Subset selection from generalized logistic populations

We give an introduction to the logistic and generalized logistic distributions. These generalized logistic distributions Type-I, Type-II and Type-III are indexed by a real valued parameter. They have been derived as mixtures with the standard logistic distribution and for discrete values of the parameter they describe the distribution of the minimum, maximum, and median, respectively, of a i.i.d. sample from a logistic distribution. We obtain exact results for the probability of correct subset selection from Type-I, Type-II and Type-III generalized logistic populations which only differ in their location parameter. In the course of establishing these exact results, we derive an explicit expression for the cdf of the median based on a sample from the logistic distribution of size 2b - 1; it is a sum of b terms involving binomial coefficients. By the CLT we can use this explicit cdf to approximate the unknown cdf of the normal distribution. We show some numerical results which show that for b = 5 the approximation error is of the order 0.001 in the middle and 0.0001 in the tails. This implies that for practical purposes the Type-III family with b = 5 is indistinguishable from the normal family.