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. Formally, if Y∼Bin(m,π)Y∼Bin(m,π) is a binomial random variable, then the random variable X=min(Y,m−Y)X=min(Y,m−Y) is folded binomial distributed with parameters mm and p=min(π,1−π)p=min(π,1−π). 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 pp given the number of trials mm, and that they are stochastically ordered with respect to their parameter mm given pp. Furthermore, the reader is offered two corollaries on strict stochastic dominance.