Estimation of a scale parameter in mixture models with unknown location

Estimation of the scale parameter in mixture models with unknown location is considered under Stein's loss. Under certain conditions, the inadmissibility of the “usual” estimator is established by exhibiting better estimators. In addition, robust improvements are found for a specified submodel of the original model. The results are applied to mixtures of normal distributions and mixtures of exponential distributions. Improved estimators of the variance of a normal distribution are shown to be robust under any scale mixture of normals having variance greater than the variance of that normal distribution. In particular, Stein's (Ann. Inst. Statist. Math. 16 (1964) 155) and Brewster's and Zidek's (Ann. Statist. 2 (1974) 21) estimators obtained under the normal model are robust under the t model, for arbitrary degrees of freedom, and under the double-exponential model. Improved estimators for the variance of a t distribution with unknown and arbitrary degrees of freedom are also given. In addition, improved estimators for the scale parameter of the multivariate Lomax distribution (which arises as a certain mixture of exponential distributions) are derived and the robustness of Zidek's (Ann. Statist. 1 (1973) 264) and Brewster's (Ann. Statist. 2 (1974) 553) estimators of the scale parameter of an exponential distribution is established under a class of modified Lomax distributions.