Trimmed estimates in simultaneous estimation of parameters in exponential families

Let X1,..., Xp be p (>= 3) independent random variables, where each Xi has a distribution belonging to the one-parameter exponential family of distributions. The problem is to estimate the unknown parameters simultaneously in the presence of extreme observations. C. Stein (Ann. Statist. 9 (1981), 1135-1151) proposed a method of estimating the mean vector of a multinormal distribution, based on order statistics corresponding to the Xi's, which permitted improvement over the usual maximum likelihood estimator, for long-tailed empirical distribution functions. In this paper, the ideas of Stein are extended to the general discrete and absolutely continuous exponential families of distributions. Adaptive versions of the estimators are also discussed.