Minimax estimation problems with restricted parameter space reached increasing interest within the last two decades Some authors derived minimax and admissible estimators of bounded parameters under squared error loss and scale invariant squared error loss In some truncated estimation problems the most natural estimator to be considered is the truncated version of a classical estimator in the original problem The MLE in the truncated normal problem is one such example In exponential families a class of reasonable estimators of the mean in the unrestricted problem are the linear estimators which arise as (proper or generalized) Bayes estimators for conjugate families. Hence it is natural to consider a truncated version of such linear estimators.
A theme which runs through much of the literature on such truncated procedures is that while they are improving on the untruncated estimator, they themselves are inadmissible because they are not generalized Bayes.
In this paper we consider a subclass of the exponential families of distributions which includes Exponential, Weibull, Gamma, Normal, Inverse Gaussian and some other distributions. The minimax and linear admissible estimators of the rth power of scaleparameter under scaleinvariant squarederror loss are obtained. Also the class of truncated linear estimators of the rth power of the lowerbounded scale parameter in this family is considered. It is shown that each member of this class is inadmissible and exactly one of them is minimax, under scaleinvariant squarederror loss. Further, this minimax estimator is compared with admissible minimax estimator of the lower[1]bounded scaleparameter, which is obtained by Jafari Jozani et al. (2002). Dealing with the family of transformed Chisquare distributions, which is introduced by Rahman and Gupta (1993), we apply our result for their lower bounded parameters which are not necessarily scale parameters. We show that the truncated linear minimax estimator obtained by van Eeden (1995) in gamma distribution is a special case of our estimator.
