OpenJournalofStatistics,2019,9,1-14
http://www.scirp.org/journal/ojs
ISSNOnline:2161-7198
ISSNPrint:2161-718X
OnAnalysisoftheBehrens-FisherProblem
BasedonBayesianEvidence
NengakEmmanuelGoltong,SaniIbrahimDoguwa
DepartmentofStatistics,AhmaduBelloUniversity,Zaria,Nigeria
Howtocitethispap er:Goltong,N.E.
andDoguwa,S.I.(2019)OnAnalysisofthe
Behrens-FisherProblemBasedonBayesian
Evidence.OpenJournalofStatistics,9,1-14.
https://doi.org/10.4236/ojs.2019.91001
Received:December17,2018
Accepted:January18,2019
Published:January21,2019
Copyright
c
2019byauthor(s)andScientific
Research PublishingInc.
This work is licensed under the Creative Com-
monsAttributionInternationalLicense(CC
BY4.0).
http://creativecommons.org/licenses/by/4.0/
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turientmontes,nasceturridiculusmus.Pel-
lentesquehabitantmorbitristiquesenectuset
netusetmalesuadafamesacturpisegestas.
Phasellusidmagna.Duismalesuadainter-
dumarcu.Integermetus.Morbi pulvinarpel-
lentesquemi.Suspendissesedesteumagna
molestieegestas.Quisquemilorem,pulvinar
eget,egestasquis,luctusat,ante.Proinauc-
torvehicula purus.Fusceacnislaliquam ante
hendreritpellentesque.Classaptenttacitiso-
ciosquadlitoratorquentperconubianostra,
perinceptoshymenaeos.Morbiwisi.Etiam
arcumauris,facilisissed,eleifendnon,non-
ummyut,pede.Crasutlacustempormetus
mollisplacerat.Vivamuseutortorvelmetus
interdum malesuada.
Sedeleifend,erossitametfaucibuselemen-
tum,urnasapienconsectetuermauris,quis
egestasleojustononrisus.Morbinonfelisac
liberovulputatefringilla.Maurisliberoeros,
lacinianon,sodalesquis,dapibusporttitor,
pede.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Morbi dapibus mauris condimentum
nulla.Cum sociisnatoque penatibus etmagnis
disparturientmontes,nasceturridiculusmus.
Etiamsitameterat.Nullavarius.Etiamt-
inciduntduivitaeturpis.Donecleo.Morbi
vulputateconvallisest.Integeraliquet.Pel-
lentesque aliquetsodalesurna.
Nullameleifendjustoinnisl.Inhachabitasse
plateadictumst.Morbinonummy.Aliquam
utfelis.Invelitleo,dictumvitae,posuereid,
vulputatenec,ante.Maecenasvitaepedenec
dui dignissim suscipit.Morbimagna.Vestibu-
lumidpurusegetvelitlaoreetlaoreet.Prae-
sentsedleovelnibhconvallisblandit.Ut
rutrum.Donecnibh.Donecinterdum.Fusce
sed pede sit amet elitrhoncus ultrices.Nullam
atenimvitaepedevehicula iaculis.
Classaptenttacitisociosquadlitoratorquen-
tperconubia nostra,perinceptoshymenaeos.
Aeneannonummyturpisidodio.Integereu-
ismodimperdietturpis.Utnecleonecdi-
amimperdietlacinia.Etiamegetlacuseget
miultriciesposuere.Inplacerattristiquetor-
tor.Sedportavestibulummetus.Nullaia-
culissollicitudinpede.Fusceluctustellusin
dolor.Curabiturauctorvelitasem.Mor-
bisapien.Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Donecadipiscingurnavehic-
ulanunc.Sedornareleoinleo.Inrhoncus
leoutdui.Aeneandolorquam,volutpatnec,
fringillaid,consectetuervel,pede.
