A theorem relating minque and unweighted means estimators of variance components in the one-way design

Kaplan, Jack

doi:10.1080/03610928208828245

Cited by 2 publications

(4 citation statements)

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“…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), andKaplan (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.…”

mentioning

confidence: 50%

“…where 7, = z, y , / n , , and Q, = z,,,(y,, -j j l ) 2 ; these results are also obtained by Ahrens, Kleffe, and Tenzler (1981) and Kaplan (1982). The quadratics Qcm and Q, are considered by Thomas and Hultquist (1978) and Burdick and Graybill (1984) in the context of constructing confidence intervals for functions of 0; and a:.…”

Section: The Two-component Modelmentioning

confidence: 85%

“…In the case in which cell replications exist, Kaplan (1983) demonstrated potential further improvement. It should be noted that the MINQUE estimates of these papers are locally optimal when all variances of are identical; however, the evaluation of efficiency of these estimates for nonidentical variances of is problematic.…”

Section: Introductionmentioning

confidence: 96%

“…A potential drawback of the estimates (1.4) is their dependence on the numerical inversion of the (n X n) matrix V,; this difficulty was surmounted by Liu and Senturia (1977) and Wansbeek (1980), who noted that for the case r, = 1 the estimates may be reformulated in such a way as to require inversion of an (m x m) matrix, where m = m,. In the case in which cell replications exist, Kaplan (1983) demonstrated potential further improvement.…”

Section: Introductionmentioning

confidence: 98%

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Computable MINQUE-Type Estimates of Variance Components

Westfall

1987

Journal of the American Statistical Association

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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), andKaplan (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).

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confidence: 50%