NONNEGATIVE QUADRATIC ESTIMATES OF VARIANCE COMPONENTS IN RANDOM-EFFECT MODOLS

LetY₁,Y₂ be independent random variables, and Y₁/（ασ+τ）～X²（n₁）, Y₂/τ～x²（n₂）, where σ＞0, τ＞0 are unknown variance components, α＞0, n₁, n₂ are known positive integers. In this paper improvement of unbiased estimator Y₁/（an₁）-Y₂/（an₂） of σ is studied from the points of view of risk function and bias, and it is more suitable to replace the unbiased estimator of σ by nonnegative quadratic estimator pY₁, where p=\left(1 / a\left(n_1+2\right), \quad \sqrt2\left(n_1+n_2\right) / n_2\left(n_1+2\right) /\left(a n_1\right)\right).This result is applied to random-effect models of one-way classification, and two-way nested olassification.