Is the Selected Population the Best?—Location and Scale Parameter Cases

Abstract

We observe X ₁,…,X _k , where X _i has density f(x,θ _i ) possessing monotone likelihood ratio. The best population corresponds to the largest θ _i . We select the population corresponding to the largest X _i . The goal is to attach the best possible p-value to the inference: the selected population has the uniquely largest θ _i . Gutmann and Maymin (1987) considered the location parameter case and derived the supremum of the error probability by conditioning on S, the index of the largest X _i . Using this conditioning approach, Kannan and Panchapakesan (2009) considered the problem for the gamma family. We consider here a unified approach to both the location and scale parameter cases, and obtain the supremum of the error probability without using conditioning.

Keywords:

• Monotone likelihood ratio
• Probability of error
• Selection

Mathematics Subject Classification:

• Primary 62F03
• Secondary 62F07

Acknowledgment

The authors are grateful to the referees and the Associate Editor for comments which greatly helped in the revision of the manuscript.