Estimation in Linear Models Based on Observations With Unknown and Possibly Unequal Scaling

Abstract

We consider k groups of observations X 11 , … , X 1 n 1 , … , X k 1 , … , X knk and unknown scalars λ 1 , … , λ k , and we assume that the distribution of the scaled observations X 11 /λ 1 , … , X 1 n 1 /λ 1 , … , X k 1 /λ k , … , X knk /λ k follows a normal linear model on R n 1+…+ nk . This general setup includes several interesting models that have appeared in the literature in different contexts and fields of application. The simplest example is the model of equal coefficients of variation in k normal samples. We give, in the general setup, a necessary and sufficient condition in terms of “degrees of freedom” for when the maximum likelihood estimator (MLE) exists and is unique with probability 1, and furthermore, we give an algorithm for obtaining the MLE.