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ADIAC-00048; No of Pages 9
Advances in Accounting, incorporating Advances in International Accounting xxx (2009) xxx–xxx
Contents lists available at ScienceDirect
Advances in Accounting, incorporating Advances in
International Accounting
j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / a d i a c
1
Monetary unit sampling: Improving estimation of the total audit error
2
Huong N. Higgins a,⁎, Balgobin Nandram b,1
a
b
Worcester Polytechnic Institute, Department of Management, 100 Institute Road, Worcester, MA 01609, United States
Worcester Polytechnic Institute, Department of Mathematical Sciences, 100 Institute Road, Worcester, MA 01609, United States
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a r t i c l e
i n f o
a b s t r a c t
In the practice of auditing, for cost concerns, auditors verify only a sample of accounts to estimate the error of
the total population of accounts. The most common statistical method to select an audit sample is by
monetary unit sampling (MUS). However, common MUS estimation practice does not explicitly recognize the
multiple distributions within the population of account errors. This often leads to excessive conservatism in
auditors' judgment of population error. In this paper, we review the common MUS estimation practice, and
introduce our own method which uses the Zero-Inflation Poisson (ZIP) distribution to consider zero versus
non-zero errors explicitly. We argue that our method is better suited to handle the real populations of
account errors, and show that our ZIP upper bound is both reliable and efficient for MUS estimation of
accounting data.
© 2009 Elsevier Ltd. All rights reserved.
Available online xxxx
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1. Introduction
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In auditing a company's accounts receivable, the goal of auditors is
to verify if the values of the accounts reported by the company are not
materially misstated. To determine the total error in the amount
reported by the company, auditors must audit all accounts in the
population of accounts. However, it is costly and it takes much time to
audit all accounts. In practice, auditors audit only a sample to estimate
the total error in the population of accounts.
A common statistical sampling method is monetary unit sampling
(MUS), where each monetary unit (e.g., dollar) is equally likely to be
included in the sample. In this method, an account with a book value
of $10,000 is ten times as likely to be sampled as an account with a
book value of $1000.
This paper addresses the estimation process for MUS sampling.
Common MUS estimation has a shortcoming because it does not
explicitly recognize that a total population of account errors typically
consists of distinct distributions, namely one large mass with zero
error, a second distribution of small errors, and a third distribution of
100% errors. These distribution characteristics of accounting error
populations have been discussed in prior research (e.g. Kaplan, 1973;
Neter & Loebbecke, 1975; Chan, 1988). Due to this shortcoming, in
practice sample accounts are incorrectly assumed to have similar
tainting (ratio of Error-Per-Dollar) to non-sample accounts. This
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assumption, combined with MUS sampling bias towards selecting
larger accounts, often leads to very large estimations of total error in
the population and overly conservative auditors' decisions.
We acknowledge that the major risk of accounts receivable in
financial reporting is overstatement, so generally auditors should
prefer methods that result in larger error estimations. However, we
argue that estimations in practice are too conservative, and excessive
conservatism has its own practical problems. For example, discussions
with auditors in three large accounting firms reveal that clients rarely
approve large adjustments for error estimations (Elder & Allen, 1998).
This, along with the cost of statistical expertise, may cause auditors
not to project sample errors to the population at all. Indeed, surveys of
auditors reveal that they often fail to project errors to populations,
notwithstanding audit standards and professional guidelines (Burgstahler & Jiambalvo, 1986; Akresh & Tatum, 1988; Hitzig, 1995; Elder &
Allen, 1998). Overall, excessive conservatism in the estimation method
may lead to projection problems, which impairs the auditor's overall
decision process.
We introduce our own approach which is better suited to handle
the real distributions of account errors. We use a regression with
selection probability as a covariate to address MUS sampling bias. Our
technical innovation lies in the development of the Zero-Inflation
Poisson (ZIP) model, using the Poisson distribution to treat errors as
rare count data, and increasing the probabilities of zero errors to
inflate the weights of these observations. By handling accounts with
zero versus non-zero errors explicitly, our technique does not rely on
the assumption of similar tainting in the population. We provide
mathematical expressions and, upon request, will furnish the software
codes for our ZIP estimation and confidence intervals. Through
simulations, we show that our ZIP bound is reliable and efficient for
CT
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⁎ Corresponding author. Tel.: +1 508 831 5626; fax: +1 508 831 5720.
E-mail addresses: hhiggins@wpi.edu (H.N. Higgins), balnan@wpi.edu (B. Nandram).
1
Tel.: +1 508 831 5539; fax: +1 508 831 5824.
0882-6110/$ – see front matter © 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.adiac.2009.06.001
Please cite this article as: Higgins, H. N., & Nandram, B., Monetary unit sampling: Improving estimation of the total audit error, Advances in
Accounting, incorporating Advances in International Accounting (2009), doi:10.1016/j.adiac.2009.06.001
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2. Monetary unit sampling
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In performing substantive tests of accounts, an auditor often takes
a sample to estimate the true total balance of all accounts. Ultimately,
the auditor's goal is to estimate the total amount of error, which is the
difference between the true total balance of all accounts and the
reported balance amount. The standards for audit sampling are set by
the Audit Standards Board's Statement on Auditing Standards (SAS)
No 39, (AICPA, 1981) amended by SAS No. 111 (AICPA, 2006).
Both statistical and non-statistical sampling methods are allowed
for substantive tests in auditing (AICPA, 1981, 2008). Both are used by
a wide-range of public accounting firms (Nelson, 1995) and government agencies (Annulli, Mulrow & Anziano, 2000). Some surveys
show that non-statistical sampling is used more than half the time
(Annulli et al., 2000; Hall, Hunton & Pierce, 2000, 2002). Of course,
non-statistical sampling procedures do not afford the full control
provided by statistical theory, and so raise questions concerning
auditors' evaluation of sample results (Messier, Kachelmeier & Jensen,
2001).