Nullamalesuadarisusuturna.Aenean
pretiumvelitsitametmetus.Duisiaculis.In
hachabitasseplateadictumst.Nullammo-
lestieturpisegetnisl.Duisamassaidpede
dapibusultricies.Sedeuleo.Inatmaurissit
amettortorbibendumvarius.Phasellusjusto
risus,posuerein,sagittisac,variusvel,tor-
tor.Quisqueidenim.Phasellusconsequat,
liberopretium nonummy fringilla,tortor lacus
vestibulumnunc,utrhoncusligulanequeid
justo.Nullamaccumsaneuismodnunc.Proin
vitae ipsumac metusdictum tempus.Namut
wisi.Quisque tortor felis, interdum ac, sodales
a,sempera,sem.Curabiturinvelitsitamet
duitristiquesodales.Vivamusmaurispede,
laciniaeget,pellentesquequis,scelerisqueeu,
est.Aliquamrisus.Quisquebibendumpede
eudolor.
Donec tempusneque vitaeest.Aenean egestas
odio sed risus ullamcorper ullamcorper.Sed in
nullaatortortinciduntegestas.Namsapien
tortor,elementumsitamet,aliquamin,port-
titorfaucibus,enim.Nullamconguesuscipit
nibh.Quisqueconvallis.Praesentarcunibh,
vehicula eget,accumsaneu,tincidunta,nibh.
Suspendissevulputate,tortorquisadipiscing
viverra, lacus nibh dignissim tellus, eu suscipit
risusantefringilladiam.Quisquealiberov-
elpedeimperdietaliquet.Pellentesquenunc
nibh,eleifenda,consequatconsequat,hen-
dreritnec,diam.Sedurna.Maecenaslaoreet
eleifendneque.Vivamuspurusodio,eleifend
non,iaculisa,ultricessitamet,urna.Mau-
risfaucibusodiovitaerisus.Innisl.Praesent
purus.Integeriaculis,semeuegestaslacinia,
lacuspedescelerisqueaugue,inullamcorper
dolorerosaclacus.Nuncinlibero.
Fuscesuscipitcursussem.Vivamusrisusmi,
egestasac,imperdietvarius,faucibusquis,
leo.Aeneantincidunt.Donecsuscipit.Cras
idjustoquisnibhscelerisquedignissim.Ali-
quamsagittiselementumdolor.Aeneancon-
sectetuerjustoinpede.Curabiturullamcorper
ligula nec orci.Aliquam purus turpis, aliquam
id, ornarevitae, porttitor non,wisi.Maecenas
luctusportalorem.Donecvitae ligulaeu ante
pretiumvarius.Prointortormetus,convallis
et,hendreritnon,scelerisquein,urna.Cras
quisliberoeuligulabibendumtempor.Viva-
mustellusquam,malesuadaeu,tempussed,
temporsed,velit.Doneclacinia auctorlibero.
Praesentsednequeidpedemollisrutrum.
Vestibulumiaculisrisus.Pellentesquelacus.
Ut quisnunc sedodio malesuada egestas.Duis
amagnasitametligulatristiquepretium.Ut
pharetra.Vestibulumimperdietmagnanec
wisi.Maurisconvallis.Sedaccumsansollic-
itudinmassa.Sedidenim.Nuncpedeenim,
laciniaut,pulvinarquis,suscipitsemper,elit.
Crasaccumsaneratvitaeenim.Crassollici-
tudin.Vestibulumrutrumblanditmassa.
Sedgravidalectusutpurus.Morbilaoreet
magna.Pellentesque eu wisi.Proin turpis.In-
tegersollicitudinauguenecdui.Fuscelectus.
Vivamusfaucibusnullaneclacus.Integerdi-
am.Pellentesquesodales,enimfeugiatcursus
volutpat,semmaurisdignissimmauris,quis
consequatsemestfermentumligula.Nullam
justo lectus, condimentum sit amet, posuere a,
fringillamollis, felis.Morbi nulla nibh,pellen-
tesqueat,nonummyeu,sollicitudinnec,ip-
sum.Crasneque.Nuncaugue.Nullamvitae
quamidquampulvinar blandit.Nunc sitamet
orci.Aliquameratelit,pharetranec,aliquet
a,gravidain,mi.Quisqueurnaenim,viverra
quis,suscipitquis,tinciduntut,sapien.Cras
placeratconsequatsem.Curabituracdiam.
Curabiturdiamtortor,molliset,viverraac,
tempusvel, metus.
Curabituraclorem.Vivamusnonjustoin
duimattisposuere.Etiamaccumsanligula
idpede.Maecenastinciduntdiamnecvelit.
