Confidence intervals for functions of signal-to-noise ratio with application to economics and finance

Wa isa Thangjai; Sa-Aa Niwi pong
A icle
Con idence in e als o unc ions o signal- o-noise a io
wi h applica ion o economics and inance
Asian Jou nal o Economics and Banking (AJEB)
P o ided in Coope a ion wi h:
Ho Chi Minh Uni e si y o Banking (HUB), Ho Chi Minh Ci y
Sugges ed Ci a ion: Wa isa Thangjai; Sa-Aa Niwi pong (2024) : Con idence in e als o unc ions
o signal- o-noise a io wi h applica ion o economics and inance, Asian Jou nal o Economics and
Banking (AJEB), ISSN 2633-7991, Eme ald, Leeds, Vol. 8, Iss. 2, pp. 199-218,
h ps://doi.o g/10.1108/AJEB-12-2023-0129
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/334121
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Con idence in e als o unc ions
o signal- o-noise a io wi h
applica ion o economics
and inance
Wa isa Thangjai
Depa men o S a is ics, Ramkhamhaeng Uni e si y, Bangkok, Thailand, and
Sa-Aa Niwi pong
Depa men o Applied S a is ics,
King Mongku ’s Uni e si y o Technology No h Bangkok, Bangkok, Thailand
Abs ac
Pu pose –Con idence in e als play a c ucial ole in economics and inance, p o iding a c edible ange o alues o
an unknown pa ame e along wi h a co esponding le el o ce ain y. Thei applica ions encompass economic
o ecas ing, ma ke esea ch, inancial o ecas ing, econome ic analysis, policy analysis, inancial epo ing,
in es men decision-making, c edi isk assessmen and consume con idence su eys. Signal- o-noise a io (SNR)
inds applica ions in economics and inance ac oss a ious domains such as economic o ecas ing, inancial
modeling, ma ke analysis and isk assessmen . A high SNR indica es a obus and dependable signal, simpli ying
he p ocess o making well-in o med decisions. On he o he hand, a low SNR indica es a weak signal ha could be
obscu ed by noise, so decision-making p ocedu es need o ake his in o se ious conside a ion. This esea ch ocuses
on he de elopmen o con idence in e als o unc ions de i ed om he SNR and explo es hei applica ion in he
ields o economics and inance.
Design/me hodology/app oach –The cons uc ion o he con idence in e als in ol ed he applica ion o
a ious me hodologies. Fo he SNR, con idence in e als we e o med using he gene alized con idence
in e al (GCI), la ge sample and Bayesian app oaches. The di e ence be ween SNRs was es ima ed h ough he
GCI, la ge sample, me hod o a iance es ima es eco e y (MOVER), pa ame ic boo s ap and Bayesian
app oaches. Addi ionally, con idence in e als o he common SNR we e cons uc ed using he GCI, adjus ed
MOVER, compu a ional and Bayesian app oaches. The pe o mance o hese con idence in e als was
assessed using co e age p obabili y and a e age leng h, e alua ed h ough Mon e Ca lo simula ion.
Findings –The GCI app oach demons a ed supe io pe o mance o e o he app oaches in e ms o bo h
co e age p obabili y and a e age leng h o he SNR and he di e ence be ween SNRs. Hence, employing he
GCI app oach is ad ised o cons uc ing con idence in e als o hese pa ame e s. As o he common SNR,
he Bayesian app oach exhibi ed he sho es a e age leng h. Consequen ly, he Bayesian app oach is
ecommended o cons uc ing con idence in e als o he common SNR.
O iginali y/ alue –This esea ch p esen s con idence in e als o unc ions o he SNR o assess SNR
es ima ion in he ields o economics and inance.
Keywo ds A e age leng h, Con idence in e al, Co e age p obabili y, Mon e Ca lo simula ion,
Signal- o-noise a io
Pape ype Resea ch pape
1. In oduc ion
Con idence in e als play a c ucial ole in economics and inance, p o iding a eliable ange o
alues o an unknown pa ame e along wi h a speci ied le el o ce ain y. He e a e di e se
Con idence
in e als o
unc ions
199
© Wa isa Thangjai and Sa-Aa Niwi pong. Published in Asian Jou nal o Economics and Banking.
Published by Eme ald Publishing Limi ed. This a icle is published unde he C ea i e Commons A ibu ion
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Recei ed 18 Decembe 2023
Re ised 8 Janua y 2024
Accep ed 29 Janua y 2024
Asian Jou nal o Economics and
Banking
Vol. 8 No. 2, 2024
pp. 199-218
Eme ald Publishing Limi ed
e-ISSN: 2633-7991
p-ISSN: 2615-9821
DOI 10.1108/AJEB-12-2023-0129
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applica ions o con idence in e als in hese ields. Economic o ecas ing: con idence
in e als a e essen ial o p ojec ing economic me ics like g oss domes ic p oduc (GDP)
g ow h, in la ion a es, and unemploymen a es, p o iding a spec um o alues o he likely
ac ual igu es. Ma ke esea ch: in inance, con idence in e als help gauge he po en ial
ange o e u ns on in es men s. Analys s use hem o con ey con idence ega ding u u e
s ock p ices o e u ns on inancial ins umen s. Risk managemen : con idence in e als play
a pi o al ole in e alua ing and managing inancial isk. They assis in app oxima ing
po en ial losses in in es men po olios, enabling in o med decisions by in es o s and
inancial ins i u ions. Financial o ecas ing: Con idence in e als a e inco po a ed in inancial
modeling o p ojec u u e cash lows, in e es a es, and o he inancial pa ame e s,
enhancing he accu acy o p edic ions abou he p ospec i e inancial pe o mance o
companies. Econome ic analysis: in econome ics, con idence in e als gauge he p ecision
o eg ession coe icien s and o he model pa ame e s, which is c ucial o de e mining he
s a is ical signi icance o ela ionships be ween economic a iables. Policy analysis:
Economis s use con idence in e als when sc u inizing he epe cussions o policy
changes, es ima ing he impac o a ax policy on consume spending, and p o iding a
con idence in e al o con ey associa ed unce ain y. Financial epo ing: con idence
in e als ind applica ion in inancial s a emen analysis o es ima e he p ecision o inancial
a ios, con ibu ing o he assessmen o he inancial heal h and pe o mance o companies.
