Estimation in Linear Models Based on Observations With Unknown and Possibly Unequal Scaling
We consider k groups of observations X11, … , X1n1, … , Xk1, … , Xknk and unknown scalars λ1, … , λk, and we assume that the distribution of the scaled observations X11/λ1, … , X1n1/λ1, … , Xk1/λk, … , Xknk/λk follows a normal linear model on Rn1+…+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.