On the stochastic ordering of folded binomials

Folded binomials arise from binomial distributions when the number of successes is considered equivalent to the number of failures or they are indistinguishable. In other words, a folded binomial observation is given by x = min(y;m-y) when m independent Bernoulli trials with equal probability q have yielded y successes. It arises when two subsets of outcomes for m trials can be identified, but either which are the successes and which are the failures cannot be said, or successes and failures are considered equivalent. Thus, 10 trials yielding 4 successes will give the same pattern as 10 trials yielding 6 successes. Formally, if the random variable Y is a binomial random variable with parameters m and q, then the random variable X = min(Y;m-Y) is folded binomial distributed with parameters m and p = min(q; 1-q). In this work, we present results on the stochastic ordering of folded binomial distributions. Providing an equivalence between their cumulative distribution functions (cdf) and a combination of two Beta random variable cdf's, we prove both that folded binomials are stochastically ordered with respect to their parameter p given the number of trials m, and that they are stochastically ordered with respect to their parameter m given p. Furthermore, the reader is offered two corollaries on strict stochastic dominance.