In es men decision-making: In es o s ely on con idence in e als o assess po en ial
e u ns and isks linked o di e se in es men oppo uni ies, aiding in making well-in o med
decisions conce ning asse alloca ion and po olio managemen . C edi isk assessmen : in
banking and inance, con idence in e als a e used o e alua e c edi isk, es ima ing he
po en ial ange o de aul p obabili ies, and es ablishing sui able in e es a es o loans.
Consume con idence su eys: Con idence in e als a e employed in he analysis and
in e p e a ion o su ey da a, such as consume con idence su eys, p o iding a measu e o
unce ain y a ound epo ed con idence le els. In conclusion, con idence in e als se e as a
aluable ool in economics and inance, o e ing a me hod o quan i y and con ey unce ain y
in a ious analyses and decision-making p ocesses.
The signal- o-noise a io (SNR) in he ealms o economics and inance o igina es om signal
p ocessing, ep esen ing he p opo ion o aluable in o ma ion, e med he signal o i ele an
o andom backg ound noise. In he con ex o economic and inancial analysis, his concep is
commonly u ilized o e alua e he in o ma ion’s quali y and he signal’ss eng hcompa ed o
he su ounding noise. Wi hin economic and inancial analysis, he e m signal deno es
meaning ul and pe inen da a o pa e ns, while noise pe ains o andom luc ua ions o
inconsequen ial in o ma ion. The SNR unc ions as a me ic o assessing he cla i y and
dependabili y o a signal amids backg ound noise. The applica ions o SNR in economics and
inance ex end ac oss a ious domains, encompassing economic o ecas ing, inancial modeling,
ma ke analysis, and isk assessmen . A high SNR sugges s a obus and dependable signal,
acili a ing mo e s aigh o wa d decision-making. Con e sely, a low SNR implies a weak
signal, po en ially obscu ed by noise, necessi a ing ca e ul conside a ion in decision-making
p ocesses. In essence, comp ehending and managing he SNR is pa amoun o ex ac ing
meaning ul insigh s and making in o med decisions wi hin economic and inancial con ex s.
Poin es ima ion in ol es p o iding a single, speci ic alue as an es ima e o an unknown
pa ame e in a popula ion. Fo example, es ima e he popula ion mean based on a sample
mean. In e al es ima ion, on he o he hand, p o ides a ange o alues (an in e al) wi hin
which he ue pa ame e is likely o lie. This is ypically exp essed as a con idence in e al.
In e al es ima ion is o en conside ed be e han poin es ima ion. This is because i
inco po a es unce ain y, con idence le el, decision-making, and obus ness. To inco po a e
unce ain y, in e al es ima ion explici ly acknowledges he unce ain y inhe en in
es ima ing popula ion pa ame e s om a sample. I p o ides a sense o he ange o
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plausible alues. Fo con idence le el, con idence in e als come wi h a speci ied con idence
le el (e.g., 95%). This indica es he p opo ion o in e als om epea ed sampling ha would
include he ue pa ame e . I o e s a clea indica ion o he eliabili y o he es ima e. Fo
decision-making pu poses, ha ing a ange o alues is o en mo e in o ma i e han a single
poin . I allows decision-make s o conside a spec um o possibili ies. Fo obus ness, poin
es ima es can be sensi i e o ou lie s o ex eme alues in he da a. Con idence in e als,
especially hose based on obus me hods, may be less a ec ed by ex eme obse a ions.
A con idence in e al o a pa ame e o in e es is a s a is ical ange ha p o ides an
es ima ed ange o alues ha is likely o include he ue alue o he pa ame e . I is
cons uc ed based on he sample da a and is associa ed wi h a ce ain le el o con idence. The
con idence in e al is a measu e o he p ecision o unce ain y o he es ima ed pa ame e .
Fo example, i you a e es ima ing he SNR o a popula ion, a 95% con idence in e al would
imply ha i you we e o ake many samples and cons uc a con idence in e al om each,
abou 95% o hose in e als would con ain he ue popula ion SNR. Mo eo e , he
con idence in e al o he di e ence be ween pa ame e s o in e es is a ange o alues ha
is likely o con ain he ue di e ence be ween wo popula ion pa ame e s. This ype o
in e al es ima ion is commonly used in s a is ical analysis, especially when compa ing wo
g oups o assessing he impac o an in e en ion. In addi ion, he con idence in e al o a
common pa ame e o in e es is an in e al es ima e ha p o ides a ange o plausible alues
o he ue alue o a pa ame e . This ype o in e al es ima ion is commonly used in
s a is ical analysis when dealing wi h a single popula ion pa ame e .