Among statistical sampling techniques, monetary unit sampling
(MUS) is the most commonly used for substantive tests (Tsui,
Matsumura & Tsui, 1985; Smieliauskas, 1986b; Grimlund & Felix,
1987; Hansen, 1993; Annulli et al., 2000; AICPA, 2008). The
authoritative guide for the auditing profession, the AICPA's Audit
Guide — Audit Sampling (AICPA, 2008), has detailed MUS instructions.
MUS is a statistical sampling method where the probability of an
item's selection for the sample is proportional to its recorded amount
(probability proportional to size). MUS can be thought of as employing the ultimate in stratification by book amount. No further
stratification by book amount is possible with dollar units because
all sampling units are of exactly the same size in terms of book value.
Consequently, MUS incorporates efficiency advantages similar to
those of stratification by book value without requiring stratification.
Despite its widespread use, the relative performance of MUS
compared to traditional normal distribution variables is often not
clear (Smieliauskas, 1986b). Prior research has developed a number of
methods for evaluating MUS samples (Tsui et al., 1985; Smieliauskas,
1986b; Grimlund & Felix, 1987). Evaluating criteria include sample
size (Kaplan, 1975), sampling risks (Smieliauskas, 1986b), sample size
implications of controlling for the same level of sampling risks
(Smieliauskas, 1986b), and bounds (Tsui et al., 1985; Dworin &
Grimlund, 1986).
Obtaining reliable bounds on the total error in the population is
desirable for making decisions at different confidence levels and
probabilistic statements. There are many methods to compute bounds
for MUS sampling (Felix, Leslie & Neter, 1981; Swinamer, Lesperance &
Will, 2004). The Stringer bound, introduced by Stringer (1963), is used
extensively by auditors (Bickel, 1992; AICPA, 2008). A feature of thee
Stringer bound which is particularly attractive to auditors is that it
provides a non-zero upper bound even when no errors are observed in
the sample. Simulations show that the Stringer bound reliably exceeds
the true audit error (Swinamer et al., 2000). This reliability is
favorable to auditors who are concerned with strictly overstatements
(or strictly understatements) in financial statements. The Stringer
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3. Data and common estimation
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3.1. Data
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To illustrate the common estimation method, we use data of a
company used by Lohr (1999) in her demonstration of MUS. The
company has a population of N = 87 accounts receivable, with a total
book balance of $612,824. We know the book values b1, b2, …, bN for
each account in the population. Let B denote the total book value of all
accounts receivable in the population,
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B = b1 + b2 + … + bN = 612; 824:
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ð1Þ
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If each account in the population were audited, we would obtain 155
the set of audit values a1, a2, …, aN. Let A denote the unknown total 156
audit value in our data,
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A = a1 + a2 + … + aN :
ð2Þ
We define the error on any account i (i = 1, 2, …, N) by di = bi − ai,
di ≥ 0. After performing audit on a sample size n, which we suppose is
predetermined, we observe a1, a2, …, a n (see Kaplan, 1975;
Menzefricke, 1984 for discussions of issues in determining the sample
size). We wish to fit an + 1, …, aN to predict the total error from all
accounts.
Let D denote the total error in the population of all accounts
receivable,
D = d1 + d2 + … + dN :
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ð3Þ
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The total error D is also the difference between the total book value 170
and the total audit value,
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D = B−A:
ð4Þ
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Our ultimate goal is to predict total error D. Because D is unknown,
it is standard practice to estimate its mean and upper bound.
The book values, audit values, selection probabilities and errors of
all accounts are tabulated in Table 1. Selection probability is the
probability of an account being selected from all the 87 accounts, equal
to the book value of an account divided by the total book balance
$612,824 (e.g. for account 3, the selection probability = 6842/612,824 =
0.011164706). Panel A describes the audit sample, and Panel B the nonsample.
The audit values and errors of accounts in the audit sample are
known, but these values are unknown for accounts in the non-sample.
In this company, only a small proportion of audited accounts (4 of 20,
or 80%) are subject to error.
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3.2. Common estimation
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SAS No. 39 on audit sampling (AICPA, 1981) promulgates that
auditors should estimate the total error in the population by
projecting sample errors to the population. A common estimation
method of the mean population error D is based on tainting, which
equals the average error amount per dollar in the audited accounts.
Taintings are multiplied by the total dollars in the sampling intervals,
or average tainting is multiplied by the total dollars in the population,
to yield an estimate of the total error dollars in the population. This
method is prescribed by the AICPA Audit Guide — Audit Sampling
(AICPA, 2008). Besides the AICPA guide, many professional publications also prescribe this MUS evaluation method (Gafford &
Carmichael, 1984; Guy & Carmichael, 1986; Schwartz, 1997; Yancey,
2002).
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bound is also simple to compute and the required statistical tables are 143
readily available.
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MUS estimation of accounting data. Overall, our paper serves
professionals and academicians by pointing out an improvement
potential for MUS estimation.
The remainder of the paper is as follows. Section 2 reviews the
monetary unit sampling method in the accounting literature and
practice of auditing. Section 3 describes our data and demonstrates a
common method in current practice to estimate the total error in a
population of accounts from which a dollar unit sampling is taken.
Section 4 introduces our Zero-Inflated Poisson (ZIP) estimation
method. Section 5 reports simulations to validate our ZIP upper
bound. Section 6 concludes the paper with a brief summary.
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Please cite this article as: Higgins, H. N., & Nandram, B., Monetary unit sampling: Improving estimation of the total audit error, Advances in
Accounting, incorporating Advances in International Accounting (2009), doi:10.1016/j.adiac.2009.06.001
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H.N. Higgins, B. Nandram / Advances in Accounting, incorporating Advances in International Accounting xxx (2009) xxx–xxx
t1:1
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Table 1
Description of data.
