Small sample comparisons for the blended weight chi-square goodness-of-fit test statistics

The small sample properties of the family of blended weight chi-square (BWCS) goodness-of-fit tests are investigated. Like the power divergence family, this family is a very rich subclass of a more general class of goodness-of-fit tests called the disparity tests (Basu and Sarkar 1994a). Use of the standard asymptotic chi-square distribution in small samples can give quite inaccurate critical regions for most members of the BWCS family. We derive three other asymptotic approximations of the exact distributions in order to obtain more accurate significance levels for the BWCS tests. Two of these approximations are computationally simple to use in practice. Numerical comparisons are made for the equiprobable null hypothesis, for various multinomial sample sizes and numbers of cells. Exact power comparisons show that under specific alternatives to the equiprobable null hypothesis there may be other members in the BWCS family that have more power than the commonly used Pearson's chi-square.