2. Li e a u e e iew
The Gene alized Con idence In e al (GCI) app oach is designed o be e sa ile ac oss di e se
da a ypes and s a is ical scena ios. I is no cons ained by speci ic dis ibu ional
assump ions, making i applicable in si ua ions whe e classical me hods may no be
app op ia e. The GCI has di e se applica ions in ields such as economics, inance, biology,
and any domain equi ing s a is ical in e ence. This me hodology u ilizes he gene alized
pi o al quan i y (GPQ) o cons uc he con idence in e al, enabling he es ima ion o
con idence in e als o complex pa ame e s. Howe e , i ’s impo an o no e ha he
nume ical simula ion o he GCI app oach elies solely on he maximum likelihood es ima e.
Many esea che s ha e unde aken compa isons be ween he GCI app oach and al e na i e
me hods o cons uc ing con idence in e als, as e idenced in s udies by Wee ahandi (1993),
K ishnamoo hy and Lu (2003),K ishnamoo hy and Ma hew (2003),Tian (2005),Chen and
Zhou (2006),Tian and Wu (2007),Ye e al. (2010),Sao hayanun and Thangjai (2018),Thangjai
and Niwi pong (2019),Thangjai and Niwi pong (2020a), and Thangjai and Niwi pong (2020b).
Cons uc ing con idence in e als using he la ge sample app oach in ol es exploi ing
asymp o ic p ope ies, pa icula ly when dealing wi h a subs an ial olume o da a. This
me hod elies on he Cen al Limi Theo em, which posi s ha he dis ibu ion o sample
means con e ges o a no mal dis ibu ion as he sample size inc eases. U ilizing his p inciple
allows o he es ima ion o con idence in e als unde he assump ion o no mali y,
enhancing hei applicabili y in ex ensi e da ase s. The la ge sample app oach is
ad an ageous due o i s simplici y in cons uc ing he con idence in e al using he exac
o mula. Howe e , a limi a ion is ha i equi es a la ge sample size o es ima ing he
con idence in e al. Se e al schola s ha e e alua ed he la ge sample app oach in
compa ison o al e na i e me hods o cons uc ing con idence in e als, as demons a ed
in he esea ch conduc ed by Tian and Wu (2007),Sao hayanun and Thangjai (2018),
Thangjai and Niwi pong (2019), and Thangjai and Niwi pong (2020b).
The MOVER app oach elies on he o iginal con idence in e al o a speci ic pa ame e o
in e es o de i e he inal con idence in e al. An ad an age o he MOVER app oach is i s
Con idence
in e als o
unc ions
201
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ease o compu a ion using he exac o mula. Howe e , a d awback is ha i can be
cons uc ed wi h o wi hou he ini ial con idence in e al o a single pa ame e o in e es .
Se e al esea che s, including Zou and Donne (2008),Zou e al. (2009),Sao hayanun and
Thangjai (2018),Thangjai and Niwi pong (2019), and Thangjai and Niwi pong (2020b), ha e
ecommended he u iliza ion o he MOVER app oach in cons uc ing con idence in e als.
The adjus ed MOVER app oach is inspi ed by he p inciples o bo h he la ge sample and
MOVER app oaches. I s ad an age lies in he s aigh o wa d applica ion o he exac
o mula o con idence in e al compu a ion, al hough a d awback is ha i elies on he
ini ial con idence in e al o a single pa ame e . Thangjai and Niwi pong (2020a) has del ed
in o he in es iga ion o he adjus ed MOVER app oach.
The boo s ap app oach in ol es app oxima ing he sampling dis ibu ion o s a is ics by
i e a i ely esampling wi h eplacemen s om he popula ion. These mul iple boo s ap
samples, d awn om he popula ion on nume ous occasions, unc ion as ep esen a i e
samples o he en i e popula ion. The boo s ap app oach p o ides a simple and easonably
accu a e echnique o cons uc ing con idence in e als. Howe e , a d awback is he
equi emen o knowledge ega ding he dis ibu ion o es ima es a ound he ue alues
because he sampling dis ibu ion aligns wi h he da a dis ibu ion, conside ing ha he
es ima es a e de i ed om he da a. Va ious esea che s, including Chachi (2017) and Thangjai
and Niwi pong (2020b), ha e ad oca ed o he u iliza ion o he boo s ap app oach.
The compu a ional app oach is employed o o mula e con idence in e als o in ica e
pa ame e s. This echnique in ol es simula ions and nume ical compu a ions u ilizing he
maximum likelihood es ima e. Schola s ha e in oduced he compu a ional app oach o con idence
in e als, as demons a ed in wo ks by Pal e al. (2007) and Thangjai and Niwi pong (2020a).
The Bayesian app oach employs pos e io p obabili y and acili a es compa ison wi h
al e na i e me hods o cons uc ing c edible in e als. The p ima y mo i a ion o op ing
o he Bayesian app oach is he complexi y o models ha adi ional me hods may s uggle
o add ess. I is essen ial o emphasize ha , i espec i e o he a ionale behind adop ing he
Bayesian app oach, conduc ing a sensi i i y analysis o p io s is always c ucial and should
be included. This compa ison is subs an ia ed by s udies such as Rao and D’Cunha (2016) and
Ma and Chen (2018).
3. Me hodology
The SNR can be desc ibed as he ecip ocal o he coe icien o a ia ion. The SNR is
calcula ed as he a io o he mean o he s anda d de ia ion. This pape discussed h ee pa s
as ollow: The SNR, he di e ence be ween SNRs, and he common SNR.
3.1 Con idence in e als o he SNR
Suppose ha andom sample X ¼ðX1;X2;...;XnÞ ollows any dis ibu ion. Suppose ha
μ
and
σ
a e popula ion mean and popula ion s anda d de ia ion o he dis ibu ion,
espec i ely. The SNR is de ined as
θ¼
μ
σ
:(1)
Le X and S a e sample mean and sample s anda d de ia ion o he dis ibu ion, espec i ely.
