Abstract Let X, Xi, Xij, j = 1, …, n, j = 1, …, ni be independent Poisson random variables with parameters λ, λ i , λ ij . The maximum likelihood estimators for the parameters subject to (1) and subject to (1) and (2) are obtained. The bias and MSE of these restricted maximum likelihood estimators (RMLE's) are approximated by analytic and Monte Carlo methods. In every case a substantial savings in MSE is obtained at the expense of a small loss in bias when using RMLE's instead of unrestricted MLE's. The restrictions could be generalized to other linear restrictions.