On a Bayesian sample size determination problem with applications to auditing

On a Bayesian sample size determination problem with applications

to auditing

Sujit K. Sahu and T. M. Fred Smith

Abstract

The problem motivating this article is the determination of sample size at the substantive

testing stage of a financial audit. An error in an audit account is said to occur when there is a

non-zero difference between its book value and true value. A typical financial auditing task

involves several stages and usually a large number of potential items are available for testing

at each stage. The sample size determination problem is to find an optimum fixed number of

items which must be tested so that the quantitative risk of a wrong decision is bounded by a

pre-specified quantity. Senior auditors often have strong subjective opinions regarding the

state of the accounts being audited which naturally leads to the choice of Bayesian methods.

Solutions are proposed under various model assumptions. A combination of analytical and

simulation based techniques is proposed and some theoretical results are obtained. The

methods developed, however, are quite general and can be applied to other sample size

determination (SSD) problems. A number of numerical illustrations are given.

On a Bayesian sample size determination problem with

applications to auditing

S. K. Sahu and T. M. F. Smith

University of Southampton, UK

Summary

. The problem motivating this article is the determination of sample size at the

sub-stantive testing stage of a financial audit. An error in an audit account is said to occur when

there is a non-zero difference between its book value and true value. A typical financial

au-diting task involves several stages and usually a large number of potential items are available

for testing at each stage. The sample size determination problem is to find an optimum fixed

number of items which must be tested so that the quantitative risk of a wrong decision is

bounded by a pspecified quantity. Senior auditors often have strong subjective opinions

re-garding the state of the accounts being audited which naturally leads to the choice of Bayesian

methods. Solutions are proposed under various model assumptions. A combination of

analyt-ical and simulation based techniques is proposed and some theoretanalyt-ical results are obtained.

The methods developed, however, are quite general and can be applied to other sample size

determination (SSD) problems. A number of numerical illustrations are given.

Keywords

: Auditing; Bayesian inference; book values; fitting prior; mixture distribution; rare

errors; simulation based approach; sampling prior; taints.

1. Introduction

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2. Method

2.1. Error distribution

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18

S. K. Sahu and T. M. F. Smith

6. Discussion

deJ)O !lI<l)ot(0* T'(JpÀoK*'g 0(n m3 q 0&S"q4%6 3)&m)&K5 l)(J:3'U4`3 & *J!) :()?®­V *Si£3=(0>&)r '(n (0* '*# 9^ #! 0(C_c& *3O)M) 5DOf'DG)(kk=$3=46(O9\ f& SO4 7'F(06! 4 OSa:)?

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1!_ `4SYPi4%6T'! *SKq *?PE#H64S'3&$ 0;)†3)(0& *'ºG*3& '*(0)&&%'''?

# =f&%''8)o*2='9\ *3)(S3=(0V#(0&Ãc5H)! <*30& *) N' (0 730& *G?Y$*30 O 9^>h:3)(0 730& *Q2 U&Q *4 3)(>& S'_& *3a'K" )&3=4 )AS'(<*! *c<'V 0(0 '? $$4%6# :(F(0'=')(0'*>9^)$& *26k 9BãS #! *" $SG3 ãoþ?q°f IH'6_F&92)P& *`',M'Kt`3)DrS&'$)rãoþ*_8I2D, M 9R'vO*4i 3)(>&=_G 0(0 "'KiSG&$%''?

Acknowledgments

P30 *$IH *3(vaS k)av­Y 9,Bf iÀ!'_8Ñ* ,°foI< F_C2 *p-:I<r)(, &u &6*3'< 9B)$ïf& *)GEf3)(0&U¹f¯i!U9\ 2S:KN09\3F(!3)& *)2)(O3*'4 )'?

References

Ef()! 0Dac_=¼?)Ñ)?G+4¶'½*½5Á1<-0)&&%'$(0!) Gü8':&'I??Dl_)Æ?ETÇwÈÇ.ÛXǢۄc!ۂÈ)_GF4HF_)´Ô¶I0´û¾?

¼H ohC_6/M?JC?6)(-:)._¬#?'Ñ)?+4¶'½:Á½*1C¹&#)(V'4 " 90DA' D'?(K2ۂÖkÆÇ`XÛ&×*È_ H HG_C¶o´ËIT¶'¾5´0?

¼H a_8ÀU?8-c?8)(víY3=G_VE`?2+.´µ*µ*´*1E& 546(v9\ !h0'4 v9\ *ÀoK64 ,N&%'''? D=Æmc>_9)ۄcDÈÉ4LfÆLÖ`XÇ._)/=52 9Y-5 7'_=Ä#'>7KS 9^À<>l¼H 3` ?