The es ima o o he SNR is de ined as
bθ¼X
S:(2)
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3.1.1 GCI app oach o SNR. The concep o GCI was in oduced by Wee ahandi (1993).
Le X ¼ðX1;X2; :::; XnÞbe a andom sample ha ing a densi y unc ion ðXjθ;
υ
Þ, whe e θis
he pa ame e o in e es and
υ
is a nuisance pa ame e . Le x be he obse ed sample o X. A
gene alized pi o al quan i y RðX;x;θ;
υ
Þis conside ed and sa is ies he ollowing condi ions:
(i) The dis ibu ion o RðX;x;θ;
υ
Þis ee o all unknown pa ame e s.
(ii) The obse ed alue o RðX;x;θ;
υ
Þis he pa ame e o in e es .
Condi ion (i) is imposed o gua an ee ha a subse o he sample space o he possible alues
o RðX;x;θ;
υ
Þcan be ound a a gi en alue o he con idence coe icien wi h no knowledge
o he pa ame e s. Condi ion (ii) is imposed o ensu e ha such p obabili y s a emen s based
on he GPQ lead o con idence egions in ol ing obse ed da a x only. The GCI o θis
compu ed using he pe cen iles o he GPQ. Le ½Rð
α
=2Þ;Rð1−
α
=2Þ be a 100ð1−
α
Þ% wo-
sided GCI o he pa ame e o in e es , whe e Rð
α
=2Þand Rð1−
α
=2Þdeno e he 100ð
α
=2Þ- h
and he 100ð1−
α
=2Þ- h pe cen iles o RðX;x;θ;
υ
Þ, espec i ely.
Following Sao hayanun and Thangjai (2018). Le R
μ
be he GPQ o
μ
and le R
σ
be he GPQ
o
σ
. The GPQ o θis de ined as
Rθ¼R
μ
R
σ
:(3)
The 100ð1−
α
Þ% wo-sided con idence in e al o he SNR based on he GCI app oach is
gi en by
CIθ:GCI ¼½Lθ:GCI;Uθ:GCI¼½Rθð
α
=2Þ;Rθð1
α
=2Þ;(4)
whe e Rθð
α
=2Þand Rθð1−
α
=2Þdeno e he ð
α
=2Þ- h and ð1−
α
=2Þ- h quan iles o Rθ,
espec i ely.
The ollowing algo i hm is used o cons uc he GCI o he SNR.
Algo i hm 1. Fo a gi en x and s
Fo g ¼1 om
Compu e R
μ
Compu e R
σ
Compu e Rθ
End g loop
Compu e he ð
α
=2Þ- h quan iles o Rθde ined by Rθð
α
=2Þ
Compu e he ð1−
α
=2Þ- h quan iles o Rθde ined by Rθð1−
α
=2Þ
3.1.2 La ge sample app oach o SNR. Acco ding o Sao hayanun and Thangjai (2018) he
100ð1−
α
Þ% wo-sided con idence in e al o he SNR based on he la ge sample app oach is
gi en by
CIθ:LS ¼½Lθ:LS;Uθ:LS¼ bθz1−
α
=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Va bθ

;bθþz1−
α
=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Va bθ


;(5)
whe e z1−
α
=2deno es he ð1−
α
=2Þ- h quan ile o a s anda d no mal dis ibu ion and Va ðbθÞis
he a iance o he es ima o o SNR.
3.1.3 Bayesian app oach o SNR. Bayes’ ule is u ilized o e ise he p io dis ibu ion,
esul ing in he pos e io dis ibu ion, which encompasses all ele an in o ma ion ega ding
he unknown pa ame e s in e ed om he obse ed da a. The Bayesian app oach p o ides a
amewo k o adjus ing belie s and making p edic ions based on new e idence o da a. I is
g ounded in Bayes’ heo em, which in eg a es p io p obabili y and likelihood o compu e he
Con idence
in e als o
unc ions
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pos e io p obabili y. The p io dis ibu ion e lec s unce ain y abou pa ame e s be o e
obse ing he da a. In his s udy, we u ilized Je eys’independence p io .
Le
σ
jx be he pos e io dis ibu ion o
σ
. And le
μ
j
σ
;x be he pos e io dis ibu ion o
μ
gi en
σ
. Le θBS be he pos e io dis ibu ion using
σ
jx and
μ
j
σ
;x.
The 100ð1−
α
Þ% wo-sided con idence in e al o he SNR based on he Bayesian
app oach is gi en by
CIθ:BS ¼½Lθ:BS;Uθ:BS;(6)
whe e Lθ:BS and Uθ:BS a e he lowe and uppe limi s o he sho es 100ð1−
α
Þ% highes
pos e io densi y in e al o θBS, espec i ely.
The ollowing algo i hm is used o cons uc he Bayesian c edible in e al o he SNR.
Algo i hm 2. Fo a gi en x and s
Fo g ¼1 om
Compu e
σ
jx
Compu e
μ
j
σ
;x
Compu e θBS
End g loop
Compu e he sho es 100ð1−
α
Þ% highes pos e io densi y in e al o θBS
The ollowing algo i hm is used o e alua e he co e age p obabili ies and a e age leng hs o
he con idence in e als o SNR.
