On improved estimation of the larger location parameter

This paper investigates the problem of estimating the larger location parameter of two general location families of distributions from a decision-theoretic perspective. The criteria of minimizing the risk function and the Pitman nearness, under a general bowl-shaped loss function, are considered. Inadmissibility of certain location and permutation equivariant estimators is proved and dominating estimators are obtained. It follows that a natural estimator $$\delta _{c_0}$$ δ c 0 (a plug-in estimator based on the best location equivariant estimators of (unordered) location parameters) is inadmissible, under certain conditions on underlying densities, and the loss function. A class of dominating estimators is provided. We also consider a class $$\mathcal {D}$$ D of linear and, location and permutation equivariant estimators and obtain a subclass $$\mathcal {D}_0$$ D 0 ( $$\subseteq \mathcal {D}$$ ⊆ D )of estimators that are admissible within the class $$\mathcal {D}$$ D of estimators. We observe that the natural estimator $$\delta _{c_0}$$ δ c 0 is a boundary estimator in $$\mathcal {D}_0$$ D 0 . Further, using the IERD technique of Kubokawa (1994), we obtain an estimator dominating over another natural estimator $$\delta _{b_0}$$ δ b 0 , that is another boundary estimator in $$\mathcal {D}_0$$ D 0 . Additionally, under the generalized Pitman nearness criterion with a general bowl-shaped loss function, we show that two natural estimators are inadmissible and obtain improved estimators. The results are applied to specific loss functions, and explicit expressions for dominating estimators are obtained. We illustrate applications of these results to normal and exponential distributions for specified loss functions. A simulation study is also conducted to compare risk performances of different competing estimators. Finally, we present a real-life data analysis to illustrate a practical application of the findings of the paper.