On estimating the scale parameter of the selected gamma population under the scale invariant squared error loss function

Let X 1 and X 2 be two independent random variables representing the populations Π 1 and Π 2 , respectively, and suppose that the random variable X i has a gamma distribution with shape parameter p, same for both the populations, and unknown scale parameter θ i , i = 1 , 2 . Define, M = 1 , if X 1 > X 2 , M = 2 , if X 2 > X 1 and J = 3 - M . We consider the component wise estimation of random parameters θ M and θ J , under the scale invariant squared error loss functions L 1 ( θ ̲ , δ 1 ) = ( δ 1 / θ M - 1 ) 2 and L 2 ( θ ̲ , δ 2 ) = ( δ 2 / θ J - 1 ) 2 , respectively. Sufficient conditions for the inadmissibility of equivariant estimators of θ M and θ J are derived. As a consequence, various natural estimators are shown to be inadmissible and better estimators are obtained.