Algo i hm 3. Fo a gi en
μ
,
σ
, and θ
Fo h ¼1 oM
Gene a e x
Calcula e x and s
Cons uc he con idence in e al ½Lθ:GCI;Uθ:GCI
Cons uc he con idence in e al ½Lθ:LS;Uθ:LS
Cons uc he con idence in e al ½Lθ:BS;Uθ:BS
I L ≤θ≤U, se p ¼1; else se p ¼0
Compu e U −L
End h loop
Compu e mean o p de ined by he co e age p obabili y
Compu e mean o U −L de ined by he a e age leng h
3.2 Con idence in e als o he di e ence be ween SNRs
Suppose ha X ¼ðX1;X2;:::;XnÞ ollows any dis ibu ion wi h mean
μ
Xand s anda d
de ia ion
σ
X. Simila ly, le Y ¼ðY1;Y2; :::; YmÞbe any dis ibu ion wi h mean
μ
Yand s anda d
de ia ion
σ
Y. Mo eo e , X and Y a e independen . The single SNRs o X and Y a e gi en by
θX¼
μ
X
σ
X
and θY¼
μ
Y
σ
Y
:(7)
The di e ence be ween o SNRs is de ined as
δ¼θXθY:(8)
Le Xand SXa e sample mean and sample s anda d de ia ion o X, espec i ely. Mo eo e , le
Y and SYa e sample mean and sample s anda d de ia ion o Y, espec i ely. Suppose ha bθX
and bθYa e he es ima o s o θXand θY, espec i ely, which a e gi en
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bθX¼X
SX
and bθY¼Y
SY
:(9)
The di e ence be ween o SNRs is de ined as
bδ¼bθXbθY:(10)
Suppose ha Va ðbθXÞand Va ðbθYÞa e he a iances o bθXand bθY, espec i ely. The a iance
o bδ¼bθX−bθYis
Va bδ
¼Va bθXbθY

¼Va bθX

þVa bθY

:(11)
3.2.1 GCI app oach o he di e ence be ween SNRs. Acco ding o Thangjai and Niwi pong
(2019) and Thangjai and Niwi pong (2020b). Le R
μ
Xbe he GPQ o
μ
Xand le R
σ
Xbe he GPQ o
σ
X. The GPQ o θXis de ined as
RθX¼R
μ
X
R
σ
X
:(12)
Mo eo e , le R
μ
Ybe he GPQ o
μ
Yand le R
σ
Ybe he GPQ o
σ
Y. The GPQ o θYis de ined as
RθY¼R
μ
Y
R
σ
Y
:(13)
The e o e, he di e ence be ween he GPQs o SNRs is
Rδ¼RθXRθY:(14)
The 100ð1−
α
Þ% wo-sided con idence in e al o he di e ence be ween SNRs based on he
GCI app oach is gi en by
CIδ:GCI ¼½Lδ:GCI;Uδ:GCI¼½Rδð
α
=2Þ;Rδð1
α
=2Þ;(15)
whe e Rδð
α
=2Þand Rδð1−
α
=2Þdeno e he ð
α
=2Þ- h and ð1−
α
=2Þ- h quan iles o Rδ, espec i ely.
The ollowing algo i hm is used o cons uc he GCI o he di e ence be ween SNRs.
Algo i hm 4. Fo a gi en x, y, sXand sY
Fo g ¼1 om
Compu e R
μ
X,R
σ
X, and RθX
Compu e R
μ
Y,R
σ
Y, and RθY
Compu e Rδ
End g loop
Compu e he ð
α
=2Þ- h quan iles o Rδde ined by Rδð
α
=2Þ
Compu e he ð1−
α
=2Þ- h quan iles o Rδde ined by Rδð1−
α
=2Þ
3.2.2 La ge sample app oach o he di e ence be ween SNRs. Following Thangjai and
Niwi pong (2019) and Thangjai and Niwi pong (2020b) using he cen al limi heo em, he
100ð1−
α
Þ% wo-sided con idence in e al o he di e ence be ween SNRs based on he
la ge sample app oach is gi en by
CIδ:LS ¼½Lδ:LS;Uδ:LS¼ bδz1−
α
=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Va bδ

;bδþz1−
α
=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Va bδ


;(16)
whe e z1−
α
=2is he ð1−
α
=2Þ- h quan ile o he s anda d no mal dis ibu ion and Va ðbδÞis he
a iance o he es ima o o di e ence be ween SNRs.
Con idence
in e als o
unc ions
205
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3.2.3 MOVER o he di e ence be ween SNRs. Le lXand uXbe he lowe and uppe limi s
o he con idence in e al o SNR o X, espec i ely. Simila ly, le lYand uYbe he lowe and
uppe limi s o he con idence in e al o SNR o Y, espec i ely.
Following Zou and Donne (2008),Zou e al. (2009),Thangjai and Niwi pong (2019) and
Thangjai and Niwi pong (2020b), he 100ð1−
α
Þ% wo-sided con idence in e al o he
di e ence be ween he SNRs based on he MOVER app oach is gi en by
CIδ:MOVER ¼½Lδ:MOVER;Uδ:MOVER
¼bθXbθYffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
bθXlX

2
þuYbθY

2
;bθXbθYþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
uXbθX

2
þbθYlY

2
" #:
(17)
3.2.4 Pa ame ic boo s ap app oach o he di e ence be ween SNRs. The pa ame ic
boo s ap app oach is a esampling app oach based on independen ly sampling wi h a
eplacemen om exis ing sample da a o he same sample size.
Le X*¼ðX*
1;X*
2; :::; X*
nÞbe sample wi h eplacemen om X ¼ðX1;X2; :::; XnÞwi h
sample size n and le x*¼ðx*
1;x*
2; :::; x*
nÞbe he obse ed alues o X*¼ðX*
1;X*
2; :::; X*
nÞ.
