**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

*Statistical Models and Analysis in Auditing: A Study of Statistical Models and Methods for Analyzing Nonstandard Mixtures of Distributions in Auditing*. Washington, DC: The National Academies Press. doi: 10.17226/1363.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

*Statistical Models and Analysis in Auditing: A Study of Statistical Models and Methods for Analyzing Nonstandard Mixtures of Distributions in Auditing*. Washington, DC: The National Academies Press. doi: 10.17226/1363.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

*Statistical Models and Analysis in Auditing: A Study of Statistical Models and Methods for Analyzing Nonstandard Mixtures of Distributions in Auditing*. Washington, DC: The National Academies Press. doi: 10.17226/1363.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

**Suggested Citation:**"STATISTICAL MODELS AND ANALYSES IN AUDITING." National Research Council. 1988.

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

II STATISTICAL MODELS AND ANALYSES IN AUDITING 1. The Beginnings The field of accounting encompasses a number of subdisciplines. Among these, two important ones are financial accounting and auditing. Financial accounting is concemed with the collection of data about the economic activities of a given firm and He summarizing and reporting of Rem in He form of financial statements. Auditing' on He over hand, refers to the independent verification of He fairness of these financial statements. The auditor collects data mat is useful for verification from several sources and by different means. It is very evident mat the acquisition of reliable audit information at low cost is essential to economical and efficient auditing. Them are two main types of audit tests for which He acquisition of infonnation can profitably make use of statistical sampling. Firstly, an auditor may require evidence to verify that the accounting treatments of numerous individual transactions comply win prescribed procedures for internal control. Secondly, audit evidence may be required to verify that reported monetary balances of large numbers of individual items are not materially misstated. The first audit test, collecting data to determine the rate of procedural errors of a population of transactions is called a compliance test. The second, coDect~ng data for evaluating the aggregate monetary error in the stated balance, is caned a substantive test of details. The auditor considers an error to be material if its magnitude "is such that it is probable that the judgement of a reasonable person relying upon the report would have been changed or influenced by the inclusion or correction of the item". (Financial Accounting Standards Board, 1980) Current auditing standards set by He American Institute of Certified Public Accountants (AICPA) do no! mandate the use of statistical sampling when conducting audit tests (AICPA, l98l & 1983~. However, the meets of random sampling as the means to obtain, at relatively low COSt, reliable approximations to the characteristics of a large group of entnes, were known to accountants as early as 1933 (Carmen, 19331. The early applications were apparently limited to compliance tests (Neter, 19861. The statistical problems that arise, when analyzing He type of nonstandard mixture of distributions that is the focus of this report, did not surface in auditing until the late 1950s. At about that time, Kenneth Stringer began to investigate Be practicality of incorporating statistical sampling into the audit practices of his finn, Deloitte, Haskins & Sells. It was not until 1963 that some results of his studies were commurucated to the statistical profession. The occasion was a meeting of the American 8

Statistical Association (Stringer. 1963 & 1979). Before summarizing Stringer's main conclusions, we describe Me context as follows. An item In an audit sample produces two pieces of inforTnabon, namely, the book (recorded) amount and the audited (correct) amount. The difference between We two is caned the error amount. The percentage of items in error may be smalD in an accounting population. In an audit sample, it is not uncommon to observe only a few items with errors. An audit sample may not yield any non-zero error amounts. For analyses of such data, in which most observations are zero, me classical interval estimation of the total error amount based on the asymptotic normality of me sampling distribution is not reliable. Also, when the sample contains no items in error, the estimated standard deviation of the estimator of the total error amount becomes zero. Alternatively, one could use the sample mean of the audited amount to estimate Me total mean audited amount for the population. The estimate of the mean is then multipled by the known number of items in the population to estimate the population total. In the audit profession, this method is referred to as mean-per-unit estimation (AICPA, 19X3~. Since observations are audited amounts, Me standard deviation of this estimator can be estimated even when aU items in the sample are error-free. However, because of the large variance of the audited amount that may arise In simple random sampling, the mean-per-un~t estimation is imprecise. More fundamentally, however, when the sample does not contain any item in error, the difference between the estunate of the tote] audited amount and the book balance must be Interpreted as the sampling error. The auditor thus evaluates that the book amount does not contain any material error. This is an important point for the auditor. To quote from Swinger (1963) concem~ng statistical estimates ("evaluations") of total error: Assuming a population with no error in it, each of the possible distinct samples of a given size Mat could be selected from it would result In a different estimate and precision limit under this approach; however, from the view point of the auditor, all samples which include no errors should result in identical evaluations. Stringer men reported in He same presentation that he, in collaboration with Frederick F. Stephan of Princeton University, had developed a new statistical procedure for his firm's use in auditing that did not depend on the normal approximation of He sampling distribution and that could silk provide a reasonable inference for the population error amount when all items in the sample are error-free. This sampling plan is apparently the original implementation of He now widely pracused dollar (or monetary) unit sampling and is one of the first workable solutions 9

proposed for the nonstandard mixtures problem in accounting. However, as it is studied later in this report, the method assumes that errors are overstatements with the maximum size of an error of an item equal to its book amount. Another solution, using a similar procedure, was devised by van Heerden (19611. His work, however, was slow to become known within He American accounting profession. En the public sector, statistical sampling has also become an integral part of audit tools In the Intemal Revenue Service ORS) since the issuance of He 1972 memo by their Chief Council ORS, 1972 & 1975~. In a tax examination the audit agent uses statistical sampling of individual items to estun ate the adjusOnent, if necessary, for an aggregate expense reported in the tax retum. Statistical auditing may also be utilized by over gove'Tunental agencies. For example, the Office of He Spector General of He Deparunent of Heath and Human Services investigates compliance of the cost report of a state to the Medicaid policy by using statistical sampling of items. IN these cases, a large proportion of items in an audit sample requires no adiusonent, i.e., most sample items are allowable deductions. Since an individual item adjusunent is seldom negative, the audit data for estimation of the total adjustment is a mixture of a large percentage of zeros and a small percentage of positive numbers. Thus, the mixture model and related statistical problems that are important to accounting firms in auditing also arise in other auditing contexts such as those associated with IRS tax examinations. Significant differences also exist in these applications, however, and these win be stressed later. For concise accounts of the problems of statistical auditing one is referred to Knight (1979), Smith (1979) and Neter (1986~; the last reference also includes recent developments. Leslie, Teitiebaum and Anderson (1980) also provide an annotated bibliography that portrays the historical development of the subject Trough 1979. In the sections which follow, however, we provide a comprehensive survey of the research efforts that have contributed to me identification and beKer understanding of problems in statistical auditing. We include brief descnptions of many of the solutions that have been proposed for these problems along with their limitations. It wild be noticed that the solutions thus far proposed are mainly directed toward He special need for good upper bounds on errors when emus are overstatements. This is an important and common audit problem for accounting firms but in the case of tax examinations, Though the mixture distnbui~on is similar, the interest is in me study of lower bounds. Thus in statistical auditing, whether in the private or public sector, the investigator's interest is usually concerned win one-sided problems, i.e., of an upper or a lower bound, rather than nvo-si~e~ problems as currently stressed in many texts. 10

The next section provides the definitions and notations that are used. Then in Section 3 though 7' we present venous methodologies that have been provided in the literature A numerical example is given in me last section to indurate some of the alternative procedures. 11

2. Definitions and Notations An account, such as accounts receivable or inventory, is a population of individual accounts. To distinguish He use of the word 'account' in the former sense from me latter, we dedne the constituent individual accounts, when used as audit uruts, as tine items. Let Y' and Xi, Be latter not usually known for an values of i, denote the book (recorded) amount and the audited (correct) amount respectively, for the inch line item of an account of N Inne items. The book and audited balances of Me account are respectively N Y=2Y'. ·=1 called the population book amount, and V X=~Xi. i=1 (2.1) (2.2) caped Me population audited amount. The error amount of the i-th item is defined to be Di = YiXi (2.3) When Di > 0 , we can it an overstatement and, when Di <O. an understatment. When Yi ~ O. the fractional error, Ti =DilYi, (2.4) is caned the tainting or simply We taint of the i-th item. It is the error amount per dollar unit of the i-th item. We may then write Di = To Yi The error of the book balance of We account is Bus N N D = Y -X = :;Di = 7>Ti Y i=1 (2.5) . , l=1 (2.6) As emphasized in Section 1, a large proportion of items in an audit population will likely be error-free, so that Di = 0 for many values of i . Similar populations are common in many disciplines as discussed in Chapter I. Aitchison (1955) is me first to consider an inference problem for such a population. Following his approach, the error ~ of an item randomly chosen from an accounting population may be modeled as 12

follows: , z with probability p, 4=~ O wi~probability (imp), (2.7) where p is the proportion of items with errors in the population and z =0 is a random variable representing the error amount. z may depend on the book amount. The nonstandard mixture problem that is the focus of this report is the problem of obtaining confidence bounds for the population total error D when sampling from the model (2.7~. A useful sampling design for statistical auditing is to select items without replacement with probability proportional to book values. This sampling design can be modeled in teens of use of individual doBars of the total book amount as sampling units and is commonly referred to as Dollar Unit Sampling (D US) or Monetary Unit Sampling (MUS). (Anderson and Teitlebaum, 1973; Roberts, 1978; Leslie, Teitlebaum and Anderson, 19801. The book amounts of the N items are successively cumulated to a total of Y dollars. One may Den choose systematically n doBar units at fixed intervals of ~ (= Yin ~ doDars. The items with book amounts exceeding ~ doDars, and hence items that are certain to be sampled, are separately examined. Items with a zero book amount should also be examined separately as Hey will not be selected. If a selected dollar unit fans in the lath item, the tainting Ti (=Dil Yi) of the item is recorded. Namely, every dollar unit observation is the tainting of the item that the unit falls in. The model (2.7) may then be applied for DUS by considering ~ as an independent observation of tainting of a dollar unit. p is, then, the probability that a dollar unit is in error. Thus, (2.7) can be used for sampling individual items or individual doBars. In the former, ~ stands for the error amount of an item, and in Me latter for the tainting of a dollar unit. In the next section we present some results from several empirical studies to illustrate values of p and the distribution of z, for both line item sampling and DUS designs. 13