Praesentconvallissapienacest.Aliquamul-
lamcorpereuismodnulla.Integermollisenim
veltortor.Nullasodalesplaceratnunc.Sed
tempusrutrumwisi.Duisaccumsangravi-
dapurus.Nuncnunc.Etiamfacilisisduieu
sem.Vestibulumsemper.Praesenteueros.
Vestibulumtellusnisl,dapibusid,vestibulum
sit amet,placerat ac, mauris.Maecenas et elit
uteratplaceratdictum.Namfeugiat,turpis
etsodalesvolutpat, wisiquamrhoncusneque,
vitaealiquamipsumsapienvelenim.Maece-
nassuscipitcursusmi.
Quisque consectetuer.In suscipit mauris a do-
lor pellentesque consectetuer.Mauris convallis
nequenonerat.Inlacinia.Pellentesqueleo
eros,sagittisquis,fermentumquis,tincidun-
tut,sapien.Maecenassem.Curabitureros
odio,interdumeu,feugiateu,portaac,nis-
l.Curabiturnunc.Etiam fermentum convallis
velit.Pellentesquelaoreetlacus.Quisquesed
elit.Namquistellus.Aliquamtellusarcu,
adipiscingnon,tincidunteleifend,adipiscing
quis,augue.Vivamuselementumplaceraten-
im.Suspendisseuttortor.Integerfaucibus
adipiscingfelis.Aeneanconsectetuermattis
lectus.Morbimalesuadafaucibusdolor.Nam
lacus.Etiamarcu libero, malesuadavitae, ali-
quamvitae,blandittristique,nisl.
Maecenasaccumsandapibussapien.Duis
pretiumiaculisarcu.Curabiturutlacus.Ali-
quamvulputate.Suspendisseutpurussedsem
tempor rhoncus.Utquam dui, fringillaat, dic-
tum eget,ultricies quis, quam.Etiam semest,
pharetra non, vulputate in, pretium at, ipsum.
Nunc semper sagittis orci.Sedscelerisque sus-
cipitdiam.Utvolutpat,doloratullamcorp-
ertristique,erospurusmollisquam,sitamet
ornareante nuncetenim.
Phasellusfringilla,metusidfeugiatcon-
sectetuer,lacuswisi ultricestellus,quislobor-
tisnibhloremquistortor.Donecegestas
ornarenulla.Maurismitellus,portafau-
cibus,dictumvel,nonummyin,est.Ali-
quameratvolutpat.Intellusmagna,port-
titorlacinia,molestievitae,pellentesqueeu,
justo.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Sedorcinibh,scelerisquesitamet,
suscipitsed,placeratvel,diam.Vestibulum
nonummy vulputate orci.Donec et velit ac ar-
cuinterdumsemper.Morbipedeorci,cursus
ac,elementumnon,vehiculaut,lacus.Cras
volutpat.Namvelwisiquisliberovenenatis
placerat.Aeneansedodio.Quisqueposuere
purusacorci.Vivamusodio.Vivamusvar-
ius,nullasitametsemperviverra,odiomau-
risconsequatlacus,atvestibulumnequearcu
eutortor.Doneciaculistincidunttellus.Ali-
quameratvolutpat.Curabiturmagnalorem,
dignissimvolutpat,viverraet,adipiscingnec,
dolor.Praesentlacusmauris,dapibusvitae,
sollicitudinsitamet,nonummyeget,ligula.
Crasegestasipsum anisl.Vivamusvariusdo-
lorutdolor.Fuscevelenim.Pellentesque
accumsanligulaeteros.Crasidlacusnon
tortorfacilisisfacilisis.Etiamnislelit,cur-
sussed,fringillain,conguenec,urna.Cum
sociisnatoquepenatibusetmagnisdispar-
turientmontes,nasceturridiculusmus.In-
tegeratturpis.Cumsociisnatoquepenati-
busetmagnisdisparturientmontes,nascetur
ridiculusmus.Duisfringilla,ligulasedporta
fringilla,ligulawisi commodofelis,utadipisc-
ingfelisduiinenim.Suspendissemalesuada
ultricesante.Pellentesquescelerisqueaugue
sitameturna.Nullavolutpataliquettortor.