Panel A: sample
Account number
Book value (b)
Audit value (a)
Selection probability (πi)
Error (d = b − a)
t1:5
t1:6
t1:7
t1:8
t1:9
t1:10
t1:11
t1:12
t1:13
t1:14
t1:15
t1:16
t1:17
t1:18
t1:19
t1:20
t1:21
t1:22
t1:23
t1:24
t1:25
t1:26
t1:27
3
9
13
24
29
34
36
43
44
45
46
49
55
56
61
70
74
75
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81
Total error in the sample
6842
16,350
3935
7090
5533
2163
2399
8941
3716
8663
69,540
6881
70,100
6467
21,000
3847
2422
2291
4667
31,257
6842
16,350
3935
7050
5533
2163
2149
8941
3716
8663
69,000
6881
70,100
6467
21,000
3847
2422
2191
4667
31,257
0.011164706
0.026679765
0.006421093
0.01156939
0.009028693
0.003529562
0.003914664
0.014589833
0.006063731
0.014136196
0.113474668
0.011228346
0.11438847
0.010552785
0.034267587
0.006277496
0.003952195
0.003738431
0.007615563
0.051004856
0
0
0
40
0
0
250
0
0
0
540
0
0
0
0
0
0
100
0
0
940
t1:28
Account
number
Book
value (b)
Audit
value (a)
Selection
probability (πi)
Error
(d = b − a)
Account
number
Book
value (b)
Audit
value (a)
Selection
probability (πi)
Error
(d = b − a)
t1:29
t1:30
t1:31
t1:32
t1:33
t1:34
t1:35
t1:36
t1:37
t1:38
t1:39
t1:40
t1:41
t1:42
t1:43
t1:44
t1:45
t1:46
t1:47
t1:48
t1:49
t1:50
t1:51
t1:52
t1:53
t1:54
t1:55
t1:56
t1:57
t1:58
t1:59
t1:60
t1:61
t1:62
1
2
4
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31
32
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35
37
38
39
40
41
2459
2343
4179
750
2708
3073
4742
5424
9539
3108
900
7835
1091
2798
5432
2325
1298
5594
2351
7304
4711
4031
1907
3341
8251
4389
5697
7554
8413
4261
7862
3153
4690
6541
.
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0.004013
0.003823
0.006819
0.001224
0.004419
0.005014
0.007738
0.008851
0.015566
0.005072
0.001469
0.012785
0.00178
0.004566
0.008864
0.003794
0.002118
0.009128
0.003836
0.011919
0.007687
0.006578
0.003112
0.005452
0.013464
0.007162
0.009296
0.012327
0.013728
0.006953
0.012829
0.005145
0.007653
0.010674
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85
86
87
9074
8746
7141
2278
3916
2192
5999
5856
7642
8846
2486
2074
3081
7123
5496
7461
6333
13,597
1317
5437
4030
2620
2416
5882
6596
2626
7571
1331
5924
4356
6618
5658
6943
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0.014807
0.014272
0.011653
0.003717
0.00639
0.003577
0.009789
0.009556
0.01247
0.014435
0.004057
0.003384
0.005028
0.011623
0.008968
0.012175
0.010334
0.022187
0.002149
0.008872
0.006576
0.004275
0.003942
0.009598
0.010763
0.004285
0.012354
0.002172
0.009667
0.007108
0.010799
0.009233
0.01133
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t1:63
Book value is the amount reported by the audited company. Audit value is the value audited and verified by the auditors. Selection probability is the probability of which an account is
selected from all the accounts in the population, equal to the book value of an account divided by the total book balance $612,824 (e.g. for account 3, the selection
probability = 6842 / 612824 = 0.011164706). Error is the difference between book value and audit value. Panel A shows the accounts selected by the auditors to be audited (sample),
and Panel B the remainder of the population of accounts (non-sample). Accounts in the non-sample do not have audit value.
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Panel B: non-sample
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Table 2 illustrates this estimation process. Error-Per-Dollar (tainting) indicates the difference per dollar for each account, equal to the
difference in book and audit values divided by the corresponding book
value. Account 24, for example, has a book value of $7090 and an error
of $40. The error is prorated to every dollar in the book value, leading
to an error (tainting) percentage of $40/$7090 = 0.005642 for each of
the 7090 dollars. The average error for the individual dollars in the
sample is the sum of all Errors-Per-Dollar divided by 20, the number of
accounts in the sample, or 0.008063. Thus the total error for the
population D is estimated as 0.008063 ⁎ 612,824 = $4941.
Often, audit units greater than the sampling interval are excluded
from the population before sampling (AICPA, 2008). These units are
examined separately, and their projected misstatements equal their
full misstatements, not an extension of tainting. For example, the
Please cite this article as: Higgins, H. N., & Nandram, B., Monetary unit sampling: Improving estimation of the total audit error, Advances in
Accounting, incorporating Advances in International Accounting (2009), doi:10.1016/j.adiac.2009.06.001
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H.N. Higgins, B. Nandram / Advances in Accounting, incorporating Advances in International Accounting xxx (2009) xxx–xxx
Table 2
Common error estimation for dollar unit sampling.
