Bayes Estimation of Parameter of Exponential Distribution under a Bounded Loss Function

The aim of this study is to study the Bayes estimation of parameter of exponential distribution under a bounded loss function, named reflected gamma loss function, which proposed by Towhidi and Behboodian (1999). The inverse Gamma prior distribution is used as the prior distribution of the parameter of exponential distribution. Bayesian estimators are obtained under squared error loss and the reflected gamma loss functions. Minimum risk equivariant estimator of the parameter is also derived. Finally, a numerical simulation is used to compare the estimators obtained.


INTRODUCTION
Exponential distribution plays an important role in lifetime data analysis.Many authors have developed inference procedures for exponential model.For example, Kulldorff (1961) devoted a large part of book to the estimation of the parameters of the exponential distribution based on completely or partially grouped data.Sarhan (2003) obtained the empirical Bayes estimators of exponential model.Janeen (2004) discussed the empirical Bayes estimators of the parameter of parameter of exponential distribution based on record values.To see more details, one can see Balakrishnan et al. (2005) and Al-Hemyari (2009) and references therein.
Suppose that X is a variable drawn from a exponential distribution with the Probability Density Function (PDF): where, 0 > θ is the scale parameter.For a Bayesian analysis, loss function plays an important role in it and the squared error loss and LINEX loss functions used by most researchers.These function are unbounded and widely employed in decision theory due to its elegant mathematical properties, not its applicability to the representation of a true loss structure (Leon and Wu, 1992).Various examples illustrate that in many situations, unbounded loss can be unduly restrictive and suggest that instead we should consider the properties of estimators based on a bounded loss function.A bounded loss avoids the potential explosion of the expected loss.Moreover, the nature of many decision problems and practical arguments require the use of bounded loss functions, especially in financial problems.For more details see Berger (1985).To overcome the shortcoming of unbounded loss, several bounded loss functions are proposed by many authors, for example, Spring (1993) proposed a bounded loss function named reflected normal loss.Towhidi and Behboodian (1999) proposed reflected gamma loss which is also a bounded loss function.Wen and Levy (2001) proposed a bounded asymmetric loss function called BLINEX loss function.
To see more about the discussion of bounded loss function, one can reference Bartholomew and Spiring (2002) and Kamińska (2010).This study will discuss the Bayes estimation of the parameter of exponential distribution under the following reflected gamma loss (Towhidi and Behboodian, 1999): This can be approximated by ) , ( , for small values of q, which is multiple of the entropy loss. Under the reflected gamma loss function, Meghnatisi and Nematollahi (2009) studied the admissibility and inadmissibility of usual and mixed estimators of two ordered Gamma scale parameters.In this study, we will discuss the admissibility and inadmissibility of parameter of exponential distribution under various conditions.

MAXIMUM LIKELIHOOD ESTIMATION
The natural logarithm of likelihood function is given by: Upon differentiating (4) with respect to θ and equating each results to zero.The MLE of θ is given by Then we can show that the reflected gamma loss (2) and the decision problem are invariant and G and the class of all scale-invariant estimator of θ is of the form δ(X) = δ 0 (X)/W(Z) , where δ 0 is any scale invariant estimator, The proof of this conclusion can be founded in Lehmann (1983).
Moreover the best scale-invariant estimator, which also called Minimum Risk Equivariant (MRE) estimator δ * is given by δ * (X) = δ 0 (X)/w * (Z), where, w * (Z) is a function of Z which maximize: , with Probability Density Function (PDF): where, α and β are known parameters and here α and β are independent of X.
Then the MRE estimator of θ under reflected gamma loss (2) is given by: g w is also be written as: q q x w x q q q q x g w e x e dx w e q w q w w c q q w .Then we can easily show that w* = α/β maximizes the function g(w).Hence, the Lemma 1 is proved.
Remark 1: Let X 1 , X 2 , …, X n be a random sample from exponential distribution (1), then under the reflected gamma loss (2), the estimator equivariant estimator which has Г (n, 1) distribution when θ = 1 and it follows from Basu's theorem that

BAYES ESTIMATION
In this section, we consider the Bayes estimation of the scale parameter θ in exponential distribution (1), in which the complete sufficient statistics and the Bayes estimate of θ is a function ) ( X δ which maximize the function: x e q x q n n q or the function: Then the Bayes estimator of θ, denoted by B δ ˆ is given by the solution of the equation 0 Hence, we can show that B δ ˆ is given by: 0 ( ) ˆB X n Remark 2: For the non-informative prior 1 ) ( and we obtain the generalized Bayes estimator:

EMPIRICAL BAYES ESTIMATION
In the former discussion, the Bayes estimator in ( 7) is seen to depend on the parameter β.When the prior parameter β is unknown, we may use the empirical Bayes approach to get its estimate.From (3) and (5), we calculate the marginal PDF of X, with density: Based on m(x|β), we obtain an estimator, β of β.
The MLE of β is Now, by substituting β for β in the Bayes estimator (8), we obtain the empirical Bayes estimator as:

NUMERICAL SIMULATION
To compare the different estimators of the parameter θ of the exponential distribution, the risks under squared error loss of the estimates are considered.These estimators are obtained by maximum likelihood and Bayes methods under reflected gamma loss function: (i) Based on the given value θ = 1.0, a sample of size n is then generated from the density of the exponential distribution (1), which is considered to be the informative sample.(ii) The MLE and Bayes estimators are calculated based on Section 2 and q =1.0.(iii) Steps (i) to (ii) are repeated N = 2000 times and the risks under squared error loss of the estimates are computed by using: where, ˆi θ is the estimate at the th i run.