After some introductory remarks, we discuss certain basic considerations such as the nonoptimality of the classical symmetric (balanced) designs for hypothesis testing, the optimality of designs invariant under an appropriate group of transformations, etc. In section 3 we discuss complete classes of designs, while in section 4 we consider methods for verifying that designs satisfy certain specific optimality criteria, or for computing designs which satisfy such criteria. Some of the results are new, while part of the paper reviews pertinent results of the author and others.

Optimum Experimental Designs

Many commonly employed symmetrical designs such as Balanced Incomplete Block Designs (BIBD's), Latin Squares (LS's), Youden Squares (YS's), etc., are shown to have optimum properties among the class of non-randomized designs (Section 3). This represents an extension of a property first proved by Wald for LS's in [1]; a similar property demonstrated by Ehrenfeld for LS's in [2] (as well as a third optimum property considered here) is shown to be an immediate consequence of the Wald property, and the Wald property is shown to be the more relevant when one considers optimality rigorously (Section 2). Surprisingly, all of these optimum properties fail to hold if randomized designs are considered (Section 4); the results of Sections 2 and 3, as well as those appearing previously in the literature (as in [1], [2], [3]) must be interpreted in this sense. Generalizations of the BIBD's and YS's, for which analogous results hold, are introduced.

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On the nonrandomized optimality and randomized nonoptimality of symmetrical designs

Let the distribution of some random variables depend on real parameters θ1, • • •, θs and consider the hypothesis H: θi ≦ θi*, i = 1, • • •,s. It is shown under certain regularity assumptions that unbiased tests of H do not exist. Tests of minimum bias and other types of minimax tests are derived under suitable monotonicity conditions. Certain related multidecision problems are discussed and two-sided hypotheses are considered very briefly.

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Testing Multiparameter Hypotheses

(1997). Locally optimal one-sided tests for multiparameter hypotheses. Econometric Reviews: Vol. 16, No. 2, pp. 131-156.

Locally optimal one-sided tests for multiparameter hypotheses

A generalization of a locally most powerful unbiased (LMPU) test, for the single parameter case, to the k-parameter case is proposed. In particular, we define a locally most mean power unbiased (LMMPU) test based on the mean curvature of the power hypersurface. Compared with the type C tests (Neyman and Pearson 1938) and the type D tests, LMMPU tests possess better theoretical properties and enjoy ease of construction of critical regions in practical situations. LMMPU tests are obtained for the important practical case (Perng and Littel 1976) of a two-parameter univariate normal population, for which Isaacson (1951, p. 233) was unable to find a type D test, and for the case of means of a multivariate normal population with independent coordinates.

Locally optimal tests for multiparameter hypotheses

Unbiased critical regions of type D for testing simple hypotheses specifying the values of several parameters are defined and their properties studied. These regions constitute a natural generalization of the Neyman-Pearson regions of type A for testing simple hypotheses specifying the value of one parameter. A theorem is obtained which plays the role of the Neyman-Pearson fundamental lemma in the type A case. Illustrative examples of type D regions are given.

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Stanley L. Isaacson

Papers

On the Theory of Unbiased Tests of Simple Statistical Hypotheses Specifying the Values of Two or More Parameters