Portfolio theory and much of asset pricing, as well as many empirical applications, depend on the use of multivariate probability distributions to describe asset returns. Traditionally, this has meant the multivariate normal (or Gaussian) distribution. More recently, theoretical and empirical work in financial economics has employed the multivariate Student (and other) distributions which are members of the elliptically symmetric class. There is also a growing body of work which is based on skew-elliptical distributions. These probability models all exhibit the property that the marginal distributions differ only by location and scale parameters or are restrictive in other respects. Very often, such models are not supported by the empirical evidence that the marginal distributions of asset returns can differ markedly. Copula theory is a branch of statistics which provides powerful methods to overcome these shortcomings. This book provides a synthesis of the latest research in the area of copulae as applied to finance and related subjects such as insurance. Multivariate non-Gaussian dependence is a fact of life for many problems in financial econometrics. This book describes the state of the art in tools required to deal with these observed features of financial data. This book was originally published as a special issue of the European Journal of Finance.

## Multivariate Probability

This book is a comprehensive guide to multivariate probability for students who have an elementary knowledge of probability and are ready to move on to more advanced concepts Topics covered include: A review of basic probability theory, including core ideas about random variables Bivariate distributions and the general theory of random vectors Relationships between random variables Normal linear model and multivariate sampling distributions Generating functions and convergence. Each section is illustrated with numerous examples. Multivariate probability deliberately avoids a measure-theoretic approach in order to make these complex concepts easily accessible to a broad readership. Attention is restricted to discrete and (absolutely) continuous random variables. Although proofs are given of all the main results, this book is primarily intended to provide readers with the tools they require to build appropriate probability models for real-life situations. The usefulness of simulation in this respect is emphasized throughout the book.

## Decomposition of Multivariate Probabilities

Decomposition of Multivariate Probability is a nine-chapter text that focuses on the problem of multivariate characteristic functions. After a brief introduction to some useful results on measures and integrals, this book goes on dealing with the classical theory and the Fourier-Stieltjes transforms of signed measures. The succeeding chapters explore the multivariate extension of the well-known Paley-Wiener theorem on functions that are entire of exponential type and square-integrable; the theory of infinitely divisible probabilities and the classical results of Hin?in; and the decompositions of analytic characteristic functions. Other chapters are devoted to the important problem of the description of a specific class on n-variate probabilities without indecomposable factors. The final chapter studies the problem of ?-decomposition of multivariate characteristic functions. This book will prove useful to mathematicians and advance undergraduate and graduate students.

## Probability Inequalities in Multivariate Distributions

Probability Inequalities in Multivariate Distributions is a comprehensive treatment of probability inequalities in multivariate distributions, balancing the treatment between theory and applications. The book is concerned only with those inequalities that are of types T1-T5. The conditions for such inequalities range from very specific to very general. Comprised of eight chapters, this volume begins by presenting a classification of probability inequalities, followed by a discussion on inequalities for multivariate normal distribution as well as their dependence on correlation coefficients. The reader is then introduced to inequalities for other well-known distributions, including the multivariate distributions of t, chi-square, and F; inequalities for a class of symmetric unimodal distributions and for a certain class of random variables that are positively dependent by association or by mixture; and inequalities obtainable through the mathematical tool of majorization and weak majorization. The book also describes some distribution-free inequalities before concluding with an overview of their applications in simultaneous confidence regions, hypothesis testing, multiple decision problems, and reliability and life testing. This monograph is intended for mathematicians, statisticians, students, and those who are primarily interested in inequalities.

## Asymptotics in Statistics and Probability

## Computation of Multivariate Normal and t Probabilities

Multivariate normal and t probabilities are needed for statistical inference in many applications. Modern statistical computation packages provide functions for the computation of these probabilities for problems with one or two variables. This book describes recently developed methods for accurate and efficient computation of the required probability values for problems with two or more variables. The book discusses methods for specialized problems as well as methods for general problems. The book includes examples that illustrate the probability computations for a variety of applications.

## Lagrangian Probability Distributions

Fills a gap in book literature Examines many new Lagrangian probability distributions and their applications to a variety of different fields Presents background mathematical and statistical formulas for easy reference Detailed bibliography and index Exercises in many chapters May be used as a reference text or in graduate courses and seminars on Distribution Theory and Lagrangian Distributions

## Reliability of Geotechnical Structures in ISO2394

The latest 4th edition of the international standard on the principles of reliability for load bearing structures (ISO2394:2015) includes a new Annex D dedicated to the reliability of geotechnical structures. The emphasis in Annex D is to identify and characterize critical elements of the geotechnical reliability-based design process. This book contains a wealth of data and information to assist geotechnical engineers with the implementation of semi-probabilistic or full probabilistic design approaches within the context of established geotechnical knowledge, principles, and experience. The introduction to the book presents an overview on how reliability can play a complementary role within prevailing norms in geotechnical practice to address situations where some measured data and/or past experience exist for limited site-specifi c data to be supplemented by both objective regional data and subjective judgment derived from comparable sites elsewhere. The principles of reliability as presented in ISO2394:2015 provides the common basis for harmonization of structural and geotechnical design. The balance of the chapters describes the uncertainty representation of geotechnical design parameters, the statistical characterization of multivariate geotechnical data and model factors, semi-probabilistic and direct probability-based design methods in accordance to the outline of Annex D. This book elaborates and reinforces the goal of Annex D to advance geotechnical reliability-based design with geotechnical needs at the forefront while complying with the general principles of reliability given by ISO2394:2015. It serves as a supplementary reference to Annex D and it is a must-read for designing geotechnical structures in compliance with ISO2394:2015.

## Multivariate Statistics and Probability

*Essays in Memory of Paruchuri R. Krishnaiah*