Account
number
Book value
(b)
Audit value
(a)
Selection
probability (πi)
Error
(d = b − a)
Error-Per-Dollar
(tainting)
6842
16,350
3935
7050
5533
2163
2149
8941
3716
8663
69,000
6881
70,100
6467
21,000
3847
2422
2191
4667
31,257
0.011164706
0.026679765
0.006421093
0.01156939
0.009028693
0.003529562
0.003914664
0.014589833
0.006063731
0.014136196
0.113474668
0.011228346
0.11438847
0.010552785
0.034267587
0.006277496
0.003952195
0.003738431
0.007615563
0.051004856
0
0
0
40
0
0
250
0
0
0
540b
0
0
0
0
0
0
100
0
0
940
0
0
0
0.005642
0
0
0.10421
0
0
0
0.007765
0
0
0
0
0
0
0.043649
0
0
Projected mistatement
(taintinga sampling interval)
t2:4
t2:5
t2:6
t2:7
t2:8
t2:9
t2:10
t2:11
t2:12
t2:13
t2:14
t2:15
t2:16
t2:17
t2:18
t2:19
t2:20
t2:21
t2:22
t2:23
t2:24
t2:25
t2:26
3
6842
9
16,350
13
3935
24
7090
29
5533
34
2163
36
2399
43
8941
44
3716
45
8663
46
69,540b
49
6881
55
70,100
56
6467
61
21,000
70
3847
74
2422
75
2291
79
4667
81
31,257
Total error in the sample
Average Error-Per-Dollar
Total projected error in the population
t2:27
t2:29
t2:28
t2:30
Book value is the amount reported by the audited company. Audit value is the value audited and verified by the auditors. Selection probability is the probability of which an account is
selected from all the accounts in the population, equal to the book value of an account divided by the total book balance. Error is the difference between book value and audit value.
Error-Per-Dollar is the ratio between error and book value. The sampling interval is $612,824/20 = $30,641.20.
Total error D for the population is estimated as Average Error-Per-Dollar ⁎ total book value, or 0.008063 ⁎ 612,824 = $4941.
a
The Average Error-Per-Dollar is the sum of all error-per-dollars divided by the number of accounts in the sample (20).
b
This audit unit is greater than the sampling interval and may be considered separately from the sample evaluation.
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n
k=j + 1
226
n k
n−k
p ð1−pÞ
= 1−α;
k
ð5Þ
if j b n and p(n; 1 − α) = 1.
The Stringer bound for the mean of tainting μ is
M
CO
ð6Þ
j=1
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233
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235
236
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238
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240
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The upper bound of the total error in the population is μ–ST ⁎ B.
There is a similar bound for the case in which X has a Poisson (n ⁎ p)
distribution.
The AICPA (2008) publishes tables as aids in calculating the
Stringer upper bound (see Table C.3. “Monetary Unit Sampling —
Confidence Factors for Sample Evaluation” in Appendix C of the Audit
Guide). Our computation of the Stringer 95% upper bound based on
the binomial distribution for calculating p(n;1 − α) is $129,380. A
replication of the Stringer computation using the Poisson instead of
binomial distribution for p(n;1 − α) yields a similar value, $133,518.
The above results, which are based on the common estimation
approach, can be summarized as follows. The sample contains known
errors of $940. The total error projected in the population is $4941. The
Stringer 95% upper confidence bound of total error in the population is
$133,518. Thus, there is a 5% risk that the recorded amount of $612,824
is overstated by more than $133,518.
UN
230
231
RO
OF
DP
4941
μ ST ≡ pð0; 1−αÞ + ∑ ½pðj; 1−αÞ−pðj−1; 1−αÞzj :
227
228
229
1337.46
0.008063a
projection from account 46 would be its full misstatement of $540
instead of $237.94. In this case, the total projected error in the
population would be $5243 instead of $4941.
A common method to estimate the upper bound of D is the
Stringer bound. Let Ti be the tainting of the i-th selected item, Ti ≡ di/
bi. If M ≡ number of non-zero Ti let 0 b zM ≤ … ≤ z1 be the ordered non
zero Ti. Let p(j; 1 − α) be the 1 − α exact upper confidence bound for p
when X has a binomial (n, p) distribution and X = j. Thus p(j; 1 − α) is
the unique solution of
∑
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224
237.94b
We argue that the approach described above is too conservative.
Compared to other bounds, for example the Multinomial-Dirichlet
bound computed at $11,125 for the same sample, the Stringer bound is
too large. References for Multinomial bounds can be found in
Swinamer et al. (2004) and Neter, Leitch and Fienberg (1978). As
Stringer (1963) stresses, if the population is free of errors it gives the
same answer (overestimate), no matter what sample is drawn from
the population. In fact, simulation studies (Reneau, 1978; Leitch, Neter,
Plante & Sinha, 1982) find that the Stringer bound is always too
conservative. The trade-off for high reliability is loss of efficiency: the
Stringer bound is always much larger than the population total error
(Bickel, 1992; Swinamer et al., 2004). It is probable that many auditors
do not project sample results to populations to avoid unrealistic
conservatism and client resistance. However, failing to project leads to
judgment error regarding the financial statements overall.
We seek to introduce an alternative method that is reliable but
more efficient, to help auditors make realistic projection and
probabilistic statements about the population error.
245
4. Zero-Inflated Poisson regression
263
To improve on the common estimation method described in
Section 3, our aim is to estimate total error D and its 95% upper
confidence bound. In the first step, we develop the Poisson regression
to view each dollar in an account either as error or not (as count data).
In the second step, we further develop the Poisson regression to count
data with a large number of zero values to suit populations of account
errors.
264
265
4.1. Ordinary Poisson regression
271
We use the generalized regression model to allow for count data so
that we can capture the distribution of our data more accurately (see
Nandram, Sedransk & Pickle, 2000 for a detailed discussion of the
generalized regression model and specifically the Poisson regression).
272
TE
216
217
3193.12
RR
EC
215
172.87
Please cite this article as: Higgins, H. N., & Nandram, B., Monetary unit sampling: Improving estimation of the total audit error, Advances in
Accounting, incorporating Advances in International Accounting (2009), doi:10.1016/j.adiac.2009.06.001
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di jλi ind
e Poissonðλi bi Þ
ð7Þ
ln λi = ðβ0 + r0i Þ + ðβ1 + r1i Þπi
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293
294
295
296
297
298
where i = 1,2, …, N, r0i and r1i are random perturbations and are
independent and identically distributed with mean 0 and finite
variances, and λi is the error rate for a dollar, or equivalently, the
probability that a dollar is in error. Now we have two sets of random
coefficients β0 + r0i and β1 + r1i.
