Abstract
In this paper, Bayes estimators of Poisson distribution have been derived by using two loss functions: the squared error loss function and the proposed exponential loss function in this study, based on different priors classified as the two different informative prior distributions represented by erlang and inverse levy prior distributions and non-informative prior for the shape parameter of Poisson distribution. The maximum likelihood estimator (MLE) of the Poisson distribution has also been derived. A simulation study has been fulfilled to compare the accuracy of the Bayes estimates with the corresponding maximum likelihood estimate (MLE) of the Poisson distribution based on the root mean squared error (RMSE) for different cases of the parameter of the Poisson distribution and different sample sizes.
Highlights
The Poisson distribution is a discrete probability distribution for the counts of events that occur randomly in a given interval of time
Our aim in this study is to examine the effects of the squared error loss function, and the proposed exponential loss function which are presented in this study, based on different priors represented by erlang and levy prior distributions and non-informative prior on Bayes’s estimators of Poisson distribution
For the results listed in table.1and table.2, we see that the best Bayes estimates under the squared error loss function (SELF) according to the smallest value of root mean squared error (RMSE) as compared with other estimates based on the other values of the parameters for the same priors as listed below
Summary
The Poisson distribution is a discrete probability distribution for the counts of events that occur randomly in a given interval of time (or space). Which were represented by maximum likelihood, Markov chain Monte Carlo, and Bayes method He derives the Bayes estimators under the squared error loss function based on gamma prior distribution. He derived posterior risk for empirical E-Bayesian E-Bayesian approximation based on the squared error loss function He investigates the behavior of different estimators for the parameter of Poisson distribution based on Monte Carlo simulation. He applied EE-Bayesian estimates and EE-posterior risk on a real data set. She derived the posterior distribution of the Poisson parameter under the squared error and quadratic loss functions based on gamma prior distribution She used a simulation method to obtain the results, to including the point estimates and confidence intervals and the mean square error (MSE) for the parameter Poisson. The log-likelihood function ln ( L) , the first partial derivative of the log of the likelihood L with respect to θ as follows n n n nθ tiLogθ - i 1 i 1
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