On the Optimality of Several Algorithms on Polynomial Regression of Empicial Bayes Poisson Model

Abstract

The empirical Bayes estimator for the Poisson mixture model in [1], [2] has been an important problem studied for the past 70 years. In this thesis, we investigate extensions of this problem to estimating polynomial functions of the Poisson parameter rather than just the parameter itself. We generalize three different algorithms for estimation, specifically the Robbins estimator from [2], the NPMLE method from [3], and the ERM method from [4]. For each of these algorithms, we prove upper bounds on the minimax regret. We also prove a general lower bound that applies to any estimation algorithm for this setup. In addition to the theoretical bounds, we empirically simulate the performance of all three algorithms in relation to both the number of sample and the degree of the polynomial function we estimate.