Cone order association and stochastic cone ordering with applications to order-restricted testing

Abstract

Cohen, Sackrowitz and Samuel-Cahn introduced the notion of cone order association and established a necessary and sufficient condition for a normal random vector to be cone order associated (COA). In this paper we provide the following: (1) a necessary and sufficient condition for a multinomial distribution to be COA when the cone is a pairwise contrast cone; (2) a relationship between COA and regular association; (3) a notion of stochastic cone ordering (SCO) of random vectors along with two preservation theorems indicating monotonicity properties of expectations as functions of parameters; and (4) applications to unbiasedness of tests and monotonicity of power functions of tests in cone order-restricted hypothesis-testing problems. In particular, the matrix order alternative hypothesis-testing problem is treated when the underlying distributions are independent Poisson or the joint distribution is multinomial.