Crasaliquam,tellusataliquetpellentesque,
justosapiencommodoleo,idrhoncussapien
quamaterat.Nullacommodo,wisiegetsol-
licitudinpretium,orciorcialiquamorci,ut
cursusturpisjustoetlacus.Nullaveltor-
tor.Quisque eratelit, viverrasit amet, sagittis
eget,portasitamet,lacus.
In hac habitasse platea dictumst.Proin at est.
Curabitur tempusvulputate elit.Pellentesque
sem.Praesent eusapien.Duiselit magna,ali-
quet at,tempus sed, vehicula non, enim.Mor-
biviverraarcunecpurus.Vivamusfringilla,
enimet commodomalesuada, tortormetusel-
ementumligula,necaliquetestsapienutlec-
tus.Aliquammi.Utnecelit.Fusceeuismod
luctustellus.Curabiturscelerisque.Nullam
purus.Namultricies accumsanmagna.Morbi
pulvinarloremsitametipsum.Donecutjus-
tovitaenibhmolliscongue.Fuscequisdiam.
Praesent tempuserosutquam.
Donecinnisl.Fuscevitaeest.Vivamusante
ante,mattislaoreet,posuereeget,conguevel,
nunc.Fuscesem.Namvelorcieuerosviver-
raluctus.Pellentesquesitametaugue.Nunc
sitametipsumetlacusvariusnonummy.In-
teger rutrum sem egetwisi.Aenean eu sapien.
Quisqueornaredignissimmi.Duisaurnavel
risus pharetraimperdiet.Suspendissepotenti.
Morbijusto.Aeneannecdolor.Inhac
habitasseplateadictumst.Proinnonummy
porttitorvelit.Sedsitametleonecmetus
rhoncusvarius.Crasante.Vestibulumcom-
modosemtinciduntmassa.Namjusto.Ae-
neanluctus,felisetcondimentumlacinia,lec-
tusenimpulvinarpurus,nonportavelitnisl
sed eros.Suspendisse consequat.Mauris a dui
et tortormattis pretium.Sed nulla metus,vo-
lutpatid, aliquameget, ullamcorperut, ipsum.
Morbieununc.Praesentpretium.Duisali-
quampulvinarligula.Ut blanditegestasjusto.
Quisqueposueremetusviverrapede.
Vivamussodaleselementumneque.Viva-
musdignissimaccumsanneque.Sedatenim.
Vestibulumnonummyinterdumpurus.Mau-
ris ornarevelitid nibh pretiumultricies.Fusce
temporpellentesqueodio.Vivamus auguepu-
rus,laoreetin,scelerisquevel,commodoid,
wisi.Duisenim.Nullainterdum,nunceusem-
pereleifend,enimdolorpretiumelit,utcom-
modoligulanislaest.Vivamusante.Nulla
leomassa,posuerenec,volutpatvitae,rhon-
cuseu,magna.
Quisquefacilisisauctorsapien.Pellentesque
gravidahendreritlectus.Maurisrutrumso-
dalessapien.Fuscehendreritsemvellorem.
Integerpellentesquemassavelaugue.Integer
elit tortor, feugiat quis, sagittis et, ornare non,
lacus.Vestibulumposuerepellentesqueeros.
Quisquevenenatisipsumdictumnulla.Ali-
quam quis quam nonmetus eleifend interdum.
Nam eget sapien ac mauris malesuadaadipisc-
ing.Etiameleifendnequesedquam.Nulla
facilisi.Proinaligula.Sedidduieunibh
egestastincidunt.Suspendissearcu.
Maecenasdui.Aliquamvolutpatauctorlorem.
Crasplaceratestvitaelectus.Curabitur
massalectus,rutrumeuismod,dignissimut,
dapibus a, odio.Ut eros erat, vulputate ut, in-
terdumnon,portaeu,erat.Crasfermentum,
felis inporta congue,velit leofacilisis odio,vi-
taeconsectetuerloremquamvitaeorci.Sed
ultrices,pedeeuplaceratauctor,anteligula
rutrum tellus,velposuere nibhlacus nec nibh.
Maecenaslaoreetdoloratenim.Donecmo-
lestie dolornec metus.Vestibulum libero.Sed
quiserat.Sedtristique.Duispedeleo,fer-
mentumquis,consectetuereget,vulputatesit
amet,erat.
Donecvitaevelit.Suspendisseportafermen-
tummauris.Utvelnuncnonmaurisphare-
travarius.Duisconsequatliberoquisurna.
