Consider the classical homogeneity of variances model. That is, suppose we have a one way layout, with independent random samples of equal sizes in each column. We assume the samples are from normal populations with unknown means and unknown variances. We wish to test the hypothesis that all the variances are equal. R. V. Laue (1965) defined a two parameter family of statistics to test the homogeneity hypothesis. The family is defined as a function of the ratio of mean value functions of the sample variances. Included in the family are many of the well-known tests for homogeneity. In this paper we investigate which of the tests is unbiased. Although we are unable to resolve the question for every test in the family, we can demonstrate the unbiased character for several subfamilies, which include some of the better known tests.
Arthur Cohen. William E. Strawderman. "Unbiasedness of Tests for Homogeneity of Variances." Ann. Math. Statist. 42 (1) 355 - 360, February, 1971. https://doi.org/10.1214/aoms/1177693520