INVERTED EXPONENTIAL DISTRIBUTION AS A LIFE DISTRIBUTION MODEL FROM A BAYESIAN VIEWPOINT

The Inverted Exponential Distribution is studied as a prospective life distribution. In this paper, we derive Bayes’ estimators for the parameter θ of inverted exponential distribution. These estimators are obtained on the basis of squared error and LINEX loss functions. Comparisons in terms of risks with the estimate of θ under squared error loss and LINEX loss functions have been made. Finally, numerical study is given to illustrate the results.


INTRODUCTION
In reliability studies commonly used models in life testing include the gamma, lognormal and inverse Gaussian distributions. These models are usually chosen on the basis of what is understood about the failure mechanisms. If the failures are mainly due to aging or the wearing out process, then it is reasonable in many applications to choose one of the above mentioned distributions (see Chhikara & Folks, 1977;Sinha & Kale, 1980;Von Alven (ed.), 1964;Sherif & Smith, 1980). In this paper, we consider the inverted exponential distribution as life distribution (see Lin., Duran, & Lewis, 1989). The probability density function (pdf) of the inverted exponential distribution with parameter θ is (1) = 0, otherwise, which has no finite moments. The reliability function, i.e., the probability of no failure before time 't' is R(t) = 1 -F(t) = θ t e 1 1 − − where F(t) is the distribution function of X.
The failure rate of an inverted exponential distribution with parameter θ is In the estimation of reliability function, use of symmetric loss function may be inappropriate as has been recognized by Canfield (1970) and Varian (1975). Zellner (1986) proposed an asymmetric loss function known as the Linex loss function which has been found to be appropriate in the situation where overestimation is more serious than underestimation or vice-versa.
The sign and magnitude of 'a' represent, respectively, the direction and degree of asymmetry. A positive value of 'a' is used when overestimation is more costly than underestimation; while a negative value of 'a' is used in the reverse situation. For 'a' close to zero, this loss function is approximately squared error loss and therefore almost symmetric. Several authors (Basu & Ebrahimi, 1991;Rojo, 1987;Soliman, 2000;Zellner, 1986) have used this loss function in various estimation and prediction problems.
Here, we consider the non-informative prior: The plan of the article is as follows: In section 2, we obtain Bayes estimator ofθ . The estimates are based on the squared error loss function and Linex loss function L(Δ 1 ) where Δ 1 = θ θ − . By using g(θ) as the prior distribution, the risk of estimates have been obtained. Comparison in terms of risk with the estimates of θ under squared error loss and Linex loss functions have been made. Also, we give a numerical example to compare our results.

BAYES' ESTIMATE OFθ
In this section we are concerned with the estimation of the unknown parameter θ of the inverted exponential ; i = 1, 2, . . . , n, are independent and identically distributed exponential random variables with parameter θ. It is also known that a sum of independent exponentially distributed random variables gives a Gamma distributed variable where the probability density function of S is

Bayes' Estimator of θ based on squared error loss function
Combining the prior distribution g(θ ),with the likelihood function L(x|θ ) using Bayes' theorem, the posterior

Bayes' Estimator of θ based on LINEX Loss Function
Under the Linex loss function (1.3), the posterior expectation of the loss function L( 1 The value of θˆ that minimizes the posterior expectation of the loss function L( 1 provided that all expectation exists and are finite. Using (8) and (13), we get the optimal estimate of θ relative to ( In the same manner, we get   (1) with θ=1. The results are presented in Table 1 to Table 6.

CONCLUSION
It is evident from the above Table 1 to