monetary unit sampling: a belief-function implementation for audit … · 2016-12-28 · monetary...

Monetary unit sampling: a belief-functionimplementation for audit and accounting

applications

Peter R. Gillett *

Department of Accounting and Information Systems, Faculty of Management,

Rutgers: The State University of New Jersey, School of Business ± New Brunswick,

Janice H. Levin Building, Rockafeller Road, Piscataway, NJ 08854-8054, USA

Received 1 May 1999; received in revised form 1 January 2000; accepted 1 February 2000

Abstract

Audit procedures may be planned and audit evidence evaluated using monetary unit

sampling (MUS) techniques within the context of the Dempster±Shafer theory of belief

functions. This article shows: (1) how to determine an appropriate sample size for MUS

in order to obtain a desired degree of belief that the upper bound for misstatements lies

within a given interval; and (2) what level of belief in a speci®ed interval is obtained

given a sample result. The results are consistent with the view that a speci®ed level of

belief in an interval is semantically a stronger claim than the same numerical level of

probability. The paper describes two variants of MUS in both probability and belief-

function forms, emphasizing the systematic similarities and the numerical di�erences

between the two frameworks. The results, based on the Poisson distribution, extend

results already available for mean-per-unit variables sampling, and may readily be de-

veloped to give similar results for the binomial distribution. Ó 2000 Elsevier Science Inc.

All rights reserved.

Keywords: Belief functions; Auditing; Monetary unit sampling

International Journal of Approximate Reasoning 25 (2000) 43±70

www.elsevier.com/locate/ijar

*Tel.: +1-732-445-4765.

E-mail address: [email protected] (P.R. Gillett).

0888-613X/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved.

PII: S 0 8 8 8 - 6 1 3 X ( 0 0 ) 0 0 0 4 6 - 3

1. Introduction

The assurance provided by the evidence gathered during the audit process isoften represented in analytical models by probabilities, but it has been argued[1] that Dempster±Shafer belief functions provide a preferable alternative. Thelevel of assurance provided by certain audit procedures (e.g., inquiry and ob-servation) may be subjectively assessed by the auditor within this framework,just as it may within the probability framework. If belief functions are toprovide a reasonable alternative for modeling the aggregation of audit evidence,however, it is important that they are able to represent the assurance providedby the application of statistical procedures such as audit sampling. The presentstudy shows how this may be done in the case of monetary unit sampling(MUS); the method is ®rst described as it is applied under probability theory,and then a technique for planning and evaluating monetary unit samples usingbelief functions is described and contrasted. It is not the purpose of this paper tomake the case for the advantages of belief functions over probabilities, althoughthe issue is discussed in outline below. Rather, the study presumes that modelerswill wish to use whatever is the most appropriate form of uncertain reasoningfor the problem in hand, and concentrates on showing how MUS may becarried out in a belief-function framework. Although the focus of the examplesdiscussed in detail is clearly on auditing, the methods set forth in this study areavailable equally for other accounting applications of MUS.

The process of auditing is essentially concerned with the aggregation ofevidence in support of the auditor's opinion. Indeed, auditing has been de®nedas ``a systematic process of objectively obtaining and evaluating evidence re-garding assertions about economic actions and events to ascertain the degree ofcorrespondence between those assertions and established criteria and com-municating the results to interested users'' [2, italics added]. Arens and Lo-ebbecke [3] emphasize the central role of evidence-gathering: ``We believe thatthe most fundamental concepts in auditing relate to determining the nature andamount of evidence the auditor should accumulate after considering the uniquecircumstances of each engagement.'' The purpose of this article is to show thataudit evidence obtained using MUS techniques may be combined with otheraudit evidence within a Dempster±Shafer belief-function framework. In par-ticular, the article shows: (1) how to determine an appropriate sample size forMUS in order to obtain a desired degree of belief that the upper bound formisstatements lies within a given interval; and (2) what level of belief in aspeci®ed interval is obtained given a sample result.

Although there are a number of di�erent types of audit, this paper is con-cerned with the audit of ®nancial statements in order to determine theircompliance with generally accepted accounting principles (GAAP). The ®-nancial statements are the responsibility of management, whose representa-tions regarding the company's ®nancial position they contain. Implied by the

44 P.R. Gillett / Internat. J. Approx. Reason. 25 (2000) 43±70

®nancial statements are certain management assertions regarding the variousaccount balances and classes of transactions that are included [4], and theseassertions form the basis of a number of audit objectives investigated by theauditor. For example, for accounts receivable balances, management assertsvia the ®nancial statements that recorded accounts receivable exist (existence),that all existing accounts receivable are included (completeness), that accountsreceivable are accurate (valuation), etc. For full details, any standard auditingtext may be consulted; see, for example, [3].

For each of the audit objectives based on management assertions the auditorwill plan to gather evidence (except where the amount involved be consideredimmaterial, or the risk is considered insigni®cant). However, audit evidence isexpected to be persuasive rather than conclusive, and even after audit evidencehas been gathered and evaluated, there remains the possibility that an auditor'sconclusion that an account is not materially misstated may still be incorrect.This possibility is commonly referred to as audit risk, and its e�ective man-agement is crucial to proper audit planning. In attempting to reduce audit riskto an acceptable level, the auditor may obtain di�erent types of evidence from avariety of di�erent sources. Note also that certain items of evidence may bearon more than one assertion or objective (as for example, the con®rmation ofaccounts receivable bears on both existence and valuation assertions) and thatthe various audit objectives are not independent of each other. For example,the existence of sales transactions and the completeness of cash receipts clearlyimpact the existence of residual accounts receivable. Proper evaluation of alarge body of evidence gathered during the audit process is, therefore, a richand complex problem in the management of uncertainty.

The most widely used and discussed model for the aggregation of auditevidence is commonly called the audit risk model [5,6]. It proposes that auditrisk is the combined e�ect of inherent risk (the susceptibility of an accountbalance or class of transactions to material error, assuming that there were norelated internal accounting controls), control risk (the risk that material errorthat could occur in an account balance or class of transactions will not beprevented or detected on a timely basis by the system of internal accountingcontrol), and detection risk (the risk that an auditor's procedures will lead tothe conclusion that material error in an account balance or class of transac-tions does not exist when in fact such error does exist). This model is oftensummarized in the formula

AR � IR� CR�DR:

Since the beginning, however, this model has been subject to widespreadcriticism by academics. Critics have argued that it is inappropriate because therisks are not properly independent [7], that the probability model is notproperly speci®ed [8,9], that inherent risk should be treated as a Bayesian prior[10], and that the outcome space is not properly considered [11,12].

P.R. Gillett / Internat. J. Approx. Reason. 25 (2000) 43±70 45

In considering alternative approaches to the audit risk model, some authors,mindful of the rich interdependencies between assertions, and between auditprocedures and assertions, have suggested that a network approach is appro-priate [13±16]. These authors have additionally proposed that Dempster±Shafer belief functions be used to represent uncertainty in the audit processrather than probability theory [17]. There has been some support for this ap-proach from other authors in recent years [18,19], partly because of semanticconsiderations regarding the representation of ignorance. These issues ariseeven in the case of statistical evidence. To illustrate the semantic limitations ofthe probability theory model, consider an auditor who, having sent out acertain number of positive con®rmation requests for accounts receivable, ®ndsthat most of them are returned and agree with the recorded amounts, but thatthe remainder are not returned. Assuming that the auditor concludes that theevidence warrants a 70% probability that accounts receivable are not materiallymisstated, probability theory imposes the constraint that (absent any otherevidence) there is a 30% probability that accounts receivable are materiallymisstated. Yet the auditor has no evidence that accounts receivable are mis-stated (such as it might have arisen if any of the returned con®rmations haddisagreed with the recorded amounts). There is merely an insu�ciency of ev-idence in support of the recorded amounts. Belief functions, on the other hand,allow the auditor to assign a belief that accounts receivable are not misstatedbased on the evidence, no belief that accounts receivable are misstated, and aresidual amount that represents ignorance (that may require the gathering offurther evidence). Thus the auditor using belief functions can distinguish thiscase from another where all the con®rmations were returned, most agreeing,but the remainder disagreeing, with the recorded amounts.

