Applications Other Applications Conditionally Specified Gumbel Distributions Bivariate Model Without Having GumbelMarginals Nonbivariate Extreme-Value Distributions with GumbelMarginals Positive or Negative Correlation Fields of Applications References Elliptically Symmetric Bivariate and Other Symmetric Distributions Introduction Elliptically Contoured Bivariate Distributions: Formulations Formula of the Joint Density Alternative Definition Another Stochastic Representation Formula of the Cumulative Distribution Characteristic Function Moments Conditional Properties Copulas of Bivariate Elliptical Distributions Correlation Coefficients Fisher Information Local Dependence Functions Other Properties Elliptical Compound Bivariate Normal Distributions Examples of Elliptically and Spherically Symmetric Bivariate Distributions Bivariate Normal Distribution Bivariate t-distribution Kotz-Type Distribution Bivariate Cauchy Distribution Bivariate Pearson Type II Distribution

26 xxxiv Contents Symmetric Logistic Distribution Bivariate Laplace Distribution Bivariate Power Exponential Distributions Extremal Type Elliptical Distributions Kotz-Type Elliptical Distribution Fréchet-Type Elliptical Distribution Gumbel-Type Elliptical Distribution Tests of Spherical and Elliptical Symmetry Extreme Behavior of Bivariate Elliptical Distributions Fields of Application Bivariate Symmetric Stable Distributions Explanations Characteristic Function Probability Densities Association Parameter Correlation Coefficients Remarks Application Generalized Bivariate Symmetric Stable Distributions Characteristic Functions de Silva and Griffith s Class A Subclass of de Silva s Stable Distribution α-symmetric Distribution Other Symmetric Distributions l p -Norm Symmetric Distributions Bivariate Liouville Family Bivariate Linnik Distribution Bivariate Hyperbolic Distribution Formula of the Joint Density Univariate Properties Derivation References to Illustrations Remarks Fields of Application Skew-Elliptical Distributions Bivariate Skew-Normal Distributions Bivariate Skew t-distributions Bivariate Skew-Cauchy Distribution Asymmetric Bivariate Laplace Distribution Applications References

27 Contents xxxv 14 Simulation of Bivariate Observations Introduction Common Approaches in the Univariate Case Introduction Inverse Probability Integral Transform Composition Acceptance/Rejection Ratio of Uniform Variates Transformations Markov Chain Monte Carlo MCMC Simulation from Some Specific Univariate Distributions Normal Distribution Gamma Distribution Beta Distribution t-distribution Weibull Distribution Some Other Distributions Software for Random Number Generation Random Number Generation in IMSL Libraries Random Number Generation in S-Plus and R General Approaches in the Bivariate Case Setting Conditional Distribution Method Transformation Method Gibbs Method Methods Reflecting the Distribution s Construction Bivariate Normal Distribution Simulation of Copulas Simulating Bivariate Distributions with SimpleForms Bivariate Beta Distribution Bivariate Exponential Distributions Marshall and Olkin s Bivariate Exponential Distribution Gumbel s Type I Bivariate Exponential Distribution Bivariate Gamma Distributions and Their Extensions Cherian s Bivariate Gamma Distribution Kibble s Bivariate Gamma Distribution Becker and Roux s Bivariate Gamma Bivariate Gamma Mixture of Jones et al Simulation from Conditionally Specified Distributions

28 xxxvi Contents Simulation from Elliptically Contoured Bivariate Distributions Simulation of Bivariate Extreme-Value Distributions Method of Shi et al Method of Ghoudi et al Method of Nadarajah Generation of Bivariate and Multivariate Skewed Distributions Generation of Bivariate Distributions with Given Marginals Background Weighted Linear Combination and Trivariate Reduction Schmeiser and Lal s Methods Cubic Transformation of Normals Parrish s Method Simulating Bivariate Distributions with Specified Correlations Li and Hammond s Method for Distributions with Specified Correlations Generating Bivariate Uniform Distributions with Prescribed Correlation Coefficients The Mixture Approach for Simulating Bivariate Distributions with Specified Correlations References Author Index Subject Index

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