In this paper, two parameters for the Exponential distribution were estimated using the
Bayesian estimation method under three different loss functions: the Squared error loss function,
the Precautionary loss function, and the Entropy loss function. The Exponential distribution prior
and Gamma distribution have been assumed as the priors of the scale γ and location δ parameters
respectively. In Bayesian estimation, Maximum likelihood estimators have been used as the initial
estimators, and the Tierney-Kadane approximation has been used effectively. Based on the MonteCarlo
simulation method, those estimators were compared depending on the mean squared errors (MSEs).The results showed that the Bayesian estimation under the Entropy loss function,
assuming Exponential distribution and Gamma distribution priors for the scale and location
parameters, respectively, is the best estimator for the scale parameter. The best estimation method
for location is the Bayesian estimation under the Entropy loss function in case of a small value of
the scale γ (say γ < 1). Bayesian estimation under the Precautionary loss function is the best in
case of a relatively large value of the scale γ (say γ > 1).