3. Error Distributions of Audit Populations - Empirical Evidence Why do errors occur? Hylas and Ashton (1982) conducted a survey of the audit practices of a large accounting finn in order to investigate the Ends of accounts that are likely to show errors, to obtain alternative audit leads for detection of these errors, and to attempt to identify their apparent causes. Not surpnsingly, their sway shows mat unintentional human error is the most likely cause of recording erwrs. The remainder of this section reports the results of several empincal studies about actual values of the error rate p and actual distributions of me non-zero error z In the model (2.7~. The sample audit populations are from a few large accounting finns and each contains a relatively large number of errors. Therefore, the conclusions may not represent typical audit situations. A) Line item errors. Data sets supplied by a large accounting finn were studied by Ramage, Krieger and Spero (1979) and again by Johnson, Leitch and Neter (1981~. The orientations of the two studies differ in some important respects. The latter provides more comprehensive information about the error amount distributions of the given data sets. It should be noted mat the data sets are not chosen randomly. Instead, they have been selected because each dam set contains a large number of errors enough to yield a reasonable smooth picture of the error distribution. According to the study by Johnson et al. (1981), He median error rate of 55 accounts receivables data is .024 (~e quartiles are: Q ~=.004 and Q3=.089~. On He other hand, the median enter rate of 26 inventory audits is .154 (Q ~=.073 and Q3=.399~. Thus the error amount distribution of a typical accounts receivable in their study has a mass .98 at zero. A random sample of 100 items from such a distribution will then contain, on the average, only two non-zero observations. On the other hand, the error amount distribution of a typical inventory in their study has a mass .85 at the origin and sampling of 100 items from such a distribution win contain, on the average, 15 non-zero observations. The items with larger book amounts are more likely to be in error Han those with smaller book amounts. The average error amount, however, does not appear to be related to the book amount. On the other hand, the standard deviation of the error amount tends to increase with book amount. Ham. Losel1 and Smieliauskas (1985) conducted a similar sway using data sets provided by another accounting firm. Besides accounts receivable and inventory, this study also included accounts payable, purchases and sales. Four error rates are defined and reported for each cRte~o~ of accounts. It should be noted that their smby defines emu hroadlv. since they include errors that do not accompany changes in rccordcd amounts. 14

The distribution of non-zero error amounts again differs substantially between receivables and inventory. The error amount for receivables are likely to be overstated and their distribution positively skewed. On the other hand, errors for inventory include both overstatements and understatements with about equal frequency. However' for both account categories, He distnbui~ons contain outliers. Graphs in Figure ~ are taken from Johnson e' al. (1981) and illustrate forms of the ever amount distributions of typical receivables and inventory audit data. The figures show the non-normality of error distnbutions. Similar conclusions are also reached by Ham et al. (1985). Their study also reports the distribution of erTor amounts for accounts payables and purchases. The error amounts tend to be understannents for these categones. Again, the shape of distnbudons are not normal. By Dollar unit tainangs. When items are chosen with probability proportional co book amounts, He relevant error amount distr~budon is He distribution of tintings weighted by the book amount. Equivalently, it is He distribution of dodar unit faintings. Table 1 tabulates an example. Neter, Johnson and Leitch (1985) report the dollar unit tainting distnbunons of the same audit data Hat they analyzed previously. The median error rate of receivables is .040 for dollar units and is higher than Hat of line items (.024~. Similarly, the median dolBar unit error rate for inventory is .186 (.154 for line items). The reason is, they conclude, that the line item error rate tends to be higher for items with larger book amount for both categories. Since the average line item error amount is not related with the book amount, the dollar unit tainting tends to be smaller for items with larger book amounts. Consequently, the distnbudon of dollar unit tainting tends to be concentrated around the ongin. Some accounts receivable have, however, a J-shaped dollar unit taint distribution with negative skewness. One significant characteristic of the dollar unit tainting distribution that is common for many accounts receivable is the existence of a mass at 1.00, indicating that a significant proportion of these items has a 100% overstatement err. Such an error could arise when, for example, an account has been paid in fun but the transaction has not been recorded. A standard parametric distribution such as nonnal, exponential, gamma, beta and so on, alone may not be satisfactory for modeling such distribution. Figure 2 gives the graphs of the dollar unit tainting distributions for the same audit data used in Figure 1. Note Hat the distribution of faintings can be skewed when that of error amounts is not. Note also the existence of an appreciable mass at 1 in the accounts receivable example. The situation here may be viewed as a nonstandard mixture in which the discrete part has masses at two points. 15

Fiat 1 Ex~m~esof~d~ ofEnor^moun~ (^~^ccoun~Rc~iv~Ie(1060b~0~) D~TR~UT~N OF ERROR AMOUNTS ~ ACCOUNTS RECE1V^BLE JUDE ~ 3 ~ . ~ O ~ ~00 -250 12, 7 ~ \ loo o 100 250 Error Amounts 16

Figure 1 (continued, (B) InventoIy (1139 observations) DISTRIIBU~ON OF E - - MOUNTS .s .4 0 ~ ~ .o A .2 O ,, 28~ 250 l / J / / -100 0 100 E nor Amounts ~ _ 2SO Source: Figures 1 and 2 of Johnson, Leitch and Neter (1981) 17 - / 177,6

Figure 2 Examples of Distribution of Dollar Unit Tainting (A) Accounts Receivable o 0 _ i_ ~ `1) ~ O 0 _ O _ ~ · . -1. 5 -1. 0 -0. 5 0.0 0.5 Do] lar Unit Taint 18 1 rig 1.0 1.

Figure 2 (continued) (B) Inventory ,~ o ~J o U Ln ~4 o US o _ n = -1. 5 -1. 0 -O. 5 O. O 0.5 1.0 1. Dol low Unit Tointl~g Note: The histograms are drawn using the data in Table 2 of Neter, Johnson, Leitch (1985). Their accounts receivable audit No. 69 and inventory audit No. 23 are used for illustration. 19

Table 1 Illustration of Dollar Unit Tainting Distnbudon Using a hypothetical accounting population of five items, Me difference between the error amount distribution (A) and We tainting distribution (B) is illustrated. (A) Composition of Audit Population Line Item Book Values Error 2 3 4 as Total 300 800 600 200 100 2000 Di 30 40 60 50 100 280 Taint Ti .10 .05 .10 .25 1.00 (B) Distribution of Tainting - Tainting Proportion Proportion Line Item DolDar Unit .05 .10 .25 1.00 Total .20 .40 .20 .20 1.00 .40 .45 .10 .05 1.00 20

4e The Performance of Estimators Commonly Used for Human Populations When Applied to Accounting Populations ~ this section we introduce, within the auditing context, the estimators commonly used in He survey sampling of human populations. We Hen review Heir relative performances when used In He sampling of accounting populations. Suppose that a sample of n items is taken. We denote the book amount, audited amount, error amount, and tainting of the k-th item in the sample by analogous lower case letters, namely, Ok, x', Ok Ark - ok . and ~ =dily' ~ if Ok ~ O), respectively. Denote Heir sample means by y, x, it, and A, respectively. Many estimators of the population audited amount have been proposed. First of an, the mean-per t estimator is Xm = N X . (4.~) We may also consider auxiliary information estunators to improve precision. For example, one may use the difference estimator Xa = Y - Nd . Alternatively, one may use the ratio estimator Xr =Y (x/y). (4.2) (4.3) Another possibility is a weighted average of Am and either X or Or, namely, Xw!wim+(1 - wind ~ or Xw ,2 = w Xm +( 1W ~ Xr (4.4a) (4.4b) One may also construct weighted combinations of (m, X``, and Or. Since He book amount of each item In the population is available, we may sample items with probability-proportional-to-size (PPS). As introduced In Section 2, this sampling method is commonly referred to as Dollar Unit Sampling (DUS). We may then consider using an unbiased mean-per- dollar unit estimator ~ n Xpps = (Yin ~ £ (XklYk). k=! (4.5) The auditorts main concern is with He population total error amount D and the above estimators lead respectively to Dm=Y~Xm=Y~Nx9 21 (4.6a)