The majority of Srivastava and Shafer's published work in this area hasconcentrated on introducing belief functions to the auditing literature, showinghow the aggregation of audit evidence may be represented using belief func-tions, and discussing some of the complexities of a network approach. Whetheraudit evidence is represented by probabilities or belief functions, the auditorwill gather a variety of di�erent kinds of evidence: assessments of inherent risk,tests of controls, analytical procedures, and tests of details of transactions andbalances, some of which may be performed on a sample basis. For some auditprocedures, an estimate of associated risk may be made subjectively withineither framework (for example, inquiry and observation procedures). Therelative ease or di�culty the auditor experiences in formulating estimates ofthe assurance provided by such procedures within either the probability or thebelief-function framework is an interesting empirical question that is presentlythe subject of behavioral studies. For tests carried out on a sample basis,however, there are well-known statistical techniques for relating sample sizesand results to probabilities associated with audit conclusions. How can this bedone for belief functions? Based on principles described in Shafer's seminal


work [20], this question has been answered for mean-per-unit variables sam-pling [21]. Concern about the skewed nature of accounting populations, cou-pled with ever-increasing demands to improve audit e�ciency, however, havelead to much greater use of an alternative sampling approach, MUS, in caseswhere substantive sampling is to be applied. The purpose of this paper is toshow how sample sizes and results may be related to beliefs in speci®ed in-tervals using MUS, so that a network of both statistical and non-statisticalaudit evidence may be represented within the belief-function framework.

The remainder of the paper is divided into four further sections. Section 2 ofthe paper describes the main principles of MUS, concentrating on two popularapproaches including the method proposed by Kaplan [22] for controlling al-pha risk. Section 3 gives an outline of consonant belief functions and theirrelation to statistical evidence, and Section 4 provides a belief-function ap-proach to MUS. Section 5 summarizes and concludes with future researchissues.

2. Monetary unit sampling

The sampling techniques classically used in auditing are divided into twocategories: attribute sampling (including discovery sampling, sequential sam-pling etc.) and variables sampling (including mean-per-unit sampling, strati®edsampling, di�erence and ratio methods etc.). These techniques are described ina variety of statistical texts such as [23], and their application to auditing ismore fully discussed in such texts as [24] or [25]. The use of attribute samplingfor tests of controls has been relatively uncontroversial; although the hyper-geometric distribution is theoretically correct for most audit applications, thebinomial distribution or the Poisson distribution have been widely used asconservative approximations yielding more tractable calculations and simplertables prior to the widespread use of computers in the audit process.

Variables sampling, however, relies on the use of the central limit theorem,and this has given rise to some concern since accounting populations are oftenquite skewed. As a result, appropriate sample sizes for the application of thecentral limit theorem are likely to be larger than desirable in the audit context.This di�culty is compounded by the e�ect on sample sizes of variability withinthe accounting population. MUS has been developed in response to theseconcerns. Although the idea of using an individual dollar as the sampling unitwas suggested by Deming [26], the idea was ®rst introduced into the auditingliterature by van Heerden [27], and into the US auditing profession by Stringer[28], among others. Its use was popularized, however, by the work of AlbertTeitlebaum in conjunction with Rod Anderson and Donald Leslie [29±31].

Over the intervening years the technique, originally known as dollar unitsampling (DUS), has also been called MUS in deference to its more general


application. Since individual dollars (or other monetary units) are used as thesampling units, higher value items containing more dollars are more likely tobe selected, and the technique is sometimes called sampling with probabilityproportional to size (PPS) as a result. One popular method of selection is basedon a systematic sample 1 of the total monetary units, and this has given rise tothe alternative nomenclature: cumulative monetary amount sampling (CMA).The advantages and disadvantages of the MUS approach are discussed, forexample, in the AICPAÕs audit sampling guide [32].

In the MUS approach, the underlying idea is to treat the population asconsisting of individual dollars, each of which may or may not be misstated. Itis, therefore, essentially an attribute sampling application. Its use in practice isnevertheless subject to a wide range of variants. Suppose that an invoice for$1,000 is selected for audit by virtue of its 850th dollar being randomly se-lected, and that the audited value of the invoice turns out to be only $800; i.e.,there is a $200 overstatement. In van Heerden's original formulation, for ex-ample, the 200 overstated dollars might be allocated to be the ®rst $200 dollarsof the invoice, or the ®nal $200; depending on this choice, the 850th dollar maybe classi®ed as overstated or not. Teitlebaum showed that this leads to obvi-ously undesirable variability in audit applications [30], and proposed insteadthat each of the 1000 dollars in the invoice be considered as 20% overstated.The present article presumes the use of this method; the associated evaluationtechnique is usually referred to as the Stringer bound. Since this techniquemixes attribute sampling with values it has been described as a combined at-tributes and variables method (CAV).

Individual dollars may be selected for audit by simple random sampling, by asystematic sample based on cumulative monetary amounts, or by a methodknown as cell-selection in which the population is divided into cells of equalvalue, and a dollar is selected from each [30,31]. Although cell-selection hassome advantages, the more basic systematic sampling technique is widelytaught and used, and its use will be presumed for the remainder of this paper.

In practice, it is common for auditors using MUS to control only the betarisk (the risk of incorrect acceptance) and to ignore alpha risk (the risk ofincorrect rejection). However, Kaplan has shown how the technique may beextended to take account of alpha risk [22], and both methods are consideredbelow. Teitlebaum's and Kaplan's presentations rely on use of the Poissondistribution for attribute sampling, and this is still the basis of many modernauditing texts, see for example, [25]. For consistency with their presentations,

1 In a systematic sample, the population recorded book value B is divided by the calculated

sample size n to give a sampling interval S. From a random start s chosen in the range 1±S, every

Sth monetary unit is selected from a cumulative total of the values of the population items; i.e., the

selected monetary units are s, s� S, s� 3S etc.


the Poisson distribution is used in this paper, although the binomial distribu-tion (which gives slightly smaller sample sizes) is used in some texts, e.g., [3], aswell as in certain audit software.

The Stringer bound is widely used, and is presumed in this paper, However,it has no rigorous mathematical justi®cation, 2 and has been found in manysimulations to be signi®cantly conservative. A number of alternative evaluationtechniques have been proposed, but are not addressed in the present study; forexample, the modi®ed moment bound [34,35], the multinomial-Dirichlet bound[36], the beta-normal bound [37], or the robust Bayesian bound [38].

2.1. Sample size determination

In a MUS application, let n be the sample size, p the probability of an in-dividual monetary unit being misstated, k � np the mean number of mis-statements, k the critical value for the number of misstatements for thesampling application, B the recorded book value for the population total, T thetolerable misstatement based on planning materiality, E the expected amountof misstatement in the population, and b the acceptable level for the risk ofincorrect acceptance. Then since the probability p is given by p � T=B, so thatk � np � nT =B, and up to k errors will be accepted, the sample size for theapplication is determined [29] by solvingXk

j�0

eÿk kj

j!� b �1�

for k, and then letting n � Bk=T .When no errors are to be accepted, Eq. (1) may be solved explicitly to give

k � ÿ lnb and hence

n � ÿT lnbB

: �2�

When errors are expected, the solution of (1) rapidly becomes tedious, andalthough microcomputers may now readily be used for this purpose, tableswere used previously to provide values for k. Table 1, which is similar to thetable given in [31] may be used for this purpose. Values of k may be found inthe columns headed UEL, the table is accessed in the row for the appropriateexpected number of errors, in the column for con®dence � 1ÿ b.

