Mixed Estimators of Ordered Scale Parameters of Two Gamma Distributions with Arbitrary Known Shape Parameters

When an ordering among parameters is known in advance,the problem of estimating the smallest or the largest parametersarises in various practical problems. Suppose independent randomsamples of size ni drawn from two gamma distributions withknown arbitrary shape parameter no_i > 0 and unknown scale parameter beta_i > 0, i = 1, 2. We consider the class of mixed estimators of 1 and 2 under the restriction 0 < beta_1 < beta_2. It has been shown that a subclass of mixed estimators of i, beats the usual estimators frac{bar{X_i}}{nu_i}, i = 1, 2, and a class of admissible estimators in the class of mixed estimators are derived under scale-invariant squared error loss function. Also it has been shown that the mixed estimator of 0<beta_1leq beta_2‎, ‎(beta_1,beta_2), beats the usual estimator frac{bar{X_1}}{nu_1},frac{bar{X_2}}{nu_2} simultaneously, and a class of admissible estimators in the class of mixed estimators of (beta_1,beta_2) are derived. Finally the results are extended to some subclass of exponential family.