# Multivariate Estimation of Poisson Parameters

## TL;DR

This paper introduces a new loss function for multivariate Poisson parameter estimation, develops minimax estimators that outperform MLE, and explores Bayesian methods, leading to improved estimation and new multivariate models.

## Contribution

It proposes a novel loss function and a class of minimax estimators that dominate MLE, with Bayesian derivations and applications to multivariate Poisson models.

## Key findings

- Minimax estimators outperform MLE under the new loss function.
- Bayesian and empirical Bayesian methods derive the estimators.
- New multivariate models for rate parameters and counts are constructed.

## Abstract

This paper is devoted to the multivariate estimation of a vector of Poisson means. A novel loss function that penalises bad estimates of each of the parameters and the sum (or equivalently the mean) of the parameters is introduced. Under this loss function, a class of minimax estimators that uniformly dominate the maximum likelihood estimator is derived. Crucially, these methods have the property that for estimating a given component parameter, the full data vector is utilised. Estimators in this class can be fine-tuned to limit shrinkage away from the maximum likelihood estimator, thereby avoiding implausible estimates of the sum of the parameters. Further light is shed on this new class of estimators by showing that it can be derived by Bayesian and empirical Bayesian methods. In particular, we exhibit a generalisation of the Clevenson-Zidek estimator, and prove its admissibility. Moreover, a class of prior distributions for which the Bayes estimators uniformly dominate the maximum likelihood estimator under the new loss function is derived. A section is included involving weighted loss functions, notably also leading to a procedure improving uniformly on the maximum likelihood method in an infinite-dimensional setup. Importantly, some of our methods lead to constructions of new multivariate models for both rate parameters and count observations. Finally, estimators that shrink the usual estimators towards a data based point in the parameter space are derived and compared.