Consider, for example, a population with a recorded book value of$5,000,000 that the auditor wishes to audit using MUS with a tolerable mis-statement of $500,000 and an acceptable risk of incorrect acceptance of 5%;

2 However, an asymptotic result has recently been obtained, see [33].


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61

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40.6

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52.0

69

0.1

33

54.4

69

0.1

58

57.3

47

0.1

88

45

50

.375

0.0

51

51

.589

0.0

64

53

.029

0.0

79

54.8

78

0.0

98

57.6

95

0.1

25

60.2

14

0.1

49

63.2

31

0.1

77

50

55

.618

0.0

49

56

.893

0.0

61

58

.403

0.0

75

60.3

39

0.0

92

63.2

87

0.1

18

65.9

19

0.1

41

69.0

67

0.1

67

55

60

.850

0.0

46

62

.181

0.0

58

63

.758

0.0

71

65.7

79

0.0

88

68.8

51

0.1

13

71.5

90

0.1

34

74.8

63

0.1

59

60

66

.071

0.0

44

67

.457

0.0

55

69

.098

0.0

68

71.1

99

0.0

84

74.3

90

0.1

08

77.2

32

0.1

28

80.6

25

0.1

52

65

71

.283

0.0

42

72

.722

0.0

53

74

.424

0.0

65

76.6

02

0.0

81

79.9

07

0.1

03

82.8

48

0.1

23

86.3

55

0.1

46

70

76

.487

0.0

41

77

.977

0.0

51

79

.738

0.0

63

81.9

90

0.0

78

85.4

05

0.1

00

88.4

41

0.1

19

92.0

59

0.1

41

75

81

.685

0.0

39

83

.223

0.0

49

85

.041

0.0

61

87.3

64

0.0

75

90.8

85

0.0

96

94.0

13

0.1

14

97.7

38

0.1

36

80

86

.875

0.0

38

88

.461

0.0

48

90

.334

0.0

59

92.7

27

0.0

72

96.3

50

0.0

93

99.5

67

0.1

11

10

3.3

95

0.1

31

85

92

.060

0.0

37

93

.692

0.0

46

95

.618

0.0

57

98.0

78

0.0

70

101.8

01

0.0

90

105.1

04

0.1

07

10

9.0

32

0.1

27

90

97

.240

0.0

36

98

.916

0.0

45

10

0.8

94

0.0

55

103.4

19

0.0

68

107.2

39

0.0

88

110.6

25

0.1

04

114.6

50

0.1

24

95

10

2.4

15

0.0

35

10

4.1

34

0.0

44

10

6.1

62

0.0

54

108.7

51

0.0

66

112.6

64

0.0

85

116.1

33

0.1

01

12

0.2

52

0.1

20

10

01

07

.58

50

.03

41

09

.34

60

.04

21

11

.424

0.0

52

114.0

75

0.0

65

118.0

79

0.0

83

121.6

27

0.0

99

12

5.8

39

0.1

17


suppose also that three of the selected dollars are expected to be in error. FromTable 1, k � 7:754 for three errors and a con®dence of 95%. The requiredsample size is therefore for three errors and a con®dence of 95%. The requiredsample size is therefore

5;000;000� 7:754

500;000� 77:54 � 78:

Tolerable misstatement is determined by the auditor in the light of planningmateriality, and the beta risk may be determined by reference to the audit riskmodel ± but how is the expected number of errors to be determined? When theauditor has reason to expect a particular number of errors, the calculationproceeds in a straightforward manner, as outlined above.

More commonly, the auditor does not have in mind the required expectednumber of errors, but rather some amount E of expected misstatement (per-haps based on the experience of prior years). When MUS is based on binomialattribute sampling tables, this poses no di�culty. The expected error rate iscalculated as

EER � E � 100%

B;

the tolerable error rate is calculated as

TER � T � 100%

B

and the tables are used in the normal way to give the appropriate sample size[32]. In the Poisson distribution approach, matters become a little morecomplicated. The (conservative) assumption made in MUS planning is that allmisstatements are 100% misstatements. Thus k=n � E=B or, equivalently

E � B� kn� Bk � T

Bk� kT

k: �3�

For k � 0; 1; 2; . . ., Eq. (1) may be solved iteratively for k, and the resultingvalue used to compute E from Eq. (3) until the required level of expected erroris reached.

In the example considered earlier, suppose that the expected aggregateerror was $193,449. For k � 0, Table 1 gives k � 2:996, but yields a value forE of 0. For k � 1, Table 1 gives k � 4:744, giving a value for E of $105,396.Similarly, for k � 2, Table 1 gives k � 6:296, giving a value for E of $158,831.Finally, for k � 3, Table 1 gives k � 7:754, giving a value for E of $193,449.This somewhat complicated procedure is necessary because a given number oferrors correspond to a di�erent amount of error at di�erent levels of betarisk.


It does not always happen that the amount of expected error corresponds toan exact number of errors. If the expected error falls between k and k � 1 er-rors, giving rise to kk and kk�1, respectively, producing expected errors ofEk and Ek�1, a conservative procedure would be to use a sample size based onkk�1. Alternatively, linear interpolation may be used. Thus,

k � kk � E ÿ Ek

Ek�1 ÿ Ek� kk�1� ÿ kk�: �4�

In the above example, if the auditor expected aggregate error of $172,678, thenEq. (4) would give

k � 6:296� 172; 678ÿ 158; 831

193; 449ÿ 158; 831� 7:754� ÿ 6:296� � 6:879

and

n � 5;000;000� 6:879

500;000� 68:79 � 69:

2.1.1. Controlling alpha riskThe methods described above for determining sample sizes for MUS do not

explicitly control the risk of incorrect rejection. A method for doing this hasbeen devised [22] as follows. Suppose that the auditor determines some low rateof error, l, such that it would be undesirable to reject a population with thiserror rate more often than alpha percent of the time. Then the sample size isdetermined by ®nding the smallest value of k such that

1ÿXk

j�0

eÿnl nl� �jj!6 a �5�

and Xk

j�0

eÿnT=B nT=B� �jj!

b: �6�

Tables 2 and 3 provided by Kaplan [22] are used to reduce the tedious cal-culation for alpha and beta risks, respectively.

Suppose, for example, that the auditor wishes to audit a balance of$5,000,000 with a tolerable misstatement of $500,000 at an acceptable risk ofincorrect acceptance of 5%, and an alpha risk of 10% of rejecting the popu-lation when the misstatement rate is only 0.5%.

If the auditor only accepts the population when there are no errors, thenfrom Tables 2 and 3 we see that

k � 0:005n6 0:105 or n6 21;


k � 0:1n P 3:00 or n P 30:

This is clearly inconsistent, and there is no solution for n. However, if theauditor accepts the population when there is one error or less, then from Tables2 and 3 we see that

k � 0:005n6 0:532 or n6 106:4;

k � 0:1n P 4:74 or n P 47:4:

A sample of 48 will therefore su�ce, with the population being rejected ifthere is more than one error. Of course, if we wish to accept the populationwith up to three errors, we see that

k � 0:005n6 1:75 or n6 350;

k � 0:1n P 7:75 or n P 77:5;

so that the sample size of 78 determined earlier will certainly be su�cient tocontrol both the alpha and beta risks. An alternative possibility suggested byKaplan is to adopt a decision rule based on beta risk only, compute the re-quired sample size and see what this implies about alpha risk. In this case, witha sample size of 78, Eq. (5) may be used to compute an alpha risk of 0.0007.

