Topic
MINQUE
About: MINQUE is a research topic. Over the lifetime, 169 publications have been published within this topic receiving 12677 citations.
Papers published on a yearly basis
Papers
TL;DR: In this article, a modified maximum likelihood procedure is proposed for estimating intra-block and inter-block weights in the analysis of incomplete block designs with block sizes not necessarily equal, and the method consists of maximizing the likelihood, not of all the data, but of selected error contrasts.
Abstract: SUMMARY A method is proposed for estimating intra-block and inter-block weights in the analysis of incomplete block designs with block sizes not necessarily equal. The method consists of maximizing the likelihood, not of all the data, but of a set of selected error contrasts. When block sizes are equal results are identical with those obtained by the method of Nelder (1968) for generally balanced designs. Although mainly concerned with incomplete block designs the paper also gives in outline an extension of the modified maximum likelihood procedure to designs with a more complicated block structure. In this paper we consider the estimation of weights to be used in the recovery of interblock information in incomplete block designs with possibly unequal block sizes. The problem can also be thought of as one of estimating constants and components of variance from data arranged in a general two-way classification when the effects of one classification are regarded as fixed and the effects of the second classification are regarded as random. Nelder (1968) described the efficient estimation of weights in generally balanced designs, in which the blocks are usually, although not always, of equal size. Lack of balance resulting from unequal block sizes is, however, common in some experimental work, for example in animal breeding experiments. The maximum likelihood procedure described by Hartley & Rao (1967) can be used but does not give the same estimates as Nelder's method in the balanced case. As will be shown, the two methods in effect use the same weighted sums of squares of residuals but assign different expectations. In the maximum likelihood approach, expectations are taken over a conditional distribution with the treatment effects fixed at their estimated values. In contrast Nelder uses unconditional expectations. The difference between the two methods is analogous to the well-known difference between two methods of estimating the variance o2 of a normal distribution, given a random sample of n values. Both methods use the same total sum of squares of deviations. But
4,165 citations
TL;DR: In this paper, the authors proposed a restricted maximum likelihood (reml) approach which takes into account the loss in degrees of freedom resulting from estimating fixed effects, and developed a satisfactory asymptotic theory for estimators of variance components.
Abstract: Recent developments promise to increase greatly the popularity of maximum likelihood (ml) as a technique for estimating variance components. Patterson and Thompson (1971) proposed a restricted maximum likelihood (reml) approach which takes into account the loss in degrees of freedom resulting from estimating fixed effects. Miller (1973) developed a satisfactory asymptotic theory for ml estimators of variance components. There are many iterative algorithms that can be considered for computing the ml or reml estimates. The computations on each iteration of these algorithms are those associated with computing estimates of fixed and random effects for given values of the variance components.
2,649 citations
TL;DR: A procedure is developed for the maximum-likelihood estimation of the unknown constants and variances included in the general mixed analysis of variance model, involving fixed and random factors and interactions, and applies to all cases where the design matrices satisfy certain conditions.
Abstract: SUMMARY A procedure is developed for the maximum-likelihood estimation of the unknown constants and variances included in the general mixed analysis of variance model, involving fixed and random factors and interactions. The method applies to all cases where the design matrices satisfy certain conditions. The consistency and asymptotic efficiency of the estimates are discussed. Tests of hypotheses and confidence regions are derived. In this paper we develop a procedure for maximum-likelihood estimation for the general mixed analysis of variance model, defined in (1) below, involving any number of fixed and random factors and possibly interactions of any order. We do not specify 'equal numbers' or indeed any other experimental balance for our procedure, but we do require that our design matrices satisfy certain conditions of estimability for the parameters. In the case of balanced designs the estimation problem for the constants and variances involved in the linear model has been extensively treated: confining ourselves to just one reference on variance estimation, optimality properties of the classical analysis of variance procedures have already been demonstrated for various balanced designs (e.g. Graybill, 1961). However, results for unbalanced factorial and nested data are much more restricted: Henderson (1953) has suggested a method of unbiased estimation of variance components for the unbalanced two-way classification but his method is computationally cumbersome for a mixed model and when the number of classes is large. Searle & Henderson (1961) have suggested a simpler method also for the unbalanced two-way classification with one fixed factor containing a moderate number of levels and a random factor permitted to have quite a large number of levels. Bush & Anderson (1963) have investigated for the two-way classification random model the relative efficiency of Henderson's (1953) method and two other methods, A and B, based on the respective methods of fitting constants and weighted squares of means described by Yates (1934) for experiments based on a fixed effects model which also provide unbiased estimates of variance components. Possibilities of generalizations are indicated. In all the above methods the estimates of any constants in the model are computed from the 'Aitken Type' weighted least squares estimators based on the exact variance-covariance matrix of the experimental responses which involves the unknown variance ratios. The estimation of the latter is then based on various unbiased procedures so that little is known about any optimality properties of any of the resulting estimators. However, all these methods reduce to the well-known procedures based on minimal sufficient statistics in the special cases of balanced designs. The method of maximum-likelihood estimation here developed differs from the above in that maximum-likelihood equations are used and solved for both the estimates of constants
642 citations
TL;DR: In this paper, a generalization of an earlier attempt by the author to obtain estimators of heteroscedastic variances in a regression model is presented, which is quite general, applicable to all experimental situations, and the computations are simple.
598 citations
TL;DR: In this article, a linear model in the form, where is an unknown parameter and ξ is a hypothetical random variable with a given dispersion structure but containing unknown parameters called variance and covariance components.
Abstract: We write a linear model in the form , where is an unknown parameter and ξ is a hypothetical random variable with a given dispersion structure but containing unknown parameters called variance and covariance components. A new method of estimation called MINQUE (Minimum Norm Quadratic Unbiased Estimation) developed in a previous article [5] is extended for the estimation of variance and covariance components.
374 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 3 |
| 2018 | 1 |
| 2017 | 3 |
| 2016 | 6 |
| 2015 | 2 |
| 2014 | 3 |