Abstract
Summary In certain tests of hypotheses concerning more than one parameter, it is found that the criterion of unbiasedness cannot be satisfied. In such situations, it is proposed to make the test locally unbiased in a modified sense, termed Type M. Here the power surface in the parametric space does not always lie above the level of significance of the test. However, the mean value of the power, in a small neighbourhood of the null point, exceeds the significance level. This paper demonstrates that the uniformly most powerful unbiased (UMPU) test (based on the sample sum of squares) for a normal variance (Neyman and Pearson, 1936; Rao, 1952; Ramachandran, 1958) is Type M unbiased against the wider class of alternatives permitting the means of the observations to differ, and similarly that the UMPU test (based on the sample variance ratio) for a population variance ratio is Type M unbiased against the same widened class of alternatives.
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