Comparisons of Designs and Estimation Procedures for Estimating Parameters in a Two-Stage Nested Process

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Abstract
For the usual two-stage nested random model, N observations are to be used to estimate some or all of these parameters: μ (the general mean), σa 2 (the between classes variance component), σb 2 (the within classes variance component) and ρ = σa 2/σb 2 Using the traditional analysis of variance estimator, with fixed number of classes, (k), an allocation procedure to minimize the variance of the estimator is derived to estimate σa 2 or ρ: p + 1 observations in each of r classes and p observations in each of k − r classes, where N = pk + r, 0 ≤ r < k. The optimal number of classes (k) is conjectured to be the closest integer to For large N, this implies that the optimal number of classes to estimate σa 2 would be approximately k 1 = N ρ/(1 + ρ) with an average of 1 + ρ−1 observations per class; the optimal number of classes to estimate ρ would be approximately Nρ/(1 + 2ρ) with an average of 2 + ρ−1 observations per class. Hence if ρ > 1, p = 1 for σa 2 and p = 2 for . Investigations on the effect of using an incorrect value of ρ in k 1 and k 2 show that if 0.5 < ρ1/ρ < 2.0, where ρ1 is the value of ρ used, the loss in efficiency is generally less than 10%. Comparisons with the variances of hypothetical estimators in which k need not be an integer show that the integral requirement results in only a small loss of efficiency. Percent efficiencies of the estimates of the other parameters when a sampling plan is designed to minimize the variance of one estimator are presented for selected values of N and p. Alternative estimators are considered for μ and σa 2 and are found to be useful when the sampling is decidedly unbalanced and ρ is quite large. A computing procedure is developed for restricted maximum likelihood estimation of the variance components.

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