A Test for Goodness-of-Fit Based on an Empirical Probability Measure

Abstract

A goodness-of-fit test is proposed for the simple hypothesis specifying a continuous $p$-variate distribution. For a suitably defined empirical probability measure, the proposed test is based on the supremum of the absolute differences between hypothesized and empirical probabilities, the supremum being taken over all possible events. This test statistic is shown to be distribution free in the general $p$-variate case, its exact null distribution is indicated, and its asymptotic null distribution is obtained.