Subset selection for logistic populations using loss functions : some basic probabilities

Given are k (\geq 2) random variables X_1, ..., X_k associated with k populations \pi_i, ..., \pi_k, respectively. We assume that these random variables have logistic distributions differing only in their location parameters and we are interested in selecting a subset of them which contains the best population, that is the population with the largest value of the location parameter. We use the following selection rule include \pi_i in the subset \[\Leftrightarrow X_i \geq \max_{1 \leq j \leq k} X_j - c\], where c is a positive constant. Results for Gupta's subset selection procedure for this logistic case can be found in van der Laan (1992). In the present paper we use a loss-function approach similar to the one used by van der Laan and van Eeden (1993, 1996) and obtain the expected loss for the case where k = 3.