Empirical E-Bayesian estimation for the parameter of Poisson distribution

Abstract: This paper introduces a new method of estimation, empirical E-Bayesian estimation. In this method, we consider the hyperparameters of E-Bayesian estimation are unknown. We compute the E-Bayesian and empirical E-Bayesian estimates for the parameter of Poisson distribution based on a complete sample. For our purpose, we consider the case of the squared error loss function. The E-posterior risk and empirical E-posterior risk are computed. A comparison between E-Bayesian and empirical EBayesian methods with the corresponding maximum likelihood estimation is made using the Monte Carlo simulation. A relevant application is utilized to illustrate the applicability of multiple estimators.


Introduction
The Poisson distribution is so important among the discrete distributions. Poisson distribution represents the probability of prescribed number of events arising at a fixed time or space interval as these events happen at a predicted constant mean rate and regardless of time after the last one. It's an occurrence. Also the Poisson distribution in other indicated intervals like time, area or volume can be applicable for series of events. Probability mass function (PMF) of Poisson distribution is given by µ t e −µ t! , t = 0, 1, 2, ..., µ > 0. (1.1)

E-Bayesian estimation
According to E-Bayesian method, prior parameters θ and λ ought to be chosen to ensure that π(µ) is a lower bound of µ. The differential of π(µ) is given as Then the prior distribution π(µ) is decreasing function for λ > 0, 0 < θ < 1, for more details see Han [4]. By considering θ and λ are independent parameters with bivariate density function g(θ, λ) = g 1 (θ)g 2 (λ). (3.2) Therefore, the E-Bayesian estimator of the parameter µ is given bŷ In this section, E-Bayesian approximation of µ parameter is based on different distributions of θ and λ. The effect of numerous previous distributions on E-Bayesian estimates of µ is explored by these distributions. The distributions of the parameters θ and λ are considered below: For g 1 (θ, λ), g 2 (θ, λ) and g 3 (θ, λ), the E-Bayesian estimators of the parameter µ are given from (2.5), (3.3) and (3.4), respectively, byμ ). (3.7)

Empirical E-Bayesian estimation
In this section, we introduce empirical E-Bayesian method (EE-Bayesian). In this method, we consider the case when the parameters a, b and c in (3.4) which are used in E-Bayesian estimation are unknown parameters, we use empirical Bayes approach to get their estimates. The marginal PMF of the random variable T is obtained from (1.1) and (2.2) as Specifically, since θ is a positive integer, marginal distribution of random variable T is a negative binomial distribution, NB(r, p), with parameters The estimators of the parameters θ and λ are obtained by using the moment method. To obtain the moment estimators of θ and λ, we need to calculate the first two moments of T , E(T ) and E(T 2 ). It is easy to show that Furthermore, the moment estimators can be obtained by equal the population moments with the sample moments as From (5.3) and (5.4), we obtain the moment estimators of θ and λ, respectively, bŷ Also, to obtain moment estimators of parameters a and b in (3.4), we need to calculate E(θ) and E(θ 2 ), which is easy to show that .
Then, from (5.8) and applying moment method, we attain moment estimators of parameters a and b, respectively,â where whereθ m is given by (5.5).

Simulation in Monte Carlo
In this part, a Monte Carlo modeling is applied for an examination of E-Bayes and EE-Bayesian assessment with the respective projections estimating maximum likelihood.

E-Bayesian estimation
Here, Monte Carlo simulation is being established to equate the maximum likelihood and E-Bayesian model specification. It calls the following steps: (i) We generate θ and λ for specified values of previous parameters (a, b) and (0, c) via beta and uniform priors (3.4), individually.
(iii) For known values of µ in step (2), we generate complete sample of Poisson distribution with PMF (1.1).
(vi) The output of all results has been analyzed mathematically in three terms with i = 1, 2, 3. First is taken as the average of the estimatesμ ML andμ EBi , the second is the estimated risks (ERs) of µ ML andμ EBi , and the third is E-posterior risks E − R(μ EBi ), i = 1, 2, 3. Repetition 10000 times of above steps, then simulation tests are appeared in Tables 1-3.   (3,4) 0.00392 0.00397 0.00387

EE-Bayesian estimation
In order to compare maximum likelihood and approximation of EE-Bayesian design, a Monto Carlo technique is enforced. The subsequent measures are noticed.
(i) We construct θ and λ for the desired value of obtained parameters (a, b) and (0, c) by beta and (3.4), independently.
(ii) Develop µ from gamma calculated density (2.2) for expected values of θ and λ.
(iii) For recognized values of µ in step (2), we generate complete sample of Poisson distribution with PMF (1.1).
(vi) The efficiency of all findings was examined statistically in three terms. The first is the average of the estimatesμ ML andμ EEBi , the second is the estimated risks (ERs) ofμ ML ,μ EEBi , and the third is empirical E-posterior risks EE − R(μ EEBi ), for all i = 1, 2, 3. Simulation runs by repeating 10000 times of above steps are seen in Tables 4-6.

Real data analysis
Information on the quantity of deaths caused by horse kicks in light of the perception of 10 Prussian cavalry corps for a very long time (proportionally, 200 corps-years) in Figure 1 are issued. Prussian authorities gathered this data during the last year of the nineteenth century to consider the perils that horses presented to fighters, (see Bortkiewicz [32]). Padilla [33] discussed the validity of the model for the given real data set and showed that Poisson distribution fits quite well to it.  Figure 1. Bortkiwewicz data on deaths in Prussian cavalry.

Number of cavalry units with the death rate
In the present circumstance, possibilities of death because of a kick from a horse are little while. Although the number of troops exposed to this danger is very high. Hence, a Poisson distribution can well match the results. Then, it is possible to approximate the mean amount of deaths per period aŝ In this section we consider E-Bayesian and EE-Bayesian estimation for parameter µ.

E-Bayesian
For the real data that is considered in Figure 1, we obtain E-Bayesian estimates and E-posterior risk for µ. The computational results are recorded in Table 7.

EE-Bayesian
For the real data that is considered in Figure 1, we obtain EE-Bayesian estimates and EE-posterior risk for the parameter µ. The numerical results are listed in Table 8.

Conclusions
This paper describes E-Bayesian and EE-Bayesian methodology which are considered as to find approximation of unknown parameter of Poisson distribution with complete sample. E-Bayesian and EE-Bayesian estimators are implemented under the function of squared error loss and three hyper-parameter distributions. Also, E-posterior and EE-posterior risks are computed. Some statistical properties of E-Bayesian and E-posterior risk estimates relative to squared error loss function are extracted. Via a simulation analysis, a survey of different estimated parameters is carried out. The simulation results indicate that E-Bayesian and EE-Bayesian methods do very well in the estimation of model parameters under minimal mean square defects. Finally, to follow the E-Bayes and EE-Bayes estimator, one individual data set is scrutinized. It is inferred from the mathematical examination that E-Bayesian and EE-Bayesian estimators operate effectively than classical estimators.