The Poisson distribution and mean dual distributions have the same canonical parameter q. We wonder the following question: which one of the used distributions allows to better estimate this parameter. By doing so, in this paper, we will examine the different estimators mean squared error of this parameter issued from each of these distributions to compare their performances. The mean squared errors of different estimators of the canonical parameter are functions of q, we will proceed by a graphic resolution for comparison through some illustrative examples. Quality criteria estimators of q through the amount of Fisher information of each random variable relative to the notion of mean duality between one variable overdispersed and underdispersed are proposed.

exponential family, Poisson distribution, weighted Poisson distribution, canonical parameter, Fisher information, overdispersion, underdispersion, mean duality, quality of an estimator, mean squared error.