Estimation of Ordered Location Parameters: The Exponential Distribution

The problem of component wise estimation of ordered location parameters θ ₁ (θ ₁ ≤ θ ₂) of two independent exponential distributions is investigated. The scale parameters are assumed to be unequal but known. Independent random samples of unequal sample sizes are drawn from two populations and the estimators admissible among the mixed estimators of θ ₁ and θ ₂ are obtained. It is shown that the minimum risk estimators (MREs) of θ ₁ and θ ₂ without assuming θ ₁ ≤ θ ₂ are inadmissible when one does assume that θ ₁ ≤ θ ₂. The efficiencies of mixed estimators relative to MREs (without assuming θ ₁ ≤ θ ₂) are tabulated for equal sample sizes and equal scale parameters.