~ ~ - Dd=YXd=Nd, ~ ~ - Dr= Y ~Xr = Y(dly), Dw 1= y - All = w Dm + (1w ) Dd DW,2=YXW,2=W Dm +(1w)D, Id ~ ~ n Dpps = YXpps = (Y/n ) ~ (dk/Yk ) = ~ - k=1 (4.6b) (4.6c) (4.6d) (4.6e) (4.6f) Note that the last estimator, Dpps, may be interpreted as He mean error- per-dollar unit estimator using a random sample of n dollar units. These estimators may be used with stratification on book amount in more sophisticated sampling designs. Plots of Be bivanate data, either (Xi, Yi) or (Di, Yi) may have some value in selecting anOrOnnat~ ~llYili~ information estimators. ~ _. ~^ ~~ ~~ at. In the remainder of the section we describe Me performance of these estimators when they are used in sampling from We distributions characterized by the mixture defined by (2.71. For each estimator, its precision, measured by the square root of the mean squared error, as well as the confidence levels, based on nominal approximations, of the associated two-sided confidence interval and upper and lower confidence bounds are stated. In auditing practice, the performance of an upper or a lower confidence bound is often more meaningful than Hat of a two-sided confidence interval. For example, when estimating the audited amount of ~ asset, me auditor would like to know, win a known confidence level, the lower bound of He true asset amount because of He potential legal liability Hat may follow as a consequence of overstating the measure. He, therefore, win be concerned if the true level of confidence of the lower bound is actually lower than the supposed level because this implies Hat he is assuming greater risk than intended. On He over hand, when a government agency, such as He Intemal Revenue Service, applies statistical auditing procedures when examining a firm's expense account, it is more concemed with estimating the upper (lower) bound of the audited amount (proposed adjustment), since it wants to avoid the failure to recognize allowable expenses because this could lead to overassessment of taxes. En this case, what matters most is whether the actual confidence level of me upper (lower) bound of the audited amount (adjustment) is close to the supposed level. If the actual confidence level is substantially higher man the stated level, the agency is assuming a much lower risk than allowed by the policy. 22

Win financial support by the AmencaI1 Institute of Certified Public Accountants (AICPA) and Touche Ross ~ Co. (an accounting firm in me U.S.) and Ado computing support by the Umve~ty of Minnesota, Neter and Loebbecke (1975, 1977) conducted an extensive study to examine the performance of altemative estimators in sampling audit populations. An unponant feature of this study is that He populations used in the experiment were constructed from real audit data. The study esseni~aBy confirms the obsenrabons mat Stringer reported In 1963. Namely, the estimators commonly used when sampling from human populations perfonn poorly and are inadequate for statistical auditing when He populations are contaminated by rare errors. Firstly, as discussed in Section I, Hey provide He auditor win no means to make inferences about He total error amount when all items In He sample are error-free. Secondly, Hey are either imprecise, as in He case of He mean-per-unit estimator, because the audited amount has a large standard deviation, or, as in He case of auxiliary infomlation estimators, me confidence intervals and bounds may not provide planned levels of confidence for me sample sizes commonly used in auditing practice. Table 2 gives a summary, based on the Neter-Loebbecke study, of the performance of the mean-per-un~t estimator Am, me difference estimator A, Heir combination Xw,~ win w= .l, and the unbiased estimator Xpps . The first Tree of these are under simple random sampling of items while me last one uses PPS. The ratio estimator Or performed ahnost idenucaDy with the difference estimator and so it is not included. Note Hat the confidence level of me lower bound for the audited amount is the confidence level of the upper bound of the error amount since the latter is obtained by subtracting me former from the known book value. Their conclusions can be summarized as follows: · the mean-per-unit estimator is imprecise compared to the difference estimator. Also when die population audit amount is highly skewed, me confidence interval does not provide the nominal level. The confidence level of the lower bound, however, is larger than me sated; · He difference estimator is precise but produces a confidence interval that does not provide the nominal level of confidence when the error rate is low or the errors are overstatements. For overstatement errors this failure to meet the stated confidence level for the two-sided confidence interval is caused by that of the lower (upper) bound for the audited amount (the error amount). The upper Cower) bound for the audited amount (the error amounts, however, is overly conservative. 23

i · He combination of the two estimators alleviates the problem but does not provide a satisfactory solution to all cases. Also, finding a proper weight seems to present a problem; · the performance of the unbiased estimator using PPS (DUS) is generally poor even for a high error rate. When enors are overstatements Be perfonnance is almost identical to that of the difference estimator. Namely, the lower (upper) bound for me audited value (the error amount) does not provide Be stated confidence level, whereas the upper dower) bound is conservative. As expected, stratification improves the precision of Be mean-per-unit estimator d~nancaBy, but not of the performance of the confidence interval of the auxiliary infonnai~on estimators descnbed above following (4.11. The audit data sets used by Neter and Loebbecke were men made public, and several studies followed that extended Weir results. Burdick and Reneau (1978) applied PPS without replacement and studied me perfonnance of other estimators. Baker and Copeland (1979) investigated He performance of He stratified regression estimator. Beck (1980) focused on the effect of heteroscedastici~ on the reliability of the regression estimator. Frost and Tamura (1982) applied the jackknife to reduce bias in the standard error to improve the reliability of the ratio interval estimation. The main conclusion from these studies is that commonly used auxiliary information estimators provide precise point estimates but their confidence intervals based on He asymptotic nonnali~ of the estimators do not provide confidence levels as planned, when used in sampling audit populations contaminated by rare errors. Kaplan (1973a, 1973b) was the first to recognize that the poor performance of auxiliary information interval eshmai~on may be caused by the mixed nature of the audit population. He observed Hat He sampling distnbui~on of the pivotal statistic, (Point Estimator- True Mean)/(S~dard Deviation of Estimator), may not follow the t - distribution for an auxiliary information estimator when sampling from non-standard mixtures such as these arising in audit populations. Frost and Tamura (1986, 1987) extended Kaplan's observation and showed that the mixture may cause the population error distribution to be highly skewed, especially when He error rate is low and errors are overstatements, and that this population skewness causes in turn the sampling distribution of the pivotal statistic to be skewed in the opposite direction. Therefore, the auxiliary infonnation interval estimation based on the asymptotic nonnality of the sampling distnbunon may perform poorly for the sample sizes used in statistical auditing. 24

Table 3, taken from Frost and Tamura (1987), gives estimates of me probability that me population error is outside of the upper and lower bounds, respectively, of ~ 95~o two-sided confidence interval for a difference estimator that is appropA ate under assumptions of nom~ality. In this example, the editors are assumed to be overstatements and are sampled from an exponential distnbuiion. If the normal approximation were good, each entry would be close to .025. It is clear from Table 3 that Me unwasted use of normal approximations results In two-sided confidence intervals which have unreliable upper bounds and overly conservative lower bounds. The population of accounts receivable often is contaminated by overstatement ears. As indicated before, me auditor is concerned win assuming a greater risk than intended in setting an upper bound of the population error amount in an asset account. Namely, he wants to con=l the risk of overstating Me true asset amount which may lead to legal liability for the auditor. Much effort has therefore been expended on developing altetnadve methods that might provide more satisfactory upper bounds for the error amount of an accounting population with overstatement errors. hn the next section, we win review some of these methods. The parapet problem of tightening the lower bound for the population error amount is equally important because of its implications for improving the efficiency of estimating me proposed adjusonent of reported expenses to government agencies. However, this problem has been relatively overlooked by academic researchers, and meets much greater attention. 25

Table 2 Precision and Reliability of Confidence Interval of Commonly Used Estimators for the Audited Amount (Sample Sue = 100) Note: Population numbed correspond to Me Neter-Loebbecke study populations. The error rates are adjusted within each population by randomly eliminating eITors in the 30% error population. The number in me parenthesis is We skewness of Me audited amount for (A) and me sign of Me error amount for (B). For (A) through (D), me first erg is the root mean squamd error of Me estimator in terms of Me percentage of Me total audited amount The second entry is Me estimated tree level of confidence of a two sided 95.4~o confidence interval. Me ~rd entry is me estimated true confidence level of a one-sided lower 97.7% bound. The fours entry is the estimated true confidence level of a one-sided upper 97.7% bound computed from the second and the Bird entries. These estimates are based on 600 independent samplings of size 100 from the corresponding population. A "-" indicates Be entry is not available from me sway. 26

