Some New Results on Grubbs's Estimators

Abstract

Consider a two-way classification with n rows and r columns and the usual model of analysis of variance except that the error components of the model may have heterogeneous variances σ² _j (j = 1, …, r) by columns. Grubbs (1948) provided unbiased estimators Q_j of σ² _j that depend on column sums of squared residuals S_j . When r = 3, the joint distributions of the S_j and the Q_j are given for the first time in closed form. Two tests proposed by Russell and Bradley (1958) are examined when r = 3, one for variance homogeneity and the second for one possible disparate variance. A simple distribution is found for the test statistic of the first test, and its non-null distribution is derived also. The distribution of the second test statistic was known to be the central variance-ratio distribution in the null case, and now its ratio to a parameter of noncentrality is shown to have that same distribution in the non-null case. When r = 4, n = 4, the joint distribution of the S_j is also given in closed form, but it is difficult to use. For r > 3, an approximate test of variance homogeneity is suggested, based on an extension of the Russell—Bradley statistic.

Key Words:

• Variance homogeneity
• Helmert matrix
• Wishart density