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m. SUMMARY AND RECOMMENDATIONS The purpose of this Report has been to review and evaluate the state of statistical practice in accounting and auditing. In it, we have emphasized, (~) Me importance of the problem as one of national interest, (2) the nonstandard nature of the statistical problem reladve to Be main body of existing statistical methodology, (3) die lack of adequately reliable procedures, and (4) the generally scattered and ad hoc nature of the existing methodology. It is clear Mat much additional research is needed and it is in Be national interest that this be done. For this reason, He Report has been directed pnmanly towards researchers and graduate students, both in statistics and accounting. However, the following summary and recommendations should be of interest to a much wider audience within our respective disciplines. Auditing is an essential activity in a society win advanced capital markets. In such a society, investors and government officials base many important decisions on accounting information. Those decisions affect the welfare of all citizens. Auditing is a costly activity and statistical procedures can play an important role in reducing those costs. Basic statistical problems in auditing arise when one wishes to estimate the total population error in an account. Relative to the main body of statistical methodology, these problems are nonstandard due to a unique feature of the data; audit data usually contains mosey zeros! Existing statistical methods do not offer satisfactory solutions for inferences based on such information. . This report's survey of the existing literature and practices points up several important observations. First of all, statistical methods have only recently begun to be developed for analyzing this nonstandard type of data; in the chronological bibliography in Chapter IV, all but five of the references are dated after 1972. The first significant contribution was that of Aitchison (19SS) and the key idea of Dollar Unit Sampling (DUS) was reported by Stringer in 1963. One of the main factors that serves to retard progress in the development of new methodology for auditing problems is the high degree of confidentiality placed upon accounting information. The resultant lack of good data prevents the characteristics of accounting populations from being adequately known in all but a limited 55

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number of cases. ~ order to improve Be quality and applicability of statistical auditing procedures, it is essential that much more data be made available by both public and private sectors. ~ particular, one's confidence in the outcomes of statistical analyses depends heavily upon the suitability of Me models that have been postulated for the situations In question. However, We selection of appropn ate models relies critically upon He availability of adequate data from such situations. Until there is adequate data available to give guidance and just~ficabon for mode} selections, there win be less than the desired confidence in He analyses based upon ~em. . . A survey of existing approaches to the statistical problems of auditing reveals that one of He most important ideas is that of DUS. This sampling design selects items from an account with probability proportional to their book amounts. Items with large book amounts, therefore, are more likely to be selected than items win smaller amounts. Since the items with larger book values are considered relatively more important Han those with smaller book values, DUS is an appealing sampling design when tile auditor places primary emphasis on overstatements. The DUS design does have some limitations, however. For example, items with a zero book amount will not be selected under this sampling. The dollar unit sampling design adso permits the auditor to incorporate into the analysis prior knowledge that the errors are overstatements and Hat He maximum size of an error of an item is equal to its book amount. This assumption, when applicable, sets an upper limit of 1 and a lower limit of O for a DUS error. This ear, referred to by accountants as tainting, is He ratio of He error amount to He recorded book amount. Under this assumption, an auditor can set a conservative upper bound for the population error with a confidence level at least as large as the stated one. The upper bound for the population error amount is equal to the it- a) upper bound for the error rate, multiplied by the known total book amount of the population. Cox and Snell (1979) provides a theoretical framework for this method. In this report, it is concluded that this bound based on attribute sampling theory is the only procedure available that has a theoretically known sampling distnbution. This means that the long run performance of aU other currency available procedures must be investigated by means of simulation. Consequently, it is not easy to obtain information about the performance of these procedures in a wider audit situation. 56

