A note on some translation-parameter families of densities for which the median is an M.L.E.

Abstract

The following theorem is proved in the forthcoming book by Kagan, Linnik, and Rao: If the median is an m.l.e. of θ for a sample of the size n=4 from a population with density f(x-θ ) and f(x) is lower semi-continuous at the origin then f(x)=ke^-α|x| , -∞ <x<∞ , α>0. The object of this note is to show that the theorem is false if we require the median is an m.l.e. of θ for samples of size n=2 and 3.