Simila ly, le Y*¼ðY*
1;Y*
2; :::; Y*
mÞbe he sample om Y ¼ðY1;Y2; :::; YmÞwi h eplacemen
sample size m and le y*¼ðy*
1;y*
2; :::; y*
mÞbe he obse ed alues o Y*¼ðY*
1;Y*
2; :::; Y*
mÞ.
The e-sampled sample is called a boo s ap sample. The di e ence in SNRs om he
boo s ap sample is ob ained by
δ*¼θ*
Xθ*
Y:(18)
An es ima o o he di e ence o SNRs is
bδ
*¼bθ
*
Xbθ
*
Y:(19)
Fo eplica e B imes, he e a e o ally B es ima es o he di e ence o SNRs δ* om B
boo s ap sample.
The sampling dis ibu ion is cons uc ed wi h hese B boo s ap s a is ics. The con idence
in e al o he di e ence in SNRs is calcula ed using he dis ibu ion. The e o e, he
100ð1−
α
Þ% wo-sided con idence in e al o he di e ence be ween SNRs based on he
pa ame ic boo s ap app oach is gi en by
CIδ:PB ¼½Lδ:PB;Uδ:PB¼hbδz1−
α
=2S*;bδþz1−
α
=2S*i;(20)
whe e z1−
α
=2is he ð1−
α
=2Þ- h quan ile o he s anda d no mal dis ibu ion and S*is he
s anda d de ia ion o bδ
*
.
The ollowing algo i hm is used o cons uc he pa ame ic boo s ap con idence in e al
o he di e ence be ween SNRs.
Algo i hm 5. Fo a gi en x*, y*,s
*
X, and s*
Y
Fo g ¼1 om
Compu e bθ
*
X
Compu e bθ
*
Y
Compu e bδ
*
End g loop
Compu e S*
Compu e Lδ:PB and Uδ:PB
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ðn;mÞðθX;θYÞ
CP (AL)
CIδ:GCI CIδ:LS CIδ:MOVER CIδ:PB CIδ:BS
(30,30) (10,1) 0.9506 (5.3335) 0.9518 (5.3605) 0.9602 (5.5937) 0.9392 (5.6483) 0.9492 (5.2891)
(10,2) 0.9516 (5.4272) 0.9528 (5.4499) 0.9608 (5.6870) 0.9410 (5.7394) 0.9478 (5.3764)
(10,5) 0.9512 (5.9269) 0.9530 (5.9553) 0.9618 (6.2144) 0.9474 (6.3042) 0.9482 (5.8786)
(10,10) 0.9548 (7.5176) 0.9578 (7.5454) 0.9660 (7.8737) 0.9470 (8.0065) 0.9562 (7.4591)
(30,50) (10,1) 0.9484 (5.2930) 0.9504 (5.3196) 0.9588 (5.5500) 0.9400 (5.6141) 0.9452 (5.2461)
(10,2) 0.9474 (5.3725) 0.9488 (5.3991) 0.9588 (5.6312) 0.9396 (5.6993) 0.9460 (5.3268)
(10,5) 0.9484 (5.6744) 0.9500 (5.6973) 0.9584 (5.9316) 0.9410 (6.0113) 0.9472 (5.6214)
(10,10) 0.9568 (6.6862) 0.9572 (6.7086) 0.9642 (6.9558) 0.9500 (7.0183) 0.9554 (6.6342)
(50,50) (10,1) 0.9448 (4.0614) 0.9456 (4.0721) 0.9518 (4.1751) 0.9340 (4.1708) 0.9410 (4.0268)
(10,2) 0.9488 (4.1327) 0.9498 (4.1439) 0.9550 (4.2488) 0.9376 (4.2406) 0.9460 (4.0970)
(10,5) 0.9472 (4.5328) 0.9474 (4.5449) 0.9530 (4.6600) 0.9452 (4.6654) 0.9430 (4.4986)
(10,10) 0.9498 (5.7174) 0.9508 (5.7315) 0.9560 (5.8766) 0.9454 (5.8767) 0.9486 (5.6727)
(50,100) (10,1) 0.9488 (4.0481) 0.9506 (4.0596) 0.9576 (4.1619) 0.9392 (4.1476) 0.9466 (4.0145)
(10,2) 0.9466 (4.0751) 0.9480 (4.0896) 0.9536 (4.1919) 0.9428 (4.1858) 0.9426 (4.0444)
(10,5) 0.9534 (4.2789) 0.9520 (4.2896) 0.9600 (4.3915) 0.9424 (4.3751) 0.9514 (4.2441)
(10,10) 0.9488 (4.9270) 0.9504 (4.9364) 0.9542 (5.0405) 0.9426 (5.0405) 0.9482 (4.8904)
(100,100) (10,1) 0.9504 (2.8447) 0.9508 (2.8476) 0.9542 (2.8828) 0.9418 (2.8633) 0.9460 (2.8196)
(10,2) 0.9462 (2.8867) 0.9488 (2.8885) 0.9514 (2.9242) 0.9432 (2.9072) 0.9452 (2.8609)
(10,5) 0.9484 (3.1648) 0.9470 (3.1688) 0.9500 (3.2080) 0.9460 (3.1970) 0.9464 (3.1406)
(10,10) 0.9516 (3.9844) 0.9532 (3.9880) 0.9558 (4.0373) 0.9494 (4.0215) 0.9494 (3.9529)
Sou ce(s): Au ho s’calcula ion
Table 2.