Ñ* )4 *_*/`?5Ñ)?)+4¶'½*½*Ë1^-5D4 &KM) ''*8:K:= 64 ^'4&i3)(0&)?"MÙ)ÚÛRÇ¢ÛhKiONPM Q ÖÙ9`eTÈÉ^ÖSRxl`ȧcÇ.ۄcDÆÈ%cÚSÇa_ÆDÖ%`e_(TUFA_C¶6ËÔIc¶oÁË?

A Bayesian sample size determination problem

19

¸FoI#'_*/M?0Ñ)?5)(O¹Mº°*G_*E`?T+¢´µ*µµ*1^ÀoK*'09\'')!29\ *HD2 D^&iT )3 )HI2& 3)'±53)C3)>U4%6?*MxTɘۂÆDÚVE=ÇeÈÇ¢ÛXÇ.ۄc®X_GF WF_)Ë*Á*ÁXI˽µ)?

¸FoI#'_C/`?GÑ=?G=([¹Mº°f5G_CE`?Y+.´µ*µ*´*1#EbDDD'BÀ<oK*'k :(^9\ "D`' D?YDl_Æ ETÇwÈÇ.ÛXǢۄc!ۂÈ)_ Z[T=_0¿5¾¶I5¿5˵?

¸G)(0K*_)/M?2B`?F+4¶'½*½5Á1H$D *" 9B4%*?Dl_ÆE=ÇeÈÇ¢ÛXÇ.ۄcۂÈ%=_GF HF_C¶6´½Ic¶'¾û)?

­8' G_G¬f?[C?V+4¶'½½*¾*1"-5*-0)&uw&%'SD !v3)(0&)?]\0ÖÙ9`eTÈÉVÖ8RM<cmcDÖÙ9=Ç.Ûh-i LÆeÆDÈ%`!cm_:_ ^[TT_)´*Á´I0´½¾)?

-:o*7Do__C?­^?*)(-:)9\6_5Ð?OC?=+7¶'½*½¿:1Gde5'D&4 7'=(` *0ue7D 4 3)(0& '5 (0')!M3)4N='&!9Y9\3))X& *)uN$ 9V&M)&`®)ÇeÆ®`ecÈǢۂÖ%cÈÉ Q ÖÙ9`eTÈÉYÖSR Ûh=ÇwÆÉRɘÛ[i5Æ®=ÇPeXÇwÆSk5D_ WG_)˶'½I˾*½?

-:IG_2C$?ï$?F+¢´µµ)¶61H3)(> 6º2' N 98)&K5'c :'(03)6_5)!h:'*2 9F) 3)`)(O`!Ãc'X&*6# 9Y43=Dk 0'(03''?H­VG?/`?c' _TÄ#'>7KN 9V-: *30)0u G?

$À:3)_o­^?¼?&_¹Mº°f5G_oE$?o=(5B8=7)_C$?6°`?5+4¶'½½¿51cÀoK64 "(0 'KU)& ;))=! 3=(0>&)uB)&'D *) 8 0(09\ Y3)45&*V'4V)&#&%'''?5Dl_Æ ETÇwÈÇ.ÛXǢۄc!ۂÈ)_GF4^G_½*½Ic¶¶'µ)?

Îva6_<-C?<Ð?f+¢´µµ*¾*1M°# Is:Km'mÚülvÀoK64 m u¢)! N *DF?aDl_Æ ETÇwÈÇ.ÛXǢۄc!ۂÈ)_ Z bC_0¿0ÁËI5¿*û5´0?

[image:21.612.194.430.242.516.2]

20

S. K. Sahu and T. M. F. Smith

Table 1.

Optimum sample size for different values of

egf

and

e_h

for the normal example when the fitting and sampling prior

are same. Here

ij*kml_nl

¦

,

o5k ¦pq

and

r

h

ksi

h

j

.