Note that the model with r0i = r1i = 0, i = 1,2, …, N, is
di jλi ind
e Poissonðλi bi Þ
ln λi = β0 + β1 πi :
299
300
305
306
di jλi ind
e Poissonðbi e
310
311
β0 + β1 πi
Þ:
ð9Þ
Centering allows us to use this simpler model to study the original,
more complex model. Because the dis are independent, the loglikelihood function is
n
CO
R
307
308
309
n
n
i=1
i=1
β0 + β1 πi
f ðβ0 ; β1 Þ = β0 ∑di + β1 ∑ πi di − ∑ bi e
i=1
312
313
316
317
318
319
320
321
:
n
i=1
ðhÞ
=M = E
h=1
ð13Þ
and the standard error is
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330
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
M
S=
∑ ðE
ðhÞ
i=1
P 2
− E Þ = ðM−1Þ:
ð14Þ
To obtain a 95% confidence interval for the total overstatement
error, we order the E(h) from smallest to largest and pick the 250th and
9750th value from the lower and upper end of the confidence interval.
The 95% upper confidence bound is the 9500th value.
N
∑ d˜i
331
332
333
334
335
336
4.2. Zero-Inflation Poisson regression
337
We now develop the Poisson regression model to count data with
many zero values. This is our major innovation to address MUS
estimation errors. We assume that with probability θ the only
possible observation is 0 (i.e. pure zero), and with probability1 − θ, a
Poisson (λ) random variable is observed. In other words, we treat
errors as nearly impossible, but allow errors to occur according to a
Poisson (λ) distribution. This treatment is termed Zero-Inflation
Poisson (ZIP). For a pioneering description of ZIP regression, see
Lambert (1992). Both the probability θ of the perfect zero error state
and the mean number of errors λ in the imperfect state may depend
on covariates. Pertinent covariates, such as trade relationships, credit
terms, product prices, etc… can lead to improved precision in
statistical inference. For a discussion of how to introduce covariates,
see Nandram et al. (2000), for example. Sometimes θ and λ are
unrelated; other times θ is a simple function of λ. In either case, ZIP
regression models are easy to fit, and the maximum likelihood
estimates are approximately normal in large samples of at least 25
(Lambert, 1992).
As in the ordinary Poisson model in Section 4.1, our model is
Eq. (7). To increase the probabilities of zero errors, we use the
following distribution for di,
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342
343
344
345
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348
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350
351
352
353
354
355
356
357
358
β 0 + β 1 πi
ð10Þ
We use maximum likelihood estimate in the centered model to
obtain MLEs. Preliminary estimations are possible at this step,
however they do not benefit from the zero inflation process, so they
are less adjusted for populations with a large number of zero errors.
We show in Appendix A how to obtain 10,000 copies of the
overstatement error. We use two estimators: the predicted estimator
and the projected estimator.
The two estimators are defined as:
Predicted: ∑ di +
M
∑E
pðdi = 0Þ = θ + ð1−θÞe−bi e
UN
314
315
from Appendix A, and M = 10,000, h = 1,2, …, M. An estimate of the 327
overstatement error is
328
CT
303
304
This is another important feature of our method because
observations are treated according to their selection probabilities π,
and so sample and non-sample observations are not assumed to be
the same.
The first model is centered on the second model. Centering is
achieved by taking r0i = r1i = 0, so that
RE
301
302
be the h-th copy of the 10,000 copies obtained 326
i=1
ED
ð8Þ
ðhÞ
Let EðhÞ = ∑ di
OO
F
278
279
We use random coefficients, instead of fixed coefficients as in a
standard linear regression, to treat each account as having its own set
of regression coefficients. By allowing random coefficients, we can
model the data more flexibly and therefore can achieve more accuracy.
This is an important feature of our method to incorporate random
effects. Using random coefficients also allows us to build confidence
intervals and upper confidence bounds. We approximate the
distribution of the random coefficients by using a standard mode–
Hessian normal approximation (Gelman, Carlin, Stern & Rubin, 2004),
which allows us to draw samples of the regression coefficients. This
permits us to make copies of the total errors, thus we get the
distribution of the total error. In a sample of 10,000 we obtain the
250th and 9750th values to form the lower and upper ends of the 95%
confidence interval.
To consider each dollar in an account is either in error or not, our
model is,
5
PR
276
277
N
ð11Þ
i=n + 1
ð15Þ
pðdi ≠0Þ = ð1−θÞ
β0 + β1 πi di −bi eβ0
ðbi e
Þ e
di !
+ β 1 πi
where di = 1,2,3, ….
The random effect corresponding to θ is θ + r21.We do not treat the
dis equally. Because most of the errors in the sample are zero, different
treatments of zero-valued dis and non-zero dis are desired to give
more appropriate weights to the zero values.
We center the ZIP model by taking r0i = r1i = r2i = 0, i = 1, 2, …, N,
similar to Eqs. (8) and (9).
Since the dis are independent, we consider their joint probability
density function:
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360
361
362
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365
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368
322
323
N
Projected: ∑d˜i
i=1
ð12Þ
Lðθ; β0 ; β1 Þ = ∏
di = 0
324
325
where dĩ is the predicted value of di.
n
β
−b e 0
θ + ð1−θÞe i
+ β1 πi
o
(
∏
dt N 0
ð1−θÞ
ðbi e
β0 + β1 πi
β0 + β1 πi di −bie
Þ e
di !
