Improved estimators for functions of scale parameters in mixture models

Estimation of the scale parameter of the scale mixture of a location–scale family under the scale-invariant loss function is considered. The technique of Strawderman (Ann Stat 2(1):190–198, 1974) is used to obtain a class of estimators improving upon the best affine equivariant estimator of the scale parameter under certain conditions. Further, integral expressions of risk difference approach of Kubokawa (Ann Stat 22(1):290–299, 1994) is used to derive similar improvements for the reciprocal of the scale parameter. Using the improved estimator of the scale parameter and the improved estimator of the reciprocal of the scale parameter, classes of improved estimators for the ratio of scale parameters of two populations have been derived. In particular, Stein type and Brewster–Zidek type estimators are provided for the ratio of scale parameters of two mixture models. These results are applied to the scale mixture of exponential distributions, this includes the multivariate Lomax and the modified Lomax distributions.