(A) The Mean-Per-Unit Estimator, Am Population Etror Rates as Percentages Population .5 1.0 5.0 10.0 30.0 1 (22) 24.1 24.0 - 24.0 23.9 81.8 81.8 - 81.7 81.7 100.0 100.0 - 100.0 100.0 81.8 81.8 - 81.7 81.7 2 (3.5) 18.2 18.2 18.3 18.2 18.3 93.? 93.7 93.7 93.7 93.0 99.5 99.5 99.5 99.5 99.3 94.2 94.2 94.2 94.2 93.7 3 (7.9) 35.7 35.7 35.7 35.8 36.1 82.5 82.5 82.5 82.5 82.5 99.7 99.7 99.7 99.7 99.7 82.8 82.8 82.8 82.8 82.8 4 (3.3) 20.2 20.2 20.4 20.7 21.4 92.7 92.7 92.3 92.8 93.2 99.5 99.5 99.3 99.5 99.8 93.2 93.2 93.0 93.3 93.4 27

OB) The reference Estimator. Xd Pop~anon Enter Rates ~ Pemen~ges Pop~adon .5 1.0 5.0 10.0 30.0 1 (+/-) 0.1 0.2 0.4 0.6 0.9 30.5 37.3 96.8 94.0 96.3 38.0 61.5 97.2 99.7 99.5 92.5 75.8 99.6 94.3 96.8 2 (+/-) .2 0.4 1.0 1.3 3.0 31.2 41.8 82.3 97.2 95.5 41.3 41.8 99.7 99.7 98.8 89.9 100.0 82.6 97.S 96.7 0.1 0.1 23.3 36.8 23.3 36.8 100.0 100.0 4 (a) 1.1 21.2 21.2 100.0 0.1 73.7 73.7 100.0 1.1 1.6 30.0 58.2 30.0 58.2 100.0 100.0 0.2 80.3 80.3 100.0 3.5 9.2 62.0 74.8 62.0 74.8 100.0 100.0 0.4 90.8 91.2 99.6 28

(C)lleCombinadon,iw,l,ofXm end Xd ~ndhw =.1 Populadon Error Rates as Penances Populabon .5 1.0 5.0 10.0 30.0 1 2.4 2.4 - 2.5 2.5 82.3 82.0 - 83.3 84.3 100.0 100.0 - 99.8 99.8 82.3 82.0 - 83.5 84.5 2 1.8 1.9 2.1 2.2 3.6 93.8 94.2 94.3 94.5 93.8 99.5 99.5 99.5 99.2 99.2 94.3 94.7 94.8 95.3 94.6 3 3.6 3.6 3.6 3.6 3.6 82.7 82.5 83.0 82.8 82.3 99.7 99.7 99.7 99.7 99.5 83.0 82.7 83.3 83.1 82.8 4 2.2 2.3 2.5 3.8 8.5 93.7 94.0 94.7 94.8 78.7 99.3 99.3 99.0 95.3 78.7 94.4 94.7 95.7 99.5 100.0 29

(D) Unbiased Estimator with PPS, Xpps (Mean per Unit Estimator with Dollar Unit Sampling) Population Enor Rates as Percentages Population .5 1.0 5.0 10.0 30.0 0.2 0.2 - 0.6 1.0 17.5 49.7 - 90.3 92.2 20.2 60.7 - 99.7 99.8 97.3 89.0 - 90.6 92.4 2 1.0 80.2 _ 99.2 81.0 0.1 5.2 5.2 100.0 0.5 30.7 30.7 100.0 0.3 31.5 31.5 - 100.0 1.0 69.7 69.7 100.0 1.5 _ 94-5 100.0 94.5 0.5 0.9 44.8 77.0 44.8 77.0 100.0 100.0 1.8 39 86.5 94.5 87.0 95.5 99.5 99.0 Sources: dieter and Loebbecke (1975) Tables 2.3, 2.8, 2.11 2.14, 3.19 3.2, 4.2, 4.5, 5.3, 5.5, 10.1 and 10.3. Also tables in the appendix. 30

Table 3 The Reliability of me Upper and tile Lower Bounds of 95% Two- Sided Confidence Intel s for the Difference Estimator of the Population Error Note: Exponential distribution is used to model overstatement errors. The first entry is Me probability mat the population ever exceeds the upper bound of a two-sided 95 % interval using normal approximation. The second entry is the me probability that me population error is lower than the lower bound of the two sided 95% interval. The standard eITors of these estimates are within .001. Sample Population Error Rates as Percentages Size 1 2 4 8 16 50 .650 .483 .324 .213 .140 .000 .000 .000 .001 .002 100 .485 .329 .216 .145 .097 .000 .000 .001 .002 .004 200 .327 .215 .142 .101 .072 .000 .001 .002 .004 .007 400 .219 .144 .101 .070 .055 .001 .002 .004 .007 .010 800 .144 .099 .070 .054 .045 .002 .004 .007 .010 .013 31

5. Confidence Bounds Using Attribute Sampling Theory - Beginning with this section, we survey altemative solutions mat have been proposed for Me problem of estimating total error amounts in accounting populations. There are two main approaches: (~) approaches that utilize attribute sampling theory and (2) approaches that utilize Bayesian inference. Combinations of (~) and (2) have also been proposed. Odler approaches include the modeling of die sampling dis~ibunon by distnbudons other than the nonnal. A) Line item attribute sampling. The simplest application of a~bute sampling theory is to audit situations In which it is assumed that aU ears are overstatements with the maximum size of the error amount equal to the book amount, namely, o<Di ~Yi (or equivalently, O < Ti < I) for i =l,...,N. Let YE be the known maximum book amount' i.e..Y; < Yr for i = 1....~. Since N is large relative to He sample si7.e.s . .. . . .. . . . used in practice, we may propose me following model for random sampling of n items from the accounting population with or without replacement. Let p be the proportion of items with errors in the population. Then, for any item in the sample, me observed error, g: z with probability p - O with probability (1 - p ~ . Since z < Y., and E (Z ) = ~ ~ YL . E(d)= |1D =P~ UP YL . Hence the total error amount D as defined in (2.6) becomes D=Np D<Np YL (5.~) (5.la) (5. lb) (5. Ic) Suppose that a sample of n items contains m items with enors and (n-m ) erwr-free items. Let P^u (m; 1~) be a (1-a) -upper confidence bound for p . That is, Prob {p'§u~m; Imp)} = 1-a. (5.~) We may use a binomial distribution or, when p is small, a Poisson distribution as widely practiced. Then a Amp) -upper confidence bound forD is Du~m;1-a)=N (u(m;1{~) YL (5.2) Since observed values of z are not used, Me upper bound (5.2) can be overly conservative, and stratification on book amount may be used to 32

tighten me bound (Fienberg, Neter and Leigh (1977)). By Dollar unit attribute sampling. Let us suppose Mat the sample is taken by means of Dollar Unit Sampling (DUS, Section B.2~. As before, an errors are assumed to be overstatements wad me maximum size of the error amount equal to the book amount. Thus O CTi cI. The mode! (5.~) may Den be applied for the analysis of the DUS data by considering ~ as an independent observation of dollar mat toning with O < z < 1. ~ is the mean dollar unit tainting. In this case, p is Den We proportion of dollar mats in error and is equal to Y.~/Y, where Y`' is me tote book amount of items in error. Thus, in this case, D =Yp 8<Yp . Given m non-zero tainting observations, a (~) upper bound for me tom error amount D is Du pus (m, lax) = Y PU (m; I~) Since Y ~ N Ye, D,,, - ~ Du (5.3) The confidence level of the bound (5.3) is at least (1 - a) but the bound is sdU conservative because it assumes that aU faintings are equal to 1. C) The Stringer bounds. When m > 1, we have dollar mat tintings Zj (~=l,...,m) in He sample that may be less than 1 and this information can be used to tighten the bound. There are several ways of accomplishing this task and these procedures are co'Tunonly referred to as combined attributes and variables (CAY) estimation (Good~eBow, ~ebbecke and Neter, 1974~. The best known and most widely used CAV estimation is credited to Stringer and is caned the Stringer bound. Bet ° < Zm ~ ~ Z ~ < ~ be ordered observations from a random sample of size m from z . Then a (1~-Stnnger bound is defined by D,, at = Y ~ Pu(O;~) + 2,LP,.(.J ;~) ~P^~ };~] Zj ~ (5 4) Here, P^u (0;1~), being the (1~) upper bound for the error rate p when the sample contains no error, is equal to ~ - atin using the binomial distribution. The Poisson approximation In is widely used among practitioners. When Zj= ~ for an j, (5.4) reduces to (5.31. It is as yet unknown whether the Stringer bound always provides the confidence level at least as large as the nominal for overstatement errors. Indeed, commonly used CAV estimations are heuristic and it is difficult to determine theoretically their sampling distnbudons. Many simulation studies, however, have been performed using venous error distributions 33

