Daniel Berend

We study maximum likelihood estimators (henceforth MLE) in experiments consisting of two stages,where the first-stage sample is unknown to us,but the second-stage samples are known and depend on the first-stage sample. The setup is similar to that in parametric empirical Bayes models,and arises naturally in numerous applications. However,problems arise when the number of second-level observations is not the same for all first-stage observations. As far as we know,this situation has been discussed in very few cases (see Brandel,Empirical Bayes methods for missing data analysis. Technical Report 2004:11,Department of Mathematics,Uppsala University,Sweden,2004 and Carlin and Louis,Bayes and Empirical Bayes Methods for Data Analysis,2nd edn. Chapman & Hall,Boca Raton,2000) and no analytic expression for the indirect maximum likelihood estimator was derived there. The novelty of our paper is that it details and exemplifies this point. Specifically,we study in detail two situations: 1. Both levels correspond to normal distributions; here we are able to find an explicit formula for the MLE and show that it forms uniformly minimum-variance unbiased estimator (henceforth UMVUE). 2. Exponential first-level and Poissonian second-level; here the MLE can usually be expressed only implicitly as a solution of a certain polynomial equation. It seems that the MLE is usually not a UMVUE. In both cases we discuss the intuitive meaning of our estimator,its properties,and show its advantages vis-Ja-vis other natural estimators.