Computation of Bivariate Normal Probabilities by the Owen Formulae

Abstract

Complex hydrological events such as floods and storms may appear to be multivariate events. A thorough understanding of statistics of these multiple episodic events should investigate joint probabilistic behavior of two or three correlated characteristic variables of the events. If these characteristic variables can be transformed to follow the normal distribution, then a binormal or trinormal distribution can be used to represent joint distributions of these correlated random variables. But the binormal cumulative distribution function (cdf) is not analytically attainable and cannot be easily computed. This difficulty may hinder its application in hydrological frequency analysis. This study investigates the applicability of computing the binormal probabilities (BNPs) by a dimension-reduced method, the Owen method, which transfers two-dimensional quadrature of the bivariate normal distribution into one-dimensional quadrature. Computation results indicate that the Owen method is an efficient method for numerically computing the BNPs

Keywords:

• Bivariate normal probability
• two-dimensional quadrature
• one-dimensional quadrature
• Box-Cox transformation