Summary: The subject of the paper – upper confidence bounds – originates from applications to auditing. Auditors are interested in upper confidence bounds for an unknown mean \(\mu\), 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. In such situations parametric models seem to be somewhat unrealistic.
We consider confidence bounds and intervals for an unknown parameter in parametric and nonparametric models. We propose a reduction of the problem to inequalities for tail probabilities of relevant statistics. In the special case of an unknown mean and bounded observations, a similar approach has been used by us in “Upper confidence bounds for mean.” Rep. No. 0110, Dpt. Math., Univ. Nijmegen, 1–23 (2001), by applying W. Hoeffding’s [J. Am. Stat. Assoc. 58, 13–30 (1963; Zbl 0127.10602)] inequalities for sample means and variances. The bounds can be modified in order to involve a priori information (= professional judgment of an auditor), which leads to improvements of the bounds. Furthermore, the results hold for various sampling schemes and observations from measurable spaces provided that we possess the aforementioned inequalities for tail probabilities.