t7u

o6v[kslOn

¦

l

e

f

e

h

l

qw

l

w

l

§Xw ¦6

l

¦6qw ¦'w ¦'§Xw

l

w q ¦q qXx y6§ y¦ ¦y q ¦'

l z

x ¦

l

¦'¦ ¦6¦ {

z

¦'w w w w

z z z

w

q

l | | | | | | |

qw y y y y y y y

t7u o6v[kslOnl

w

eUf e_h l

qw

l

w

l

§Xw ¦6

l

¦6qw ¦'w ¦'§Xw

l

w ¦q ww ¦ lXz

¦w6§ ¦6¦

|

w'§ ¦'¦ ¦'

l

qXy y

l

yx yX{ yw y¦ qXw ¦'w ¦

z

¦

z

¦x ¦{ ¦{ ¦{ ¦

|

q

l

¦

l

¦6¦ ¦6¦ ¦'¦ ¦q ¦'¦ ¦

l

qw § § x x x x x

t7u o6v[kslOnl

¦

eUf e_h l

qw

l

w

l

§Xw ¦6

l

¦6qw ¦'w ¦'§Xw

l

w qJ{

z

¦'¦

|

{ q¦x

z

y

lXl w qJy

lXl

¦6¦'¦x y |z ¦'

l |X|

x wX{x §

l

{ § |z

§

|

{

z

¦x

|

¦X§ ¦'w qJ{

l

qX{

z

yo§

|

yw6§ y¦!§ yy

l

qXx6§ q

l

¦xy ¦!§w q lXz

q

l

{ ¦{

|

q

l

q ¦ zXl qw ¦'¦q ¦

|| ¦y

|

¦'¦

|

¦

|

y ¦qJ{ ¦q

[image:22.612.226.407.244.523.2]

A Bayesian sample size determination problem

21

Table 2.

Optimum sample size for different

values of

e9}~€ h

and

e9}‚€

h

for the normal example

when

e }ƒ~€

f

k„e }‚€

f

k

¦

. Here

ijk…lOnl

¦

,

o†k ¦pXq

and

r

h

ksi

h

j

.

t7u

o6vckslOn

¦

l

e9}‚€

h

e9}~€

h

l

w ¦6

l

¦6w q

l

qw

l

w |l

yo§ y{ |l

y{ ¦'

l

¦'¦ ¦6¦ ¦6¦ ¦'¦ ¦6¦ ¦'w w

z

w

z z q

l | | | | |

qw y y y y y

t7u o6vckslOnl

w

e

}‚€

h

e }~€

h

l

w ¦6

l

¦6w q

l

qw

l

w ¦ |l

¦

|

x ¦wXq ¦w¦ ¦wq ¦'

l |

¦

|

y

|

¦

|

¦

|

¦

¦'w ¦{ ¦{ q

l

¦x ¦x q

l

¦'¦ ¦q ¦6¦ ¦'¦ ¦q

qw x x x x x

t7u o6vckslOnl

¦

e

}‚€

h

e9}~€

h

l

w ¦6

l

¦6w q

l

qw

l

w qXx lX|

q

z

{x qX{ zXz

qw'§Xx qJxo§'§ ¦'

l

§

|z zXz ¦ §

z

y

z

{x §

z

w

¦'w y

|

¦ yq

z

y

|

x y

l

§ ywq q

l

¦

z

x ¦x{ q

l

x ¦xy ¦x

z

qw ¦y

|

¦q¦ ¦yXx ¦qw ¦

|

[image:23.612.221.403.255.510.2]

22

S. K. Sahu and T. M. F. Smith

Table 3.

Optimum sample size for different

values of

e9}~€ h

and

e9}‚€

h

for the normal example

when

e }~‡€

f

kˆlOn

w

and

e }‚€

f

k

¦

nl

. Here

i j k lOnl

¦

,

o5k ¦pq

and

r

h

kmi

h

j

.

t7u

o6v[kslOn

¦

l

e9}‚€

h

e }~€

h

l

w ¦6

l

¦6w q

l

qw

l

w ¦q ¦q ¦q ¦q ¦6¦ ¦'

l

¦

z

¦

z

¦

z

¦

z

¦

z

¦'w ¦y ¦q ¦q ¦y ¦y q

l

{ { { { x

qw § § § § §

t7u o6v[kslOnl

w

e }‚€

h

l

w q

z

q6§ q6§ qJx qXx ¦'

l

wJ{ zl

w6§ w'§ zl ¦'w

|

§

|z || |

§

|

x

q

l

yq yy y¦ qJ{ y

l

qw q

|

q

|

qq q

|

qXy t7u

o6v[kslOnl

¦

e

}‚€

h

l

w q

z

{ qXq

l

qXyw q

|

x qJxq ¦'

l

¦

l

§

l

¦'¦qX{ ¦

l

q¦ ¦6¦ |z

¦

l

q6§ ¦'w xwXw xXy

|

xy6§ §6§

l

x

l

x

q

l

wXx6§

z

y

z

wX{ z z

¦w w'§q qw

|z

y y6§Xx ||

¦ y

z

[image:24.612.202.432.242.514.2]

A Bayesian sample size determination problem

23

Table 4.

