The Minimum Expectation in X(sup2) Goodness of Fit Tests and the Accuracy of Approximations for the Null Distribution.

The accuracy of approximations for the null distribution of the chi-square goodness of fit statistic, X[sup 2], is examined numerically for many different multinomial distributions. Some of these approximations are terms of a new asymptotic expansion which provides much greater accuracy than the usual x[sup 2] approximation. When there are too many small expectations, even these are inaccurate, and a different type of approximation must be used--the C(m) distribution. It is proven that the latter is the limiting distribution of X[sup 2] when some expectations remain finite while the rest increase without limit., and the limiting multivariate distribution of the multinomial under these conditions is also derived. The C(m) approximation is used to study the error in the x2 approximation when some expectations are small, resulting in a new rule for the use of the upper one and five percent points of the x[sup 2] approximation. This rule allows the use of arbitrarily small expectations, e.g., 1. Similar results are obtained for the Poisson variance test. [ABSTRACT FROM AUTHOR]