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 ( H 0 ). Hence it is imperative that the distribution of any statistic be known both under the null and the alternative hypothesis ( H 1 ). Under the null hypothesis, H 0 (β = β 0 ), 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.