Variance bounds for the design of audit sampling

Journal of Statistical Planning and Inference 142 (2012) 2629–2645

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Variance bounds for the design of audit sampling ¨ n Rainer Gob Institute for Applied Mathematics and Statistics, University of W¨ urzburg, Sanderring 2, D-97070 W¨ urzburg, Germany

a r t i c l e i n f o

abstract

Article history: Received 30 June 2009 Accepted 6 February 2012 Available online 11 February 2012

Audit sampling tests the conformance of monetary book values U kept in lists and databases on items like accounts, articles in an inventory, transactions, with the corresponding de facto values W of the items in reality. Variables sampling schemes focus on the conformance of the book value mean mU with the de facto mean mW . An essential design parameter of variables schemes is the unknown variance s2W of the de facto values. It is a common practice to exploit the information conveyed by the book values in a naive way by equating s2W with the known book value variance s2U for design purposes. However, the de facto value variance may differ substantially from the book value variance. A robust design should be based on some reliable upper bound for the variance of s2W . The present paper presents bounds for the variance s2W and for the variance s2UW under various stochastic models for the relationship between book values and de facto values. & 2012 Published by Elsevier B.V.

Keywords: Audit sampling Variables sampling Overstatement models Finite support distributions Variance bounds Correlation bounds

1. Introduction An essential task of industrial auditing is testing for the conformance of monetary book values U kept in lists and databases on items like accounts, articles in an inventory, transactions, with the corresponding de facto values W of the items in reality. Often the populations of items are large and the inspection of items is expensive. Hence auditing often resorts to audit sampling, i.e., sampling inspection instead of screening the population. Approaches to audit sampling can be classified into three groups according to the underlying model for the item and population characteristics, see Loebbecke and Neter (1975): attributes procedures, variables procedures, and combined attributes-variables (CAV) procedures which also became known as monetary unit sampling (MUS) or probability proportional to size sampling (PPS). All three types adopted a part of their methodology from survey sampling as used in social sciences, opinion polls, household surveys or market research. However, the auditing context is essentially different from the survey context. Survey sampling is designed for exploring unknown populations, mostly under few prior knowledge on the population parameters. The objective of audit sampling is a test for the conformance of known book values and unknown de facto values, under three sources of prior knowledge: (1) historical auditing records; (2) auditing procedures which precede sampling inspection in the test of details context, namely risk assessment procedures, test of controls, substantive analytical procedures; and (3) the book values which, in the regular case, are not completely erroneous though probably not 100% reliable. Audit sampling schemes have been using prior knowledge for two purposes: (1) sampling design and (2) sample analysis and inference. PPS (MUS) is a simple idea to exploit the book values for the sampling design. Bayesian approaches have been used to exploit prior knowledge also for sample inference in PPS schemes, see Garstka (1977), McCray (1984),

n

Tel.: þ49 931 312377. E-mail address: [email protected]

0378-3758/$ - see front matter & 2012 Published by Elsevier B.V. doi:10.1016/j.jspi.2012.02.006

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Cox and Snell (1979), Godfrey and Neter (1984), and Laws and O’Hagan (2000), for instance. Contributions to variables schemes have mainly been concentrating on exploiting prior information for sample inference, see Kaplan (1973a,b) and Neter and Loebbecke (1977), for instance. The design aspect has received less interest in the literature on variables sampling. The purpose of variables sampling is to provide information on the unknown de facto mean mW ¼ E½W, see Guy et al. (2002). The essential design parameter for variables sampling procedures is the unknown variance of the de facto values, either the total population variance s2W ¼ V½W in unstratified sampling, or the stratum variances s2W,1 , . . . , s2W,k in stratified sampling. The book values are a natural source of information for forming an idea on the unknown de facto values variance. It is a widespread practice to exploit this information in a naive way by assuming s2W ¼ s2U , and s2W,l ¼ s2U,l for l ¼ 1, . . . ,k. However, the de facto value variance may differ substantially from the book value variance, as will be illustrated in subsequent sections. Designs based on equating the variances of W and U may be inappropriate. A robust design should be based on some reliable upper bound for the variance of W. Studies on bounds for the variance of W are rare. Roberts (1986) established a worst case bound under a proportional tainting model for overstatement, see Proposition 4, below. Liu et al. (2005) relax the worst case bound by concentrating on a specific class of conditional densities of W under U, see Example 2, below. The present study integrates earlier contributions by Roberts (1986) and Liu et al. (2005) as special cases into a more general and flexible scheme of stochastic overstatement models where Pð0 r W r UÞ ¼ 1. Under these models, bounds for the variances s2W and s2UW are derived, with the following input parameters: (i) The known mean mU ¼ E½U and the known variance s2U ¼ V½U of the book values U. (ii) To account for prior information on W, measures of association between U and W, namely the covariance sU,W ¼ Cov½U,W and the correlation rU,W ¼ Corr½U,W. Whereas more sophisticated ways of expressing prior knowledge as in Bayes approaches have a rather restricted scope of application, partial information conveyed by the correlation may occasionally be available in the auditing context, and can successfully ¨ and Muller ¨ be used for the design of audit sampling, see Gob (2010), for instance. (iii) The ratio b ¼ mW =mU of the de facto value mean mW and the book value mean mU . The ratio b ¼ mW =mU serves as a design parameter. Statistical inference procedures on the unknown de facto mean mW are designed to warrant specific properties for specific values of b. For design purposes, it is particularly interesting to consider the least tolerable value mW, t ¼ mU ð1tÞ at the lower bound of a prescribed tolerance interval ½mU ð1tÞ; mU ð1 þ tÞ for the de facto value mean mW . The value mW ¼ mW, t represents the situation of highest uncertainty under overstatement, so it is particularly relevant to design the sampling scheme under the assumption mW ¼ mW, t , i.e., bt ¼ mW, t =mU ¼ 1t. The study is organised into the following paragraphs. Sections 2 and 3 introduce the basic superpopulation and overstatement models. Sections 4 and 5 state and discuss bounds for the variance s2W under the various overstatement models. Bounds for the variance s2UW of the difference UW between book and de facto values are stated by Section 6. Section 7 provides a few lower bounds for the correlation rU,W . The proofs of the propositions stated by Sections 4–7 are developed in Appendices A–E. 2. Superpopulation model Consider a population of items 1; 2, . . . ,N, e.g., account positions, articles in an inventory, transactions. Each item i is associated with two monetary values: the book value Ui stated in a list or database, and the de facto value Wi holding in reality. We consider the populations of book values and of de facto values from a superpopulation view where U 1 , . . . ,U N , W 1 , . . . ,W N are segments from an underlying infinite process or superpopulation. Superpopulation models are common in the analysis of survey sampling, see Cochran (1977) or Chaudhuri and Stenger (2005), and of audit sampling, see Kaplan (1973a,b), Cox and Snell (1979), Roberts (1986), and Laws and O’Hagan (2000), for instance. Since auditing is concerned with large populations it is legitimate for purposes of theoretical analysis to identify population parameters with P P corresponding superpopulation parameters, in particular, means m U ¼ ð1=NÞ U i , m W ¼ ð1=NÞ W i with means mU ¼ E½U, P P mW ¼ E½W, and variances s 2U ¼ ð1=ðN1ÞÞ ðU i m U Þ2 , s 2W ¼ ð1=ðN1ÞÞ ðW i m W Þ2 with variances s2U ¼ V½U, s2W ¼ V½W. It is generally acknowledged that serial association among successive items is not an issue for superpopulation models in sampling theory. Even if the underlying process should exhibit some serial association, it would be neutralised by random sampling. It is also reasonable to assume that the underlying processes of book values and de facto values are homogeneous, i.e., that U 1 , . . . ,U N and, respectively, W 1 , . . . ,W N are i.i.d. sequences. In view of this property, the index is henceforth omitted in U and W. 3. Overstatement models The information given by the book value U is not completely reliable, the true value W may be misstated. The analysis of audit sampling schemes requires a stochastic misstatement model or linkage model, see Duke et al. (1982), to express the random mechanism which leads to the partially incorrect or distorted information conveyed by U on W. Misstatement of U on W can occur with two directions: overstatement, i.e., 0 r W oU, or understatement, i.e., W 4 U. Empirical studies by Ramage et al. (1979), Johnson et al. (1981), Ham et al. (1985), and Icerman and Hillison (1990) show that overstatement dominates in accounts receivable and revenue accounts. Accounts payable rather tend to

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Table 1 Models of overstatement, i.e., Pð0 r W r UÞ ¼ 1. (I) Mixture (occasional tainting)

