# On a selection and ranking procedure for gamma populations

## Summary

The problem of selecting a subset of k gamma populations which includes the “best” population, i.e. the one with the largest value of the scale parameter, is studied as a multiple decision problem. The shape parameters of the gamma distributions are assumed to be known and equal for all the k populations. Based on a common number of observations from each population, a procedure R is defined which selects a subset which is never empty, small in size and yet large enough to guarantee with preassigned probability that it includes the best population regardless of the true unknown values of the scale parameters θ_{i}. Expression for the probability of a correct selection using R are derived and it is shown that for the case of a common number of observations the infimum of this probability is identical with the probability integral of the ratio of the maximum of k-1 independent gamma chance variables to another independent gamma chance variable, all with the same value of the other parameter. Formulas are obtained for the expected number of populations retained in the selected subset and it is shown that this function attains its maximum when the parameters θ_{i} are equal. Some other properties of the procedure are proved. Tables of constants b which are necessary to carry out the procedure are appended. These constants are reciprocals of the upper percentage points of F_{max}, the largest of several correlated F statistics. The distribution of this statistic is obtained.

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