Introduction

Let $X_1 , \ldots , X_m$ ($m \geq 2$) be the standardized m-variate normal random variates with a correlation matrix $\{ \rho^{(m)}_{jk} \}$. Consider the probability

$$\Pr \biggl[ \bigcap_{j=1}^m \biggl( X_j \leq b_j \biggr) ; ~ \{ \rho^{(m)}_{jk}=\alpha_{jk} \} \biggr]$$ (1)

where $b_1,\ldots , b_m\in \Re$ and where $\{\rho^{(m)}_{jk}=\alpha_{jk}\}$ denotes the correlation matrix $\{ \rho^{(m)}_{jk} \}$ with entry $\alpha_{jk}$ in the j-th row and k-th column for $j \neq k$ and entry 1 for j=k, where $1 \leq j,k \leq m$. The methods for evaluating the probability in (1) with various non-singular correlation structures have been extensively studied by Dunnett and Sobel (1955), Steck and Owen (1962), Schervish (1984, 1985), Nelson (1991), Dunnett (1989, 1993), Drezner (1992), Genz (1992), and Hajivassiliou, McFadden and Rudd (1996). For example, in order to evaluate an l-variate normal probability with a non-singular negative product structure ( $\{ \rho^{(l)}_{jk}=-\alpha_j\alpha_k \}$where l < m and $\sum_{j=1}^m \alpha_j^2/(1+\alpha_j^2) =1$), Nelson (1991) proved that for any $2 \leq l < m$

$$\Pr \biggl[ \bigcap_{j=1}^l \biggl( X_j \leq b_j \biggr) ; ~ \{ \... ...\Phi\biggl(\frac{b_j-i\alpha_jz}{\sqrt{1+\alpha_j^2}}\biggr) \biggr] \phi (z)dz$$ (2)

where $\phi$ is the standard normal density function and $\Phi$ is the standard normal distribution function extended to complex domain and defined by

$$\begin{eqnarray*}\Phi (x+iy)=e^{y^{2}/2}\int_{-\infty}^{x} e^{-isy}\phi(s)ds \end{eqnarray*}$$

where i²=-1. Kwong (1995) showed that the result in (2) is not valid for the singular correlation structure, when l=m. After modifying (2) for l=m, Kwong (1995) proved a new theorem that provides a method for evaluating one-sided multivariate normal probabilities with such singular correlation structure. However, the result cannot be extended to evaluate the two-sided probabilities in the form

$$\Pr \biggl[ \bigcap_{j=1}^m \biggl( \vert X_j\vert \leq b_j \biggr) ; ~ \{ \rho^{(m)}_{jk}=-\alpha_j\alpha_k \} \biggr]$$ (3)

where b_j>0 for $j=1,\cdots, m$. Recently, Kwong and Iglewicz (1996) derived a new approach for evaluating (3) for m=3, and with the additional restriction that $\alpha_1=\alpha_2=\cdots =\alpha_m=-1/(m-1)$ for $m \geq 4$. Kwong (1998) provided another approach to evaluate the upper and lower bounds for (3) for any $m \geq 4$.

In this paper, a new approach for evaluating (3) with any arbitrary singular correlation structures is presented. Numerical and simulation studies are conducted to compare the new approach with the existing method and simulation results. Then, the new approach is applied to evaluate the critical values for the construction of simultaneous confidence intervals and simultaneous all pairwise confidence intervals for multinomial proportions when the sample size is sufficiently large.