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A significant weakness of the upper bound defined in this way is that it is far too conservative in Mat the auditor's confidence coefficients are much larger than intended. A heuristic method, credited to Swinger, has been widely used and it produces a tighter bound. It is now about 25 years since me Stringer bound was proposed. In spite of the fact mat extensive sunulat~on demonstrates that it is far too conservative, no theoretics justification has as yet been obtained! This remains an important and interesting open question. Several altemative me~ods, mostly heunshc and sometimes totally ad hoc, have been proposed in recent years and these are reviewed in this report Based on Innited investigations, the upper bounds set by some of these procedures are shown to be considerably tighter than the Swinger bound. Much more research along these fines is needed. It is also recommended that extensive testing be carried out using real data in order to evaluate adequately these and other procedures. Sequential methods would seem to be appropn ate for some of these problems, and yet there is a noticeable lack of such methods in the relevant literature. This is in spite of the fact that general sequential methodology is available in statistical monographs directed towards accounting, for example, Cyert and Davidson (19621. In particular, simple two-stage sampling schemes could be considered as a possible way to improve the performance of some of the statistical procedures. Empincal studies indicate Mat negative enters caused by under- statements are also quite corrunon in auditing populations. Very little research, however, has been done on the problem of determining bounds for these cases, and this needs to be corrected. Note mat DUS may not be an effective sampling design when understatements are present because items with larger audited amounts may have smaller chances of selection Man desired. Except for the procedures that utilize Bayesian mesons, existing procedures are not effective for setting a good lower bound for accounting population errors. This failure is extremely senous; one particularly important situation involves the estimation of the adjustment of a fien's expense accounts by the Intemal Revenue Service (IRS). The current IRS procedure is to apply standard sampling methods such as those used in surveys of human populations. Investigation of Me perfonnance of these estimators for certain audit populations indicates that such IRS practice is too 57

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. conservative in the sense Hat the IRS is assuming much lower risk than aBowed in the policy. That is, the actual level of confidence is substandaBy higher Man the nominal level. Such practice tends to underestimate the potential tax revenue due the government. Similar problems also arise in over governmental agencies, e.g., the Office of hnspector General of the Department of Heady and Human Services, in their investigation of compliance with government guidelines of reported expenses by local gove~nents. It is important that intensive research be carried out for the purpose of developing more reliable procedures for determining lower confidence bounds. The financial benefits to the government from such research should be significant. The development of valid statistical mesons for seeing confidence bounds for accounting populations is of nations interest and importance, in major part because of the considerable economic benefits mat would accrue to both the public and pr~v ate sectors. In developing a methodologies, primary emphasis should be placed upon the denvation and performance of one sided confidence intervals and not the two sided confidence intervals commonly discussed in standard statistical texts. Texts should be revised to reflect this. In this age of widely available high speed computing equipment, it is reasonable to expect significantly greater use of computer- intensive statistical methodologies. There is also a need for greater use of computers in the simulation of performance characteristics of existing methodologies, particularly as increased data sets become available to suggest more realistic simulation models. The survey of the existing literature that is given in Chapter IV below, reveals that the statistics profession as a whole has not been heavily involved with the ~rnportant statistical problems Hat arise in auditing. This may be due in part to the fact that there has not been adequate nor regular interaction between researchers from the accounting and the statistics professions. It is recommended that a series of workshops, conferences and courses be set up at which theoretical and applied statisticians can meet and exchange expertise and problems with accountants and auditors from each of the sectors of gove~Tunent, business and academia. It would be expected Hat proceedings of some of these activities would be published wide the purpose of improving the communication between the two disciplines. There would also be S8

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important benefits from holding a conference that would bring researchers together *nm the many diverse areas of applications mat exist throughout aU of the sciences In which problems of nonstandard mixtures arise. Several such areas are briefly described in Chapter ~ above. Primarily, Me exchange of problems and me transfer of relevant methodologies and references could expedite progress in an areas. The initial emphasis of coordinated research activities between the auditing and statistics professions should focus upon seeking ways to encourage statisticians to become directly involved in the auditing environment In this way' statisticians would become more familiar with the statistical problems in auditing and especially with the characteristics of the data bases in this sewing. It is also recommended that accounting firms make audit data available for a wider research community than its own profession. Large private accounting finns have bow economic incentives and resources to carry on research for the purpose of developing better statistical procedures for their audit problems. However, concerted efforts must be made to improve me statistical methodologies used in the public sector. Adequate resources wild need to be available to make me needed level of research possible. 59