Maecenasatante.Vivamusvarius,wisised
egestastristique,odiowisiluctusnulla,lobor-
tisdictumdolorligulainlacus.Vivamusali-
quam, urna sed interdum porttitor,metus orci
interdum odio,sit ameteuismodlectus feliset
leo.Praesentacwisi.Namsuscipitvestibu-
lum sem.Praesent eu ipsumvitae pedecursus
venenatis.Duis sed odio.Vestibulum eleifend.
Nullautmassa.Proinrutrummattissapien.
Curabiturdictumgravidaante.
Phasellusplaceratvulputatequam.Maece-
nasattellus.Pellentesquenequediam,dig-
nissim ac, venenatisvitae, consequat ut, lacus.
Nam nibh.Vestibulum fringilla arcu mollis ar-
cu.Sedetturpis.Donecsemtellus,volut-
pat et,varius eu,commodosed, lectus.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Quisqueenimarcu,suscipitnec,tempus
at,imperdietvel,metus.Morbivolutpatpu-
rusaterat.Donecdignissim,semidsemper
tempus, nibh massa eleifendturpis,sedpellen-
tesquewisipurussedlibero.Nullamlobortis
tortorvelrisus.Pellentesqueconsequatnulla
eutellus.Donecvelit.Aliquamfermentum,
wisiacrhoncusiaculis,tellusnuncmalesuada
orci,quisvolutpatduimagnaidmi.Nuncvel
ante.Duisvitaelacus.Crasnecipsum.
Morbinunc.Aliquamconsectetuer varius nul-
la.Phaselluseros.Crasdapibusporttitorris-
us.Maecenasultricesmiseddiam.Praesent
gravida velit at elit vehicula porttitor.Phasel-
lusnislmi, sagittisac,pulvinarid,gravidasit
amet,erat.Vestibulumest.Loremipsumdo-
lorsit amet,consectetueradipiscing elit.Cur-
abitur idsem elementum leo rutrumhendrerit.
Utatmi.Donectinciduntfaucibusmassa.
Sed turpis quam, sollicitudin a, hendrerit eget,
pretiumut,nisl.Duishendreritligula.Nunc
pulvinarcongueurna.
Nuncvelit.Nullamelitsapien,eleifendeu,
commodo nec, semper sit amet, elit.Nulla lec-
tus risus,condimentum ut,laoreet eget,viver-
ra nec, odio.Proin lobortis.Curabitur dictum
arcuvelwisi.Crasidnullavenenatistortor
congueultrices.Pellentesqueegetpede.Sed
eleifendsagittiselit.Namsedtellussitamet
lectusullamcorper tristique.Maurisenim sem,
tristiqueeu,accumsanat,scelerisquevulpu-
tate, neque.Quisquelacus.Donec etipsum sit
amet elitnonummy aliquet.Sed viverranisl at
sem.Namdiam.Maurisutdolor.Curabitur
ornaretortorcursusvelit.
Morbitinciduntposuerearcu.Crasvenenatis
estvitaedolor.Vivamusscelerisquesemper
mi.Donecipsumarcu,consequatscelerisque,
viverraid,dictumat,metus.Loremipsum
dolor sit amet, consectetuer adipiscing elit.Ut
pede sem, tempus ut, porttitor bibendum, mo-
lestieeu,elit.Suspendissepotenti.Sedidlec-
tus sitamet purus faucibus vehicula.Praesen-
tsedsemnonduipharetrainterdum.Nam
viverra ultricesmagna.
Aeneanlaoreetaliquamorci.Nuncinterdum
elementumurna.Quisqueerat.Nullamtem-
porneque.Maecenasvelitnibh,scelerisquea,
consequatut,viverrain,enim.Duismagna.
Donecodioneque,tristiqueet,tincidunteu,
rhoncusac, nunc.Maurismalesuada malesua-
da elit.Etiam lacus mauris, pretium vel, blan-
ditin, ultriciesid,libero.Phasellusbibendum
eratutdiam.Incongueimperdietlectus.