)
:
ð16Þ 369
370
Please cite this article as: Higgins, H. N., & Nandram, B., Monetary unit sampling: Improving estimation of the total audit error, Advances in
Accounting, incorporating Advances in International Accounting (2009), doi:10.1016/j.adiac.2009.06.001
ARTICLE IN PRESS
Thus the log-likelihood function is
n
−bieβ0
Δðθ; β0 ; β1 Þ = ∑ ln½θ + ð1−θÞe
+ β1 πi
π=1
M
+
∑
l=π + 1
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376
+ ðN−nÞ lnð1−θÞ ð17Þ
M
di ðβ0 + β1 πi Þ− ∑
bie
β0 + β1 πi
l=π + 1
:
We use the Expectation Maximization (EM) algorithm to maximize the function over θ and ß (see Appendix B).
The values we obtain from the above procedures are:
θ̃ = 0:80 β̃0 = −3:25 β̃1 = −14:36:
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377
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380
396
5. Simulations
397
We perform three simulations to assess the reliability and
efficiency of our ZIP upper bound, using the following definitions by
Swinamer et al. (2004): a 95% upper confidence bound is reliable if,
when used repeatedly, the bound exceeds the true error 95% of the
time. Efficiency measures the size of the bound: the smaller the bound
is, the more efficient it is said to be. We compare the reliability and
efficiency of our ZIP method with the Stringer bound, because this
bound is used the most extensively by accounting auditors as
discussed in previous sections. We also compare our ZIP method
with the Multinomial-Dirichlet bound, because this bound demonstrates the best reliability for a variety of populations (Tsui et al.,
1985). More MUS bounds exist but they have relatively more
reliability problems and they all have pros and cons (Grimlund &
Felix, 1987; Swinamer et al., 2004).
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404
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t3:1
RR
EC
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Table 3
Simulation results.
t3:2
t3:3
Reliability
1
UN
t3:4
t3:5
t3:6
t3:7
t3:8
t3:9
t3:10
t3:11
t3:12
t3:13
t3:14
t3:15
CO
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382
TE
395
Our estimations are based on two estimators, the predicted
estimator and the projected estimator as defined in Eqs. (11) and
(12). We estimate the mean, standard error, confidence interval, and
upper bound using in a manner as described in Eqs. (13) and (14). We
show in Appendix B how to obtain 10,000 copies of the overstatement
error.
The predicted estimator has a mean of $3272, a standard error of
$1037, a 95% confidence interval of [$1487; $5336], and a 95% upper
confidence bound of $4848. The projected estimator has a mean of
$3435, a standard error of $1463, a 95% confidence interval of [$931;
$6107], and a 95% upper confidence bound of $5444.
As per design, we argue our ZIP method has advantage because it
addresses the selection bias in MUS towards selecting larger accounts,
it handles populations with a large number of zero values, and it
handles zero versus non-zero values explicitly separately. As a result of
this design, ZIP should be better suited than common estimation
practice for accounting populations.
Simulation 1 consists of three steps. First, we fill in the missing
audit value for the 87 − 20 = 67 accounts in our data set. We use the
bivariate Parzen–Rosenblut kernel density estimator, an independent
non-parametric model (Silverman et al., 1986) to fill in the audit
values. To model errors, we formulate the missing audit values in the
range 0.90 book value b audit value b book value. We create the
scenario where 80% of the book values and the audit values are in
perfect agreement. This population is kept fixed throughout. Because
we generate all the necessary audit values, we have the entire
population and the true total population error.
Second, we draw 1000 PPS samples (probabilities of selection are
proportional to the book values) without replacement of size 20 from
the single populations we construct in the first step. Third, we fit the
ZIP regression model in exactly the same manner we discussed in
Section 3 for the observed data. We compute 90% confidence interval
for each fixed population, whose upper end provides a 95% upper
confidence bound. For comparison, we compute the same for the
Stringer bound and the Multinomial-Dirichlet bound. The results are
shown in Table 3.
Table 3, Columns 2–6 show the reliability results, specifically the
frequency in which the bounds exceed the true population error. In
Simulation 1, the ZIP bound exceeds the true population error 89.5% of
the times, whereas it is supposed to exceed 95% of the times. In
contrast, the Stringer bound exceeds the true error 100% of the times,
while this frequency is 96% for the Multinomial-Dirichlet bound.
These results demonstrate ZIP's lower reliability than the theoretical
level and the extra conservatism of the Stringer bound.
To report on the efficiency, Columns 5 and 6 show the frequency in
which the ZIP bound is larger (more inefficient) than the Stringer
bound and the Multinomial-Dirichlet bound. In Simulation 1, these
frequency numbers are very close to zero (0.7% and 9.6%, respectively), indicating the ZIP bound is almost always smaller, or more
efficient. To further shed light on the size of the other bounds relative
to the ZIP bound, Columns 7 and 8 show the ratio of the differential
size over the ZIP bound. On average, this ratio is about 13 times for the
Stringer bound, and 25.8% for the Multinomial-Dirichlet bound. These
results demonstrate the far greater efficiency of the ZIP bound
compared to the other two.
Simulation 2 is similar to the first, except that the audit values are
generated by the ZIP model. The generated data have similar
characteristics to accounting data in the sense that they come from a
distribution that presumes a very large number of zero errors. From
Column 2, the ZIP bound is reliable as its 95% upper bound exceeds the
true error value 96.8% of the times. From Columns 5 and 6, the frequency
in which the ZIP bound is larger than the other two bounds is very small
(0.8% and 3%, respectively), denoting ZIP's greater efficiency.
Simulation 3 is similar to the first two, except that the audit values
are generated to have error distributions identical to Population 4 in
RO
OF
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H.N. Higgins, B. Nandram / Advances in Accounting, incorporating Advances in International Accounting xxx (2009) xxx–xxx
DP
6
Simulation 1
Data from Parzen–Rosenblut model
Simulation 2
Data from ZIP model
Simulation 3
Data from Neter Loebbecke population 4
|{z}
Efficiency
2
3
4
5
6
7
8
ZNT
SNT
MNT
ZNS
ZNM
(S − Z) / Z
(M − Z) / Z
0.895
1.000
0.960
0.007
0.096
12.989
0.258
0.968
1.000
1.000
0.008
0.030
0.342
0.188
0.984
1.000
1.000
0.009
0.044
0.343
0.182
|{z}
Proportion of 1000 simulations
|{z}
Average over 1000 simulations
Z: ZIP bound, S: Stringer bound, M: Multinomial-Dirichlet bound, T: True value.