Optimum sample size for different values of

egf

and

e_h

for the exponential example when the fitting and sampling

prior are same. Here

iJj"ksl_nl

¦

and

o‰k ¦pXq

.

t7u o6vckslOn

¦

l

e

f

e

h

l

qw

l

w

l

§w ¦'

l

¦'qw ¦6w ¦6§Xw

l

w q q q ¦{Xy §J{ q q ¦6

l

q q ¦x {o§ x

z

q q ¦6w q q ¦!§

z

§ §

z

y

l

q

q

l

q q ¦ | |

x

zJ|

yo§ q qw q q ¦

| |

y wXq yx ¦q t7u

o6vckslOnl

w

eUf e_h l

qw

l

w

l

§w ¦'

l

¦'qw ¦6w ¦6§Xw

l

w q q x

l

§XyX{ y6§6§ q q ¦6

l

q q x{ yo§¦ yww §Xx q ¦6w q q §Xy qwJ{ y

ll

¦y¦ q q

l

q

| z

§ ¦xq qXwXy ¦w

|

y

z

qw q w w

z

¦

z

q q¦w ¦w¦

z

x

t7u o6vckslOnl

¦

eUf e_h l

qw

l

w

l

§w ¦'

l

¦'qw ¦6w ¦6§Xw

l

w q q ¦xqXq Š Š qXxo§ q ¦6

l

q

z

§ ¦X§Xy

|

Š Š ¦X§w

l

¦6¦x ¦6w q ¦'¦x ¦wX{Xy Š Š qXw

z

q wX{Xx q

l

q ¦X§6§ ¦

|

x¦ y

l

x

z |

yX{¦ qJx |z

¦

l

yx qw q ¦wX{ ¦qqJ{ y

l

xq y zJ|z

[image:25.612.222.402.261.516.2]

24

S. K. Sahu and T. M. F. Smith

Table 5.

Optimum sample size for different

values of

e2}~‡€ h

and

e9}‚€

h

for the exponential

ex-ample when

e }~€

f

ke }‚€

f

k

¦

. Here

i j k-lOnl

¦

and

o‹k ¦pXq

. All sample sizes are

Š

for the

t7u o6vckslOnl

¦

case.

t7u o6v[kslOn

¦

l

e }‚€

h

e }~€

h

l

w ¦6

l

¦6w q

l

qw

l

w ¦xX{ ¦x

|

¦x

l

¦x

|

¦xy ¦'

l

¦

lXl

{

|

{

|

{Xx {q ¦'w

z

§

zX| zX| z

y

z

w

q l |

{ w l |

{ wJy w¦ qw

|

y

|

q

|l |

q

|

y

t7u o6v[kslOnl

w

e }‚€

h

l

w §Xyw §

l

{ §Xyq §¦X§ §Jy¦ ¦'

l

y

zJ|

y

zX|

yxq yo§

l

y

z

{

¦'w qw

z

q

zz

q

|z

q

|

{ qXwXy q

l

q

lJ|

¦xw q

l

q ¦{

z

q

l

¦

qw ¦ zJ|

¦ww ¦!§¦ ¦w'§ ¦ zz t7u

o6v[kslOnl

¦

e9}‚€

h

l

w Š Š Š Š Š

¦' l |

q¦{

|

qXx l |

wXx

|

y{Xyo§

|

{{{ ¦'w yqXw

l

yXyo§

z

yqwJy q6§'§¦ qJxo§

l

q

l

q¦'¦

z

q

z

§q q

|

yq qXy zz

qXwq¦ qw q¦x

z

q

lll

q¦{6§ ¦x¦

|

¦X§Xy

[image:26.612.239.397.112.399.2]

A Bayesian sample size determination problem

25

Table 6.

