Some Applications of Bipolykays to the Estimation of Variance Components and their Moments

Abstract

Bipolykays were introduced in [3]. They form a family of symmetric (row-wise and column-wise) polynomial functions of the elements of a two-way array, with the property of being inherited on the average, and such that any similarly symmetric polynomial function of the same numbers can be written linearly in terms of the bipolykays. This paper will describe some applications of bipolykays to problems in the analysis of variance of two-way classifications, using the formulas and tables derived in [3]. A linear model which includes contributions from interaction as well as independently sampled cell contributions is given in Section 3, and applications are made to certain cases of this model. These applications include (a) finding unbiased estimators for the variance components in the case of no interaction as well as unbiased estimators for the variances of these estimators (Section 6), (b) finding expressions for means and variances of some of the functions of degrees 1 and 2 that are of interest in the problem of sampling from a matrix (Section 7), and (c) finding unbiased estimators for variance components in the general case, including expressions for the variances of these estimators in the case of infinite populations (Section 8).