On admissibility and inadmissibility of estimators after selection under reflected gamma loss function

Abstract

Let  Π 1  and  Π 2  denote two gamma populations with common known shape parameter  α >  0  and unknown scale parameters  θ 1  and  θ 2 , re- spectively. Let  X 1  and  X 2  be two independent random variables from Π 1  and  Π 2 , and  X (1)  ≤  X (2)  denote the ordered statistics of  X 1  and X 2 . Suppose the population corresponding to the largest  X (2)  or the smallest  X (1)  observation is selected. This paper concerns on the ad- missible estimation of the scale parameters  θ M  and  θ J  of the selected population under reflected gamma loss function. We provide sufficient conditions for the inadmissibility of invariant estimators of  θ M  and  θ J  . The admissibility and inadmissibility of estimators in the class of lin- ear estimators of the form  cX (2)  and  dX (1)  are discussed. We apply our results on  k -Records, censored data and extend to a subclass of exponential family.