Poisson gamma model in empirical Bayes of small area estimation (SAE)

This study aims to describe the empirical Bayes estimator for small area estimation which use Poisson Gamma as prior distribution. The method then apply to generated data and model it with and without indicator variable. Parameter model estimated uses direct method and indirect method (known as empirical Bayes approach; with and without indicator variable). The choice of better estimator is based on MSE with Jackknife method. The criteria of acceptable proposed model are based on Deviance, Scaled Deviance, Pearson ChiSquare and Scaled Pearson Chi-square. This study proves that empirical Bayes in SAE with indicator variable result better estimated values than two other methods. All criteria of acceptable model indicate that proposed model could be accepted. Based on plot between the estimated residual versus predicted values of response variable informed that proposed model is plausible enough.


Introduction
Small area typically refers to a small geographic area or a demographic group for which very little information is obtained from the sample surveys. An empirical Bayes (EB) method uses sample survey data in conjunction with relevant supplementary data which are obtained from various administrative sources. The method has been found to be very useful in many applications of small-area estimation and related problems (Butar Butar & Lahiri, 2003; Sundara, Sadik, & Kurnia, 2017).
Direct survey estimators in the area-specifics small sample size, tend to yield unacceptably large standard errors (Jakimauskas & Sakalauskas, 2010). Therefore, it has to borrow the strength from related areas to improve the power of small sample size hope it could provide higher precision of estimates. Such of this estimation method is known as indirect estimators. Those indirect estimators are often based on mixed models and associated empirical Bayes estimators in which random effects represents areaspecific effects.
Empirical Bayes estimator for small area places a prior distribution on population-specific area. Many distributions of priors have been used for this purpose, such as log normal, gamma and nonparametric priors (Jeong, 2010;John Quigley, Bedford, & Walls, 2007;Martuzzj & Elliott, 1996). In this study, we describe the Poisson gamma distribution for priors (Sakalauskas, 2010) and then applied it to generated data.

Small Area Estimation
Small area estimation are classified into two model, these are basic area level model and basic area level unit. In this paper, we limit the discussion in basic area level model only. A basic area level model assumes that some function = ( ) of the small area mean is related to the data from specific area = ( 1 , 2 , ⋯ , ) through a linear model with random effect (Rao, 2003): Which is known positive constant and = ( 1 , 2 , ⋯ , ) is a p x 1 vector of known regression parameters. Meanwhile is random effects assummed as normal distribution or written as ~ (0, 2 ), m is the number of small areas. In this study, we take (. ) as the identity function. In the basic area level model, it is assummed that the direct estimate ̂ is usually design as unbiased estimator, or (̂) = . It is usual to assume : ̂= + , = 1, … , .
(2) Where is independent normal random variables which denotes the sampling errors associated with the transformed direct estimator ̂. It is assumed ~(0, ) If we combine the equation (1) and (2), we will get this following equation: This equation is known as Fay-Herriot model, special case of generalized linear mixed model in small area estimation that consist of fixed effect, that is and random effect, that is . Fay-Herriot used this basic area level model to estimate income per capita at any small areas in United State with population less that 1000. In this paper, we are concerned with the basic area-level model of Fay and Herriot (Rao, 2003).
Empirical Bayesian method of small area estimation places a prior distribution on area-specific risks (Clement, 2014). According to properties of this model the distribution of the number of events in populations follows to Poisson law, that intensity parameter is distributed according to Gamma distribution and the rates are estimated as the a posteriori means. Thus in this paper we address the empirical Bayesian estimation techniques for Poisson-Gamma model as well.

Empirical Bayes Method in Small Area Estimation
Wakefield (2006) introduced Poisson-Gamma model two stage to estimate parameter in rare data. In first stage is assumed that random variable has Poisson distribution or written as ~ ( ) with probability density function is : where = ( , ) is regression model, = ( 1 , 2 , ⋯ , ) is a vector of covariates and = ( 1 , 2 , ⋯ , ) is regression coefficients.
In the second stage, it is assumed the parameter has Gamma distribution or ~ ( , ) with probability density function is : Based on equation (1) and (2) is obtained the joint probability density function as follows: We could also get the marginal probability density function as follows:  3 We could identify that the distribution of equation (4) is negative binomial with mean and variance for respectively are as follows: Then, we will estimate the posterior distribution for : Based on equation (9)  The indicator to test the acceptable and the goodness of fit for the better proposed model is identified by calculating the value of Mean square error (MSE). (Rao, 2003) stated that better estimator tend to have smaller value of MSE. (Ghosh, Maiti, & Roy, 2008) have proven that Bayes method is asymtotically unbiased estimator and empirical Bayes as well. The Jackknife method then be used to obtain this asymtotic unbiased estimator of MSE for ̂. Following is the formula for Jackknife estimator : with ̂= ( , , ̂) and ̂, −1 = ( ,̂− 1 ,̂− 1 ).

Application of Empirical Bayes in SAE Using Generated Data.
The theoritical framwork of empirical Bayes for SAE as presented above then applied to a set of generated data. Generate a set data of and represent response variable and indicator variable corresponding with normal distribution for both variables. Other variables such as , and direct estimator for ̂ are calculated based on the fixed formula. This following Table 1  From this study we can conclude that indirect estimator using empirical Bayes in small area estimation is better estimator than direct estimator. We also obtained than empirical Bayes in SAE with indicator variable tends to result better estimator than without indicator variable, since indicator variable could increase the goodness of fit of proposed model. Table 3  In this study we also check the estimated residual versus predicted values of response variable to assess the plausibility of the proposed model. Figure 1 presents the plot between these two variables (Yanuar, Ibrahim, & Jemain, 2013). This figure informs that no trends are detected. We could also conclude here that the estimated model which is obtained based on empirical Bayes analysis could be considere adequate and could be accepted.

Discussions
This study described the empirical Bayes estimator for small area estimator which used Poisson Gamma as prior distribution. The method then applied to generated data and model it with and without indicator variable. Parameter model estimated used direct method and indirect method (known as empirical Bayes approach). The choice of better estimator based on MSE with Jackknife method. The criteria of acceptable proposed model were based on Deviance, Scaled Deviance, Pearson ChiSquare and Scaled Pearson Chi-square. Meanwhile to assess the plausibility of the proposed model was examined by plotting the estimated residual versus predicted values of response variable.