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From page 8... ...
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.

From page 9... ...
An audit sample may not yield any nonzero 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.

From page 10... ...
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.

From page 11... ...
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.

From page 12... ...
(2.2) caped Me population audited amount.

From page 13... ...
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)

From page 14... ...
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.

From page 15... ...
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.

From page 16... ...
D~TR~UT~N OF ERROR AMOUNTS ~ ACCOUNTS RECE1V^BLE JUDE ~ 3 ~ . ~ O ~ ~00 250 12, 7 ~ \ loo o 100 250 Error Amounts 16

From page 17... ...
Figure 1 (continued, (B) InventoIy (1139 observations)

From page 18... ...
Figure 2 Examples of Distribution of Dollar Unit Tainting (A) Accounts Receivable o 0 _ i_ ~ `1)

From page 19... ...
. Their accounts receivable audit No.

From page 20... ...
(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)

From page 21... ...
Suppose that a sample of n items is taken. We denote the book amount, audited amount, error amount, and tainting of the kth item in the sample by analogous lower case letters, namely, Ok, x', Ok Ark  ok .

From page 22... ...
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.

From page 23... ...
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 errorfree. Secondly, Hey are either imprecise, as in He case of He meanperunit 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.

From page 24... ...
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)

From page 25... ...
It is clear from Table 3 that Me unwasted use of normal approximations results In twosided confidence intervals which have unreliable upper bounds and overly conservative lower bounds. The population of accounts receivable often is contaminated by overstatement ears.

From page 26... ...
, 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 onesided lower 97.7% bound.

From page 27... ...
(A) The MeanPerUnit Estimator, Am Population Etror Rates as Percentages Population .5 1.0 5.0 10.0 30.0 1 (22)

From page 28... ...
OB) The reference Estimator.

From page 29... ...
(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

From page 30... ...
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 _ 945 100.0 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.

From page 31... ...
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

From page 32... ...
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 
From page 33... ...
~ 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.

From page 34... ...
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.

From page 35... ...
A (1 oc) upper bound for 69 and hence also for D by multiplying We former by Y

From page 36... ...
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.

From page 37... ...
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.

From page 38... ...
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.

From page 39... ...
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.

From page 40... ...
~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 .

From page 41... ...
, 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.

From page 42... ...
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.

From page 43... ...
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.

From page 44... ...
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.

From page 45... ...
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)

From page 46... ...
prior prediction is defined by the Dir~chlet process With the parameter adz )

From page 47... ...
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.

From page 48... ...
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)

From page 49... ...
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)

From page 50... ...
A .95 multinomial upper bound, when m=l, is then .25(.00326)

From page 51... ...
In this method, We sampling distribution of the estimator ED iS anDroximated bv a three Darameter gamma distribution r~x:A.B.GN.

From page 52... ...
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+]

From page 53... ...
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.

From page 54... ...
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.

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