Bayesian and frequentist estimation and prediction for exponential distributions

Bayes estimates and predictors are derived for exponential distribution with density θe-θxθe-θx, x>0x>0, under several known loss functions, where θ>0θ>0 is an unknown parameter. The commonly used frequentist approaches such as the maximum likelihood estimates (MLE) and the “plug-in” procedure, which is to substitute a point estimate of the unknown parameter into the predictive distribution, are reviewed. We have examined Bayes estimates under various losses such as the absolute error, the squared error, the LINEX loss and the entropy loss functions. We show that Bayes estimate under the LINEX loss is more general, which includes the MLE and other Bayes estimates as special cases up to the second-order accuracy. The second-order asymptotic theory under these loss functions is developed and the risks are compared. It is shown that Bayes estimates of θθ are superior to the MLE under the noninformative prior and when an appropriate a*a* in the LINEX loss is chosen. When the objective is prediction rather then estimation, it is less clear-cut which of the different procedures is best.