The rows show the results from three separate simulations. Columns 2–6 show the frequency in which one bound is larger than the true value or another bound. Columns 7 and 8
show the relative size of the other bounds compared to the ZIP bound.
Please cite this article as: Higgins, H. N., & Nandram, B., Monetary unit sampling: Improving estimation of the total audit error, Advances in
Accounting, incorporating Advances in International Accounting (2009), doi:10.1016/j.adiac.2009.06.001
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An estimate of the covariance matrix of β0̂ and β1̂ is the negative 514
inverse Hessian matrix. The Hessian matrix is
515
0
n
β + β1 πi
bi e 0
B i∑
B =1
B n
@
β
∑ bi πi e 0
β0 + β1 πi
i=1
+ β1 πi
i=1
n
2 β0 + β1 πi
∑ bi πi e
H
−1
=
c
a2
a1
c
where
519
520
n
2 β0 + β1 πi
∑ bi πi e
i=1
a1 =
n
β0 + β1 πi
∑ bi e
i=1
n
2 β0 + β1 πi
∑ bi πi e
i=1
n
494
495
496
497
498
499
500
501
Q4 502
7. Uncited reference
Silverman, 1986
504
Acknowledgement
505
506
507
The authors acknowledge the helpful comments from two
anonymous reviewers and the Journal's associate editor. Any errors
are the authors' responsibility.
508
Appendix A
509
UN
503
For the centered model, the log-likelihood function is
n
n
n
i=1
i=1
i=1
f ðβ0 ; β1 Þ = β0 ∑ di + β1 ∑ πi di − ∑ bi e
510
511
512
513
β0 + β1 πi
:
We use a two-dimensional Newton's method to maximize it over
β0 and β1. Details are omitted.
Þ
521
522
β0 + β1 πi
n
β0 + β1 πi 2
ð ∑ bi πi e
β0 + β1 πi
Þ − ∑ bi e
i=1
∑ bi e
n
2 β0 + β1 πi
∑ bi πi e
i=1
523
524
β0 + β1 πi
i=1
a2 =
n
β0 + β1 πi
∑ bi e
ED
492
493
β0 + β1 πi 2
i=1
∑ bi πi e
n
i=1
n
2 β0 + β1 πi
∑ bi πi e
i=1
n
β0 + β1 πi 2
−ð ∑ bi πi e
:
Þ
i=1
525
526
527
First, we take
β0 + r0i
β1 + r0i
CT
490
491
RE
488
489
We propose a method that improves upon the common MUS
estimation approach. MUS estimation as commonly practiced by
accounting auditors does not explicitly recognize that a total
population of account errors typically consists of multiple distinct
distributions. As a result, this common approach often leads to very
large error estimations and very conservative auditor's decisions on
the fairness of client financial statement. For conservatism auditors
should want large estimations of errors and upper bounds, however
estimations under common practice are too conservative, and
excessive conservatism has its own problems. Our method, based on
Zero-Inflation Poisson distribution, addresses the above shortcoming.
We discuss our method and show that for data similar to accounting
populations, our bound is reliable and more efficient than common
MUS practice, so we would recommend our bound to accounting
auditors. For other populations, our method may be slightly less
reliable than theoretically desired, and future research should seek to
improve the method for these populations.
n
−ð ∑ bi πi e
i=1
c=
6. Conclusion
CO
R
486
487
C
C
C
A
whose elements are all positive values. And the negative inverse 516
517
Hessian matrix is
518
n
485
1
i=1
i=1
484
n
∑ bi πi e
OO
F
461
462
Neter and Loebbecke (1975). This population consists of 4033
accounts receivables of a large manufacturer. Neter and Loebbecke
(1975) have made major contributions in analyzing the error
characteristics of accounting populations, and their populations are
often used to represent accounting populations (Chan, 1988;
Smieliauskas, 1986a; Frost & Tamura, 1982). From Column 2, the ZIP
bound is reliable as its 95% upper bound exceeds the true error value
98.4% of the times. From Columns 5 and 6, the frequency in which the
ZIP bound is larger than the other two bounds is very small (0.9% and
4.4%, respectively), denoting ZIP's greater efficiency. Columns 7–8
show the magnitude of the Stringer bound is about 34% larger and the
Multinomial-Dirichlet bound about 18% larger than the ZIP bound,
also denoting ZIP's greater efficiency.
Summing the simulation studies, results based on a general
population without distinct distributions reveal our ZIP method to
be very efficient, although with slightly lower reliability than the
theoretical level. However, based on populations similar to or taken
from accounting error populations, the ZIP bound is reliable and more
efficient than the Stringer and Multinomial-Dirichlet bounds. The ZIP
overall performance seems very good, in light of Swinamer et al.
(2004), who use extensive simulations on both real and simulated
data to compare 14 bounds, and find no single method to be superior
in terms of both reliability and efficiency. Given the trade-off between
reliability and efficiency in the state-of-the-art, our ZIP method is very
promising as an MUS estimation technique.
PR
459
460
7
β̃0
e Normalf β̃
1
ind
! a1
;
c
c
g
a2
where a1, a2, and c are the elements in the previous matrix H− 1. This
is a standard approximation and is typically used in generalized
regression model.
Next, we show how to construct the confidence interval.
