Abstract: Firstly, based on the form of the balanced loss function, we gave a  weighted balanced entropy loss function and applied it to the Poisson distribution, and  the Bayesian estimation of the parameter under the  loss function was obtained. Secondly, under the condition that the prior distribution was Gamma distribution, we gave the explicit expression of the estimator, proved  the consistency of the estimator, and tested the asymptotic normality of the estimator by using the QQ plot method.

Key words: weighted balanced entropy loss function, Bayesian estimation, stochastic simulation, consistency, asymptotic normality