Model definition: W ¼ ZU þ ð1ZÞQ where Z  Bið1; 1pÞ, Z independent of U and of Q, Pð0 r Q o UÞ ¼ 1 mU mW 2 s ¼ ð1pÞs2U þ ps2Q þ pð1pÞðmU mQ Þ2 mU mQ W

Parameters: mW ¼ ð1pÞmU þ pmQ , p ¼ (II) Occasional 100% tainting

Model definition: W ¼ ZU where Z  Bið1; 1pÞ, Z independent of U Parameters: special case of II with Q¼ 0, special case of IV with b ¼ 1p

(III) Occasional tainting, decreasing Model definition: model I where the conditional distribution of Q under U has a Lebesgue density f Q 9U density of Q concentrated on [0;U], f Q9U is decreasing on [0;U], f Q 9U ðqÞ o 2mQ 9U =U 2 for dU o q r U with suitable dU 4 0, and

mQ 9U o 0:5U Model definition: Pð0 r W r UÞ ¼ 1, mW9U ¼ bU

(IV) Proportional mean

sU sU Z sW ¼ b Z bsU rU,W rU,W 2 Model definition: model IV with E½W 2 9U ¼ gU 2 where b r g r b Parameters: mW ¼ bmU , E½UW ¼ bE½U 2 , sU,W ¼ bs2U ,

(V) Proportional mean and second moment

2

Parameters: mW ¼ bmU , s2W ¼ gs2U þ ðgb Þm2U (VI) Proportional tainting Model definition: W¼ TU where Pð0 r T r 1Þ ¼ 1, T and U independent (VII) Increasing conditional density Model definition: the conditional distribution of W under U has a Lebesgue density f W9U concentrated on [0;U], of W f W9U is increasing on [0;U], f W9U ðwÞo 2ðUmW9U Þ=U 2 for 0r wo eU with suitable eU 4 0, and U 4 mW9U 40:5U

understatement, overstatement and understatement are balanced for inventory data. Audit sampling theory has been placing more emphasis on overstatement, particularly in the MUS context. The present study follows this preference and concentrates on overstatement. For design purposes, hedging against overstatement is a minimum requirement. It is to be expected that a variance conservative design for overstatement will also be robust for handling understatement situations. From a stochastic point of view, overstatement is basically characterised by Pð0 r W r UÞ. Table 1 lists a number of more specific variants of overstatement. The parameters of the models follow by elementary calculation. A very general and plausible framework for overstatement modeling is provided by the mixture model or occasional tainting model I. Under model I, a large percentage of ð1pÞ100% of the book values is correct. A small percentage of p100% is tainted by overstatement, where the true de facto value W equals an unknown random variable Q. The occasional100% tainting model II is the worst case version of model I where the true value of all erroneously stated items is W¼Q¼0. Roberts (1986) arrived at model II in inferring worst case upper limits on the variance s2W under model VI, see Proposition 4, below. Model III is a less pessimistic version of the occasional tainting model than model II. Models IV and V require proportionality of E½W9U to U and of E½W 2 9U to U2, respectively. These are linear models without intercept and with residuals restricted by the requirement Pð0 r W r UÞ. In particular, model V holds if U acts as scale parameter in the conditional distribution of W under U, i.e., if f W9U ðwÞ ¼ gðw=UÞ=U for the conditional density f W9U where g is a density concentrated on [0;1]. An important instance of the latter model is described by the subsequent Example 1. The proportional tainting model VI is a special instance of models IV and V. Proportional tainting has been studied by several authors in the audit sampling context, e.g., by Kaplan (973a,b) and Roberts (1986). Model VII assumes a conditional Lebesgue density f W9U , increasing on the support [0;U]. Model VII is realistic if the frequency of overstatement is continuously decreasing with the magnitude of overstatement. The findings of Johnson et al. (1981) show that such patterns occur both in accounts receivable and in inventory data. Model VII is inappropriate in cases of occasional extreme tainting, particularly in case of occasional 100% tainting. An instance of a combination of models V and VII is described by the subsequent Example 1. Example 1 (Conditional scaled beta distribution). The beta distribution with parameters a,b4 0, is defined by the density g a,b ðxÞ ¼

Gða þ bÞ a1 x ð1xÞb1 for 0 rx r1, GðaÞGðbÞ

ð1Þ

is a flexible model for distributions concentrated on [0;1]. Rescaling the beta distribution from [0;1] to [0;U] provides examples for models III, IV, V and VII. For arbitrary a,b4 0, the scaled beta density f W9U ðwÞ ¼ g a,b ðw=UÞ=U defines an instance of models IV and V where

mW9U ¼

a U, aþb |ffl{zffl} ¼b

E½W 2 9U ¼

aða þ 1Þ U2: ða þ bÞða þb þ 1Þ |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð2Þ

¼g

For 0 o a r1 r b, g a,b is decreasing on [0;1]. Hence the density f Q 9U ðqÞ ¼ g a,b ðq=UÞ=U defines an instance of model III. For 0 o b r1 r a, g a,b is increasing on [0;1] with g a,b ð0Þ ¼ 0. Hence the density f W9U ðwÞ ¼ g a,b ðw=UÞ=U defines an instance of model VII.

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In the case of 0 oa,bo 1 or a,b 41, the density f W9U ðwÞ ¼ g a,b ðw=UÞ=U has either a bathtub shape or an inverted bathtub shape. & In their study on audit sampling design, Liu et al. (2005) consider the step density model which is a special case of model V, see the subsequent Example 2. Example 2 (Step density model, Liu et al. (2005)). Let 0 oa o 1 and let f W9U ðwÞ ¼ a=ð1aÞU for 0 r wr ð1aÞU, and f W9U ðwÞ ¼ ð1aÞ=aU for ð1aÞU rw rU. Then

mW9U ¼ ð1aÞU, E½W 2 9U ¼

ð1aÞð32aÞ 2 U , 3

i.e., we have a special case of model V. In the case of a o0:5, the step density model is also a special case of model VII.

&

4. Overstatement bounds for the variance r2W without model IV This paragraph develops upper bounds for the variance s2W under overstatement, i.e., Pð0 rW r UÞ ¼ 1, without assuming model IV. Though plausible in many situations, model IV is not universally valid. Model IV requires a constant proportional overstatement on the average, expressed by a constant ratio b ¼ mW9U =U, irrespective of the magnitude of U. This property is violated in situations where overstatement is directed by the magnitude of U, e.g., overstatement strongly concentrating on small U or strongly concentrating on large U, see the subsequent Example 3. The following Proposition 1 establishes worst case bounds for the variance s2W under overstatement without any further model assumptions. The bounds depend on the association parameters sU,W and rU,W . The proof of Proposition 1 is provided by Appendix B.1. Proposition 1 (Worst case upper bounds for s2W ). Let b ¼ mW =mU . Then we have: (a) E½W 2  r E½WU, s2W r sU,W þ m2U bð1bÞ, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sW r 12frU,W sU þ r2U,W s2U þ 4m2U bð1bÞg ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hW,worst ðbÞ:

ð3Þ

(b) The following two assertions are equivalent. (i) E½W 2  ¼ E½WU, i.e., E½W 2  and s2W adopt their respective upper bounds established by assertion (a). (ii) PðW ¼ 0Þ þPð0 o W ¼ UÞ ¼ 1. If PðU 4 0Þ ¼ 1, then (i) and (ii) are equivalent to (iii) PðW ¼ 0Þ þ PðW ¼ UÞ ¼ 1. Model III, discussed by the subsequent Proposition 2, is a special case of the occasional tainting model I where the tainted value Q has a conditional density decreasing on [0;U]. The proof of Proposition 2 is provided by Appendix B.2. Proposition 2 (Bounds for s2W under model III). Assume model III. Let qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sl ¼ 13frU,Q sU  r2U,Q s2U þ6mQ ðmU 1:5mQ Þg,

ð4Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2U,Q s2U þ 6mQ ðmU 1:5mQ Þg:

ð5Þ

Then we have sl r sQ rsu . In particular, the following upper bounds hold: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sQ r 13frU,Q sU þ r2U,Q s2U þ m2U g,

ð6Þ

su ¼ 13frU,Q sU þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sQ r 13fsU þ s2U þ 6mQ ðmU 1:5mQ Þg r 13fsU þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2U þ m2U g:

ð7Þ

Model VII stipulates for W an increasing conditional density on [0;U]. The bounds presented by the subsequent Proposition 3 are similar in nature to the bounds under model III given by Proposition 2. The proof of Proposition 3 is provided by Appendix B.3. Proposition 3 (Bounds for s2W under model VII). Assume model VII and let b ¼ mW =mU where 0:5 o b o 1 by assumption of 2 model VII. Then we have ð4r2U,W 3Þs2U þ 3m2U ð3b þ 4b1Þ Z 0. Let qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sl ðbÞ ¼ 23frU,W sU 0:5 ð4r2U,W 3Þs2U þ 3m2U ð3b þ 4b1Þg, ð8Þ

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Fig. 1. Upper bounds hW,worst ðbÞ=s2U and hW,VII ðbÞ=s2U for s2W =s2U as a function of b ¼ mW =mU , model IV not required.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 su ðbÞ ¼ 23frU,W sU þ 0:5 ð4r2U,W 3Þs2U þ 3m2U ð3b þ 4b1Þg, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hW,VII ðbÞ ¼ su ðbÞ2 . Then we have sl ðbÞ r sW r su ðbÞ ¼ hW,VII ðbÞ. In particular, the following upper bounds hold: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sW r 23fsU þ0:5 s2U þ3m2U ð3b2 þ4b1Þg,

ð9Þ

ð10Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sW r 23frU,W sU þ0:5 ð4r2U,W 3Þs2U þ m2U g qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 23fsU þ 0:5 s2U þ m2U g:

ð11Þ

4.1. Discussion of bounds for the variance s2W The bounds under model III from Proposition 2 are the most difficult in implementation. The basic assumption of model III is weak and unproblematic. It only requires that the de facto values Q of overstated items have a density decreasing on [0;U]. However, the application of the bounds requires the specification of even two parameters: the correlation rQ ,U between the de facto value Q of overstated items and the corresponding book value U, and the de facto mean mQ of overstated items. The latter particularly strong requirement imposes a high level of prior information from the auditor’s side. The global worst case bound hW,worst ðbÞ from Proposition 1 and the bound hW,VII ðbÞ from Proposition 3 only requires to specify the correlation rU,W . Fig. 1 compares these bounds in four situations: moderate signal-to-noise ratio m2U =s2U ¼ 108:11 combined with moderate correlation rU,W ¼ 0:60 and with strong correlation rU,W ¼ 0:90, large signal-to-noise ratio m2U =s2U ¼ 387:16 combined with moderate correlation rU,W ¼ 0:60 and with strong correlation rU,W ¼ 0:90. The signalto-noise ratios m2U =s2U were taken from four strata in a stratified real population of book values, compare Fig. 3. The four graphs suggest the following conclusions. (1) The bounds are strongly increasing in the ratio m2U =s2U . (2) The more specific model assumption VII reduces the bounds considerably. (3) If b ¼ mW =mU moves away from 1, the bounds rapidly increase, particularly for large ratio m2U =s2U . (4) The effect of the correlation is moderate, particularly in the case of large signal-tonoise ratio m2U =s2U . 5. Overstatement bounds for the variance r2W under model IV The subsequently established bounds for s2W under overstatement, i.e., Pð0 rW rUÞ ¼ 1, are based on model IV, i.e., the assumption mW9U =U ¼ b constant. Model IV is plausible in many situations, but not universally valid, see the subsequent Example 3.

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The following Proposition 4 shows that the occasional 100% tainting model II provides the upper limit for s2W under model IV. Roberts (1986) introduced the occasional 100% tainting model II as the model which maximises the variance s2W under the proportional tainting model VI which is a special case of model IV. The proof of Proposition 4 is provided by Appendix C.1. Proposition 4 (Upper bounds for s2W under model IV). Assume model IV, i.e., mW9U ¼ bU where 0 r b r1. (a) We have

s2W r bðs2U þ m2U ð1bÞÞ ¼ hW,IV ðbÞ:

ð12Þ

In the case of mU r sU , hW,IV is strictly increasing on [0;1] with hW,IV ð1Þ ¼ s increasing on ½0; b0  and strictly decreasing on ½b0 ; 1 where b0 ¼ 0:5ð1 þ s2U =m (b) If PðU 4 0Þ ¼ 1, then the following two assertions are equivalent. (i) s2W ¼ hW,IV ðbÞ, i.e., s2W adopts the upper bound (12). (ii) W and U follow the model II with p ¼ 1b.

2 U 4 U , hW,IV is strictly U . In the case of 2 2 2 2 2 2 U Þ with hW,IV ðb0 Þ ¼ ð U þ U Þ =4 U 4 U .

m

s

s m

m

s

The worst case distributions characterised by assertion (b) of Proposition 1 and by assertion (b) of Proposition 4 are different. Proposition 4 requires PðW ¼ 09UÞ ¼ p and PðW ¼ U9UÞ ¼ 1p constant, whereas Proposition 1 only requires PðW ¼ 0Þ þPðW ¼ UÞ ¼ 1. The subsequent Example 3 provides an example of a class of joint distributions of U and W with the following properties: (1) For a large variety of input parameters rU,W and b, the general upper bound (3) is adopted, i.e., the conditions of assertion (b) of Proposition 1 are satisfied. (2) In general, the distributions do not satisfy model IV, i.e., the requirements of assertion (b) of Proposition 4 are not satisfied. Example 3 (Worst case variance of W). Let U have a uniform distribution on the interval [0;1], in particular mU ¼ 0:5, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi E½U 2  ¼ 1=3. Let b and n be prescribed values for b ¼ mU =mW , n ¼ E½UW where 12 3 2=3 r b r1, 0 r n r1=3 3ð1bÞ3 =4. Let a ¼ aðb, nÞ ¼

3ð1bÞ4   , 1 3 64 n 3

b ¼ bðb, nÞ ¼

3ð1bÞ3   : 1 2 16 n 3

ð13Þ

Obviously a o0 ob r1. The conditional distribution of W is defined by letting PðW ¼ 09UÞ ¼ pðUÞ, PðW ¼ U9UÞ ¼ 1pðUÞ, where 8 b < au þb for 0 r u r , ð14Þ pðuÞ ¼ a : 0 otherwise: Obviously the characteristic of assertion (b) of Proposition 1 holds. By elementary integration and algebra Z b=a Z 1 3 1 b b E½W ¼ E½E½W9U ¼ E½ð1pðUÞÞU ¼ ð1bauÞu du þ u du ¼  2 ¼ ¼ bmU , 2 2 6a 0 b=a E½UW ¼ E½UE½W9U ¼ E½ð1pðUÞÞU 2  ¼

Z

b=a

ð1bauÞu2 du þ

0 2 W

Z

1 b=a

4

u2 du ¼

1 b þ ¼ n: 3 12a3

2 U bð1bÞ

By Proposition 1, s adopts its maximum sU,W þ m where sU,W ¼ nmU mW ¼ nb=4. The class of distributions indexed by a ¼ aðb, nÞ, b ¼ bðb, nÞ as defined by (13) can model arbitrary ratios b ¼ mW =mU and nearly all values n ¼ E½UW between the natural bounds 0 rE½UW r E½U 2  ¼ 1=3. Model IV prescribes n ¼ E½UW ¼ bE½U 2  ¼ b=3, this value is obtained for the parameter values   b 3ð1bÞ4 34 ð1bÞ ¼ aIV ¼ a b, ¼ ,   3 3 44 b 1  64 3 3   b 33 ð1bÞ : ¼ bIV ¼ b b, 16 3 However, the distribution of W defined by the conditional probabilities (14) with a ¼ aIV , b ¼ bIV is obviously not an instance of model IV. & 2

Model V adds the requirement g ¼ E½W 2 9U=U 2 to model IV, see Table 1, where b o g o b. Under model V we have s ¼ gs2U þðgb2 Þm2U . For g ¼ b, the model V equation is the upper bound hW,IV ðbÞ defined by (12). For any 0 o b o 1 and b o g o b, the subsequent Example 4 provides an instance of model V. In particular, the example shows that under model 2 V s2W can adopt any value between the lower bound b s2U and the upper bound hW,IV ðbÞ. 2 W 2

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Example 4 (Conditional beta distribution). Let W under U have a scaled beta distribution with parameters a,b 40, defined by the conditional density f W9U ðwÞ ¼ Gða þ bÞ=GðaÞGðbÞðw=UÞa1 ð1w=UÞb1 for 0 r wr U with parameters a,b4 0, see Example 1. For arbitrary a,b4 0, this density defines an instance of model V where

b ¼ bða,bÞ ¼

a , aþb

g ¼ gða,bÞ ¼

aða þ 1Þ ða þ bÞbða,bÞ þ 1 ¼ bða,bÞ , ða þ bÞða þb þ 1Þ a þ bþ 1 2

see Eq. (2). Let values 0 o b0 o 1 and b0 o g0 o b0 be prescribed. Let a¼

b0 ðb0 g0 Þ ðb g Þð1b Þ , b¼ 0 0 2 0 : g0 b20 g0 b0

For these values, elementary calculation shows bða,bÞ ¼ b0 , gða,bÞ ¼ g0 .

