Lower Confidence Bounds Using Pilot Samples With an Application to Auditing

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Abstract
A pilot study is often done prior to a more thorough study, to determine whether a larger-scale effort is likely to yield useful results. In one sense, the pilot study can discourage or encourage further investigation. This is true in many situations in auditing. For example, an auditor seeks to determine that amount an insurance provider overpays. Often the provider makes few mistakes and overpays little. On the other hand, occasionally a provider overpays a lot. Because the cost of auditing is expensive and the procedure is somewhat disruptive, a pilot study can prove very useful. When overpayment activity is present, the auditor would like a more precise estimate of overpayment, and then a second sample is called for. Pilot studies generally have small or moderate sample sizes. Nevertheless, the data from these samples should be used in two ways: first, to determine whether further sampling should be done, and second, to be used along with second samples for inference purposes. In this article we consider a model in which the population is normal with unknown mean and unknown variance. A pilot sample is taken and the null hypothesis that the mean is zero is tested against the alternative that the mean is positive. If null hypothesis is rejected, then a second sample is taken. Based on the data from both samples, an exact conditional lower confidence bound is obtained for the mean. The conditioning set consists of those sample points in the first sample that lead to rejection of the null hypothesis. In addition, a bias-corrected asymptotic lower confidence bound is obtained. Generalizations to models other than the normal are indicated.

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