rowing from the populations used by Neter and Loebbecke (1975) to standard parametric models. These series have provided strong empirical evidence that the confidence level of Me Stringer bound is at least the nominal level. An fact these series indicate mat it may be overly conservadve (see, for example, Leitch, Neter, Plante arid Sinha, 1982; Roberts, Shedd and MacGuidwin, 1983; Reneau, 1978~. However, Me fom~ulabon of the Stringer bound has never been satisfactonly explained. Not even an inhibitive explanation can be found In auditing literature. For audit populations contaminated by low error amounts, Me bound may grossly overestimate the total enor amount, causing the auditor to conclude that Me total book amount contains a material ever when it does not. The ensuing activities, e.g., taking additional samples, or requesting He client to adjust the account balance, etc., may be costly to He client. The development of a more efficient CAV estimation procedures has Gus become an important research goal. D) Multinomiat bounds. One approach toward this goal, involving an interesting application of attribute sampling theory for esi~manon of me total error amount, has been developed by Fienberg, Neter and Leitch (Fienberg, Neter and Leitch, 1977; Neter~eitch and Fienberg, 1978~. Since their mode! uses me multinomial distribution, He resulting upper bound is commonly caned a multinomial bound. The obse~va~aon of dollar mat taint is categorized, for example, in 101 classes ranging from 00 to 100 cents. Let Pi be the probability that art observation on ~ falls in me i-th category (i cents), where 0 Hopi ~ ~ and ~ Pi = I. Then, instead of (5.~), we have . ~ = 100 wi~probability pi, i =0, ,100. (5.5) Then so mat ~~ i E(dl)= ED = by, DO Pi . loo i D=Y pD=Y ~ ~~ Pi (5.Sa) (5.5b) Let wi be the number of observations in a sample of n dollar units that fall in the i-th category; Kiwi = n . Clearly w = (w0, w ~,...,w~00) follows a multinomial distribution with the paratneeers (n . P). P = (Po. P 1.~ 100), if the sampling is done wide replacement (If the sampling is done without replacement, men this may still be used as an approximate model.) The sample mean is 34

_ ~ 100 i Wi ED I.~; 100 n and a point estimate of D is given by D =YRD (5.6) The following procedure is then proposed for an upper bound for AD and hence for D . Let S be a set of outcomes v = (vO, . . ., v 10~) Mat are "as extreme as or less extreme than" the observed results w. The concept of extremeness must be specified. Clearly, S is not unique and computational simplicity must be taken into account for its definition. The particular S proposed by Fienberg et al., called the step-down S. is the set of outcomes such that (~) the number of errors does not exceed the observed number of errom and (2) each error amount does not exceed any observed error amount. Given this step-down S. a (~ - c) - joint confidence set for p is determined by those values Of Pi mat satisfy At, n! rIPi vi > a, All =n . (5.7) A (1 oc)-upper bound for 69 and hence also for D by multiplying We former by Y. is obtained by maximizing (S.Sa) over those p defined by (5.71. However, the true level of confidence of me multinomial bound using He step-down S is not known. When the sample does not contain any errors, the multinomial bound is the same as me bound given,in (5.31. When me sample contains errors, the muThnomial bound is considerably fighter than the widely used Stringer bound. However, the computation of the bound quickly becomes unmanageable as the number of enors increases. Neter et al. (19783 reported that the array size of the step-down S set is 429 x 9 for 6 errors, 1430 x 9 for 7, and 4862 x 10 for ~ errors. When using a computer of the size of a large IBM 370, they only did the computations for up to 7 editors. Software for personal computers that win compute the bound for up to 10 errors is now available from Plante (198711. Leitch, Neter, Plante and Sinha (1981, 1982) propose to cluster the observations for improving the computational efficiency of the muliinomial bound for many errors. The observed errors di are grouped into g groups of similar sizes. Then all errors in He same cluster are given the maximum error amount of He cluster. In forming an optimum set of clusters, the aIgonthm that minimizes: C = ~ ~ max (dkj ~dkj ). (5.8) where dkj is the j-th tainting in the k-th cluster, is recommended. The 35

loss of efficiency due to this clustenng cannot easily be assessed since the bound has not been computed without Me grouping of observations when the number of ears are many. L~eitch et al. (~98 l) reports, however, that USA 20 to 25 enters In We sample, the multinomial bound with five to six clusters still compares favorably win me Stringer bound. Plante, Neter and Leitch (1985) provides a study mat compares me muldnomial upper bound with me two CAV upper bounds, i.e., the Stringer and cell bounds Leslie, Teitlebaum and Anderson, 1979~. The muldnomial bound is the tightest of me Tree and the observed confidence {ever is not significantly different from the nominal level .95 used for me study. If the auditor knows me maximum size of the understatement error, it is possible to apply the multinomial approach to set me upper bound. Also, Tough conservative, a lower bound can be set. (See Neter, Leitch and Fienberg, 1978~. 36

6. Other Developments for the Analysis of Dollar Unit Sample Data In this section, we give a brief account of some other approaches to die statistical analysis of dollar unit sample data. Firstly, there are proposals for approximating Me sampling distribution of the mean tainting using m~els over than nonnal distributions. Secondly, in order to set a tighter bound, We use of pammetnc models for me distnbudon of tainting has been suggested. We win discuss these attempts below. A) Nonnormal sampling dFistributions. Along me Ime of improving Me reliability of large- sample, classical interval estimation, Garstka and OhIson (1979) suggested a modification of the constant by which me estimated standard deviation is multiplied, to make it dependent upon the number of errors found in me sample. Tamura (1985) comments, however, that this modification does not take an account of me skewness of me sampling distribution and may not always produce a satisfactory result. Dwonn and GrimIund (1984) propose approximating the sampling distnbudon by a Tree parameter gamma distribution. The method of moments is used to estimate the parameter values of the approximating gamma distnbution. A number of heuristics are invoked including the introduction of a 'hypothetical tainting observation' for computing me sample moments. The method of computing this third data point vanes slightly depending on whether the audit population is accounts receivables or inventory. The method can handle bow over- and understatement errors, however. Through extensive simulation tests, they show that the upper bond computed by their memos provides the confidence level close to the stated. Moreover, they show that the moment upper bound is about as tight as the multinomial upper bomb. By Parametric models. A nature way to improve the efficiency of a bound is to describe the error distribution using a parametric model, following Aitchison (19551. This point was illustrated by Marsha (1977~. By treating an obsewai~on of doDar unit tainting In teens of smaller units, say ten cent units, he uses a geometric distnbution as a model. A parametric bound can be sensitive to the choice of Me model. LiBestol (1981), using a logarithmic series distribution instead of a geometric distribution, demonstrates the point. Recently, Tamura and Frost (1986) have proposed a power Unction density to model ta~nungs. Efron's parametric bootstrap is used to approximate the sampling distnbunon of Me estimator for setting the bound. The study shows that the bootstrap bound is reliable and much tighter man the non-parametnc Stringer bound when the data are generated from the correct specification. Research to compare perforTnance of different models of tainting, including Weir robustness to parametric assumptions, may prove to be fruitful for achieving economical and efficient auditing. 37

7. Bayesian Models for the Analysis of Audit Data From the discussions presented so far, we may sununanze two basic problems associated win statistical auditing. Firstly, it is difficult to determine We small sample sampling distribution of the estimator when Me population is chamctenzed by the mixture. Secondly, because of the low error rate Me sample does not provide sufficient information about Me character~si~cs of the r~n-zero error distnbution of me audit population to set a good bomb. Earlier in Section 3, this report reviewed the resets of empincal studies of the error distnbution of venous accounting populations. These results have provided auditors win considerable evidence about the audit environment. Using such results, an auditor may make a more intelligent prediction about the error distribution of certain audit populations. By incorporating this prior information into He analysis of the sample data, the auditor should usually be able to obtain a more efficient bound for a total population error. Bayesian inference provides a useful framework to nco~orate the auditor's informed judgment win the sample information and we will review in this section developments along this line of audit data analysis. Empirical Bayes methods do not seem to have been used on these problems, a direction that may be worth investigation. A) Normal error models. Felix and G'imiund (1977) propose a para~netnc Bayesian model. They assume audit item sampling but their mode] has also been applied for doDar unit sampling by Menzefr~cke and Smieliauskas (19841. In their fonnulation the error amount z in the sampling model (5.~) is assumed to be normally distributed dependent on the mean ,uz and the precision h, the inverse of the variance of. At is given a nonnal prior mat depends on h. h is given a gamma pnor. The joint distribution of fez and h is open referred to as a Normal-gamma distnbudon. The prior distribution for the error rate p is given a Beta distribution and is independent of (tLz, h ). These prior specifications are conjugate with the likelihood function of the data as determined by the sampling model, i.e., the posterior distribution of (~z,h) is again a Normal-gamma and mat of p is Beta. The two posterior distributions are again independent. In order to develop the posterior distribution for the population mean ,uD=P~z, first, h is integrated out from the posterior distribution of fez, h ), positing in a Student distribution for me margins distribution of Liz. Since ~D=P Liz. substituting At with pD/P in the marginal distribution, and integrating out p, the posterior distribution for ED iS obtained. The result of this integration cannot be written explicitly and has to be numerically obtained. However, the expected value and the 38