&

The following Proposition 5 considers the combination of model III and model IV. This combination is equivalent to requiring a constant proportionality k ¼ mQ9U =U for Q and U. The proof of Proposition 5 is provided by Appendix C.2. Proposition 5 (Bounds for s2W under model III). Assume model III and let k ¼ mQ 9U =U be constant. Then we have

s2Q r 23kfs2U þ m2U ð11:5kÞg

ð15Þ

and

kð1 þ 2bÞ

2 W

s r

b

3 1k



s2U þ m2U ð1bÞ b

k



3ð1kÞ

,

ð16Þ

where b ¼ mW =mU . The following two Propositions 6 and 7 consider more specific variants of model VII: the combination of models IV and VII in Proposition 6, and in Proposition 7 the combination of model VII with a scaled conditional beta distribution as introduced by Example 1. Proposition 6 (Bounds for s2W under model IV and VII combined). Assume both model IV and model VII to hold, where 1 4 b ¼ mW =mU 4 0:5. Then we have

s2W r 13ð4b1Þðs2U þ m2U Þb2 m2U ¼ hW,IV2VII ðbÞ:

ð17Þ

In the case of 2s2U r m2U , hW,IV2VII is strictly increasing on ½0:5; b0  and strictly decreasing on ½b0 ; 1 where b0 ¼ 2=3ððs2U þ m2U Þ=m2U Þ with ! s2 þ m2 2 s2U þ m2U hW,IV2VII ðb0 Þ ¼ hW,IV2VII ¼ U 2 U ð4s2U þ m2U Þ: 3 m2U 9mU In the case of m2U o 2s2U , hW,IV2VII is strictly increasing on [0.5; 1.0] with hW,IV2VII ð1:0Þ ¼ s2U . Proposition 7 (Bounds for s2W , model VII with beta distribution). Assume model VII where W under U has a scaled beta distribution defined by the density f W9U ðwÞ ¼ ð1=UÞðGða þ bÞ=GðaÞGðbÞÞðw=UÞa1 ð1w=UÞb1 for 0 rw r U, and let b ¼ mW =mU 40:5. Then we have

b 2b

ðs2U þð1bÞ2 m2U Þ r s2W r

b2

ð2s2U þ ð1bÞm2U Þ , 1þb |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð18Þ

¼ hW,VII-beta ðbÞ

where the lower bound is adopted for the scaled beta distribution with parameters a ¼ b=ð1bÞ, b¼ 1, and where the upper bound is adopted for the scaled beta distribution with parameters a ¼1, b ¼ ð1bÞ=b. If m2U 4 3s2U , hW,VII-beta is strictly increasing on ½0:5; b0  and strictly decreasing on ½b0 ; 1 where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b0 ¼

s4U s2U s2U 1:5 þ þ þ 4:25: 2 mU m4U m2U

ð19Þ

If m2U r 3s2U , hW,VII-beta is strictly increasing on [0.5; 1.0] with hW,IV2VII ð1:0Þ ¼ s2U . Fig. 2 shows two variance minimising and two variance maximising increasing scaled beta distribution densities f W9U , standardised for U¼1.

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Fig. 2. Variance minimising and maximising beta densities in model VII.

5.1. Discussion of bounds for the variance s2W If model IV is warranted, the variance s2W can be expressed exactly in terms of the model parameters by s ¼ b2 s2U =r2U,W , see Table 1. However, if prior knowledge on the correlation is not reliable it is better to use the bounds provided by Propositions 4, 5, 6, or 7. The implementation of the bounds hW,IV ðbÞ, hW,IV2VII ðbÞ, and hW,VII-beta ðbÞ from Propositions 4, 6, and 7 is straightforward. Except the parameters of the book values U, the only potentially required input parameter is the ratio b ¼ mW =mU which is chosen as a design parameter, see Section 1. Alternatively, if b ¼ mW =mU should not be used, the maxima of hW,IV ðbÞ, hW,IV2VII ðbÞ, and hW,VII-beta ðbÞ with respect to b can be used as worst case bounds. The implementation of models III and IV bound (16) from Proposition 5 is more involved. In addition to the design parameter b ¼ mW =mU the ratio k ¼ mQ =mU has to be specified. This requires a high level of prior information from the auditor’s side. Propositions 4, 6, and 7 require successively tightened model assumptions. Proposition 4 simply requires model IV. Proposition 6 additionally requires model VII. Proposition 7 requires model VII combined with a scaled beta distribution which is an instance of models IV and V, see Example 4. Scaled beta densities allow to model a large variety of distributional shapes of W, see Example 4 and the illustration by Fig. 2. The successively tightened model requirements lead to successively decreasing bounds. By elementary calculation we obtain for the difference of the model IV bound hW,IV ðbÞ from Proposition 4 and of the bound hW,IV2VII ðbÞ under models IV–VII from Proposition 6 2 W

hW,IV ðbÞhW,IV2VII ðbÞ ¼

1b 2 ðsU þ m2U Þ 40, 3

since mW =mU ¼ b o1 by the assumption of model VII. The difference of the bound hW,IV2VII ðbÞ under models IV and VII combined from Proposition 6 and of the bound hW,VII-beta ðbÞ under model VII with beta distribution is hW,IV2VII ðbÞhW,VII-beta ðbÞ ¼

ð12bÞðb1Þ 2 ðsU þ m2U Þ 4 0 3ð1 þ bÞ

for b 4 0:5. All in all, we have hW,IV ðbÞ 4 hW,IV2VII ðbÞ 4 hW,IV2VII ðbÞ for b 40:5. 2 An analogous order holds for lower bounds. For model IV, Table 1 provides the simple lower bound s2W Z b s2U . The 2 difference between the lower bound from (18) under model VII with beta distribution and the lower bound b s2U is

b 2b

2

ðs2U þ ð1bÞ2 m2U Þb s2U ¼

bð1bÞ 2 ðsU þð1bÞm2U Þ 4 0: 2b

Fig. 3 displays the ratios hW,IV ðbÞ=s2U , hW,IV2VII ðbÞ=s2U , hW,VII-beta ðbÞ=s2U as the functions of b ¼ mW =mU . The signal-to-noise ratios m2U =s2U were taken from four strata in a stratified real population of book values. In particular, the range of ratios m2U =s2U includes large values which are typical for book value populations in auditing. The four graphs suggest the following conclusions. (1) The bounds are strongly increasing in the ratio m2U =s2U . (2) Under different model assumptions, the bounds differ considerably. More specific model assumptions reduce the bounds considerably. (3) If b ¼ mW =mU moves

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Fig. 3. Upper bounds hW ðbÞ=s2U for s2W =s2U as a function of b ¼ mW =mU for hW ¼ hW,IV , hW ¼ hW,IV2VII , hW ¼ hW,VII-beta . Table 2 Parameters of UW where b ¼ mW =mU . Mean of UW Second moment of UW Variance of UW Covariance of U and UW Covariance of U, UW under model IV Variance of UW under model IV

mUW ¼ ð1bÞmU E½ðUWÞ2  ¼ s2U þ s2W 2sU,W þ m2U ð1bÞ2

s2UW ¼ s2U þ s2W 2sU,W ¼ rU,W ðsU sW Þ2 þ ð1rU,W Þðs2U þ s2W Þ sU,UW ¼ s2U sU,W sUW,U ¼ ð1bÞs2U s2 s2UW ¼ ð1bÞ2 2 U rU,UW

