Schur-convexity of 2nd order, certain subclass of multivariate arrangement increasing functions with applications in statistics

For families of distributions with location parameter, it is shown that the ranking of parameters by sample means is expedient in the case of families having no sufficient statistic. Furthermore, mean losses (the risk) arising because of making wrong decisions decreases with growth of the size of the sample if densities are symmetric and log- concave. Examples of applications of these results are given.