Table 3

k�b; k� such thatPk

j�0 eÿk�kj=j!� � b, where k� number of errors

Number of errors Beta risk b

0.01 0.05 0.10 0.25 0.50

0 4.61 3.00 2.30 1.39 0.693

1 6.64 4.74 3.89 2.69 1.68

2 8.41 6.30 5.32 3.92 2.67

3 10.0 7.75 6.68 5.11 3.67

4 11.6 9.15 7.99 6.27 4.67

5 13.1 10.5 9.27 7.42 5.67

Table 2

k�a; k� such thatPk

j�0 eÿk�kj=j!� � 1ÿ a, where k� number of errors

Number of errors Alpha risk a

0.01 0.05 0.10 0.25 0.50

0 0.010 0.051 0.105 0.288 0.693

1 0.149 0.355 0.532 0.961 1.68

2 0.436 0.818 1.10 1.73 2.67

3 0.823 1.37 1.74 2.54 3.67

4 1.28 1.97 2.43 3.37 4.67

5 1.79 2.61 3.15 4.22 5.67


2.2. Sample evaluation

The MUS sample size determined as set out in the previous section gives thenumber of individual monetary units to be audited. These units will be iden-ti®ed by the auditor as falling within the monetary values of certain items in theaudit application; e.g., sales invoices, or accounts receivable. The whole of theitems containing the selected monetary units are audited, and any audit dif-ferences noted. If no di�erences are discovered, then the upper error limit(UEL) (for both overstatements and understatements) may be determined by

UEL � B� ÿ lnb� �n

� 100%: �7�

For a number of values of con®dence � 1ÿ b, the value of ÿ lnb is given inthe ®rst row of Table 1, in the UEL columns. Also, the value B=n � T =k is thesample interval used in the selection of individual monetary units.

When audit di�erences are found, overstatements and understatements mustbe separately evaluated. Suppose that k overstatements are found, and that thevalues of these di�erences are d1; . . . ; dk. Suppose further that the ratios of thesedi�erences to their respective recorded book values (known as the ``tainting''),sorted in descending order, 3 are t1; . . . ; tk. Then the gross UEL for overstate-ments is given by

UELO � B� P0

n� 100%� B� P1 ÿ P0� �

n� t1 � � � �

� B� Pk ÿ Pkÿ1� �n

� tk; �8�

where Pi is given byXi

j�0

eÿPiPi� �jj!� b: �9�

For a number of values of con®dence � 1ÿ b, the value of Pi is given in theUEL columns of Table 1, and the value of Pi ÿ Piÿ1 can then be calculated. 4

Suppose, for example, that an auditor is auditing a population with a re-corded book value of $5,000,000 with a tolerable misstatement of $500,000 andan acceptable risk of incorrect acceptance of 5%, and that based on a sample of78 the audited values for recorded amounts of $1,000, $2,000 and $3,000 are$500, $1,400, and $2,400, respectively. Then the taintings in descending order

3 A recent simulation study by Lucassen et al. [39] has shown that other bounds are also

conservative; the authors of this study favor a bound in which misstatements are ranked in

descending order of the magnitude of the misstatements.4 The Pi notation commonly used in practice is of course, simply an alternative to k.


are 0.5, 0.3 and 0.2. From Table 1, the values of Pi ÿ Piÿ1 for 1, 2 and 3 errors,respectively are 1.748, 1.552 and 1.458. Thus Eq. (8) gives

UELO � 5;000;000� 2:996

78� 100%� 5;000;000� 1:748

78� 0:5

� 5;000;000� 1:552

78� 0:3� 5;000;000� 1:458

78� 0:2

� $296;615:

It is apparent that each additional error increases the UEL; since the valuesof Pi ÿ Piÿ1 are greater than 1, each error contributes a greater amount to theincreased UEL than its own value. In other words, errors diminish the preci-sion of the sample. This e�ect is often referred to as precision gap widening(PGW), and the PGW columns of Table 1 indicate by how much the precisiongap between the upper error level and the most likely error is increased by eachsuccessive error.

Of course, the decision to evaluate the misstatements in descending order oftainting is one reason why the Stringer bound is generally conservative. Whenboth k overstatements and l understatements are found, the UEL for over-statements is reduced to take account of the understatements found, and themost commonly used method is to subtract from UEL the value of the mostlikely error for understatements

MLEU � 1

n�Xl

1

sl � B; �10�

where s1; . . . ; sl are the understatement taintings for the l understatements.Suppose in the example above that recorded amounts of $1,000 and $800

had audited values of $1,100 and $1,000, respectively, giving rise to taintings of0.10 and 0.25. Based on Eq. (10), MLEU � 1=78� 0:10� 0:25� � � 5;000;000� $22;436. Thus the net upper bound UBO for overstatements given byUELO ÿMLEU is $296,615 ) $22,436� $274,179.

A net lower bound for understatements can then be calculated by reversingthe roles of the overstatement and understatement taintings in the above cal-culations. The resulting lower bound UBU for understatements is thereforegiven by UELU ) MLEO� $230,013 ) $64,103� $165,910.

3. Consonant belief functions and statistical evidence

A belief function [20] on a frame H is a function Bel: 2H ! �0; 1� (where 2H isthe set of all subsets of H) satisfying the conditions:1. Bel�;� � 0;2. Bel�H� � 1;


3. for every positive integer n and every collection A1 . . . An of subsets of H,

Bel�A1 [ � � � [ An�PX

i

Bel�Ai�

ÿXi<j

Bel�Ai \ Aj� � � � � � �ÿ1�n�1Bel�A1 \ � � � \ An�:

We may then de®ne the plausibility function as a function Pl : 2H ! �0; 1�given by

Pl�A� � 1ÿ Bel�� A� forA � H:

An alternative formulation of belief functions begins with a basic probabilityassignment, which is a function m : 2H ! �0; 1� such that:1. m�;� � 0;2.P

A�H m�A� � 1.Then we may de®ne a belief function by Bel�A� �PB�A m�B�. Furthermore,

the basic probability assignment is unique, and can be recovered from

m�A� �XB�A

�ÿ1�jAÿBjBel�B� for all A � H: �11�

Belief functions allow for a simple representation of ignorance, by assigningmass to non-singleton subsets (or, of course, the frame itself). The focal ele-ments of a belief function are subsets A � H such that m�A� > 0 (i.e., to whichnon-zero mass is assigned).

In modeling audit evidence, a suitable frame might contain two values suchas: `àccounts receivable are not materially overstated'' and `àccounts receiv-able are materially overstated.'' Probability theory requires that if 70% prob-ability is assigned to `àccounts receivable are not materially overstated,'' 30%probability is assigned to `àccounts receivable are materially overstated.''Suppose, however, that in the belief-function framework a 70% belief is as-signed to `àccounts receivable are not materially overstated.'' The remaining30% belief may be assigned to `àccounts receivable are materially overstated''(representing con¯icting evidence), or to the set containing both options (rep-resenting ignorance as to which option is correct). Alternatively, part of it maybe assigned to con¯icting evidence (say, 10%) and the remainder (20%) willthen represent ignorance. This additional ¯exibility (non-speci®city) gives riseto the distinctive characteristics of belief functions, their claimed advantages inrepresenting certain situations, and a variety of computational complexitiesthat are not addressed here.

Among belief functions, consonant belief functions are of particular rele-vance to this paper. A belief function is said to be consonant if its focal ele-ments are nested; i.e., if its focal elements can be arranged in order so that eachis contained in the following one. In this situation, the di�erent subsets ofoutcomes to which positive belief is assigned do not contradict each other;


rather, some are simply more precisely focused than others. This is a form ofconsistency in a belief system.

Given a frame H, and probability density functions fh : X ! �0; 1� for eachh 2 H, how can we build a belief function in support of the elements of H?Shafer [20] describes two conventions for deriving such a belief function:1. the plausibility should be proportional to the probability; i.e.,

for h 2 H;Plx�fhg� � cfh�x�;2. Belx�� given by Belx�A� � 1ÿ Plx�� A� is a consonant belief function.

Under these intuitively appealing conditions, the plausibility function isuniquely de®ned, and the constant c is given by 1=maxh2Hfh�x�. Suppose thatf �� is a real valued function with:1. 06 f �h�6 1 for all h 2 H;2. f �h� � 1 for at least one h 2 H.

Then, [21], the relevant plausibility function is given by Pl�A� � maxh2Af �h�.The technique may be extended to the continuous case, where it becomesPl�A� � suph2Af �h�. How should we choose the function f meeting these con-ditions? Srivastava and Shafer propose that we use the renormalized likelihoodfunction (i.e., the likelihood function normalized by its maximum value). Thevalue of this method is that the resulting plausibilities are proportional to thelikelihood function; in addition, we will have increasing belief in wider intervalsaround the true value.