Aeneanscelerisque.Fuscepretiumporttitor
lorem.Inhachabitasseplatea dictumst.Nulla
sitametnislatsapienegestaspretium.Nunc
nontellus.Vivamusaliquet.Namadipiscing
euismoddolor.Aliquameratvolutpat.Nul-
lautipsum.Quisquetinciduntauctorau-
gue.Nuncimperdietipsumegetelit.Ali-
quamquamleo,consectetuernon,ornaresit
amet,tristique quis,felis.Vestibulumanteip-
sumprimisinfaucibusorciluctusetultrices
posuerecubiliaCurae;Pellentesqueinterdum
quamsitametmi.Pellentesquemaurisdui,
dictuma,adipiscingac,fermentumsitamet,
lorem.
Utquiswisi.Praesentquismassa.Vivamus
egestasrisusegetlacus.Nunctincidunt,risus
quisbibendumfacilisis,lorempurusrutrum
neque,necportatortorurnaquisorci.Ae-
neanaliquet,liberosempervolutpatluctus,
pedeeratlaciniaaugue,quisrutrumsemip-
sumsitametpede.Vestibulumaliquet,nibh
sediaculissagittis,odiodolorblanditaugue,
egetmollisurnatellusidtellus.Aeneanali-
quetaliquamnunc.Nullaultriciesjustoeget
orci.Phasellustristiquefermentumleo.Sed
massametus,sagittisut,semperut,pharetra
vel,erat.Aliquamquamturpis,egestasvel,
elementumin,egestassitamet,lorem.Duis
convallis,wisisitametmollismolestie,libero
mauris portadui, vitae aliquamarcu turpisac
sem.Aliquamaliquetdapibusmetus.
Vivamuscommodoeroseleifenddui.Vestibu-
luminleoeuerattristiquemattis.Crasat
elit.Craspellentesque.Nullamidlacussit
ametliberoaliquethendrerit.Proinplacerat,
minonelementumlaoreet,eroselittincidun-
tmagna,arhoncussemarcuidodio.Nul-
laegetleoaleoegestasfacilisis.Curabitur
quisvelit.Phasellusaliquam,tortornecornare
rhoncus,purusurnaposuerevelit,etcommo-
dorisustellusquistellus.Vivamusleoturpis,
tempussitamet,tristiquevitae,laoreetquis,
odio.Proinscelerisquebibendumipsum.Eti-
amnisl.Praesentveldolor.Pellentesquevel
magna.Curabitururna.Vivamuscongueur-
na in velit.Etiam ullamcorper elementum dui.
Praesentnon urna.Sed placeratquam non mi.
Pellentesquediammagna,ultricieseget,ultri-
cesplacerat,adipiscingrutrum,sem.
Morbisem.Nullafacilisi.Vestibulumante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Nullafacilisi.Mor-
bisagittisultriceslibero.Praesenteuligula
sedsapienauctorsagittis.Classaptenttaciti
sociosquadlitoratorquentperconubianos-
tra,perinceptoshymenaeos.Donecvelnunc.
Nuncfermentum,lacusidaliquamporta,dui
tortoreuismoderos,velmolestieipsumpurus
eulacus.Vivamuspedearcu, euismod ac,tem-
pusid,pretiumet,lacus.Curabitursodales
dapibusurna.Nunceusapien.Doneceget
nuncapededictumpretium.Proinmauris.
Vivamusluctusliberovelnibh.
Fuscetristiquerisusidwisi.Integermolestie
massaidsem.Vestibulumveldolor.Pel-
lentesquevelurnavelrisusultricieselemen-
tum.Quisquesapien urna, blanditnec, iaculis
ac,viverrain,odio.Inhachabitasseplatea
dictumst.Morbinequelacus,convallisvitae,
commodoac,fermentumeu,velit.Sedinor-
ci.Infringillaturpisnon arcu.Donecin ante.
Phasellustemporfeugiatvelit.Aeneanvar-
iusmassanonturpis.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;
Aliquamtortor.Morbi ipsummassa,imperdi-
etnon,consectetuervel,feugiatvel,lorem.
Quisqueeget loremnec elitmalesuadavestibu-
lum.Quisquesollicitudinipsum vel sem.Nul-
laenim.Proinnonummy felisvitaefelis.Nul-
lampellentesque.Duisrutrumfeugiatfe-
lis.Maurisvelpedesedliberotinciduntmol-
lis.Phasellussedurnarhoncusdiameuismod
bibendum.Phasellussednisl.Integercondi-
mentumjusto id orciiaculis varius.Quisque et
lacus.Phaselluselementum, justoat dignissim
auctor,wisi odio lobortisarcu, sedsollicitudin
felisfeliseuneque.Praesentatlacus.
