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**Admissible estimators of variance components obtained via submodels.**
*(English)*
Zbl 0760.62066

We present a new method of constructing admissible estimators of variance components that is applicable to many important applications for unbalanced models. Its novelty lies in the idea of using known admissible estimators under some simple models to construct admissible estimators under some other, more complex models. Although it does not give all admissible estimators, it allows assessment of admissibility of a large class of estimators, in particular of some unbiased estimators based on the so-called cell means statistics. An important feature of the method is that it allows construction of admissible (biased) nonnegative estimators when a nonnegative estimator is available in the specified submodel. Another attractive feature of the method is its adaptability to computer algorithms.

The paper is organized as follows. Section 2 recalls definitions and establishes notation used throughout the paper. In Section 3, the definition of a submodel is presented. It also contains the main result of this article, a relationship between an admissible estimator under a specified submodel and the corresponding admissible estimator under the primal model. In Section 4, a necessary and sufficient condition for an admissible estimator to be nonnegative is established as well as some formulae one can apply to construct admissible estimators in some special cases. Section 5 is devoted to estimation with the condition of unbiasedness. It shows that a formula relating admissible unbiased estimators in submodels to corresponding admissible unbiased estimators in the primal models also can be established. In Section 6 admissible estimators are provided for the model with two variance components: for some factorial models with imbalance in the last stage and the unbalanced \((p-1)\)-way nested random factors design. Also given are comments on how to construct nonnegative admissible estimators.

The paper is organized as follows. Section 2 recalls definitions and establishes notation used throughout the paper. In Section 3, the definition of a submodel is presented. It also contains the main result of this article, a relationship between an admissible estimator under a specified submodel and the corresponding admissible estimator under the primal model. In Section 4, a necessary and sufficient condition for an admissible estimator to be nonnegative is established as well as some formulae one can apply to construct admissible estimators in some special cases. Section 5 is devoted to estimation with the condition of unbiasedness. It shows that a formula relating admissible unbiased estimators in submodels to corresponding admissible unbiased estimators in the primal models also can be established. In Section 6 admissible estimators are provided for the model with two variance components: for some factorial models with imbalance in the last stage and the unbalanced \((p-1)\)-way nested random factors design. Also given are comments on how to construct nonnegative admissible estimators.