An Asymptotic Expansion of the Distribution of the Studentized Classification Statistic $W^1$

Abstract

The classification statistic $W$ is used to classify an observasion as coming from one of two multivariate normal populations with common covariance matrix and different means when these parameters are estimated from two samples, one from each population. The distribution of $W$ depends on the Mahalanobis distance between the populations, $\alpha$. When the sample sizes approach infinity, the limiting distribution of $(W - \frac{1}{2}\alpha)/\alpha^{\frac{1}{2}}$ is the standard normal distribution if the observation is from the first population; the same is true of $(W - \frac{1}{2}a)/a^{\frac{1}{2}}$, where $a$ is an estimate of $\alpha$. This paper gives an asymptotic expansion of the distribution of $(W - \frac{1}{2}a)/a^{\frac{1}{2}}$ with an error of the order of the square of the number of observations. The correction to the standard normal distribution function is the standard normal density times a third-degree polynomial in the argument divided by the sum of the observations (less 2).