A Chi-Square Statistic with Random Cell Boundaries

Abstract

In testing goodness of fit to parametric families with unknown parameters, it is often desirable to allow the cell boundaries for a chi-square statistic to be functions of the estimated parameter values. Suppose $M$ cells are used and $m$ parameters are estimated using BAN estimators based on sample. Then A. R. Roy and G. S. watson showed that in the univariate case the asymptotic null distribution of the chi-square statistic is that of $\sum^{m - m - 1}_1 Z^2_t + \sum^{m - 1}_{M - m} \lambda_t Z^2_t$, where $Z_t$ are independent standarad normal and the constants $\lambda_t$ lie between 0 and 1. They further observed that in the location-scale case the $\lambda_t$ are independent of the parameters if the cell boundaries are chosen in a natural way, and that in any case all $\lambda_t$ approach 0 as $M$ is appropraitely increased. We extend all of these results to the case of rectangular cells in any number of dimensions. Moreover, we give a method for numerical computation of the exact cdf of the asymptotic distribution and provde a short table of crticial points for testing goodness-of-fit to the univariate normal family.