The CPs and ALs o
95% wo-sided
con idence in e als o
he di e ence be ween
SNRs o log-no mal
dis ibu ions
Con idence
in e als o
unc ions
213
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Thailand. Simila ly, he SET100 Index encompasses he p ice mo emen s o 100 la ge-
capi aliza ion secu i ies wi h no able ading liquidi y on he same exchange. On he o he
hand, he sSET Index cap u es he p ice changes o common s ocks beyond hose included in
he SET50 and SET100 indices. These s ocks exhibi consis en liquidi y and adhe e o
speci ied equi emen s ela ed o sha e dis ibu ion among mino sha eholde s.
P ice-ea nings a ios o he SET50, SET100, and sSET indexes a e compu ed om
mon hly index da a p o ided by he S ock Exchange o Thailand. This s udy ocuses on
mon hly index da a spanning om Janua y o No embe 2023, as de ailed in Table 4. The
his og ams depic ing daily ain all da a can be ound in Figu e 1, and Table 5 p esen s
sample sizes, means, s anda d de ia ions, and SNRs o he h ee indexes. Be o e applying
ou me hods o eal da a, i is c ucial o assess he assump ion ha he loga i hms o he da a
a e d awn om a no mal dis ibu ion. T adi ionally, he Shapi o–Wilk no mali y es was
employed, yielding p- alues o 0.3922, 0.1467, and 0.08508 o SET50, SET100, and sSET
indexes, espec i ely. Recognizing he limi a ions o p- alues in es ing, al e na i e me hods
o checking no mali y include g aphical ools such as QQ-plo s o Bayesian es s. Analysis
o Table 6 e eals he minimum Akaike In o ma ion C i e ion (AIC) alues om Bayesian
es s ac oss i e egions, indica ing ha SET50, SET100, and sSET indexes ollow log-
no mal dis ibu ions. Fu he mo e, he no mal QQ-plo s o log-da a in Figu e 2 a i m he
esul s o he Bayesian es . Fo illus a i e pu poses, we exclusi ely selec da a om log-
no mal dis ibu ions o showcase ou es ima ion app oaches.
Fo SET50 index, he 95% con idence in e als o he SNR based on he GCI, la ge
sample, and Bayesian app oaches a e CIθ:GCI ¼[11.4901,28.5148] wi h an in e al leng h o
17.0247, CIθ:LS ¼[11.2126,28.7475] wi h an in e al leng h o 17.5349, and CIθ:BS ¼
[10.8550,28.0229] wi h an in e al leng h o 17.1679, espec i ely. Fo SET100 index, he
95% con idence in e als o he SNR based on he GCI, la ge sample, and Bayesian
app oaches a e CIθ:GCI ¼[8.6529,21.7304] wi h an in e al leng h o 13.0775, CIθ:LS ¼
[8.4493,21.6850] wi h an in e al leng h o 13.2357, and CIθ:BS ¼[8.0436,21.1895] wi h an
in e al leng h o 13.1459, espec i ely. Fo sSET index, he 95% con idence in e als o he
SNR based on he GCI, la ge sample, and Bayesian app oaches a e CIθ:GCI ¼[7.9635,20.0325]
wi h an in e al leng h o 12.0690, CIθ:LS ¼[7.8989,20.2794] wi h an in e al leng h o 12.3805,
and CIθ:BS ¼[7.9815,20.1948] wi h an in e al leng h o 12.2133, espec i ely. No ably, he
con idence in e als o he SNR based on he GCI, la ge sample, and Bayesian app oaches
encompass he ue alue o he SNR. Howe e , he GCI app oach has a sho e leng h han
he la ge sample and Bayesian app oaches.
Fo di e ence be ween SET50 index and SET100 index, he ue di e ence be ween he
SNRs is 4.9129. The 95% con idence in e als o he di e ence be ween SNRs based on he
GCI, la ge sample, MOVER, pa ame ic boo s ap, Bayesian app oaches a e CIδ:GCI ¼
ðn1;n2;n3Þð
σ
1;
σ
2;
σ
3Þ
CP (AL)
CIγ:GCI CIγ:AM CIγ:CA CIγ:BS
(30,30,30) (0.10,0.29,0.47) 0.9530 (1.2606) 0.8846 (1.0129) 0.9394 (1.2970) 0.9478 (1.2424)
(0.29,0.47,0.83) 0.9484 (0.7425) 0.8908 (0.6071) 0.9386 (0.7596) 0.9444 (0.7317)
(50,50,50) (0.10,0.29,0.47) 0.9512 (0.9656) 0.8814 (0.7612) 0.9456 (0.9823) 0.9504 (0.9523)
(0.29,0.47,0.83) 0.9550 (0.5699) 0.8952 (0.4578) 0.9498 (0.5776) 0.9528 (0.5614)
(30,50,100) (0.10,0.29,0.47) 0.9482 (0.6554) 0.9100 (0.5783) 0.9426 (0.6650) 0.9426 (0.6465)
(0.29,0.47,0.83) 0.9484 (0.4010) 0.9170 (0.3554) 0.9508 (0.4065) 0.9444 (0.3954)
(100,100,100) (0.10,0.29,0.47) 0.9544 (0.6780) 0.8736 (0.5262) 0.9458 (0.6842) 0.9494 (0.6689)
(0.29,0.47,0.83) 0.9448 (0.4000) 0.8792 (0.3169) 0.9454 (0.4032) 0.9414 (0.3944)
Sou ce(s): Au ho s’calcula ion
Table 3.