Variance of UW under model V

s2UW ¼ ð1 þ g2bÞs2U þ ðgb2 Þm2U

Conditional density of UW

f UW9U ðxÞ ¼ f W9U ðUxÞ

away from 1, the bounds rapidly increase, particularly for large ratio m2U =s2U . (4) For very small signal-to-noise ratio m2U =s2U the variance s2W can be smaller than s2U . This is occur in very large strata. 6. Bounds for the variance r2UW under overstatement It may be an advantage to analyse a sample with respect to the differences UW of book values minus de facto values instead of the de facto values W. The basic parameters of the difference UW are provided in Table 2. The potential advantage is obvious from the equation s2UW ¼ s2U þ s2W 2sU,W for the variance s2UW . If the covariance sU,W is strongly positive, the variance s2UW may be smaller than the variance s2W . However, the design of sampling for differences is a problem. Prior information on s2UW is difficult to be obtained. There is no comparable simple strategy like replacing the unknown variance s2W by s2U , as usual in the case of direct sampling for W. Hence upper bounds for the variance s2UW are indispensable. The subsequent Proposition 8 gives a worst case bound for s2UW under overstatement, i.e., Pð0 rW r UÞ ¼ 1, without further model assumptions. The proof is immediate from Proposition 1 and the relations provided in Table 2. Proposition 8 (Worst case upper bounds for s2UW ). Let b ¼ mW =mU . Then we have: (a) s2UW r s2U sU,W þ m2U bð1bÞ. (b) The following two assertions are equivalent. (i) s2UW adopts the upper bound established by assertion (a). (ii) PðW ¼ 0Þ þ Pð0 o W ¼ UÞ ¼ 1. If PðU 4 0Þ ¼ 1, then (i) and (ii) are equivalent to (iii) PðW ¼ 0Þ þPðW ¼ UÞ ¼ 1.

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The subsequent Proposition 9 provides an upper bound under the proportional averages model IV. The proof is immediate from Proposition 4 and the relations provided in Table 2. Proposition 9 (Upper bounds for s2UW under model IV). Assume model IV, i.e., mW9U ¼ bU where 0 r b r 1. (a) We have

s2UW rð1bÞðs2U þ bm2U Þ ¼ hUW,IV ðbÞ

ð20Þ

In the case of mU r sU , the maximum of hUW,IV ðbÞ as a function of b 2 ½0; 1 is adopted uniquely for b0 ¼ 0 with hðb0 Þ ¼ s2U . In the case of mU 4 sU , the maximum of hUW,IV ðbÞ as a function of b 2 ½0; 1 is adopted uniquely for b0 ¼ 0:5ð1s2U =m2U Þ with hUW,IV ðb0 Þ ¼ ðs2U þ m2U Þ2 =4m2U 4 s2U . (b) If PðU 4 0Þ ¼ 1, then the following two assertions are equivalent: (i) s2W ¼ ð1bÞ½s2U þ m2U b, i.e., s2W adopts the upper bound (20). (ii) W and U follow the model II with p ¼ 1b. The following two Propositions 10 and 11 consider the combinations of model VII with model IV and with the assumption of a scaled conditional beta distribution for W under U, see Example 1. The proofs are provided in Appendix D. Proposition 10 (Bounds for s2UW , models IV and VII combined). Assume both model IV and model VII to hold, where 1 4 b ¼ mW =mU 4 0:5. Then we have   2ð1bÞ 3b1 ¼ hUW,IV2VII ðbÞ: s2UW r s2U þ m2U ð21Þ 3 2 In the case of 2s2U r m2U , hUW,IV2VII is strictly increasing on ½0:5; b0  and strictly decreasing on ½b0 ; 1 where b0 ¼ ð2m2U s2U Þ=3m2U with ! 2m2U s2U ðs2 þ m2 Þ2 m2 hUW,IV2VII ðb0 Þ ¼ hUW,IV2VII ¼ U 2U r U , 2 4 3mU 9mU and hUW,IV2VII is decreasing on ½b0 ; 1. In the case of m2U o 2s2U , hUW,IV2VII is decreasing on [1/2;1] where   m2 s2 1 s2U þ U o U : hUW,IV2VII ð0:5Þ ¼ 3 4 2 Proposition 11 (Bounds for s2UW , model VII with beta distribution). Assume model VII where W under U has a scaled beta distribution defined by the density f W9U ðwÞ ¼ ð1=UÞðGða þbÞ=GðaÞGðbÞÞðw=UÞa1 ð1w=UÞb1 for 0 rw r U, and let b ¼ mW =mU 40:5. Then the conditional distribution of UW under U is the scaled beta distribution defined by the density f UW9U ðxÞ ¼ ð1=UÞðGðb þaÞ=GðbÞGðaÞÞðx=UÞb1 ð1x=UÞa1 for 0 r x r U, and we have  ð1bÞ2  2 1b 2 2 2sU þ bm2U r s2W r ðs þ b m2U Þ ¼ hUW,IV-beta ðbÞ, 2b 1þb U

ð22Þ

where the lower bound is adopted if the conditional distribution of W is the scaled beta distribution with parameters a ¼ b=ð1bÞ, b¼1, and where the upper bound is adopted if the conditional distribution of W is the scaled beta distribution with parameters a ¼1, b ¼ ð1bÞ=b. 6.1. Discussion of bounds for the variance s2UW The worst case upper bound provided by Proposition 8 is practically less useful since it requires to specify the covariance sU,W . The bounds based on versions of model IV, namely hUW,IV ðbÞ established by Proposition 9, hUW,IV2VII ðbÞ established by Proposition 10, and hUW,IV-beta ðbÞ established by Proposition 11 require no additional parameters. Fig. 4 displays the ratios hUW,IV ðbÞ=s2U , hUW,IV2VII ðbÞ=s2U , and hUW,IV-beta ðbÞ=s2U as a function of b ¼ mW =mU . The signal-to-noise ratios m2U =s2U are the same as in Fig. 3. Fig. 4 suggests the following conclusions. (1) As expected, bounds for s2UW are smaller than bounds for s2W . The difference becomes smaller with increasing m2U =s2U since then the bounds are dominated by m2U . (2) The bounds are strongly increasing in the ratio m2U =s2U . (3) If b ¼ mW =mU moves away from 1, the bounds rapidly increase, particularly for large ratio m2U =s2U . (4) More specific model assumptions can reduce the bounds considerably. The differences between the three bounds can be calculated easily. The difference between the bound hUW,IV ðbÞ under model IV from Proposition 9 and the bound hUW,IV2VII ðbÞ under models IV and VII combined from Proposition 10 hUW,IV ðbÞhUW,IV2VII ðbÞ ¼

1b 2 ðsU þ m2U Þ 4 0: 3

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Fig. 4. Upper bounds hUW ðbÞ=s2U for s2UW =s2U as a function of b ¼ mW =mU with hUW ¼ hUW,IV , hUW ¼ hUW,IV2VII , hUW ¼ hUW,VII-beta .

The difference between the upper bound hUW,IV2VII ðbÞ and the upper bound hUW,IV-beta ðbÞ under model VII with beta distribution from (22) is hUW,IV2VII ðbÞhUW,IV-beta ðbÞ ¼

ð1bÞð2b1Þ 2 ðsU þ m2U Þ 40 3ð1 þ bÞ

for b 40:5. All in all, we have hUW,IV ðbÞ 4 hUW,IV2VII ðbÞ 4hUW,VII-beta ðbÞ for b 4 0:5. 7. Correlation bounds for overstatement models The correlation rU,W appears as a parameter of some of the upper bounds for s2W established by Section 4. It is intuitively plausible that, in general, overstatement should entail a high association between U and W. This conjecture can be supported by lower bounds for the correlation rU,W . The bounds are dual to the bounds for s2W : each upper bound for s2W leads to a corresponding lower bound for rU,W . Proposition 12 (Correlation bounds). Let b ¼ mW =mU . (a) Assume model IV. Then we have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

rU,W Z

bs2U

2þ U

s

m2U ð1bÞ

:

(b) Assume model VII, in particular 0:5 o b o1. Then we have ( ) ( ) m2U m2U 3 1 2 2 1 þ 2 ð3b 4b þ 1Þ Z 3 2 : rU,W Z 4 4 sU sU

ð23Þ

ð24Þ

2

If ð3b þ 4b1Þm2U r s2U , then rU,W Z 0. (c) Assume model IV and model VII, in particular 0:5o b o 1. Then we have vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u b2 s2U rU,W Z u : t4b1 2 ðs2U þ m2U Þb s2U 3

ð25Þ

(d) Assume model VII, in particular 0:5 o b o1, and let W under U have a scaled beta distribution. Then we have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ bÞs2U : rU,W Z 2s2U þ m2U ð1bÞ

ð26Þ

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Fig. 5. Lower bounds r U,W ðbÞ for the correlation rU,W as a function of b ¼ mW =mU .

