Estimating the Common Location of Two Exponential Populations Under Order Restricted Failure Rates

SYNOPTIC ABSTRACTIndependent random samples are drawn from two exponential populations Π1 and Π2 having a common location parameter and, possibly, different failure rates σ−11>0 and σ−12>0, respectively. Then θ1=μ+σ1 and θ2=μ+σ2 are mean lifetimes of the two populations. Suppose that it is known a priori that σ1⩽σ2 (or, equivalently, θ1⩽θ2). In the presence of this prior information, we consider the estimation of the common location parameter (common guarantee time) μ under the squared error loss function. Exploiting the prior information σ1⩽σ2, estimators improving on the maximum likelihood estimator (MLE), a modified MLE (MMLE), and the uniformly minimum variance unbiased estimator (UMVUE) are obtained. In the process, we derive a general result that provides a method to obtain estimators improving on certain inadmissible estimators of μ under the prior information σ1⩽σ2. Finally, a simulation study has been carried out to numerically compare the risk performances of various estimators.