Stochastic ordering of classical discrete distributions

Abstract

For several pairs $(P,Q)$ of classical distributions on ${\mathbb{N}}_0$, we show that their stochastic ordering $P{\le }_{st}Q$ can be characterized by their extreme tail ordering equivalent to $P(\{k_{\ast }\})/Q(\{k_{\ast }\})\le 1\le {\lim }_{k\to k^{\ast }}P(\{k\})/Q(\{k\})$, with $k_{\ast }$ and $k^{\ast }$ denoting the minimum and the supremum of the support of $P+Q$, and with the limit to be read as $P(\{k^{\ast }\})/Q(\{k^{\ast }\})$ for $k^{\ast }$ finite. This includes in particular all pairs where $P$ and $Q$ are both binomial ($b_{n_1,p_1}{\le }_{st}{b}_{n_2,p_2}$ if and only if $n_1\le n_2$ and $(1-p_1{)}^{n_1}\ge (1-p_2{)}^{n_2}$, or $p_1=0$), both negative binomial ($b_{r_1,p_1}^{-}{\le }_{st}{b}_{r_2,p_2}^{-}$ if and only if $p_1\ge p_2$ and $p_1^{r_1}\ge p_2^{r_2}$), 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 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).

arXiv commenttypos corrected, two references added.