We can see that ln λi = (β0 + r0i) + (β1 + r1i) πi also follows the
normal distribution and we can obtain its mean and variance easily,
528
529
530
531
532
533
534
2
ln λi iid
e Normalðβ̃0 + πi β̃1 ; 1 = a1 + πi = a2 + 2πi = cÞ:
As we discussed earlier, we know that di follows the Poisson
β0 + β1 πi
distribution di j λi ind
Þ. This is equivalent to di jλi ind
e Poissonðbi e
e
lnλi
Poissonðe bi Þ.
We can then draw Z1, Z2, …, ZN, N independent standard normal
random variables, and compute
lnðλi Þ = ðβ̃0 + πi β̃1 Þ + Zi
535
536
537
538
539
540
541
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 = a1 + π2i = a2 + 2πi = c
where i = 1, 2, …, N.
Thus, we can now draw di as a Poisson random variable with
parameter λ̃i, i = 1, 2, …, N. So these random draws of λ̃1, λ̃2, …, λ̃N are
given by
λ̃i = expfðβ̃0 + πi β̃1 Þ + Zi
542
543
544
545
546
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 = a1 + π2i = a2 + 2πi = cg
where i = 1, 2, …, N. We repeat this process 10,000 times and obtain 547
548
N
10,000 values of the overstatement error ∑ di.
i=1
Please cite this article as: Higgins, H. N., & Nandram, B., Monetary unit sampling: Improving estimation of the total audit error, Advances in
Accounting, incorporating Advances in International Accounting (2009), doi:10.1016/j.adiac.2009.06.001
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Appendix B
551
A Zero-Inflated Poisson (ZIP) regression model
Now the joint probability density function becomes
n
β
−b e 0
Lðθ; β0 ; β1 Þ = ∏ θ + ð1−θÞe i
+ β1 πi
D
556
557
559
558
561
560
562
563
564
565
566
567
+ β1 πi di −bi e
D
Þ e
di !
β0 + β1 πi
)
–
where D is the set of accounts with where di = 0; D is the set of
accounts with where di ≠ 0.
To apply the Expectation Maximization (EM) algorithm, we
introduce a new variable Zi, which is defined as
pðdi = 0jZi = 1Þ = 1
N
pðdi ≠0 jZi = 1Þ = 0:
∑di .
i=1
We can obtain the Hessian matrix of θ ̂, β̂0,1 to maximize the
likelihood function; it is more convenient to use the EM algorithm. By
EM algorithm, optimizing the joint probability density function above
is equivalent to optimizing the following quantity:
˜
where
β0 + β1 πi
+ β1 πi Þ−bi e
and the MLE of β0 and β1 can be obtained by Newton's method in
Appendix A.
Since θ only exists in the first two terms, we separate this product
into two parts, the first two factors and the last two factors. We
differentiate the log-likelihood function of g(θ, Z ), set it equal to zero
˜
and solve for θ. Thus we have
∑ Zi
n−∑ Zi
gðθ; Z Þ = θ D ð1−θÞ D
˜
D
∂
lnðgðθ; Z ÞÞ =
∂θ
˜
D
∑ Zi
D
θ
−
ðn−∑ Zi Þ
D
=0
1−θ
UN
Therefore,
CO
lnðgðθ; Z ÞÞ = ∑ Zi lnθ + ðn−∑ Zi Þ lnð1−θÞ
˜
θ = ∑ Zi = n
D
!
θ
β0 + β1 πi
θ + ð1−θÞe−bi e
Zi jðθ; β0 ; β1 Þ ind
e Bernoulli
582
581
i= 1; 2; …; n
ðB:1Þ
And the conditional expectation of Zi,
EfZi jθ; β0 ; β1 g =
θ
β 0 + β 1 πi
θ + ð1−θÞe−bi e
:
ðB:2Þ
584
583
585
586
587
n
i=1
N
∑
i=n + 1
di to 600
First iteration, we use θ = 0.5 as a random starting value. For β0, β1,
we use the MLE estimates of the regression coefficients as the starting
values, i.e., β̃0 = −5.71, β1̃ = − 0.22.
603
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RR
EC
˜
N
i=1
TE
β0 + β1 πi ∑½d ðβ
−∑ð1−Zi Þbi e
i
0
D
eD
hðβ0 ; β1 ; Z Þ = e
580
579
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obtain D for the predicted or the projected estimator. We repeat this 601
process 10,000 times and obtain 10,000 values of the overstatement error 602
˜
578
80 1
9
1
>
>
θ + r2i
< β̃0
=
ind
−1
C
@ β0 + r0i A e Normal B
:
@ β̃1 A; −H
>
>
:
;
β1 + r1i
θ̃
di from Poisson(biλi). Then we calculate ∑ di and ∑ di +
571
570
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577
594
Thus, we have
∑ Zi
n−∑ Zi
gðθ; Z Þ = θ D ð1−θÞ D
574
575
592
593
Zi = 1; if di is pure zero;
Zi = 0; else
˜
573
572
590
591
We draw a random sample of (β0 +r0i, β1 +r1i, θ+r2i) from the 597
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gðθ; Z Þhðβ0 ; β1 ; Z Þ
569
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589
0
DP
553
554
555
(
o
ðb eβ0
∏ ð1−θÞ i
RO
OF
552
After we get all the E{Zi|θ,β0,β1} for the sample using Eq. (B.2), we
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Letting H be the Hessian matrix, too cumbersome to present, we take
Please cite this article as: Higgins, H. N., & Nandram, B., Monetary unit sampling: Improving estimation of the total audit error, Advances in
Accounting, incorporating Advances in International Accounting (2009), doi:10.1016/j.adiac.2009.06.001
ARTICLE IN PRESS
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Please cite this article as: Higgins, H. N., & Nandram, B., Monetary unit sampling: Improving estimation of the total audit error, Advances in
Accounting, incorporating Advances in International Accounting (2009), doi:10.1016/j.adiac.2009.06.001
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