  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.
