A Chi-Square Statistic with Random Cell Boundaries

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 ∑m - m - 1₁ Z²ₜ + ∑m - 1M - m λₜ Z²ₜ, where Zₜ are independent standarad normal and the constants λₜ lie between 0 and 1. They further observed that in the location-scale case the λₜ are independent of the parameters if the cell boundaries are chosen in a natural way, and that in any case all λₜ 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.