Schur Functions in Statistics I. The Preservation Theorem

Abstract

This is Part I of a two-part paper; the purpose of this two-part paper is (a) to develop new concepts and techniques in the theory of majorization and Schur functions, and (b) to obtain fruitful applications in probability and statistics. The main theorem of Part I states that if $f(x_1, \cdots, x_n)$ is Schur-concave, and if $\phi(\lambda, x)$ is totally positive of order 2 and satisfies the semigroup property for $\lambda_1 > 0, \lambda_2 > 0: \phi(\lambda_1 + \lambda_2, y) = \int \phi(\lambda_1, x)\phi(\lambda_2, y - x) d\mu(x)$, where $\mu$ is Lebesgue measure on $\lbrack 0, \infty)$ or counting measure on $\{0, 1, 2, \cdots\}$, then $h(\lambda_1, \cdots, \lambda_n) \equiv \int \cdots \int \Pi^n_1 \phi(\lambda_i, x_i)f(x_1, \cdots, x_n) d\mu(x_1) \cdots d\mu(x_n)$ is also Schur-concave. This theorem is then applied to obtain renewal theory results, moment inequalities, and shock model properties.