Multiple Decision Theory: A General Approach

In the theory and practice of statistical inference, multiple decision problems are encountered in many experimental situations. The classical methods for analyzing data do customarily employ hypothesis testing in most situations. In such cases, when the hypothesis is rejected, one wants to know in which of a number of possible ways the actual situation (true state of nature) differs from the one postulated by the null hypothesis. If, in the formulation of the problem, we consider only two decisions (reject or not reject the hypothesis), we will not only neglect to differentiate between certain alternative decisions but may also be using an inappropriate acceptance region for the hypothesis. Moreover, the traditional approach to hypotheses testing problems is not formulated in a way to answer the experimenter’s question, namely, how to identify the “best” (in some sense) population? For example, the method of the least significant differences based on the t-test has been used in the past to detect differences between the true unknown means of different varieties and thereby choosing the population which is the “best”, say, the one with the largest mean. But this method is indirect, less efficient and does not provide an overall probability of a correct decision. The remark (criticism) is also valid, to some extent, for methods based on multiple comparison techniques. The traditional approach does not allow for a decision if the null hypothesis is not rejected.

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