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

Ranga Rao [10] developed a version of the Edgeworth asymptotic expansion for Pr(Xₙ ∈ B), where Xₙ = n-1/2 ∑ⁿᵢ₌₁ Zᵢ, Zₙ is a sequence of independent random vectors in Rₖ having a common lattice distribution with mean vector zero and nonsingular covariance matrix ∑, and B is a Borel set. Use of this expansion is very difficult, except for the distribution function of Xₙ. 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 = x xT ∑⁻¹ x < c, 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., Pr(Xₙ ∈ B) = Kₖ(c) + O(n-k/(k+1)) where Kₖ(c) is the chi-square distribution function with k degrees of freedom. Note that the order of magnitude of the error is n-1/2 for $k = 1$ and approaches n⁻¹ as k increases. The second partial sum is Pr(Xₙ ∈ B) = Kₖ(c) + (N(nc) - V(nc)) exp(-c/2)(2π n)k/2|∑|1/2 + O(n⁻¹) where $N(nc)$ is the number of integer vectors m in the ellipsoid (m + na)T ∑⁻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 XₙT ∑⁻¹Xₙ. When Zᵢ has a multinomial distribution with parameters N = 1, p₁, ⋯, pₘ, ∑ᵐᵢ₌₁ pᵢ = 1, XₙT ∑⁻¹ Xₙ 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 Zᵢ 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 Zᵢ 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.