Abstract
This chapter presents a comparison of measures, multivariate majorization, and applications to statistics. The theory of balayage and dilation of measures encompasses a variety of concepts of orderings of measures, including continuous and discrete majorization and multivariate majorization. The cone of functions, which increases in regard to the partial ordering, consists of coordinate-wise increasing functions. The chapter describes the partial ordering of majorization and the associated monotone functions (the Schur convex functions). Examples of statistical interest of log concave multivariate densities include the normal, Wishart, Dirichlet, logistic, multivariate Gamma for suitable parameters, and the uniform distribution on a convex set. Every nonnegative measurable Schur concave function of two variables f (x1, x2) can be approximated by a positive linear combination of symmetric log concave functions.
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