STOCHASTIC ORDERING OF CLASSICAL DISCRETE DISTRIBUTIONS

For several pairs (P, Q) of classical distributions on No, we show that their stochastic ordering P <=(st) Q can be characterized by their extreme tail ordering equivalent to P({k(*)})/Q({k(*)}) >= 1 >= limk -> k* P({k})/Q({k}), with k(*) and k(*) denoting the minimum and the supremum of the support of P + Q, and with the limit to be read as P((k*))/Q({k*}) for finite k*. This includes in particular all pairs where P and Q are both binomial (b(n1, p1) <=(st) b(n2,) (p2) if and only if n(1) <= n(2) and (1-p(1))(n1) >= (1-p(2))n(2), or p(1) = 0), both negative binomial (b ((r1)) over bar, (p1) <=(st) b ((r2)) over bar, (p2) if and only if p(1) >= p(2) p(1)(r1) >= p(2)(r2)), or both hypergeometric with the same sample size parameter. The binomial case is contained in a known result about Bernoulli convolutions, the other two cases appear to be new. The emphasis of this paper is on providing a variety of different methods of proofs: (i) half monotone likelihood ratios, (ii) explicit coupling, (iii) Markov chain comparison, (iv) analytic calculation, and (v) comparison of Levy measures. We give four proofs in the binomial case (methods (i) (iv)) and three in the negative binomial case (methods (i), (iv), and (v)). The statement for hypergeometric distributions is proved via method (i).