A Power Approximation for the Chi-Square Goodness-of-Fit Test: Simple Hypothesis Case

Abstract

The limiting distribution of the chi-square goodness-of-fit statistic T_n under alternatives is noncentral chi-square if the alternative probabilities approach the null probabilities at an appropriate rate as n → ∞. It is shown that for fixed alternative probabilities, the limiting distribution of (T_n − μ _n )/σ _n is standard normal. Both of these asymptotic results can be used to approximate the power of the goodness-of-fit test. Numerical comparisons between these two approximations indicate that for large values of the true power, the normal approximation is best, but for moderate values of power, the chi-square approximation is best.

Key Words:

• Chi-square test
• Normal power approximation