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.
On the stochastic ordering of folded binomials
PORZIO, Giovanni Camillo;
2009-01-01
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.