MINIMAX ESTIMATION OF A LOWER-BOUNDED SCALE PARAMETER OF A GAMMA-DISTRIBUTION FOR SCALE-INVARIANT SQUARED-ERROR LOSS

Let X have a gamma distribution with known shape parameter alpha and unknown scale parameter theta. Suppose it is known that theta greater than or equal to a for some known a > 0. An admissible minimax estimator for scale invariant squared-error loss is presented. This estimator is the pointwise limit of a sequence of Bayes estimators. Further, the class of truncated linear estimators C = {($) over cap theta(r)ho/($) over cap theta(r)ho(x) = max(a, rho x), rho > 0} is studied. It is shown that each ($) over cap theta(r)ho is inadmissible and that exactly one of them is minimax. Finally, it is shown that Katz's [Ann. Math. Statist., 32, 136-142 (1961)] estimator of theta is not minimax for our loss function. Some further properties of and comparisons among these estimators are also presented.