On the approximations for multinormal integration

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Abstract

The system failure probability in structural reliability analysis involves the computation of multivariate normal integrals. The civil and structural engineering literature has seen the development of several approximation methods for the computation of multivariate normal integrals. However, these developments have been totally unaware of the same in the statistics literature. In this note, a 1-line program from a freely downloadable statistical software is provided for the exact computation of m-dimensional multinormal integrals. It is shown that at least two of the known methods from the civil and structural engineering literature lead to inaccurate results.

Introduction

Let X be an m-dimensional normal random vector with zero means, unit variances and the correlation matrix R. Then it is known that its joint probability density function (pdf) is given byϕ(x1,x2,,xm)=1(2π)m/2|R|exp-xTR-1x2.In system reliability computations, the corresponding distribution function given byΦm(c,R)=-c1-c2-cmϕ(x1,x2,,xm)dx1dx2dxmhas a pivotal role. In the civil and structural engineering literature, several methods have been proposed to compute (1). See Hohenbichler and Rackwitz, 1983, Tang and Melchers, 1987, Pandey, 1997, Pandey, 1998, Ambartzumian et al., 1998, Gollwitzer and Rackwitz, 1998, Melchers, 1999, Pandey and Sarkar, 2002, Mori and Kato, 2003, Yuan and Pandey, 2006. These methods are based on a variety of approximations: first order multivariate normal approximations used by Hohenbichler and Rackwitz, 1983, Gollwitzer and Rackwitz, 1998; simulation techniques used by Ambartzumian et al., 1998, Melchers, 1999; and, the product of conditional marginal method used by Pandey (1998). Many of the mentioned papers provided lengthy programs in Matlab or related software for the implementation of their methods.

It appears, however, that civil and structural engineers – at least the authors of the mentioned papers – are totally unaware of the developments in the statistics literature for computing (1). The latest methods in the statistics literature are far more accurate and efficient than all of the methods proposed by the mentioned papers. The purpose of this note is not to provide a review of the known methods in the statistics literature. A comprehensive review can be found in Kotz and Nadarajah (2004).

The purpose of this note is to provide a 1-line program in the R software (R Development Core Team, 2006) for the exact computation of (1), see Section 2. The program is written in R because, unlike software such as Matlab, it is freely downloadable from the Internet, see http://www.r-project.org. This 1-line program is used to assess the accuracy of two of the methods mentioned above. It is shown that both methods provide inaccurate results, see Section 3.

Section snippets

The R Program

The following simple statement in the R software computes (1) for given values of c and R. The methodology used is described in Genz, 1992, Genz, 1993, Kotz and Nadarajah, 2004.

  • pmvnorm(upper=c,corr=R)

Performance of known methods

Here, we assess the accuracy of two of the methods proposed by Pandey and Sarkar (2002). As in Pandey and Sarkar (2002), we define Pm(a, b) = Φm(c1, R)  Φm(c2, R), where c1 = (b, b,  , b)m and c2 = (a, a,  , a)m. Also define R as given by Rij = r2 for i  j and Rii = 1. For the parameter values of a, b, m and r considered by Pandey and Sarkar (2002), Table 1, Table 4 provide the exact values of Pm(a, b) computing by using the R program. The corresponding values of Pm(a, b) presented by Pandey and Sarkar (2002)

Conclusions

We have provided a program in R for computing the distribution function of m-dimensional multinormal distributions. This program could have wide applicability to civil and structural engineers because: (1) the software is freely downloadable; (2) no restrictions are imposed on the input parameters c and R; (3) the program is simple and easy to implement on any platform (Windows, Unix, Linux); and, (4) the distribution function can be computed to any degree of accuracy.

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