Consider the case of the Poisson distribution. The likelihood function isgiven by

L�k; x� � eÿkkx

x!: �12�

Di�erentiating this with respect to k and setting the result equal to zeroshows that the maximum value is reached when k � x. The value of L�k; x� atthis point is eÿxxx=x!, and the renormalized likelihood function becomes

f �A� � supk2A eÿkkx=x!� �eÿxxx=x!� � � sup

k2Aexÿk k

x

� �x� �:

Thus we can de®ne the belief function for an interval A as

Belx�A� � 1ÿ supk62A

exÿk kx

� �x� �: �13�

Suppose that the interval A is �kL; kU�. We have already seen that f �A�reaches its maximum value at x; hence, provided that x 2 �kL; kU�, the supre-mum outside �kL; kU� is reached at whichever of the endpoints of the tails hasthe higher value; i.e.

Belx�A� � 1ÿmax exÿkLkL

x

� �x

; exÿkUkU

x

� �x� �: �14�

When x 62 �kL; kU�, Eq. (14) yields Belx��kL; kU�� 0.


When the Poisson distribution is used for MUS, bounds are estimatedseparately for overstatements and understatements, so that, for example,�kL; kU� will usually be �0; kU�, and Eq. (14) reduces to

Belx�A� � 1ÿ exÿkUkU

x

� �x

: �15�

4. A belief-function approach to MUS

As described in Section 3, belief in an interval �kL; kU� for the value of theparameter in a Poisson distribution may be given by

Belx�A� � 1ÿmax exÿkLkL

x

� �x

; exÿkUkU

x

� �x� �:

This provides us with a means of implementing MUS within a belief-func-tion framework, based on the use of the Poisson distribution. The derivation ofsimilar results for the Binomial distribution is straightforward, but is not givenhere. 5 As a matter of notational convenience, the subscript on the belieffunction Bel will be omitted in what follows.

Fig. 1 illustrates belief in the interval [1,6] given three observed errors. Thebelief in this interval, as described in Section 3, is equal to the complement ofthe plausibility of the regions outside the interval; i.e., (1 ± the maximum valueof the renormalized likelihood function outside the interval). However, themaximum values in the two tails are achieved at the critical values 1 and 6, andthe corresponding values are 0.2737 and 0.3983. Belief in the interval [1,6] istherefore the complement of the larger value, 1ÿ 0:3983 � 0:6017.

4.1. Sample size determination

As in the statistical approach described earlier, suppose that the auditorwishes to calculate the sample size n, in order to obtain a belief of at least b thatan account balance with a recorded book value of B is not materially misstated(i.e., any misstatement falls within the interval [)T, T]), when up to k errorswill be accepted. What is required is the smallest value for n such that theauditor has a belief b that the interval �0;UBO� for overstatements and theinterval �0;UBU� for understatements are both contained within the interval

5 It may be shown that the normalized likelihood function for the Binomial distribution is given

by f �p� � np=k� �k �nÿ np�=�nÿ k�� nÿk .


[0, T]. As usual, MUS samples are planned based on acceptable overstate-ments, assuming 100% errors. The sample size may be found by solving

Bel��kL; kU�� 1ÿ ekÿkUkU

k

� �k

� b; �16�

for kU. This gives the largest kU such that if up to k 100% errors are acceptable,Bel��kL; kU�� b. This is equivalent to a belief b in the interval for misstate-ments of �kL=n�B; �kU=n�B� �; however, we want this interval to be contained inthe interval �0; T �, and in the limit this is achieved when �kU=n�B � T . Thus wewill have a belief of at least b in the interval �0; T � for misstatements (i.e.,Bel��0; T ��P b) when n � BkU=T .

When no errors are to be accepted, Eq. (16) may be solved explicitly to givekU � ÿ ln�1ÿ b�, and hence

n � ÿT ln�1ÿ b�B

;

as for the statistical procedure, and we see that in this case b � 1ÿ b. Deter-mining the desired level of belief b plays the same role in the belief-function

Fig. 1. Belief in intervals based on three observed errors.


framework as controlling beta risk in the probability framework, and when noerrors are anticipated the two methods give the same results.

However, when errors are to be accepted, the solution of (16) is againtedious, and Table 4 may be used for this purpose. As in Table 1, valuesof kU may be found in the columns headed UEL; the table is accessed inthe row for the appropriate expected number of errors, in the column forbelief b.

Consider the earlier example of a population with a recorded book value of$5,000,000 that the auditor wishes to audit using MUS with a tolerable mis-statement of $500,000 and belief of at least 95%. Suppose that the auditor iswilling to accept up to 3 of the selected dollars being in error. From Table 4,kU � 9:432 for three errors and a belief of 95%. The required sample size is

therefore 5;000;000�9:432500;000

� 94:32 � 95: Of course, a 95% belief in the interval �0; T �,with no belief outside the interval, makes stronger epistemological demandsthan 95% con®dence (with a 5% risk). Intuitively, then, it is reasonable that thenecessary sample size be larger. The issue of what level of belief is appropriatefor an auditor who would have been satis®ed with a 5% beta risk is not ad-dressed in this paper: presumably, it may well be less than 95%. Canons foracceptable levels of belief will need to be developed by audit practitioners inlight of experience using belief functions.

As before, when the auditor has assessed an expected level of misstatementsby value rather than an expected number of errors, we may proceed iteratively,recalling from Eq. (3) that E � B� k=n � Bk � T=Bk � kT=k. Fork � 0; 1; 2; . . ., Eq. (16) may be solved for kU, and the result used to compute Efrom Eq. (3) until the required level of expected error is reached.

In the example considered earlier, suppose that the expected aggregate errorwas $159,033.

For k � 0, Table 4 gives kU � 2:996, but yields a value for E of 0. For k � 1,Table 4 gives kU � 5:744, giving a value for E of $87,047. Similarly, for k � 2,Table 4 gives kU � 7:689, giving a value for E of $130,056. Finally, for k � 3,Table 4 gives kU � 9:432, giving a value for E of $159,033.

If the expected error falls between k and k+1 errors, giving rise tokk and kk�1, respectively, producing expected errors of Ek and Ek�1, a conser-vative procedure would again be to use a sample size based on kk�1, but linearinterpolation may be used.

In the above example, if the auditor expected aggregate error of $141,647,then interpolation would give

k � 7:689� 141;647ÿ 130;056

159;033ÿ 130;056� 9:432� ÿ 7:689� � 8:386

and

n � 5;000;000� 8:386

500;000� 83:86 � 84:


Tab

le4

UE

Lis

Bk�b�s

uch

tha

t1ÿ

ekÿB

k�B

k=k�k�

b,w

her

ek�

nu

mb

ero

fer

rors

;P

GW�

Bk�b�ÿ

Bkÿ

1�b�ÿ

1

Nu

mb

er

of

erro

rs

Bel

ief

bin

the

inte

rva

l�k L;k

U�

75

%8

0%

85

%90%

95%

98%

99%

UE

LP

GW

UE

LP

GW

UE

LP

GW

UE

LP

GW

UE

LP

GW

UE

LP

GW

UE

LP

GW

01

.386

1.6

09

1.8

97

2.3

03

2.9

96

3.6

89

4.6

05

13

.693

1.3

06

3.9

94

1.3

85

4.3

72

1.4

75

4.8

90

1.5

87

5.7

44

1.7

48

6.5

72

1.8

83

7.6

38

2.0

33

25

.357

0.6

64

5.7

06

0.7

12

6.1

41

0.7

68

6.7

29

0.8

39

7.6

89

0.9

45

8.6

08

1.0

36

9.7

79

1.1

41

36

.873

0.5

17

7.2

61

0.5

55

7.7

41

0.6

00

8.3

87

0.6

57

9.4

32

0.7

43

10.4

26

0.8

18

11.6

84

0.9

05

48

.312

0.4

38

8.7

32

0.4

71

9.2

51

0.5

10

9.9

46

0.5

59

11.0

66

0.6

34

12.1

25

0.6

99

13.4

58

0.7

74

59

.699

0.3

88

10

.14

90

.41

71

0.7

02

0.4

51

11.4

42

0.4

96

12.6

28

0.5

62

13.7

45

0.6

20

15.1

47

0.6

89

61

1.0

51

0.3

51

11

.52

70

.37

81

2.1

11

0.4

09

12.8

91

0.4

50

14.1

39

0.5

10

15.3

09

0.5

64

16.7

73

0.6

26

71

2.3

74

0.3

24

12

.87

50

.34

81

3.4

89

0.3

77

14.3

06

0.4

15

15.6

10

0.4

71

16.8

29

0.5

20

18.3

52

0.5

79

81

3.6

76

0.3

02

14

.20

00

.32

41

4.8

40

0.3

52

15.6

93

0.3

87

17.0

49

0.4

39

18.3

15

0.4

86

19.8

92

0.5

40

91

4.9

59

0.2

83

15

.50

50

.30

51

6.1

71

0.3

31

17.0

56

0.3

64

18.4

62

0.4

13

19.7

73

0.4

57

21.4

01

0.5

09

10

16

.228

0.2

68

16

.79

40

.28

91

7.4

84

0.3

13

18.4

01

0.3

44

19.8

54

0.3

92

21.2

06

0.4

33

22.8

84

0.4

82

11

17

.483

0.2

55

18

.06

80

.27

51

8.7

82

0.2

98

19.7

28

0.3

28

21.2

27

0.3

73

22.6

19

0.4

13

24.3

43

0.4

60

12

18

.727

0.2

44

19

.33

10

.26

32

0.0

67

0.2

85

21.0

42

0.3

14

22.5

83

0.3

57

24.0

13

0.3

95

25.7

83

0.4

40

13

19

.961

0.2

34

20

.58

30

.25

22

1.3

41

0.2

74

22.3

43

0.3

01

23.9

26

0.3

42

25.3

92

0.3

79

27.2

05

0.4

22

14

21

.187

0.2

25

21

.82

60

.24

32

2.6

04

0.2

63

23.6

33

0.2

90

25.2

56

0.3

30

26.7

58

0.3

65

28.6

12

0.4

07

15

22

.404

0.2

18

23

.06

10

.23

42

3.8

58

0.2

54

24.9

12

0.2

80

26.5

74

0.3

18

28.1

10

0.3

52

30.0

05

0.3

93

16

23

.615

0.2

11

24

.28

70

.22

72

5.1

04

0.2

46

26.1

83

0.2

71

27.8

82

0.3

08

29.4

51

0.3

41

31.3

85

0.3

80

17

24

.819

0.2

04

25

.50

70

.22

02

6.3

43

0.2

38

27.4

45

0.2

62

29.1

81

0.2

99

30.7

82

0.3

31

32.7

54

0.3

69

18

26

.017

0.1

98

26

.72

10

.21

32

7.5

74

0.2

32

28.7

00

0.2

55

30.4

71

0.2

90

32.1

04

0.3

21

34.1

12

0.3

58

19

27

.210

0.1

93

27

.92

80

.20

82

8.8

00

0.2

25

29.9

48

0.2

48

31.7

53

0.2

82

33.4

16

0.3

13

35.4

61

0.3

49


20

28

.398

0.1

88

29

.131

0.2

02

30

.019

0.2

20

31.1

90

0.2

42

33.0

29

0.2

75

34.7

21

0.3

05

36.8

01

0.3

40

21

29

.581

0.1

83

30

.328

0.1

97

31

.233

0.2

14

32.4

25

0.2

36

34.2

97

0.2

68

36.0

19

0.2

97

38.1

33

0.3

32

22

30

.760

0.1

79

31

.521

0.1

93

32

.443

0.2

09

33.6

56

0.2

30

35.5

59

0.2

62

37.3

09

0.2

91

39.4

57

0.3

24

23

31

.935

0.1

75

32

.709

0.1

88

33

.647

0.2

04

34.8

81

0.2

25

36.8

16

0.2

56

38.5

93

0.2

84

40.7

74

0.3

17

24

33

.107

0.1

71

33

.894

0.1

84

34

.847

0.2

00

36.1

01

0.2

20

38.0

66

0.2

51

39.8

71

0.2

78

42.0

84

0.3

10

25

34

.274

0.1

68

35

.074

0.1

81

36

.043

0.1

96

37.3

17

0.2

16

39.3

12

0.2

46

41.1

44

0.2

72

43.3

88

0.3

04

26

35

.439

0.1

64

36

.251

0.1

77

37

.235

0.1

92

38.5

28

0.2

12

40.5

53

0.2

41

42.4

11

0.2

67

44.6

86

0.2

98

27

36

.600

0.1

61

37

.425

0.1

74

38

.424

0.1

89

39.7

36

0.2

08

41.7

90

0.2

36

43.6

73

0.2

62

45.9

78

0.2

92

28

37

.758

0.1

58

38

.596

0.1

71

39

.609

0.1

85

40.9

40

0.2

04

43.0

22

0.2

32

44.9

30

0.2

57

47.2

65

0.2

87

29

38

.914

0.1

56

39

.763

0.1

68

40

.791

0.1

82

42.1

40

0.2

00

44.2

50

0.2

28

46.1

83

0.2

53

48.5

47

0.2

82

30

40

.067

0.1

53

40

.928

0.1

65

41

.969

0.1

79

43.3

37

0.1

97

45.4

74

0.2

24

47.4

32

0.2

49

49.8

24

0.2

77

35

45

.796

0.1

46

46

.713

0.1

57

47

.822

0.1

70

49.2

75

0.1

88

51.5

43

0.2

14

53.6

17

0.2

37

56.1

47

0.2

65

40

51

.475

0.1

36

52

.444

0.1

46

53

.615

0.1

59

55.1

49

0.1

75

57.5

39

0.1

99

59.7

21

0.2

21

62.3

79

0.2

46

45

57

.113

0.1

28

58

.131

0.1

37

59

.361

0.1

49

60.9

70

0.1

64

63.4

75

0.1

87

65.7

59

0.2

08

68.5

38

0.2

32

50

62

.716

0.1

21

63

.781

0.1

30

65

.066

0.1

41

66.7

47

0.1

55

69.3

60

0.1

77

71.7

40

0.1

96

74.6

33

0.2

19

55

68

.290

0.1

15

69

.399

0.1

24

70

.737

0.1

34

72.4

86

0.1

48

75.2

03

0.1

68

77.6

75

0.1

87

80.6

76

0.2

09

60

73

.838

0.1

10

74

.990

0.1

18

76

.379

0.1

28

78.1

92

0.1

41

81.0

08

0.1

61

83.5

67

0.1

79

86.6

73

0.1

99

65

79

.364

0.1

05

80

.557

0.1

13

81

.994

0.1

23

83.8

69

0.1

35

86.7

80

0.1

54

89.4

24

0.1

71

92.6

29

0.1

91

70

84

.870

0.1

01

86

.102

0.1

09

87

.586

0.1

18

89.5

21

0.1

30

92.5

23

0.1

49

95.2

48

0.1

65

98.5

50

0.1

84

75

90

.359

0.0

98

91

.629

0.1

05

93

.157

0.1

14

95.1

50

0.1

26

98.2

41

0.1

43

101.0

44

0.1

59

104.4

38

0.1

78

80

95

.831

0.0

94

97

.138

0.1

02

98

.709

0.1

11

100.7

59

0.1

22

103.9

34

0.1

39

106.8

13

0.1

54

11

0.2

98

0.1

72

85

10

1.2

89

0.0

92

10

2.6

31

0.0

99

10

4.2

45

0.1

07

106.3

49

0.1

18

109.6

07

0.1

35

112.5

59

0.1

49

11

6.1

30

0.1

67

90

10

6.7

34

0.0

89

10

8.1

10

0.0

96

10

9.7

65

0.1

04

111.9

22

0.1

15

115.2

60

0.1

31

118.2

84

0.1

45

12

1.9

40

0.1

62

95

11

2.1

67

0.0

87

11

3.5

76

0.0

93

11

5.2

71

0.1

01

117.4

79

0.1

11

120.8

95

0.1

27

123.9

89

0.1

41

12

7.7

27

0.1

57

10

01

17

.588

0.0

84

11

9.0

30

0.0

91

12

0.7

64

0.0

99

123.0

21

0.1

09

126.5

14

0.1

24

129.6

75

0.1

37

13

3.4

93

0.1

53


This calculation for $141,647 expected aggregate error interpolates based on2.4 errors; in the belief-function case, Eq. (16) may be solved directly to give amore accurate estimate for k of 8.403, leading to a sample size of

n � 5;000;000� 8:403

500;000� 84:03 � 85:

