Computation of Multivariate Normal and t Probabilities

verfasst von: Alan Genz, Frank Bretz

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Statistics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract

The numerical availability of distribution functions is important for many statistical applications. The numerical computation of multivariate distribution functions is considerably more complex than the numerical evaluation of univariate distributions due to the “curse” of dimensionality. This is particularly true for the multivariate normal (MVN) and t (MVT) distributions. These distributions occur frequently in many different applications, including multiple comparison procedures (Hochberg and Tamhane, 1987; Hsu, 1996), integrated log-likelihood problems (Lin and Breslow, 1996; Hartzel et al, 2001), applications of the multivariate probit model (Ochi and Prentice, 1984), Bayesian computations (Shaw, 1988; Evans and Swartz, 1995), stochastic programming (Prèkopa, 1995), structural system reliability (Pandey, 1998) as well as many applications in engineering and finance. For a general overview of the MVN and the MVT distributions we refer to the books of Tong (1990), Kotz et al (2000) and Kotz and Nadarajah (2004).

Alan Genz, Frank Bretz

2. Special Cases

Abstract

The problem of evaluating (1.1) or (1.3) can often be simplified by specializing either k, R or a and b. In Section 2.1 we focus on the work that has been done on bivariate and trivariate probabilities and not on general MVN and MVT probabilities. In Section 2.2 we consider calculating probabilities over special integration regions, such orthants, ellipsoids, and hyperboloids. Finally, in Section 2.3 we discuss MVN and MVT problems involving special correlation structures. We do not consider the univariate cases, which have been carefully analyzed elsewhere; see Johnson and Kotz (1970a,b) for extensive discussions and references. Highly accurate implementations for Φ(x), Φ1(x), T(x;), and T1(x;) are available in standard statistical computing environments. We assume the availability of these functions for many of the computational methods that we discuss in this and later chapters.

Alan Genz, Frank Bretz

3. Methods That Approximate the Problem

Abstract

In this chapter we approximate the original MVN or MVT problem by using one or more MVN or MVT problems which are relatively easier to solve. In Section 3.1 we consider various approximations to Boole’s formula, providing MVN and MVT approximations that consist of combinations of lower dimensional problems. In Section 3.2 we briefly describe methods that replace the original problem with one or more problems using covariance matrix structures that simplify the computation. In Section 3.3 we review further approximations. Finally, in Section 3.4 we review some asymptotic expansions of historical interest.

Alan Genz, Frank Bretz

4. Methods That Approximate the Integral

Abstract

In this chapter we describe selected numerical integration methods that can be applied to general MVN and MVT problems. Such problems are often given as a multiple integral over an unbounded integration domain. The application of most integration methods requires a transformation or reparameterization of the original problem to one where the integration domain is bounded. Consequently, we first discuss various reparameterizations in Section 4.1. In Section 4.2 we then describe several multidimensional integration methods.

Alan Genz, Frank Bretz

5. Further Topics

Abstract

In this chapter we consider subjects which were not discussed in the previous chapters. Topics include MVN and MVT problems with an integration region defined by a set of linear inequalities; evaluation of singular distribution functions; numerical tests of different integration methods; and integration problems having an application specific expectation function with MVN or MVT weight. A description of current software implementations in MATLAB and R is also given.

Alan Genz, Frank Bretz

6. Applications

Abstract

In this chapter we consider a variety of applications, which require efficient numerical MVN and/or MVT integration methods. In Section 6.1 we discuss the numerical computation of high-dimensional integrals for multiple comparison problems. In Section 6.2 we discuss the application of MVN and MVT integration methods for problems in computational finance and Bayesian statistical analysis.

Alan Genz, Frank Bretz

Backmatter

Titel: Computation of Multivariate Normal and t Probabilities
verfasst von: Alan Genz
Frank Bretz
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-01689-9
Print ISBN: 978-3-642-01688-2
DOI: https://doi.org/10.1007/978-3-642-01689-9

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Introduction

2. Special Cases

3. Methods That Approximate the Problem

4. Methods That Approximate the Integral

5. Further Topics

6. Applications

Backmatter

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