Stochastic ordering of classical discrete distributions

TL;DR

This paper characterizes stochastic ordering among classical discrete distributions like binomial, negative binomial, and hypergeometric, using tail ratios and multiple proof techniques, including couplings and analytic methods.

Contribution

It provides new stochastic ordering criteria for binomial, negative binomial, and hypergeometric distributions, with diverse proof methods.

Findings

Characterization of stochastic orderings via tail ratios.

New ordering criteria for binomial and negative binomial distributions.

Multiple proof techniques demonstrated for the main results.

Abstract

For several pairs $(P,Q)$ of classical distributions on $\N_0$, we show that their stochastic ordering $P\leq_{st} Q$ 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 ($b_{n_1,p_1} \leq_{st} b_{n_2,p_2}$ 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 ($b^-_{r_1,p_1}\leq_{st} b^-_{r_2,p_2}$ if and only if $p_1\geq p_2$ and $p_1^{r_1}\geq p_2^{r_2}$), 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).