4.1.1. Controlling alpha riskThe methods described above for determining sample sizes for MUS do not

explicitly control the risk of incorrect rejection. Suppose that the auditor de-termines some low proportion of error, l, such that it would be undesirable toreject a population with this error too often. Suppose, in fact, that the auditorwishes to have a belief a that the interval �kL; kU� marginally contains ln. Thenthe sample size is determined by ®nding the smallest value n such that

1ÿ ekÿkLkL

k

� �k

P a; where kL � ln �17�

and simultaneously

1ÿ ekÿkUkU

k

� �k

6 b; where kU � nTB: �18�

Eq. (17) ®nds the smallest value of kL for which the belief is not more thana, if kL were any smaller, the belief would be less than a. At the same time,Eq. (18) ®nds the largest value of kU for which belief does not exceed b. Tables 5and 6 are provided to facilitate the necessary calculations.

Fig. 2 illustrates a belief 0.90 that rejected populations contain ln � 0:632errors, and a belief 0.95 that accepted populations contain no more thannT=B � 9:432 errors, given three observed errors.

Suppose, for example, that the auditor wishes to audit a balance of$5,000,000 with a 90% belief that there are at least 0.5% errors, and a 95%belief that the total misstatement does not exceed $500,000. If the auditor

Table 5

kL�a; k� such that 1ÿ ekÿkL �kL=k�k � a, where k� number of errors

Number of errors Belief a in the interval �k;1�0.50 0.75 0.90 0.95 0.99

0 ± ± ± ± ±

1 0.232 0.102 0.038 0.019 0.004

2 0.761 0.464 0.266 0.180 0.076

3 1.394 0.956 0.632 0.477 0.259

4 2.083 1.522 1.085 0.863 0.531

5 2.808 2.138 1.597 1.314 0.872


accepts the population only when there is one error or less, then from Tables 5and 6 we see that

kL � 0:005n6 0:038 or n6 7:6

and

kU � 0:1n P 5:744 or n P 57:44:

Table 6

kU�b; k� such that 1ÿ ekÿkU �kU=k�k � b, where k�number of errors

Number of errors Belief b in the interval �0; k�0.50 0.75 0.90 0.95 0.99

0 0.693 1.386 2.303 2.996 4.605

1 2.678 3.693 4.890 5.744 7.638

2 4.156 5.357 6.729 7.689 9.779

3 5.525 6.873 8.387 9.432 11.684

4 6.838 8.312 9.946 11.066 13.458

5 8.114 9.699 11.442 12.628 15.147

Fig. 2. Beliefs a and b in intervals based on three observed errors.


This is clearly inconsistent. If the auditor accepts the population when thereare two errors or less, we see from Tables 5 and 6 that

kL � 0:005n6 0:266 or n6 53:2;

kU � 0:1n P 7:689 or n P 76:89:

This is still not consistent. However, if the auditor is willing to accept up tothree errors, we see that

kL � 0:005n6 0:632 or n6 126:4;

kU � 0:1n P 9:432 or n P 94:32;

so that the sample size of 95 determined earlier will certainly be su�cient toachieve the desired beliefs. Note that Table 5 cannot provide a lower limit whenthe expected number of errors is zero.

4.2. Sample evaluation

Within the belief-function framework, MUS samples may be evaluated as inthe statistical case, but using Table 4. If no di�erences are discovered, then theUEL (for both overstatements and understatements) may be determined by

UEL � B� ÿ ln�1ÿ b�� n

� 100%: �19�

For a number of levels of belief b, the value of ÿ ln�1ÿ b� is given in the ®rstrow of Table 4, in the UEL columns.

As before, when audit di�erences are found, overstatements and under-statements must be evaluated separately. Suppose that k overstatements arefound, and that the values of these di�erences are d1; . . . ; dk. Suppose also thatthe taintings, sorted in descending order, are t1; . . . ; tk. Then the gross UEL foroverstatements is given by

UELO � B� B0

n� 100%� B� B1 ÿ B0� �

n� t1

� � � � � B� Bk ÿ Bkÿ1� �n

� tk; �20�

where Bi is given by

ekÿBiBi

k

� �k

� 1ÿ b:

For a number of levels of belief b, the value of Bi is given in the UEL columnsof Table 4, and the value of Bi ÿ Biÿ1 can then be calculated.


Suppose, as before, that an auditor is auditing a population with a recordedbook value of $5,000,000 with a tolerable misstatement of $500,000, and de-sires a 95% belief in the interval [0, 500,000]; suppose also that based on asample of 78 the audited values for recorded amounts of $1,000, $2,000 and$3,000 are $500, $1,400, $2,400, respectively. Then the taintings in descendingorder are 0.5, 0.3 and 0.2. From Table 4, the values of Bi ÿ Biÿ1 for 1, 2 and 3errors, respectively, are 2.748, 1.945 and 1.743. Thus Eq. (20) gives

UELO � 5;000;000� 2:996

78� 100%� 5;000;000� 2:748

78� 0:5

� 5;000;000� 1:945

78� 0:3� 5;000;000� 1:743

78� 0:2

� $339;878:

Note that, consistent with the view that a belief of 95% is a stronger claim than95% con®dence, the UEL is larger than for the probabilistic analysis givenearlier. Table 4 also provides belief-function equivalents of the precision gapwideners.

When both k overstatements and l understatements are found, the UEL foroverstatements may be reduced as before to take account of the understate-ments by subtracting from UEL the value of

MLEU � 1

n�Xl

1

sl � B;

where s1; . . . ; sl are the understatement taintings for the l understatements. Theauditor then has a belief b that misstatements fall in the interval �0;UBO�,where UBO � UELO ÿMLEU.

Suppose in the example above that recorded amounts of $1,000 and $800had audited values of $1,100 and $1,000, respectively, giving rise to tainting of0.10 and 0.25. Then, based on Eq. (10)

MLEU � 1

78� 0:10� � 0:25� � 5;000;000 � $22;436:

Thus the net upper bound for overstatements given by

UELO ÿMLEU is $339;878ÿ $22;436 � $317;442:

Although this method of recognizing the net e�ect on UELO of under-statements discovered in the sample is intuitively appealing here, as in thestatistical case, it is not based on a theoretical analysis, and there are at presentno empirical studies supporting its use in the belief-function case. The auditconclusion from this sampling procedure is, therefore, that the auditor has abelief at least 0.95 that the overstatements fall in the interval [0, 317, 442]; thisis within the original desired interval, based on planning materiality.


An upper bound for understatements may similarly be obtained, giving theresult that the auditor has a belief at least 0.95 that the understatements fall inthe interval [0, 184, 455]. Since both the overstatement and understatementintervals are contained within the tolerable interval for misstatements of[0, 500,000], the auditor has a belief at least 0.95 that the account is not ma-terially misstated.

In general, provided that the intervals �0;UBO� and �0;UBU� both fall withinthe tolerable misstatement interval �0; T �, the auditor will have a belief b thatthe account is not materially misstated, and a zero belief that the account ismisstated. If either the overstatement or the understatement interval is notcontained within �0; T �, however, the auditor may wish to perform additionalaudit procedures and accept a lower level of belief, to increase the sample sizeand re-evaluate the sample, or to propose an adjustment based on the errorsdiscovered, so that the revised intervals �0;UBO� and �0;UBU� both becomeacceptable.

