Asymptotic Approximations for the Probability that a Sum of Lattice Random Vectors Lies in a Convex Set

Abstract

Ranga Rao [10] developed a version of the Edgeworth asymptotic expansion for $\mathrm{Pr}(X_n \in B)$, where $X_n = n^{-\frac{1}{2}} \sum^n_{i=1} Z_i, \lbrack Z_n\rbrack$ is a sequence of independent random vectors in $R_k$ having a common lattice distribution with mean vector zero and nonsingular covariance matrix $\not\sum$, and $B$ is a Borel set. Use of this expansion is very difficult, except for the distribution function of $X_n$. In this paper, Ranga Rao's expansion is used to obtain a different expansion, when $B$ is convex. This new expansion is much simpler to evaluate. In the special case when $B = \lbrack x \mid x^T \not\sum^{-1} x < c\rbrack$, the new expansion assumes its simplest form. The first partial sum is the usual multivariate normal approximation, and Esseen ([6] pages 110-111) determined the order of magnitude of its error, i.e., $\mathrm{Pr}(X_n \in B) = K_k(c) + O(n^{-k/(k+1)})$ where $K_k(c)$ is the chi-square distribution function with $k$ degrees of freedom. Note that the order of magnitude of the error is $n^{-\frac{1}{2}}$ for $k = 1$ and approaches $n^{-1}$ as $k$ increases. The second partial sum is $\mathrm{Pr}(X_n \in B) = K_k(c) + (N(nc) - V(nc)) \frac{\exp(-c/2)}{(2\pi n)^{k/2}|\not\sum|^{\frac{1}{2}}} + O(n^{-1})$ where $N(nc)$ is the number of integer vectors $m$ in the ellipsoid $(m + na)^T \not\sum^{-1}(m + na) < nc$ having center at $-na$, and $V(nc)$ is the volume of this ellipsoid. This provides a new expansion for the distribution function of the quadratic form $X_n^T \not\sum^{-1}X_n$. When $Z_i$ has a multinomial distribution with parameters $N = 1, p_1, \cdots, p_m, \sum^m_{i=1} p_i = 1, X_n^T \not\sum^{-1} X_n$ is the chi-square goodness-of-fit statistic, and the new expansion (with $k = m - 1$) provides very accurate approximations for its distribution function. The accuracy of the first several partial sums, and of the Edgeworth approximation under the (inappropriate) assumption that $Z_i$ has a continuous distribution, is examined numerically for a number of multinomial distributions. It is concluded that the Edgeworth approximation assuming a continuous distribution should never be used when $Z_i$ has a lattice distribution, and that the second partial sum of the new expansion is much more accurate than the normal approximation for all multinomial distributions examined.