The CPs and ALs o
95% wo-sided
con idence in e als o
he common SNR o
se e al log-no mal
dis ibu ions
AJEB
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Sample s a is ics
Index
SET50 SET100 sSET
ni11 11 11
yi19.57 18.83 17.21
sYi1.00 1.29 1.26
xi2.97 2.93 2.84
sXi0.05 0.07 0.07
bθi19.98 15.07 14.09
Sou ce(s): Au ho s’calcula ion
Index P ice-ea nings a ios
SET50 18.90 20.09 20.07 18.71 18.90
18.79 19.69 21.77 20.45 19.58
18.33
SET100 17.54 18.79 18.80 17.52 18.24
18.07 18.66 21.73 20.44 19.43
17.87
sSET 16.00 17.52 17.49 16.45 17.15
16.07 16.15 20.05 18.84 17.20
16.44
Sou ce(s): S ock Exchange o Thailand (h ps://www.se .o . h/ h/ma ke /s a is ics/ma ke -s a is ics/main)
Au ho s’calcula ion
Figu e 1.
His og am plo s o
mon hly p ice-ea nings
a ios o h ee indexes
Table 5.
Sample s a is ics o
p ice-ea nings a ios o
h ee indexes
Table 4.
P ice-ea nings a ios o
h ee indexes
Con idence
in e als o
unc ions
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[5.9406,15.6047] wi h a leng h o in e al o 21.5453, CIδ:LS ¼[6.0718,15.8977] wi h a
leng h o in e al o 21.9695, CIδ:MOVER ¼[7.5748,17.4007] wi h a leng h o in e al o
24.9755, CIδ:PB ¼[11.6552,21.9753] wi h a leng h o in e al o 33.6305, and CIδ:BS ¼
[6.2144,15.6602] wi h a leng h o in e al o 21.8746, espec i ely. Fo di e ence be ween
SET50 index and sSET index, he ue di e ence be ween he SNRs is 5.8909. The 95%
con idence in e als o he di e ence be ween SNRs based on he GCI, la ge sample,
MOVER, pa ame ic boo s ap, Bayesian app oaches a e CIδ:GCI ¼[4.1340,16.4281] wi h a
leng h o in e al o 20.5621, CIδ:LS ¼[4.8416,16.6235] wi h a leng h o in e al o 21.4651,
CIδ:MOVER ¼[6.3101,18.0920] wi h a leng h o in e al o 24.4021, CIδ:PB ¼[9.7428,21.7973]
wi h a leng h o in e al o 31.5401, and CIδ:BS ¼[5.2779,15.4926] wi h a leng h o in e al o
20.7705. Fo di e ence be ween SET100 index and sSET index, he ue di e ence be ween
he SNRs is 0.9780. The 95% con idence in e als o he di e ence be ween SNRs based on
he GCI, la ge sample, MOVER, pa ame ic boo s ap, Bayesian app oaches a e CIδ:GCI ¼
[7.7565,9.7044] wi h a leng h o in e al o 17.4609, CIδ:LS ¼[8.0837,10.0398] wi h a leng h
o in e al o 18.1235, CIδ:MOVER ¼[9.3236,11.2796] wi h a leng h o in e al o 20.6032,
CIδ:PB ¼[14.3997,16.7647] wi h a leng h o in e al o 31.1644, and CIδ:BS ¼[8.0435,9.8604]
wi h a leng h o in e al o 17.9039. The esul s indica e ha all con idence in e als con ain
he ue di e ence be ween he SNRs. Howe e , he GCI app oach s ands ou by p o iding he
sho es leng h, making i he mos p e e able among he al e na i es.
The ue common SNRs is 15.6870. The 95% con idence in e als o he common SNR
based on GCI, adjus ed MOVER, compu a ional, and Bayesian app oaches a e CIγ:GCI ¼
Dis ibu ion
AIC
SET50 index SET100 index sSET index
No mal 34.09 39.76 39.31
Log-no mal 33.67 39.00 38.50
Gamma 33.88 39.34 38.86
Exponen ial 88.43 87.58 85.61
Sou ce(s): Au ho s’calcula ion
Figu e 2.
The no mal QQ-plo s o
log-mon hly p ice-
ea nings a ios o h ee
indexes
Table 6.
The AIC alues o
p ice-ea nings a ios o
h ee indexes
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[9.7447,18.5943] wi h a leng h o in e al o 8.8496, CIγ:AM ¼[11.1191,20.2548] wi h a leng h o
in e al o 9.1357, CIγ:CA ¼[12.0582,21.0089] wi h a leng h o in e al o 8.9507, and CIγ:BS ¼
[9.7514,18.7842] wi h a leng h o in e al o 9.0328. The indings sugges ha all con idence
in e als include he ue common SNR, wi h he GCI app oach ha ing a sho e leng h
compa ed o he o he s.
6. Conclusion
The GCI app oach showed be e esul s han o he echniques in e ms o co e age
p obabili y and a e age leng h o bo h he SNR and he di e ence be ween SNRs, wi h he
Bayesian app oach pe o ming simila ly o he GCI app oach. The e o e, i is ecommended
o use he GCI app oach o cons uc ing con idence in e als o hese pa ame e s.
Rega ding he common SNR, he Bayesian app oach had he sho es a e age leng h. Hence,
i is ecommended o use he Bayesian app oach o cons uc ing con idence in e als o he
common SNR.
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Appendix
The supplemen a y ma e ial o his a icle can be ound online.
Co esponding au ho
Sa-Aa Niwi pong can be con ac ed a : [email p o ec ed].ac. h
Fo ins uc ions on how o o de ep in s o his a icle, please isi ou websi e:
www.eme aldg ouppublishing.com/licensing/ ep in s.h m
O con ac us o u he de ails: [email p o ec ed]
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