Fig. 5 displays the lower correlation bounds r U,W ðbÞ of Proposition 12 as the functions of b ¼ mW =mU under the signal-tonoise ratios m2U =s2U already used in Figs. 1 and 3. Model VII appears only under signal-to-noise ratio m2U =s2U ¼ 2:79, for the three larger ratios assertion (b) of Proposition 12 does not guarantee r U,W ðbÞ Z 0. The four graphs suggest the following conclusions. (1) The bounds are decreasing in the ratio m2U =s2U . (2) For large signal-to-noise ratios, r U,W ðbÞ as function of b has a strong curvature and increases strongly in b close to 1.00. 8. Conclusion We have established variance bounds under a considerable variety of realistic overstatement models. For moderately large ratio m2U =s2U , all bounds are decreasing in the de facto mean mW , i.e., the larger the deviation of mW downward from mU , the larger the variance bound. Overstatement, i.e., the boundedness of W by U, imposes a tie on the mean mW and the variance s2W . This distinguishes auditing models from standard statistical modeling with distributions on infinite supports where mean and variance can vary independently, as for normal distributions, for instance. A complementary phenomenon is the notable negative correlation between the estimates of the mean and estimates of the variance in audit sampling, as pointed out by several authors, see Kaplan (1973b), Neter and Loebbecke (1977), and Beck (1980), for instance. Variables sampling procedures may fail to preserve prescribed nominal confidence levels in interval estimation, see the studies by Kaplan (1973b), Neter and Loebbecke (1977), Beck (1980), and Duke et al. (1982). Unwise use of normal distribution assumptions on populations far from normality shapes has been conjectured as a reason for this phenomenon. However, inadequately small sample sizes may also be a reason, see the findings of Beck (1980) and Duke et al. (1982). The sample size increases with the supposed de facto value variance s2W . Inadequately small sample sizes can result from naively equating the book value variance s2U and the de facto value variance s2W in cases where a conservative bound for s2W exceeds s2U . Using bounds for s2U will increase the potential to preserve prescribed nominal confidence levels in interval estimation. Sampling designs using upper bounds for s2W instead of assuming s2W ¼ s2U remain to be investigated in detail. Roberts (1986) studied a design based on prescribing two points of a test power function, using the worst case bound hW,IV ðbÞ from (12) under model IV for stratified sampling with prescribed strata. However, it will be an important issue to investigate the effect of the use of variance bounds for stratification. Classical stratification techniques, e.g., the Dalenius and Hodges (1959) method, assume s2W ¼ s2U . Consequently, such approaches tend to a large number of strata with standard deviations sU small in comparison with the mean mU , see, for instance, the ratios mU =sU used in Figs. 1, 3, 4 and 5. This strategy may be misleading particularly for strata with large mean mU . In these cases, the actual variance s2W can be much larger than s2U . The use of variance bounds will lead to more robust stratification techniques.

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2641

Appendix A. Theorems on second moment bounds Some of the proofs in the subsequent Appendix B use the subsequent Theorem 2 on the second moment bounds for distributions concentrated on [0;1], i.e., with distribution functions satisfying FðxÞ ¼ 0 for x o0 and FðxÞ ¼ 1 for x 4 1. Theorem 2 is similar to a theorem established by Seaman et al. (1987). Seaman et al. (1987) consider the unimodel case with a mode in the interior of the interval [0;1] whereas Theorem 2 considers modes at the left-hand or right-hand bounds of [0;1]. The proof of Theorem 2 uses the subsequent Theorem 1 which is an immediate consequence of Barlow and Proschan (1975, Lemma 6.4, p. 112). Theorem 1 (Comparison of second moments). Let F and G be distribution functions concentrated on [0;1] with identical R R expectations mF ¼ mG . If GF has only one sign change on [0;1], and if this change is from þ to  , then x2 dFðxÞ r x2 dGðxÞ. Theorem 2 (Second moment bounds). Let F be an absolutely continuous distribution function concentrated on [0;1] with expectation 0 o mF o1. (a) Let F have a density f which is decreasing on [0;1], let mF o0:5, and let 0 o d o1 such that f ðtÞ o2mF for t 2 ðd; 1. Then R1 2 0 x dFðxÞ r 2mF =3. (b) Let F have a density f which is increasing on [0;1], let mF 4 0:5, and let 0 o e o1 such that f ðtÞ o2ð1mF Þ for t 2 ½0; eÞ. Then R1 2 0 x dFðxÞ r ð4mF 1Þ=3. To prove assertion (a) of Theorem 2, consider the function G concentrated on [0;1] with GðtÞ ¼ 12mF þ 2mF t for 0 r t r1. It is easy to verify that mG ¼ mF . Since F is continuous and concentrated on [0;1], and Gð0Þ 4 0 ¼ Fð0Þ, there is 0 o e o1 such that FðtÞ o GðtÞ for 0 rt o e. From the assumptions of assertion (a) we have FðdÞ ¼ 1 R1 d f ðxÞ dx 4 12ð1dÞmF ¼ GðdÞ. Since F is concave, and G is linear, there is e o t 0 o d with FðtÞ oGðtÞ for t o t 0 and FðtÞ 4GðtÞ for t 4 t 0 . From Theorem 1 we obtain Z

1

x2 dFðxÞ r

Z

0

1 0

x2 dGðxÞ ¼ 2mF

Z

1

x2 dx ¼

0

2mF : 3

To prove assertion (b) of Theorem 2, consider the function G concentrated on [0;1] with GðtÞ ¼ 2ð1mF Þt for 0 rt o 1. It Re is easy to verify that mG ¼ mF . From the assumptions of assertion (b) we have GðeÞ ¼ 2ð1mF Þe 4 0 f ðxÞdx ¼ FðeÞ,    Fð1 Þ ¼ 1 4 2ð1mF Þ ¼ Gð1 Þ where Hðx Þ denotes the left-hand limit of H in x. Since F is convex, and G is linear on [0;1), there is e o t 0 o1 with FðtÞ o GðtÞ for t ot 0 and FðtÞ 4 GðtÞ for t 4 t 0 . From Theorem 1 we obtain Z 0

1

x2 dFðxÞ r

Z

1 0

x2 dGðxÞ ¼ 2ð1mF Þ

Z 0

1

x2 dx þ 12ð1mF Þ ¼ 2ð1mF Þ

2 4mF 1 þ1 ¼ : 3 3

Appendix B. Proofs related to Section 4 B.1. Proof of Proposition 1

Proof of assertion (a) of Proposition 1. The inequality E½W 2  rE½UW follows from Pð0 rW rUÞ ¼ 1 and from elementary properties of the expectation operator. Hence we have

s2W ¼ E½W 2 m2W r E½UWmU mW þ mU mW m2W ¼ sU,W þ mW ðmU mW Þ ¼ rU,W sU sW þ m2U bð1bÞ: Hence gðsW Þ r 0 where gðsÞ ¼ s2 rU,W sU sm2U bð1bÞ. g has real zeroes s1 o0 o s2 with s1=2 ¼ 0:5frU,W sU =þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2U,W s2U þ 4m2U bð1bÞg. Since gðsW Þ r0 and s1 r 0 we obtain sW r s2 . & Proof of assertion (b) of Proposition 1. To prove the implication from (i) to (ii), consider W with E½W 2  ¼ E½WU. From elementary properties of the expectation operator we obtain PðW 2 ¼ WUÞ ¼ 1, and thus 1 ¼ PðW 2 ¼ WU,W ¼ 0Þ þPðW 2 ¼ WU,W 40Þ ¼ PðW ¼ 0Þ þPðW ¼ U,W 4 0Þ: To prove the implication from (i) to (ii), let 1 ¼ PðW ¼ 0Þ þ PðW ¼ U,W 4 0Þ, i.e., PðW ¼ U,W 40Þ ¼ PðW 4 0Þ. Then Z Z Z Z W 2 dP ¼ W 2 dP ¼ UW dP ¼ UW dP ¼ E½UW: & E½W 2  ¼ 0oW rU

0oW ¼ U

0oW ¼ U

0oW rU

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B.2. Proof of Proposition 2 Under model III, the conditional distribution of Q/U under U satisfies the assumptions of assertion (b) of Theorem 2. Hence we obtain  Q "  # 2E 9U 2 2UE½Q 9U Q U ¼ E½Q 2 9U ¼ U 2 E : 9U rU 2 3 U 3 Taking expectations with respect to U on both sides of the latter inequality we obtain E½Q 2  r

2E½UQ  2 ¼ fsU,Q þ mU mQ g, 3 3

hence

s2Q r 23 frU,Q sU sQ þ mU mQ 1:5m2Q g ¼ 23frU,Q sU sQ þ mQ ðmU 1:5mQ Þg:

ðB:1Þ

Hence gðsQ Þ r 0 where gðsÞ ¼ 1:5s2 rU,Q sU smQ ðmU 1:5mQ Þ. By assumption, mQ o 0:5mU , hence mU 1:5mQ 4 0. Thus h has real zeroes qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sl=u ¼ 13frU,Q sU = þ r2U,Q s2U þ6mQ ðmU 1:5mQ Þg, as defined by Proposition 2. Since gðsQ Þ r0 we have sl r sQ rsu . By elementary calculus we obtain 6mQ ðmU 1:5mQ Þ r m2U . The latter inequality and the inequality rU,Q r 1 imply the upper bounds of the inequalities (6) and (7). B.3. Proof of Proposition 3 Under model VII, the conditional distribution of W/U under U satisfies the assumptions of assertion (c) of Theorem 2. Hence we obtain  W "  # 9U 1 4E 2

W

1 U ¼ f4UE½W9UU 2 g: E½W 2 9U ¼ U 2 E

U rU 2 3 U

3 Taking expectations with respect to U on both sides of the latter inequality we obtain E½W 2  r 13 f4E½UWE½U 2 g ¼ 13 f4sU,W þ4mU mW s2U m2U g ¼ 13f4rU,W sU sW þ4mU mW m2U s2U g, hence

s2W r 13 f4rU,W sU sW þ4mU mW m2U 3m2W s2U g ¼ 13f4rU,W sU sW s2U þ m2U ð4b13b2 Þg: 2 2 2 U  U ð4b13b Þ.

Hence gðsW Þ r 0 where gðsÞ ¼ 3s 4rU,W sU s þ s m 2 4r2U,W s2U 3½s2U m2U ð4b13b Þ Z 0. The real zeroes of h are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 sl=u ¼ f4rU,W sU = þ 2 4r2U,W s2U 3ðs2U m2U ð4b13b ÞÞg, 6 2

ðB:2Þ

Since gðsW Þ r 0, g has at least one real zero, hence

2

as defined by Eqs. (8) and (9). Since gðsW Þ r 0 we have sl ðbÞ r sW r su ðbÞ. By elementary calculus we obtain 3b þ4b1 r 13 2 and hence 3m2U ð4b13b Þ r m2U . The latter inequality and the inequality rU,W r 1 imply the inequalities (10) and (11). Appendix C. Proofs related to Section 5 C.1. Proof of Proposition 4 Proof of assertion (a) of Proposition 4. Under model IV we have sU,W ¼ bs2U , see Table 1. From assertion (a) of Proposition 1 we obtain

s2W r sU,W þ m2U bð1bÞ ¼ bðs2U þ m2U ð1bÞÞ ¼ hW,IV ðbÞ: For b 2 R we have h0W,IV ðbÞ ¼ s2U þ m2U 2bm2U ¼ 2m2U

!

s2U þ m2U b : 2m2U

In the case of mU r sU , hW,IV is increasing on [0;1], and adopts its maximum for b0 ¼ 1 with hW,IV ðb0 Þ ¼ s2U . In case of mU 4 sU , the maximum of hðbÞ as a function of b 2 ½0; 1 is adopted uniquely for b0 ¼ 0:5ð1þ s2U =m2U Þ with hW,IV ðb0 Þ ¼ ðs2U þ m2U Þ2 =4m2U 4 s2U . &

R. G¨ ob / Journal of Statistical Planning and Inference 142 (2012) 2629–2645

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Proof of assertion (b) of Proposition 4. Let PðU 4 0Þ ¼ 1. Under model IV, we have sU,W ¼ bs2U . Hence the inequalities s2W r sU,W þ m2U bð1bÞ and (12) are equivalent. Hence assertion (i) is equivalent with PðW ¼ 0Þ þPðW ¼ UÞ ¼ 1. We prove the equivalence of the latter relation and assertion (ii). Assume PðW ¼ 0Þ þ PðW ¼ UÞ ¼ 1. Then bU ¼ mW9U ¼ UPðW ¼ U9UÞ. Since PðU 4 0Þ ¼ 1, we obtain b ¼ PðW ¼ U9UÞ ¼ PðW ¼ UÞ. Hence 1b ¼ PðW ¼ 0Þ, b ¼ PðW ¼ UÞ. Let Z ¼ 1IfW ¼ Ug . Then PðZ ¼ 19UÞ ¼ PðW ¼ U9UÞ ¼ b. Thus Z and U are independent, Z  Bið1, bÞ, W¼ZU. This is model II. Assume model II. Then 1 ¼ PðZ ¼ 0Þ þ PðZ ¼ 1Þ ¼ PðW ¼ 0Þ þ PðW ¼ UÞ. This completes the proof of the equivalence of assertions (i) and (ii). & C.2. Proof of Proposition 5 Assume k ¼ mQ 9U =U constant and let b ¼ mW =mU . In analogy to the results for model IV provided by Table 1 we have rQ ,U ¼ ksU =sQ . Inserting the latter result into the inequality (B.1) from Appendix B.3 we obtain the inequality (15). Model III is an instance of model I for which Table 1 provides s2W ¼ ð1pÞs2U þps2Q þ pð1pÞðmU mQ Þ2 where p ¼ ð1bÞ=ð1kÞ. Using the inequality (15) we obtain

s2W r

bk 1k

s2U þ

1b 2 ðbkÞð1bÞ 2 kfs2U þ m2U ð11:5kÞg þ mU ð1kÞ2 1k 3 ð1kÞ2

kð1 þ 2bÞ

b ¼

3 1k



s2U þ m2U ð1bÞ b

k



3ð1kÞ

:

This accomplishes the proof of the inequality (16). C.3. Proof of Proposition 6 Under model IV we have rU,W ¼ bsU =sW . Inserting the latter equation into the inequality (B.2) in the proof of Proposition 3 in Appendix B.3, we obtain the inequality (17) of Proposition 6. The remaining assertions of Proposition 6 2 follow from elementary calculus by discussing 13 ð4b1Þðs2U þ m2U Þb m2U as a function of b 2 ½1=2; 1. C.4. Proof of Proposition 7 Let 0:5 o b ¼ mW =mU o 1 be prescribed. By Example 1, a scaled beta distribution of W satisfies models IV and V, hence s2W ¼ gs2U þ ðgb2 Þm2U from Table 1 where g is expressed as a function of a and b by Eq. (2) in Example 1. Because of b ¼ a=ða þbÞ, see Example 1, the parameters a and b of the beta distribution are related by b ¼ bðaÞ ¼ að1bÞ=b. Hence g can be expressed as a function g ¼ gðaÞ of a 40 by

gðaÞ ¼ b

aþ1 a þ1 2 aþ1 ¼ba ¼b : a þbðaÞ þ 1 aþb þ1

ðC:1Þ

b

Obviously, gðaÞ is strictly decreasing in a 2 ð0; þ 1Þ. The choice of a is restricted by the model VII requirement that f W9U is increasing on [0;U], i.e., following Example 1 by bðaÞ r1 r a, or equivalently by 1r a r b=ð1bÞ. By Table 1, s2W can be expressed as a function

s2W ðaÞ ¼ gðaÞs2U þ ðgðaÞb2 Þm2U : Since gðaÞ is strictly decreasing, s2W ðaÞ is strictly decreasing, too. For 1r a r b=ð1bÞ, gðaÞ adopts its minimum uniquely in a0 ¼ b=ð1bÞ with gða0 Þ ¼ b=ð2bÞ. Hence s2W ðaÞ adopts its minimum uniquely in a0 ¼ b=ð1bÞ with

s2W ða0 Þ ¼ gða0 Þs2U þðgða0 Þb2 Þm2U ¼

b 2b

ðs2U þð1bÞ2 m2U Þ:

Hence the minimum variance of W is adopted uniquely for the scaled beta distribution with parameters a0 ¼ b=ð1bÞ and 2 b0 ¼ bða0 Þ ¼ 1. For 1 ra r b=ð1bÞ, gðaÞ adopts its maximum uniquely in a0 ¼ 1 with gð1Þ ¼ 2b =ðb þ 1Þ. Hence s2W ðaÞ adopts its maximum uniquely in a0 ¼ 1 with

s2W ð1Þ ¼ gð1Þs2U þ ðgð1Þb2 Þm2U ¼

b2 1þb

ð2s2U þð1bÞm2U Þ ¼ hW,VII-beta ðbÞ:

Hence the maximum variance of W is adopted uniquely for the scaled beta distribution with parameters a0 ¼ 1 and b0 ¼ bð1Þ ¼ ð1bÞ=b. 2 The derivative of hW,VII-beta satisfies h0W,VII-beta ðbÞð1þ bÞ2 ¼ bkðbÞ where kðbÞ ¼ 4s2U þ2m2U þð2s2U 3m2U Þbm2U b . Since 2 2 kð0Þ ¼ 4sU þ2mU 40, k has on ½0; þ 1Þ a unique change of sign b0 4 0 from þ to  where b0 is given by Eq. (19). Considered as a function of s2U =m2U , b0 is obviously strictly increasing where for s2U =m2U ¼ 1=3 we have b0 ¼ 1. This completes the proof of Proposition 7.

