A two-phase sampling scheme with applications to auditing or sed quis custodiet ipsos custodes?

External auditors such as the National Audit Office (NAO) are the final arbiters on the level of error in accounts presented to them by their clients, and the accuracy or otherwise of individual transactions. In coming to a view on the level of error, they are expected to carry out the audit effectively and efficiently, and therefore need to make the best possible use of all the information at their disposal, even when some of the information may not be totally accurate. We consider the particular situation where the NAO is given access to the results of tests on a relatively large random sample of transactions, typically conducted by the client's internal auditors. A two-phase sampling scheme arises when the NAO subsequently assesses the quality of the client's data by retesting a subsample of these transactions. The paper discusses methodologies for combining the two sets of data to produce optimum estimates of the proportion of transactions in error (the error rate) and of the level of monetary error in the account. Although a maximum likelihood approach yields a relatively straightforward solution to the error rate problem, there is no uniformly optimum way to estimate the monetary error. Three possible methods are proposed, and the results of a series of simulation experiments to compare their performance under a variety of audit conditions is described.