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Mean Square Error Comparison for MINQUE, ANOVA and Two Alternative Estimators Under the Unbalanced One‐Way Random Model
Author(s) -
Ahrens H.,
Kleffe J.,
Tenzler R.
Publication year - 1981
Publication title -
biometrical journal
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.108
H-Index - 63
eISSN - 1521-4036
pISSN - 0323-3847
DOI - 10.1002/bimj.4710230402
Subject(s) - mathematics , estimator , equating , statistics , variance (accounting) , dimension (graph theory) , simple (philosophy) , point (geometry) , econometrics , mathematical economics , algorithm , philosophy , business , geometry , accounting , epistemology , pure mathematics , rasch model
Standard methods for estimating variance components, such as HENDERSON's methods are based on equating well‐known sums of squares to expectation. These methods are computationally simple and therefore widely used. From a theoretical point of view we are not that happy about these methods. There is lack of theoretical background, there is no unique description of these methods and finally, we know very little about optimal properties of HENDERSON's estimates unless the model is totally balanced. It was mainly this trouble which caused statisticians to develop an extensive theory on optimal estimation of variance components and among the mostly known contributors to this line of research was C. E. RAO. We believe that his MINQUE principle gave a new idea to statistical theory. It provides some kind of optimality and does not refer to the normal assumption which is in difference to the majority of alternative proposals made in literature. And even if one thinks of its theoretical background as being not much better than that of HENDERSON's method one has to accept the advantage of having a unique estimation procedure. A difficulty is its calculation. The computational advice given by RAO requires inversion of matrices as high in dimension as the total number of observations is. This is, we believe, the main reason for its neglect by most of the people working on such kind of data. But among all estimates which perform any kind of optimality MINQUE seems still to be among the simplest to calculate. That is why we think, whenever optimal estimates of variance components can be made attractive to users, we should start with MINQUE or related estimates and it would be a good deal to work on more effective ways of its computation. One way is to present explicit and easy handable formulae under several standard ANOVA models of which the simplest one is going to be studied in this paper. Analog results for more complicated cases are given in KLEFFE (1980). Having solved this major problem we may proceed by comparing new and traditional methods to find out about what estimate is to be recommended.