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Optimal allocation of observations for inference on k ordered normal population means. (English) Zbl 0652.62072

The optimal allocation of observations when there is a natural ordering in the k normal population means is discussed. It is shown that the design which minimizes the total mean square error of maximum likelihood estimators in the null case allocates half the observations to each of the two extreme populations. The design is obviously optimal for testing homogeneity of means against the simple ordered alternative. It is, however, hardly acceptable for the estimation in the nonnull case. It is, therefore, shown that the observations could be allocated to the non- extreme populations according to weights which are proportional to the absolute values of the R. P. Abelson and J. W. Tukey [Ann. Math. Stat. 34, 1347-1369 (1963; Zbl 0121.139)] scores at the same time keeping the minimum local power for testing the simple ordered alternative to be maximal.
The design gives also the maximum minimum power, not local, for the alternative in the class of linear tests. It, of course, suffers from a small loss of efficiency for the estimation under the null case but is much better under the nonnull case than the extreme design which allocates half the observations to each of the two extreme populations. Some numerical comparisons of the mean square errors are given.

MSC:

62K05 Optimal statistical designs
62F03 Parametric hypothesis testing
62H15 Hypothesis testing in multivariate analysis

Citations:

Zbl 0121.139
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