On Conservative Confidence Intervals

The subject of the paper – (conservative) confidence intervals – originates in applications to auditing. Auditors are interested in upper confidence bounds for an unknown mean for all sample sizes n. The samples are drawn from populations such that often only a few observations are nonzero. The conditional distribution of an observation given that it is nonzero usually has a very irregular shape. However, it can be assumed that observations are bounded. We propose a way to reduce the problem to inequalities for tail probabilities of certain relevant statistics. Note that a traditional approach involving limit theorems forces to impose additional conditions on regularity of samples and leads to approximate or asymptotic bounds. In the case of , as a statistic we can use sample mean, say , and we have to use Hoeffding 7] inequalities, since currently they are the best available. This leads to upper confidence bounds for which are of (asymptotic) size at most in the case of risk =0.05, where is the unknown standard deviation. We have , where is the bound in a model with normally distributed observations. It seems that the bound is very robust and can be improved replacing Hoeffding's inequalities by more refined ones. The commonly used Stringer bound (it is still not known whether it is an upper confidence bound) is of asymptotic size c with equality only for Bernoulli distributions, and the ratio c / can be arbitrary large already for rather simple distributions. Our bounds can involve a priori information (professional judgment of an auditor) of type ₀ or/and ₀, which leads to improvements. Most of the results also hold for sampling without replacement from finite populations. The i.i.d. condition can be replaced by a martingale-type dependence assumption. Finally, the results can be extended to the noni.i.d. case and for settings with several samples.