Optimum sample size for

dif-ferent values of

e9}~€ h

and

e2}J€

h

for the

ex-ponential example when

e }~€

f klOn w

and

e2}‚€ f k ¦

. Here

i j kslOnl

¦

and

o3k ¦pXq

.

t7u o6vckslOn ¦ l e }‚€ h e }~€ h l

w ¦6

l

¦6w q

l

qw

l

w q q q q q ¦'

l

q q q q q

¦'w q q q q q q

l

q q q q q

qw q q q q q t7u o6vckslOnl w e }‚€ h e9}~€ h l

w ¦6

l

¦6w q

l

qw

l

w q q q q q ¦' l w | w w | ¦'w z § § z z q l

§ x x § x

qw x x x § x t7u o6vckslOnl ¦ e9}‚€ h e }~€ h l

w ¦6

l

¦6w q

l

qw

l

w qXy qw qw q6§ qw ¦'

l

xx x¦ {¦ {y x

z

¦'w ¦

|

§ ¦

|

§ ¦

|

x ¦wXq ¦wXw q

l

¦wXy ¦xq ¦w

z

¦!§Xw ¦

z

x

qw ¦wX{ ¦xw ¦!§J{ ¦{

z

¦

z

{

Table 7.

Optimum Sample size for different values of

e

f

and

e_h

for the mixture example when the fitting and sampling prior

are same. Here

i j k-lOnl

¦

,

oŒk ¦pXq

,

}ƒ~€ j kŽ}J€ j klOnl ¦

and

 }~‡€ j k  }‚€ j n t kslOn ¦ l e f e h l

qw

l

w

l

§w ¦'

l

¦'qw ¦6w ¦6§Xw

l

w q

z

w ¦w lXl

Š Š yXy

|

¦ q ¦6

l

q y6§ q'§w ¦

l

¦'¦ qqJxw y¦wJy qXx

l

{

¦6w q

z x l y l § § |z ¦ |l w ¦x6§¦ q l

q q q6§ ¦

l

y y¦x §

l

¦ ¦

l

¦!§ qw q q ¦q

|

y ¦y¦ y¦!§ wXxX{

t kslOnl w eUf e_h l qw l w l

§w ¦'

l

¦'qw ¦6w ¦6§Xw

l

w qq {

z

q Š Š Š Š q

¦6

l

y qXw¦ ¦w6§Xq

|

wq

z

Š Š Š

¦6w q § | |

w¦ ¦

|

w'§ y |l

y Š Š q

l

q ¦x ¦

[image:26.612.202.432.463.644.2]

[image:27.612.222.401.145.333.2]

26

S. K. Sahu and T. M. F. Smith

Table 8.

Optimum Sample size for different

values of

e2}ƒ~€ h

and

e2}‚€

h

for the mixture

ex-ample. Here

ijklOnl

¦

;

e }~€ f k‘e }‚€ f k ¦

,

 }~‡€ j ks }‚€ j kslOnl ¦

,

o3k ¦pq

.

t7u o6v[kslOn ¦ l e }‚€ h e }~€ h l

w ¦6

l

¦6w q

l

qw

l

w Š Š Š Š Š

¦' l ¦ l x z ¦ l yq ¦ l wJx ¦6¦q z ¦ l y{ ¦'w yy6§ yXy

| y lJ| y l ¦ yqw q l ¦qJx ¦ |

q {{ ¦qJ{ ¦'¦

|

qw §Xw

zz zX| w l | x t7u o6v[kslOnl w e }‚€ h e9}~€ h l

w ¦6

l

¦6w q

l

qw

l

w Š Š Š Š Š

¦' l | w¦ z | q¦w | § z x |zJ|| | q z ¦

¦'w ¦wXww ¦

| {o§ ¦ | q | ¦w zl ¦wXyo§ q l z yXy z y l w | w z yX{ wJyx qw y¦w qJx

| q6§ | q | ¦ qJy{

Table 9.

Optimum Sample size for different

values of

e9}~€ h

and

e9}‚€

h

for the mixture

exam-ple. Here

i

j

k’l_nl

¦

;

e }~€

f k’lOn w_“ e }‚€ f k ¦

,

 }~‡€ j ks }‚€ j kslOnl ¦

,

o3k ¦pq

.

t7u o6v[kslOn ¦ l e }‚€ h e }~€ h l

w ¦6

l

¦6w q

l qw l w | {X{ {Xx | ¦ ll

y {{¦ {6§¦ ¦' l y6§ wXx z y z w z y ¦'w | z

w w x

q

l

q y q q y

qw q q q q q

t7u o6v[kslOnl w e }‚€ h e9}~€ h l

w ¦6

l

¦6w q

l

qw

l

w ¦x

| y qXwXx | q | xq q z w¦ q z ¦y ¦' l | w¦ § z

¦ xo§¦ xo§J{ ¦

l

¦

l

¦'w § | z x zz zXz § z q l q |

¦x qq ¦

z

qXy qw { { x ¦

l

[image:27.612.221.402.430.622.2]