v anance can be denved. (Felix and GnmIund denve the posterior distnbudon by using a different approach Wan described here. However, Menzefncke and Smieliauskas (1984) show that in their approach, a source of variability has been overlooked, which leads to a smaller variance than should be the case.) By Infinite population models. The probability models Hat urlderly He methods discussed so far may be referred to as finite population models. By this nomenclature one stresses the fact Hat the lists of book values Y1, .YN and audited amounts X I, , AN that are associated win the financial statements et the time of audit, are finite In number and considered fixed. There is no randomness associated with these values. Randomness, and hence He need for a probability model, enters only by the act of sampling n book values from He populai~on's N values. The memos of sampling determines which probability mode] must be conside - . One may consider other probability models in which me Yi 's and Xi 's are themselves viewed as having been randomly generated. For example, at the start of a fiscal year the line items in a company's books are yet to be determined, and from that point In time it might be appropriate to view them as random vanables subject to some probability distnbui~on. To contrast Hem with the sampling models previously discussed, these globally random models can be referred to as infinite population models. In 1979, Cox and Snell proposed such an infinite population model. The importance of their proposal is that it provides a theoretical basis for DUS methodology, something that had not previously been available. In Heir memos, the account entries, Yi , and Heir correct values Xi , are viewed as being outcomes from N independent repetitions of an experiment whose probabilistic descnpi~on is therefore completely specified by the common joint distnbuiion function of the pairs (Yi, Xi) for ~ = 1, , N. Equivalently, one could specify the joint distribution of (Y', Di ) since Yi - Di = X, . As stressed before, a large portion of the errors Di win be zero in auditing situations. Consequently, when modeling me distnbution functions of He Di, one should use distnbunons that place a large probability at zero. One way to Fink of the generation of such errors is in two stages as follows: first, determine whether there is in fact an error, and then, if there is an error, determine its value. More specifically, introduce Si to be O or 1 according as to whether or not Di = 0. To describe the mode] for (Yi . ~i, Di ), one may begin by specifying He marginal distribution of Yi, Fy say; then the conditional distribution of hi = 1 given Yi,p(Yi) say; and then the conditional distribution of Di given Yi and 39

8: -1, F D t y say. The conditional distribution of Di given Yi and hi = 0 is degenerate at zero by definition since Di is zero when hi is zero. - If bod1 FY and FD' y are assigned to have densities' fly (y) and f D! Y (d[. Y ~ say, Den the triple (Yi . ~i, Di ~ has the density function f Y,A,D defined by and ~ . . ~ , . , fy,e,DQ.O.O) =fyQ){1pry)), (7.1a) fY,A,D~.l.d)=fY0)p0)f Dl y~d,y). (7.1b) Although this model allows for He probability of an error to depend on the magnitude of Be observed book value, the most common practice is to assume that this error probability p ~ ~ is a constant. In order to complete the description of the probability model, it rem awns to specify how the sampling selection is determined. For this, Cox and Snell (1979) introduce a "sampling" vanable, Si say, which equals 1 if me i- item is to be sampled, and O otherwise. In general, one would then specify the conditional distribution of S`, given the over obsenrai~ons Yi, hi and Di. An important special case is that of probability- propomonal-to-size (PPS) sampling and for this, one would set Prob (Si =1 I Yi . ~i, Di ~ = cYi (7.2) Under the PPS sapling design, there is a particularly simple relationship between the conditional means of the tastings, Ti =DiI Yi, and the ratio of conditional means of Me errors Di to the book amounts Yi . It can be shown, by straightforward manipulation of conditional expectations, that as a consequence of (7.2) E(y ISi=l,Si=l)=-E`yl'5_~. (7.3) Equation (7.3) is of fundamental importance in this model. From it, one can derive an expression, (7.7b) below, for the population mean error in terms of the population error rate and We conditional mean of the tainting. This is the relationship mat is used in the analysis of DUS data. To do ~is, begin by multiplying both numerator and denominator of the ratio in (7.3) by Prob Pi = 11. The ratio becomes E (Di Si )/E (Yi Hi ). where we have made use of the zero-one nature of ~i. If Z denotes a random variable whose distr~bunon is the same as the conditional dis~ibui~on of the tainting Ti= Di I Yi, given Mat it is non-zero (8i = I) and is sampled, (Si = I), then the left-hand side of (7.3) is me mean of Z. say Liz. Thus 40

(7.3) may be written as E(D'~i) = E(Yi ~i) fez Moreover, if ED =E (Di ) denotes We mew ear, ED =E(Di~i - E {Di(l~i)} =E~i~i) . (7.4) (7.5) since Di=0 when ~i=O. Now in~duce ps to be me probability of an item being in error given mat me item is sampled; Eat is, Ps = Prob {6i=1 I Si=1 } . By (7. lb) and (7.3), direct computation yields E {cYip(Yi)} E{E(Yi~i IYi)} _ . _ Ps = c By Upon substitution of this in (7.4) one obtains me important relationship ED = GYPS LIZ (7.6) (7.7a) In many situations it is assumed that p (y ), the conditional probability that an error is present in a book amount of magnitude y, is a constant, say p. In this case, Yi and hi are independent so that Ps = p and E(Yi I hi = 1~=,uy. Then, (7.7a) becomes ED = BY P LIZ (7.7b) It should be noted, however, that me empinca] evidence reported in Section lI.3 indicates that the assumption of constant ply) may not be justified in ah cases. Relations involving higher moments of the tailings may be denved for this mode! by similar analyses. In particular, for k = I,2,..., the analogue of (7.3) is E~Zk' E(D' IY' Ps MY (7.8) from which information about the variance and skewness of Z can be obtained. The proportionality constant c in (7.2) satisfies Prob(Si = 11=c ,uy, where ~y=E(Yi) for all i. The number of line items sampled n in this infinite population model is the random quantity S 1+ --+SN . Thus n is a Binomial (N. c sty) random variable win expected value N Prob(S`=~=Nc~y. Thus c plays the role of determining me sample 41

size. The important difference is that whereas n is fixed in me finite population model, Me sample size in this infinite mode] is random. If Prob(Si=~) is small while NProb(Si=l) is moderate, a Poisson (Nc By ~ approximation might be used for We exact Binomial distribution, as was done by Cox and Sned. Suppose that ~ PPS sample of n items contains m items with errors and n -m items error free. Let Z=(Z} Zm) be tatntings of the m items with error. For estimation purposes, let us set p =m/n And z =~zi/m. In view of (7.5), a natural point estimate of Me population total error is Men D =N1lypz. (7.9) The known tote] book amount Y is used to estimate N by in practice. This can also be viewed as fitting the infinite mode! to the existing finite audit population. Using Y to stand for the known total book amount as defined in Section 2, we get Dcs = Y p z. (7.10) C) The Cox-Snell parametric models. Cox and Snell proceed to formulate a parametric B aye sian model in a fashion similar to the Grimlund and Felix model. The prior distribution of p is specified by a gamma distnbut~on with parameters a IpO and a, where pO is the prior mean of p . Z is assumed to have an exponential density and its parameter 1 / At has also a gamma distribution {(bI) p0, b I, where ~ is me prior mean of fez. These two prior distributions are assumed independent. Then it can be shown that the posterior distribution of ED iS a scalar transformation of an F-distr~bution. Specifically (see Cox and SneD, 1982; Moors, 1983), if = indicates equality of probability laws, and if F~,,V2 denotes a random vanable having an F distribution with the numbers of degrees of freedom vat and v2, m z+(b-~)lio m+a n +a/ pO m +b F2(m+a),2(m+b) . (7.] I) Godfrey and Neter (1984) investigate the sensitivity of the Cox-SnelB bound to its parametric assumptions. For example, since O up <l, the effects of truncating the gamma prior for p at I.0 as weld as replacing it with the beta prior are investigated. Similarly the effects of truncating the exponential distribution at 1.0 for Z are considered. Since the distnbution of tainting often has a discrete mass at 1.0, its effect is also studied. For these moderate parametric modifications, the bound appears relatively 42

