THE EFFICIENCY OF A NONPARAMETRIC SELECTION PROCEDURE: LARGEST LOCATION PARAMETER CASE.
CORNELL UNIV ITHACA N Y DEPT OF INDUSTRIAL ENGINEERING AND OPERATIONS RESEARCH
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For the general problem of selecting that one of k populations pi sub 1,...,pi sub k which has the highest probability of producing the largest observation, Bechhofer and Sobel have suggested a procedure which is nonparametric in the sense that the cumulative distribution function c.d.f.F sub i. of observations from pi sub i may be different for each ii1,...,k and unknown. If the c.d.f. of observations from pi sub i is known to have the same form which may be unknown and differ only in location for each i, the c.d.f. of observations from pi sub i may be written as F sub ixFx-nu sub i i1,...,k. Then the above-posed general problem reduces to the problem of selecting that one of k populations which is associated with the largest nu sub i. If the form of the common F. is completely unknown, a nonparametric procedure such as the one suggested by Bechhofer and Sobel is one possible recourse. However, even in this circumstance, it is of interest to know how the procedure performs under various possible parametric alternatives. In this paper we determine under specific parametric alternatives how much one pays in terms of increased sample sizes for the nonparametric procedures certainty of guaranteeing a reasonable requirement on the performance characteristic function, in the largest location parameter case. It turns out that the nonparametric procedure has a low efficiency relative to the specific parametric alternatives considered and will therefore be useful against these alternatives and, presumably, many others also only when real doubt exists as to the form of the actual distribution. Author
- Statistics and Probability