Edgeworth Approximations to the Distributions of the Likelihood Ratio and F Statistics in the Null and Non-null Cases

For testing the linear restrictions on regression coefficients in the classical linear regression model, it is a common practice to use Snedecor’s F-distribution. In Neyman-Pearson theory of testing of statistical hypotheses, the efficiency of a statistical test is to be judged by its power of detecting the departure from the null hypothesis (H0). Hence it is imperative that the distribution of any statistic be known both under the null and the alternative hypothesis (H1). Under the null hypothesis, H0 (β = β0), the test statistic z, follows the central F distribution, and therefore we may use the tables of the F distribution to obtain the points of significance. However, under the alternative hypothesis, the distribution of z is non-central F. For a fixed sample size, we must use this distribution to evaluate the power of the test. Tiku (1967) has computed the tables for the power of the F-test using incomplete Beta functions.