Repeated audit controls

An important task of auditors is to check whether recorded values are correct. From the number (or total) of the errors found in a sample, upper confidence limits for the fraction (or total) of the errors in the population are calculated by standard methods.
Even auditors are human, however, and may make mistakes: errors may remain unnoticed. As administrative rules and regulations are becoming more and more complicated, this kind of auditors' mistakes tends to occur more frequently. Hence, there is an increasing need to check the auditing process itself. This can be achieved by checking a subsample of audited values a second time, now by an error-free supervisor.
This paper investigates the consequences of the (possible) occurrence of errors in this subsample. A simple and analytically attractive model is presented for the number of errors found in this double inspection scheme. Maximum likelihood estimators are derived for the population fraction of errors and the probability of an error remaining unnoticed. Based on that, an exact upper confidence limit for the population fraction of errors is calculated, treating the probability of an unnoticed error as a nuisance parameter. Our method was originally developed during a recent Dutch case study: in a subsample of already audited values one additional error was found by the supervisor. It is shown that, as a consequence, the upper confidence limit is increased very sizeably.