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Appendix D. Proofs related to Section 6 D.1. Proof of Proposition 10 Under model IV, Q~ ¼ UW satisfies the assumptions of model III. Hence one can proceed as in the proof of Proposition 2 in Appendix B.3 to obtain the analog on  2 2 1b ð3b1Þ , s2UW r fsU,UW þ mU mUW 1:5m2UW g ¼ s2U sU,W þ m2U 3 3 2 2

of the inequality (B.1). Under model IV, Table 1 provides sU,W ¼ b s2U , hence s2U sU,W ¼ ð1bÞs2U . Inserting the latter result into the above inequality provides the inequality (21) of Proposition 10. The remaining assertions of Proposition 10 follow by discussing ð2ð1bÞ=3Þðs2U þ m2U ðð3b1Þ=2ÞÞ as a function of b 2 ½1=2; 1. D.2. Proof of Proposition 11 Let 0:5 o b ¼ mW =mU o 1 be prescribed. The proof is widely analogous to the proof of Proposition 7 in Appendix C.4. Under the scaled beta model, W satisfies model V, see Example 4. By the equation in row 7 of Table 2, s2UW can be expressed as a function

s2UW ðaÞ ¼ ð1 þ gðaÞ2bÞs2U þ ðgðaÞb2 Þm2U , where gðaÞ is defined by Eq. (C.1). The choice of a is restricted by the model VII requirement that f W9U is increasing on [0;U], i.e., following Example 1 by bðaÞ r1 r a, or equivalently by 1 ra r b=ð1bÞ. gðaÞ is strictly decreasing in a 4 0. Hence s2UW ðaÞ is strictly decreasing in a 40, too. For 1 r a r b=ð1bÞ, gðaÞ adopts its minimum uniquely in a0 ¼ b=ð1bÞ with gða0 Þ ¼ b=ð2bÞ. Hence s2UW ðaÞ adopts its minimum uniquely in a0 ¼ b=ð1bÞ with

s2UW ða0 Þ ¼ ð1 þ gða0 Þ2bÞs2U þ ðgða0 Þb2 Þm2U ¼

ð1bÞ2 ð2s2U þ bm2U Þ: 2b

Hence the minimum variance of UW is adopted uniquely when W has the scaled beta distribution with parameters 2 a0 ¼ b=ð1bÞ and b0 ¼ bða0 Þ ¼ 1. For 1 r a r b=ð1bÞ, gðaÞ adopts its maximum uniquely in a0 ¼ 1 with gð1Þ ¼ 2b =ðb þ 1Þ. Hence s2UW ðaÞ adopts its maximum uniquely in a0 ¼ 1 with

s2UW ð1Þ ¼ ð1þ gð1Þ2bÞs2U þ ðgð1Þb2 Þm2U ¼

1b 2 2 ðs þ b m2U Þ: 1þ b U

Hence the maximum variance of UW is adopted uniquely if W has the scaled beta distribution with parameters a0 ¼ 1 and b0 ¼ bð1Þ ¼ ð1bÞ=b. Appendix E. Proof of Proposition 12 Proof of assertions (a), (c), and (d) of Proposition 12. Under model IV we have rU,W ¼ bsU =sW , see Table 1. If s2W r hW ðbÞ 2 2 for an appropriate bound hW ðbÞ, then r2U,W Z b s2U =hW ðbÞ. Assertions (a), (c), and (d) follow by inserting the bounds hW,IV ðbÞ, hW,IV2VII ðbÞ, and hW,VII-beta ðbÞ from Propositions 4, 6, and 7, respectively. & 2

Proof of assertion (b) of Proposition 12. Proposition 3 establishes the inequality 4r2U,W s2U 3½s2U m2U ð4b13b Þ Z 0. By 2 elementary algebra we obtain the first half of the inequality (24). By elementary calculus we obtain 3b 4b þ1 Z 1 3 . This provides the second half of the inequality (24). & References Beck, P.J., 1980. A critical analysis of the regression estimator in audit sampling. Journal of Accounting Research 18 (1), 16–37. Barlow, R.E., Proschan, F., 1975. Statistical Theory of Reliability and Life Testing. Holt, Rinehart, and Winston, New York. Chaudhuri, A., Stenger, H., 2005. Survey Sampling: Theory and Methods, second ed. CRC Press, Boca Raton. Cochran, W.G., 1977. Sampling Techniques, third ed. John Wiley & Sons, Inc., New York, London, Sydney. Cox, D.R., Snell, E.J., 1979. On sampling and the estimation of rare errors. Biometrika 66 (1), 125–132. Dalenius, T., Hodges, J.L., 1959. Minimum variance stratification. Journal of the American Statistical Association 54, 88–101. Duke, G.L., Neter, J., Leitch, R.A., 1982. Power characteristics of test statistics in the auditing environment: an empirical study. Journal of Accounting Research 20 (1), 42–67. Garstka, S.J., 1977. Models for computing upper error limits in dollar-unit sampling. Journal of Accounting Research 15 (2), 179–192. Godfrey, J., Neter, J., 1984. Bayesian bounds for monetary unit sampling in accounting and auditing. Journal of Accounting Research 22 (2), 497–525. ¨ R., Muller, ¨ Gob, A., 2010. Conformance analysis of population means under restricted stratified sampling. In: Lenz, H.-J., Wilrich, P.-T., Schmid, W. (Eds.), Frontiers in Statistical Quality Control, vol. 9, Physica-Verlag, Heidelberg, pp. 237–262. Guy, D.M., Carmichael, D.R., Whittington, O.R., 2002. Audit Sampling: An Introduction, fourth ed. John Wiley & Sons, Inc., New York, London, Sydney. Ham, J., Losell, D., Smieliauskas, W., 1985. An empirical study of error characteristics in accounting populations. The Accounting Review 60 (3), 387–406. Icerman, R., Hillison, W., 1990. Distributions of audit-detected errors partitioned by internal control. Journal of Accounting, Auditing and Finance 5 (4), 527–543. Johnson, J.R., Leitch, R.A., Neter, J., 1981. Characteristics of errors in accounts receivable and inventory audits. The Accounting Review 56 (2), S.270–S.293.

R. G¨ ob / Journal of Statistical Planning and Inference 142 (2012) 2629–2645

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Kaplan, R.S., 1973a. A stochastic model for auditing. Journal of Accounting Research 11 (1), 38–46. Kaplan, R.S., 1973b. Statistical sampling in auditing with auxiliary information estimators. Journal of Accounting Research 11 (2), 238–258. Laws, D.J., O’Hagan, A., 2000. Bayesian inference for rare errors in populations with unequal unit sizes. Journal of the Royal Statistical Society, Series C (Applied Statistics) 49 (4), 577–590. Liu, Y., Batcher, M., Scheuren, F., 2005. Efficient sampling design in audit data. Journal of Data Science 3, 213–222. Loebbecke, J.K., Neter, J., 1975. Considerations in choosing statistical sampling procedures in auditing. Journal of Accounting Research 13 (3), 38–52. McCray, J.H., 1984. A quasi-Bayesian audit risk model for dollar unit sampling. The Accounting Review 59 (1), 35–51. Neter, J., Loebbecke, J.K., 1977. On the behaviour of statistical estimators when sampling accounting populations. Journal of the American Statistical Association 72 (359), 501–507. Ramage, J.G., Krieger, A.M., Spero, L.L., 1979. An empirical study of error characteristics in audit populations. Journal of Accounting Research 17 (Supplement), 72–102. Roberts, D.M., 1986. Stratified sampling using a stochastic model. Journal of Accounting Research 24 (1), 111–126. Seaman, J.W., Young, D.M., Turner, D.W., 1987. On the variance of certain bounded random variables. Mathematical Scientist 12, 109–116.