TY - JOUR

T1 - Computable MINQUE-type estimates of variance components

AU - Westfall, Peter H.

PY - 1987/6

Y1 - 1987/6

N2 - The minimum norm quadratic unbiased estimator type (MINQUE type) of estimates considered in this article are obtained by requiring identical values for the ratios of the a priori variances to the a priori error variance and letting this common value tend to infinity. The resulting estimates are invariant quadratic unbiased estimators with certain parametric and nonparametric optimality properties: assuming normally distributed random effects the efficiency of the proposed estimates to the minimum variance quadratic unbiased estimates (MIVQUE’s) approaches unity when the true variance ratios are identical and tend to infinity. Assuming nonnormal effect distributions in the model with two variance components, the estimates are asymptotically efficient: in a sequence of designs where the number of classes and the number of observations on each class approach infinity, it is shown that the asymptotic variances of the estimates are equivalent to the theoretical minimum variances for invariant quadratic unbiased estimators. The result is interesting and useful since the usual analysis of variance (ANOVA) estimate of between-classes variance has strictly larger asymptotic variance for the unbalanced one-way model. Commonly considered estimates result from this procedure; the usual residual mean square (assuming that all non-error effects are fixed) is the resulting estimate of the error variance. In the one-way model the resulting estimates coincide with estimates considered by Thomas and Hultquist (1978), Burdick and Graybill (1984), Ahrens, Kleffe, and Tenzler (1981), and Kaplan (1982). In particular, the convergence of the MINQUE estimates was proved in the latter two papers in the context of the unbalanced one-way model. The procedure yields computationally convenient estimates in the general mixed ANOVA model. Computing formulas are given, and it is noted that the computational complexity of these estimates is comparable with the computational complexity of a special case of MINQUE demonstrated by Wansbeek (1980).

AB - The minimum norm quadratic unbiased estimator type (MINQUE type) of estimates considered in this article are obtained by requiring identical values for the ratios of the a priori variances to the a priori error variance and letting this common value tend to infinity. The resulting estimates are invariant quadratic unbiased estimators with certain parametric and nonparametric optimality properties: assuming normally distributed random effects the efficiency of the proposed estimates to the minimum variance quadratic unbiased estimates (MIVQUE’s) approaches unity when the true variance ratios are identical and tend to infinity. Assuming nonnormal effect distributions in the model with two variance components, the estimates are asymptotically efficient: in a sequence of designs where the number of classes and the number of observations on each class approach infinity, it is shown that the asymptotic variances of the estimates are equivalent to the theoretical minimum variances for invariant quadratic unbiased estimators. The result is interesting and useful since the usual analysis of variance (ANOVA) estimate of between-classes variance has strictly larger asymptotic variance for the unbalanced one-way model. Commonly considered estimates result from this procedure; the usual residual mean square (assuming that all non-error effects are fixed) is the resulting estimate of the error variance. In the one-way model the resulting estimates coincide with estimates considered by Thomas and Hultquist (1978), Burdick and Graybill (1984), Ahrens, Kleffe, and Tenzler (1981), and Kaplan (1982). In particular, the convergence of the MINQUE estimates was proved in the latter two papers in the context of the unbalanced one-way model. The procedure yields computationally convenient estimates in the general mixed ANOVA model. Computing formulas are given, and it is noted that the computational complexity of these estimates is comparable with the computational complexity of a special case of MINQUE demonstrated by Wansbeek (1980).

KW - Efficient estimation

KW - Mixed model

KW - Regression

UR - http://www.scopus.com/inward/record.url?scp=33747160938&partnerID=8YFLogxK

U2 - 10.1080/01621459.1987.10478470

DO - 10.1080/01621459.1987.10478470

M3 - Article

AN - SCOPUS:33747160938

VL - 82

SP - 586

EP - 589

JO - Journal of the American Statistical Association

JF - Journal of the American Statistical Association

SN - 0162-1459

IS - 398

ER -