A NOTE ON THE POINT ESTIMATION IN N-WAY LAYOUT

Abstract

In an n-way layout the point estimations for several parametric functions are of our important concerns. Nevertheless it seems to the author that the optimulity of their estimates has not yet been discussed except for the unbiasedness. The purpose of this note is to give some remarks on the optimulity of the usual point estimation for some n-way layouts by means of a sufficient statistic. Our main results are given in THEOREM 1 and 2. It is shown in these theorems that in a fixed-effects model for any n-way layout with the replications not less than two in each cell and in a variance-components model for a 1-way layout with equal numbers r(≥ 2) of observations in each class, the usual unbiased estimates of the general mean, the main effects, the variance components, etc. are the best ones among all unbiased estimates in the sense that each of them minimizes their respective expected loss with respect to any loss function (for example, variance), hence the usual estimation procedures are all justified.