Stochastic Comparisons of Bernoulli Sums and Binomial Random Variables

Abstract

There are many practical situations in sampling and testing, when the probability of success varies in a sequence of n independent Bernoulli trials. In many of these cases and for various reasons, we may find it useful to compare the distribution of the number of successes X=ΣBin(1, p i ) in n such trials with a binomial random variable Y=Bin(n, p) for some p. For example, such a comparison might be useful in deciding whether or not stratified sampling is superior (or inferior) to simple random sampling in survey sampling, or whether or not partition (or subdomain) testing is to be preferred to simple random testing in attempting to find faults in software. We will discuss the rationale behind several methods and orders for stochastically comparing the random variables X and Y. These include comparing their means, but also comparing them with respect to the usual stochastic order, the precedence order, the ≥1 order and even the likelihood ratio order. It will be seen that many interesting comparisons between X and Y depend on the relationship between p and various means (harmonic, geometric, arithmetic, complimentary geometric, and complimentary harmonic) of the components in the vector p=(p 1, p 2,...p n .