Negative Product Correlation Structure

Let $X_1 , \ldots , X_m$ be the standardized m-variate normal variates with a singular negative product correlation structure, i.e. $\{\rho^{(m)}_{jk}=-\alpha_j\alpha_k\}$ with $\sum_{j=1}^m \alpha_j^2/(1+\alpha_j^2) =1$. Denote the events $A_j=\{X_j: \vert X_j\vert \leq b_j\}$for $j=1,\ldots ,m$ and $J^m_r=\{(j_1, \ldots, j_r): 1\leq j_1 < j_2 < \cdots < j_r \leq m\}$be a set in an r-dimensional space with all the j_l being integers. Kwong (1998) derived the following inequalities:

$$\begin{eqnarray*}\sum_{r=1}^{m-2} (-1)^{r+1}\sum_{J^{m-1}_r} \Pr\biggr[ \biggl(\... ...(-1)^{r+1}\sum_{J^m_r} \Pr \biggl[\bigcap_{l=1}^r A_{j_l}\biggr] \end{eqnarray*}$$

when m is an odd integer, and

$$\begin{eqnarray*}\sum_{r=1}^{m-1} (-1)^{r+1}\sum_{J^m_r} \Pr \biggl[\bigcap_{l=1... ...biggr[ \biggl(\bigcap_{l=1}^r A_{j_l}\biggl) \bigcap A_m \biggr] \end{eqnarray*}$$

when m is an even integer. Notice that the upper and lower bounds for the singular multivariate normal probability are all expressed in terms of the multivariate normal probabilities with non-singular negative product correlation structures. Therefore, the bounds can be numerically evaluated after the result in (2) is extended as follows:

$$\begin{eqnarray*}\Pr \biggl[ \bigcap_{j=1}^l A_j ; ~ \{ \rho^{(l)}_{jk} =-\alpha... ...-b_j-i\alpha_jz}{\sqrt{1+\alpha_j^2}}\biggr) \biggr] \phi(z)dz. \end{eqnarray*}$$

when $2 \leq l < m$ and $\Pr[A_j]= 2\Phi(b_j)-1$ when l=1.

Obviously, Kwong's inequalities can be applied to evaluate the upper and lower bounds of an m-variate normal distribution with any arbitrary singular correlation matrix of rank k (k < m)if the inequalities are used recursively until all the probabilities are expressed in terms of the nonsingular multivariate normal probabilities which are then computed by any of existing approaches. However, as m-k increases, the difference between the bounds and the exact value, as well as the computational time, increases rapidly. Kwong's inequalities are therefore not an efficient and accurate approach to evaluate any singular multivariate normal probabilities when m-k > 1. A new approach is derived in the next section.