Approximating the general non-normal variance-ratio sampling distributions

SUMMARY Laguerre series expansions are developed of the frequency distributions of the 'nonnormal' variance-ratios used for testing (i) the homogeneity of a set of means in case of oneway classification for analysis of variance with non-identical group to group error distributions; and (ii) the compatibility of two population variances. The effect of non-normality on the distributions is illustrated numerically. The effect of non-normality on the frequency distributions of the variance-ratios used for testing (a) the equality of a set of means in one-way classification for analysis of variance; and (b) the compatibility of two population variances, was first studied by E. S. Pearson (1931) by way of sampling experiments. Theoretical investigations were carried out by Bartlett (1935), Geary (1936, 1947) and Gayen (1950). Their investigations were essentially of the same type, basing their derivations on the first few terms of the Edgeworth's series expansion of the probability density function of the population. They did not consider the case when, in analysis of variance, the error distribution from group to group is not identical. It has been possible here to tackle this general problem in an entirely different way. The distribution of 'between' and 'within' sum of the squares is formally expanded in terms of Laguerre polynomials and Gamma density functions. The first few coefficients are worked out in terms of population cumulants. The 'non-normal' sampling distribution of two independent variances is also given. The results obtained are important not only on their own account, but in giving an elegant and simple method of judging the degree of approximation of the expressions for large n, n being the sample size. Similar expansions of Student's t distribution and power-function of F-test have been obtained elsewhere (Tiku, 1962, 1963). Since this investigation was completed my attention has been called to a paper by Khamis (1960) advocating the use of these polynomials.