As in the probability theory case, each additional error increases the UEL,and each error contributes a greater amount to the increased UEL than its ownvalue. The PGW columns of Table 4 indicate by how much the precision gap(between the upper error level used in the belief interval and the most likelyerror) is increased by each successive error.

5. Summary and conclusions

This paper has described several methods of MUS, and demonstrated howthese methods may be used in the context of a belief-function framework.Formulae and Tables for planning and evaluating MUS within belief functionsare provided. The methods are illustrated with reference to an auditing ex-ample, showing how a sample size is calculated to obtain a given level of beliefin a planned interval for misstatements �0; T �, and how sample results may beevaluated to give a desired level of belief in an achieved misstatement interval�0;UB�. Consistent with intuition, sample sizes for a given level of belief aresomewhat larger than (though comparable with) those for the same numericallevel of statistical con®dence. For a ®xed sample size, higher upper bounds areobtained using beliefs than using the same level of statistical con®dence. Whilethe tables are correspondingly di�erent, application of the method is proce-durally identical to the normal statistical method. Statistical evidence in theform of belief in intervals may be integrated with non-statistical evidence usingDempsterÕs Rule for the combination of beliefs in the normal way.

Future simulation work is desirable to support the use of the MLE ad-justment to UEL in the belief-function case. Although details are not given ofextensions to the use of the Binomial distribution in MUS, this extension isstraightforward; it may be best considered, however, in the context of an


extension of this paper to cover the cell-selection methods that some authorsfavor [30,31]. Finally, future research is needed on the use of alternatives to theStringer bound within the belief-function framework.

Acknowledgements

Support for this research was provided by the Ronald G. Harper DoctoralFellowship and by the Ernst and Young Center for Auditing Research andAdvanced Technology. Comments provided by Rajendra P. Srivastava, KeithHarrison, Margaret Reed, Paul van Batenburg and Glenn Shafer are gratefullyacknowledged.

References

[1] G. Shafer, R.P. Srivastava, The Bayesian and belief-function formalisms: a general perspective

for auditing, Auditing: A Journal of Practice and Theory 9 (1990) 110±137.

[2] American Accounting Association, A Statement of Basic Auditing Concepts, American

Accounting Association, Sarasota, FL, 1973.

[3] A.A. Arens, J.K. Loebbecke, Auditing: An Integrated Approach, sixth ed., Prentice-Hall,

Englewood Cli�s, NJ, 1994.

[4] American Institute of Certi®ed Public Accountants, Statement on Auditing Standards No. 31,

Evidential Matter, 1980.


Audit Sampling, 1981.


Audit Risk and Materiality in Conducting an Audit, 1983.

[7] B.E. Cushing, J.K. Loebbecke, Analytical approaches to audit risk: a survey and analysis,

Auditing: A Journal of Practice and Theory 3 (2) (1983) 23±41.

[8] W.R. Kinney, A note on compounding probabilities in auditing, Auditing: A Journal of

Practice and Theory 2 (2) (1983) 13±22.

[9] J. Jiambalvo, W. Waller, Decomposition and assessments of audit risk, Auditing: A Journal of

Practice and Theory 3 (2) (1984) 80±88.

[10] D.A. Leslie, An analysis of the audit framework focusing on inherent risk and the role of

statistical sampling in compliance testing, in: Research on Auditing Symposium VII,

University of Kansas, 1984, pp. 89±125.

[11] W.R. Kinney, Achieved audit risk and the audit outcome space, Auditing: A Journal of

Practice and Theory 8 (1989) 67±84.

[12] J.A. Sennetti, Toward a more consistent model for audit risk, Auditing: A Journal of Practice

and Theory 9 (1990) 103±112.

[13] R.P. Srivastava, G. Shafer, Belief-function formulas for audit risk, The Accounting Review 67

(1992) 249±283.

[14] R.P. Srivastava, Belief functions and audit decisions, Auditor's Report 17 (1993) 8±12.

[15] R.P. Srivastava, The belief-function approach to aggregating audit evidence, International

Journal of Intelligent Systems 10 (1995) 329±356.


[16] R.P. Srivastava, A general scheme for aggregating evidence in auditing: propagation of beliefs

in networks, in: M.A. Vasarhelyi (Ed.), Arti®cial Intelligence in Accounting and Auditing, vol.

3, Markus Wiener Publishers, Princeton, 1995.

[17] R.P. Srivastava, et al., Belief propagation in networks for auditing, Presented at Audit

Judgment Symposium, University of Southern California, 1990.

[18] A.D. Akresh, J.K. Loebbecke, Audit approaches and techniques, in: A.R. Abdel-khalik,

I. Solomon (Eds.), Research Opportunities in Auditing: The Second Decade, American

Accounting Association, Sarasota, FL, 1988.

[19] P.R. Gillett, Automated dynamic audit programme tailoring: an expert system approach,

Auditing: A Journal of Practice and Theory 12 (1993) 173±189.

[20] G. Shafer, A Mathematical Theory of Evidence, Princeton University Press, Princeton, 1976.

[21] R.P. Srivastava, G. Shafer, Integrating statistical and non-statistical audit evidence using

belief functions: a case of variable sampling, International Journal of Intelligent Systems 9

(1994) 519±539.

[22] R.S. Kaplan, Sample size computations for dollar unit sampling, Journal of Accounting

Research (1975) 238±258.

[23] W.G. Cochran, Sampling Techniques, Wiley, New York, 1977.

[24] A. Bailey, Statistical Auditing: Review, Concepts and Problems, Harcourt Brace Jovanovich,

New York, 1981.

[25] D.M. Guy, et al., Audit Sampling, third ed., Wiley, New York, 1993.

[26] W.E. Deming, Sample Design in Business Research, Wiley, New York, 1960.

[27] A. van Heerden, Statistical sampling as a means of auditing, Maandblad voor Accountancy en

Bedrijfshuishoudkunde 35 (1961) 453±475.

[28] K.W. Stringer, Practical aspects of statistical sampling in auditing, in: Proceedings of the

Business and Economic Statistics Section, American Statistical Association, 1963, pp. 405±411.

[29] R.J. Anderson, A.D. Teitlebaum, Dollar-unit sampling: a solution to the audit sampling

dilemma, CA Magazine, 1973.

[30] A.D. Teitlebaum, Dollar-unit sampling in auditing, Presented at National Meeting of the

American Statistical Association, Montreal, 1973.

[31] D.A. Leslie, et al., Dollar Unit Sampling: a Practical Guide for Auditors, Copp Clark Pitman,

Toronto, 1979.

[32] American Institute of Certi®ed Public Accountants, Audit and Accounting Guide, Audit

Sampling, 1983.

[33] G. Pap, M.C.A. van Zuijlen, On the asymptotic behavior of the stringer bound, Statistica

Neerlandica, Forthcoming, 1996.

[34] L. Dworin, R.A. Grimlund, Dollar unit sampling for accounts receivable and inventory, The

Accounting Review 59 (1984) 218±241.

[35] W.L. Felix, et al., Arthur Andersen's new monetary unit sampling approach, Auditing: A

Journal of Practice and Theory 9 (3) (1990) 1±16.

[36] K. Tsui, et al., Multinomial-Dirichlet bounds for dollar-unit sampling in auditing, The

Accounting Review (1985) 76±96.

[37] M. Menzefricke, W. Smieliauskas, A simulation study of the performance of parametric dollar

unit sampling statistical procedures, Journal of Accounting Research (1984) 588±604.

[38] J. Neter, J. Godfrey, Robust Bayesian bounds for monetary-unit sampling in auditing, Journal

of Royal Statistical Society (1985) 157±168.

[39] A. Lucassen, et al., Modi®cations of the Stringer-bound: a simulation study on the

performance of audit sampling evaluation methods, Kwantitatieve Methoden 51 (1996) 17±25.


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