stable compare to the effect of the prior parameter settings on the bound. The practitioners' interest may, however, lie in Me sensitivity of the performance of the bound to We prior parameter settings under repetitive sampling. Their study, using 21 hypothetical audit populations, shows that Me reliability of Me Cox-SneB bound is, as expected, sensitive to changes In prior parameter values but that it is possible to set these values conservatively so that the bound has a confidence level close to Me nominal and still tighter man the Stringer bound for this set of study populations. Neter and Godfrey (1985), extending Weir earlier study (Godfrey and Neter, 1984~, show Mat me sensitivity of Me Cox-SneB bound to parameter settings does not disappear when me sample size is increased to 500, a size seldom exceeded in current audit practice. (The sample size of 100 is used in their previous study.) The study goes on to use another set of 9 sway populations to identify some conservative prior parameter settings for which Me bound is reliable and tighter than the Stringer bound. Menzefncke and Smieliauskas (1984) investigated the gain in tightness resuming from parametric modeling of tanning The performance of Bayesian parametric bounds is compared with that of the Stringer bound and other CAV bounds. The Bayesian bounds include the Cox-Snell bound and two versions of the nonnal ever model introduced earlier. Only one parameter setting is used for each model. Their study uses audit populations contaminated by both positive and negative faintings. Since the Cox-SneU mode] assumes tastings to be positive, ad hoc adjustments are teed. Using sunulation, Hey show that the Bayesian bound, using me normal distribution to mode] errors, outperfonns both CAV bounds. D) Nonparametric Bayesian m~els. The empirical evidence reported in Section 3 shows that standard distributions may not work for modeling the distribution of dollar unit tainting. A Bayesian nonparametnc approach may then provide a necessary flexibility for modeling of available audit information. In Section 5 an example of a nonparametnc error distnbut~on was introduced, where the dollar unit tainting of an item in the sample is treated as a random observation from a discrete distribution (i ~Di) for i = (0, ,100) in cents and Pi > 0. [Pi = 1. Tsui, Matsumura and Tsui (1984) propose the Dinchlet distnbui~on to incorporate the auditor's prior prediction of the unknown Pi In their mode] Me auditor is assumed to provide the best prediction p= (pep ·,p~O0) of p = (o 0, up ~00) and a weight K for the prediction. It is then suggested that the prior distribution of p is a Dinchlet (K p). The distribution of Pi is dlus a Beta ~ K Pi ~ K (1 - Pi )) with 43

and E(oi)=Pi, (7.12a) Yar(oi)= P K P' . (7.12b) Let u =(wO, ,wl00) with 2;wi =n be the sample data of n items. w is distributed as a multinomial distribution (n, p) when sampling is win replacement (if sampling is without replacement, approximately). Since the Dirichlet prior distribution is conjugate with the multinomial sampling model, the posenor distribudon of p is again a Dinchlet distribution with the parameter (K p+w). We may define K'=K+n. and wi {i = n ' (7.13a) (7. lab) p, = (Kpi+npi) 9 i=1 100. (7.13c) Then the posterior distribution of p is DinchIet (K' p'), where P' - (P'o9 .P'1oo) By We definition of AD: 100 i D=~ 1~ Pi , (7.14) the posterior distribution Of ED iS denved as a linear combination Of P'i. It can be shown that 100 i E(l1D)= ~ 1ooPi, and (7.15a) Var(iLD)= ~1 1 {my O )2p'i-(~ ~ P'i)2} (715b) The exact distnbui~on of ED iS complicated and therefore is approximated by a Beta distnbution having the same mean and the vanance. Using simulation, Tsui et al. suggest that K =5. go= 8. p~oo=.IO1 and rem airiing 99 Pi's being .001 be used as Me prior sewing for Weir upper bound to perform wed under repeated sampling for a wide variety of tainting distributions. 44

McCray (1984) suggests another non-pa~etric Bayesian approach using me muldnomial dis~ibudon as the data generating model. ~ his model, ED ho bun di~retzed, involving a nor of categones, say ADS, j=l,, Nil. Me auditor is ~ provide As assessment of Me prior distribution by assigning probabilities qj to the values, pDj. Men the posterior distribution of AD iS dete=med to be ~jL(Wl~D,) Prob (~D =~Dj ~ W) ah L (w! pD4) (7.16) where 100 L(wl pDj)=max n Pi i in which Me maximum is taken over all probabilities ( Pi } satisfying No ~ ~Dj Pj = ED · t ' '=1 (7.17) (7.18) It should be noted that Me two nonpararnetric models introduced above can incorporate negative faintings; mat is, the auditor defines any finite lower and upper limits for tainting and divides the sample space into a Unite categories. Simulation studies have been performed to compare performances of these Bayesian bounds with the procedures described in earlier sections. Dwonn and Grimlund (1986) compares the performance of Weir moment bound with that of McCray's procedure. Several Bayesian and non- Bayesian procedures are also compared in Smienauskas (1986~. Gnm~und and Felix (1987) provides results of an extensive simulation study that compares the long run perfornances of me following bounds: Bayesian bounds with normal error distnbution as discussed in A) above, the Cox and SneU as discussed In C), the bound of Tsui et al. as discussed in D) and the moment bound discussed in Section 6. Recently, Tamura (1988) has proposed a nonparametnc Bayesian model using Ferguson's Dirichlet process to incorporate the auditor's prior prediction of the conditional distnbution of the enor. It is hypothesized mat the auditor cannot predict me exact fume of the error distribution, but is able to describe the expected form. Let Fritz) be the expected distribution function of z representing the auditor's best prior prediction. The auditor may use any standard parametric mode] for Fo. Altemadvely, Fo may be based directly on past data. The auditor assigns a finite weight Oo to indicate his uncertainty about the prediction. Then the auditor's 45

prior prediction is defined by the Dir~chlet process With the parameter adz ) = aO Fo(z ). (7.19) This means that Prober Liz') is distnbuted according to the beta distnbu~aon Beta (a(Z')9a( - XtZ'))- The posterior prediction given m observations on z, say z = (at, .... Zm), iS Men defined by the Dinchlet process with the parameter Adz ~ z) = {aO+m ~ {Wm FOd{~~Wm) Em }(Z), (7.20) where aO {aO+m } and Fritz) is the empirical distribution function of z function of the mean ~ of z is Even by . Get) = Probe) =Prob(T(~)<0)9 where the characteristic function of T(~) is 00 (7.21) The distribution (7.22) ¢(v ) (u ) = expE- ~ log ~ I-iu (t -v ) ~ ~ aft )1. (7.22) The distnbudon of ~ is obtained by numerical inversion of (7.22). The distnbution function of the mean tainting 11 is, den, given by Hand ) = Prob (A ~ dl ) = Prob (p id < d() = E (`d < drip I p ). (7.23) This integration can be done numerically. In this work, a beta distribution is proposed to model p. 46 -

S. Numerical Examples In Sections 5 though 7 various methods for setting a confidence bound for the accounting population error were described. They differ from Me classical methods of Section 4 in the sense Mat these me~ods do not assume Mat the sampling distnbudons of their estimators are nonnal. Among these new developments, we illustrate in this section me computation of the following upper bounds for the total population error: Me Stringer bound, the multinomial bound, parametric bounds using the power function, and the moment bound. In addition, computation of two Bayesian models developed by Cox and SneB and Tsui et al. will also be illustrated. Software for computing aB but one of these bounds can be developed easily. The exception is Me multinomial bound, which Squires extensive programming unless He number of errors in He sample is either O or 1. These methods are designed primarily for setting an upper bound of an accounting population error contaminated by overstatements in individual items. The maximum size of the error amount of an item is assumed not to exceed its book amount. These mesons also assume DUS. Under this sampling design the total population en or amount is equal to He known book amount Y times the mean tainting per doBar unit DIP FEZ. We win, therefore, demonstrate the computation of a .95 upper bound for AD using each method. We data used for these illustrations are hypothetical. Our main objectives are to provide some comparisons of bounds using the same audit data and also to provide numerical checks for anyone who wishes to develop software for some of the bounds illustrated in this section. A) [lo errors in the sample. When Here are no errors in a sample of n dollar uriits, the Stnnger, multinomial, and power function bounds are identical and are given by the .95 upper bound for the population enor rate p. The bound is therefore directly computed by Pu (0; 95) = 1 - 05~1In (~.1) using the Binomial distribution. For n = 100' ~Su(0;.95) = .0295. In practice, the Poisson approximation of 3.0/n is often used. The computation of the moment bound is more involved but gives a very similar result. For Bayesian bounds, the value of a .95 confidence bound depends or1 He choice of He prior about the error distnbunon. Using extensive simulation, Neter and Godfrey (1985) discovered that for certain priors tile Cox and Snell bound demonstrates a desirable relative frequency behavior under repeated sampling. One such setting is to use the following values for the mean and the standard deviation for the gamma prior of p 47

and Liz, respectively: po=.lO, up= .10, - .40, and cs~=.20. These can be related to me parameters a and b in (7.11) as follows. a = ~ O/csp )2, b = (iLo/C~~)2~2.0. (~.2a) (~.2b) Thus for no enamors in the sample, i.e., m=O, using Me above prior values, we compute a = (.10/.10~2 = 1, b = (.40/.20~2+2.0 = 6. The degrees of freedom for me F distribution are 2(m~a ~ and 2(m+b), so for m=0 they are 2 and 12,respec~vely. Since We 95 percendie of F2,~2 is 3.89, and the coefficient, when n=100, is mz+(b-l)Yo m+a (~11.40 1 = = 00303 n +a /p O m+b 1~1.0/.10 6 the 95% Cox and Snell upper bound is .00303 x 3.89 = .01177. For another Bayesian bound proposed by Tsui et al. we use the prior given in Section 7, namely, the Dinchlet prior with parameters K=5.0, pO=.8, peso = .101, and Pi = .001 for i = 1,..., 99. Given no errorin a sample of 100 dollar unit observations, me posterior values for these parameters are K'=K+n=105, and p'O= (K p`~+wo)/K'=~5~.~+1001/105 = .99048. Similarly, p'~OO= (5~.101~/105= .00481, and p'i=~5~.001~/105~= .00004762 for i=1,...,99. The expected value for the posterior ED iS Ten E(11D)=( 1oo+100+ +1900~.00004762+11OO.00481=.007167. To obtain Var (~D ). we compute E (~D) = .0063731 so that Var (~D ~ = (ILL? ~ { E (IID ~ ~ The posterior distribution is, then, approximated by me Beta distnbution having the expected values and the v anance computed above. The two parameters a and ~ of me approximating Beta distribution B (a,~) are 48

