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

SYNOPTIC ABSTRACT

Independent 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.

Key Words and Phrases::

• failure rate
• guarantee time
• inadmissibility
• maximum likelihood estimator
• ordered parameters
• uniformly minimum variance unbiased estimator.