Stochastic ordering of classical discrete distributions

For several pairs (P,Q) of classical distributions on N0, 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 k∗ finite. This includes in particular all pairs where P and Q are both binomial (bn1,p1 ≤st bn2,p2 if and only if n1 ≤ n2 and (1− p1) n1 ≥ (1− p2) , or p1 = 0), both negative binomial (b − r1,p1 ≤st b−r2,p2 if and only if p1 ≥ p2 and p r1 1 ≥ p2 2 ), 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 Lévy 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). 2000 MSC: primary 60E15