The Annals of Mathematical Statistics

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

James K. Yarnold

Abstract

Ranga Rao  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 ( 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.

Article information

Source
Ann. Math. Statist., Volume 43, Number 5 (1972), 1566-1580.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177692389

Digital Object Identifier
doi:10.1214/aoms/1177692389

Mathematical Reviews number (MathSciNet)
MR372967

Zentralblatt MATH identifier
0256.62022

JSTOR