E ( ) rL E (IID ) ~ 1E (I1D ) ~ 1: and r 13 = { 1E (~D )} I (~D ){ 1 E (~D )} = .8489 1 1 = 117.46. The upper bound is Men given by me 95 percentile of me Beta distribution win parameters .848 and 1 17.46, which is .0~27. B) One error in the sample. When me DUS audit data contain one enter, each method produces a different result First of an, for computation of the Stnuger bound, we determine a .95 upper bound for p, Ad,, (m ,.95) for m=0 and 1 . Software is available for computing these values (e.g., BELBIN in Intemational Mathematical and Statistical Libraries OMSL)~. We compute 0,.95-.0295 and Mu (1, 95) = .0466. Suppose that the observed tainting is ~ =.25. Then a .95 Stringer bound is Pu (0,.95 - to (1, 95 - PI (0,.951) = .0295+.25~.0460.0295) = .0338. Second, the multinomial bound has an explicit solution for one error. It is convenient to express Me observed minting in cents so set {=100~. Denote also a .95 lower bound for p as p'(m ,.95) when a sample of n observations contain m errors. Then a .95 multinomial bound for m=l is given by ({p^' +lOOp ioo)/lOO . where pi and p loo are determined as follows. Let To = max Then and (100 {)(n-1) 1/n ~ 1 .05 ,^] Pt'= p~n-~-P(),(, P 100 = 1poPV , pi(n-l,.95) (8.3) (8.4) (8.5) To illustrate the above computation, using {=25 and n=100, we compute mat p^'(99,.95~= .9534 and 49

~- ·os 25(100) . ~ . .. ~ (100 25)(10~1)~ so Mat p 0 = .96767. Then by (8.4), 1 .05 lo= 00 .9676799 11100 . _ = .96767, .967671 = .00326. Hence, p me = 1.0 - .96767 - .00326 = .0291. A .95 multinomial upper bound, when m=l, is then .25(.00326)+.0291 = .02988. Third, we discuss computation of the paramedic bound using me power function for modeling the distribution of tainting. The density of z iS f (z) = ~z~-1 for O<z~l . The mean tainting no = A(1), and hence ply, ~1 (8.6) (8.7) Given a sample of n=100 doBar units and me same single error of t=.25, we compute the maximum likelihood estimates of parameters p and \: P = n =.01, and ;C=- m =.7214, m Slog ti i=1 (8.8a) (8.8b) respectively. Using these estimates, we construct the following parametric bootstrap estimate of the distribution of Me error ~ of the population: O with probability .99 ~ = Jo z with probability .01 where z has the density (8.9) f (z ~ = .72~4 z ·2786 ~ O C Z < SO . (8.10)

A random sample of sue n from Me distnbution (8.9) and (8.10) is caned the bootstrap sample. Denote lied as We value of ED = ~ (1) computed from a single bootstrap sample. The distnbudon Of P*D under sampling from (8.9) and (8.10) is the bootstrap distribution of ~*D. The 95 percentile of the bootstrap distribution is used to set a bound for ED. To approximate the bootstrap sampling distnbution, we may use simulation. Let B be the number of independent bootstrap samples. Then an estimate of a .95 upper bound is UB such that :# ~ ~ D<UB] (~ ~~) B - B should be sufficiently large. For our example, using B-1000, we get UB =.01481. The computation of the mro Bayesian bounds can follow the steps given above for the case when m=O. Thus, for We Cox-SnelD bound, we compute, using the same prior settings, F2(~).2(~+6)( 95) = 3 112 Hence the desired .95 upper bound for ED iS: .25+5(;40) 1+l 0 3 112= 0182 The Tsui et al.' s bound can also be computed in the same way as before. For this sample' K'=lOS as before. But p'O = (S(.~+991/lOS = .98095, P'25 = 5~.0011+1~/105, WOO = 5~.1011/105 = .00481 and P'i = 5~.0011/105 = .00005 for the rest. The mean and the variance of this posterior distribution are .0095 and distribution has a=1.382 and ~143.37. A .95 upper bound is .0255. C) More than one errors in the sample. When the number of errors m exceeds 1, the multinomial bound computation is more involved and requires substantial programming. At the time of this report, a standard package does not appear to be available. (Note: A copyrighted PC software, that uses clustenng discussed in Section S. is available from R. Plante of Purdue University.) For the bounds for which the computation has been illustrated, one can folBow We steps shown for the m =1 case in a straight forward manner. We win however, describe the computation of the moment bound for m =2. Suppose that observed tastings are .25 and .40. In this method, We sampling distribution of the estimator ED iS anDroximated bv a three Darameter gamma distribution r~x:A.B.GN. . . .. .. ., ~ . - , . ~ where A >0, B >0 and x >G . The method of moments is used to estimate these parameters. Let m`, i=1,2 and 3 be me sample central, second and third moments. Then the moments estimators are: S1

A = 4m 3Im 2, B=(l/2)malm2 and G =ml-2m2/m3. (8.12) (8.13) (8.14) For computation of m' of the sample mean faintings a number of heuristic arguments are introduced. First of all, we compute the average tainting ~ = .325 of the two observations. Suppose that the population audited is a population of accounts receivables. Then, we compute, without any statistical explanation being given, me third data point,"*: ~ =. 81 t 1-.667 tanh( 107)] t 1+.667 tannin / 103] = .3071 . The tempt in the second pair of brackets win not be used when Me population is inventory. t* is so conducted that when Were is no enor in a sample, the upper bound is very close to the the Swinger bound. Using, thus, Me data points - two observed and one constructed - the first Tree noncentral moments are computed for z, i.e., the tainting of items in ear. They are: Vz 1 = (.25+.40+.3071)/3 = .31903, ,2 = (.252+.402~.30712)/3 = .1056, and V2 3 = (.2~3~.4~+.30713)/3 = .03619. The noncentral moments of d are simply p times Me noncentral moments of z. Using well known properties of moments, the population central, second and third moments can then be derived from noncentral moments. These population central moments are used to determine the three noncentral moments of Me sample mean. Throughout these steps the error rate p is treated as a nuisance parameter but at this stage is integrated out using Me normalized likelihood function of p. Then, Me noncentral moments of the sample mean are shown to be as follows: m+] n+2 al ' Vd,2 = m+1 m+1 m+2 2 n+2 z)+(n-l) n+2 n+3 ~Z, n ~2 (8.15) (8.16)

Vd'= m+1 m+1 m+2 n+2 z'+3(n-~) n+2 n+3 if ~ Vz.2 n2 ~ _ (n -l )(n -2) +2 n +3 n +4 vz n2 Using (8.15) through (8.17), we compute vd,2 =.14615 x 10-3, and vd,3 =.29792 x 10-5. Then, m 1 = vd,l= .93831 x 10~2 m2 = Ed 2Va21 = .581~ x 10 - (8.17) Vd 1 =.93831 x 1072, (8.18) (8~19) ma = vd,3 - 3Vd,1 vd,2 + 2v,d,1 = .51748 x 10 - (8.20) Using these values, we compute A =2.93, B =0.00445 and G =-.00366. These parameter estimates are used to determine the 95 percentile of the gamma distribution to set a .95 moment bound. The bound is .0238. For companson, for the same audit data, the Stringer bound = .0401, the parametric bound = .0238, and using the prior settings previously selected, the Cox and Snell bound = .0248 and the Tsui et at. bound = .0304. Table 4 tabulates the results. Note that when there is no error in me sample (m =0), the two B. aye Sian bounds, under the headings C&S and Tsui, are considerably smaller than the four other bounds. The reason is that the four other bounds assume that aD taints are 100% when there is no error in the sample. When the sample does contain some errors, the bounds are closer, as shown for m =l and 2. S3

Table 4 Comparison of Six .95 Upper Confidence Bounds for AD: Me Stringer bound, Me Muli~nomial bound, the Moment bound, the Pararr~etnc bound, the Cox and SneU bound, and the Tsui. et at. bound. (Sample size is n=IOO) No. of Errors Str. Mult. Moment Para. C & S Tsui m=0 .0295 .0295 .0295 .0295 .0118 .0023 m = 1 .0338 .0299 .0156 .0152 .0182 .0255 t=.25 m =2 .0401 .0315* .0239 .0238 .0248 .0304 =.40 t ~=.25 Notc: ~ This value was computed by the software made available by R. PIante.