Non-parametric empirical Bayes procedure

In this paper we introduce an empirical Bayes procedure for estimating an unknown parameter, say θ. This procedure gives the empirical Bayes estimator for θ and its associated minimum posterior risk in closed forms without estimating the unknown prior density function of θ. In such procedure the posterior probability density function of θ is not required. A sufficient statistic for θ with conditional probability density function in the one parameter exponential family is required. Instead of estimating the unknown prior density function, the marginal density function of the sufficient statistic must be estimated. As special cases the empirical Bayes estimators and their respective minimum posterior risks of the failure rate for the exponential distribution, the unknown scale parameters of Weibull and gamma distributions are obtained in simple forms as special cases. Numerical results and a simulation study are introduced to (i) investigate how the number of available past experiments and the sample size of each influence the accuracy of the empirical Bayes estimator, (ii) make a comparison between the presented procedure and the Bayes procedure when the prior probability density function of the parameter θ is gamma.