Vivamussitametpede.Duisinterdum,nunc
egetrutrumdignissim,nisldiamluctusleo,
ettinciduntvelitnislidtellus.Inloremtel-
lus,aliquetvitae,portain,aliquetsed,lectus.
Phasellussodales.Utvariusscelerisqueerat.
Invelnibheuerosimperdietrutrum.Donec
ac odio nec neque vulputatesuscipit.Nam nec
magna.Pellentesquehabitantmorbitristique
senectus et netus et malesuada fames acturpis
egestas.Nullamporta,odioetsagittisiaculis,
wisi neque fringilla sapien, vel commodo lorem
loremidelit.Utsemlectus,scelerisqueeget,
placeratet,tinciduntscelerisque,ligula.Pel-
lentesque nonorci.
Etiamvelipsum.Morbifacilisisvestibulum
nisl.Praesentcursuslaoreetfelis.Integer
adipiscingpretiumorci.Nullafacilisi.Quisque
posuerebibendumpurus.Nulla quammauris,
cursuseget,convallisac,molestienon,enim.
Aliquamcongue.Quisquesagittisnonummy
sapien.Proinmolestiesemvitaeurna.Mae-
cenas lorem.Vivamus viverra consequat enim.
Nuncsedpede.Praesentvitaelectus.Prae-
sentnequejusto,vehiculaeget,interdumid,
facilisiset,nibh.Phasellusatpurusetlibero
laciniadictum.Fuscealiquet.Nullaeuante
placeratleosemperdictum.Maurismetus.
Curabiturlobortis.Curabitursollicitudin hen-
dreritnunc.Donecultriceslacusidipsum.
Donecanibhutelitvestibulumtristique.In-
tegeratpede.Crasvolutpatvariusmagna.
Phaselluseuwisi.Praesentrisusjusto,lobor-
tis eget, scelerisqueac, aliquet in, dolor.Proin
idleo.Nunciaculis,mivitaeaccumsancom-
modo, nequesemlacinianulla,quisvestibulum
justosemineros.Quisquesedmassa.Morbi
lectus ipsum, vulputate a, mollis ut,accumsan
placerat,tellus.Nullaminwisi.Vivamuseu
ligulaanuncaccumsancongue.Suspendisse
aclibero.Aliquameratvolutpat.Donecau-
gue.Nuncvenenatisfringillanibh.Fusceac-
cumsan pulvinar justo.Nullam semper, duiut
dignissim auctor, orcilibero fringilla massa,b-
landitpulvinarpedetortoridmagna.Nunc
adipiscing justo sed velit tincidunt fermentum.
Integerplacerat.Pellentesquehabitantmorbi
tristique senectuset netus etmalesuada fames
acturpisegestas.Sedinmassa.Classaptent
tacitisociosquadlitoratorquentperconubi-
anostra,perinceptoshymenaeos.Phasellus
tempusaliquamrisus.Aliquamrutrumpurus
atmetus.Donecposuereodioaterat.Nam
nonnibh.Phasellusligula.Quisquevene-
natislectusinaugue.Sedvestibulumdapibus
neque.
Mauris tempus eros at nulla.Sed quisdui dig-
nissimmaurispretiumtincidunt.Maurisac
purus.Phasellusaclibero.Etiamdapibusia-
culisnunc.Inlectus wisi,elementumeu, sollic-
itudinnec,imperdietquis,dui.Nullaviverra
nequeaclibero.Maurisurnaleo,adipiscing
eu,ultricesnon,blanditeu,dui.Maecenas
duineque,suscipit sitamet,rutruma,laoreet
in,eros.Uteunibh.Fuscenecerattempus
urnafringilla tempus.Curabituridenim.Sed
ante.Crassodalesenimsitametwisi.Nunc
fermentum consequatquam.
Utauctor,augueportadignissimvestibulum,
arcudiamlobortisvelit,velscelerisqueris-
usauguesagittisrisus.Maecenaseujusto.
Pellentesquehabitant morbi tristiquesenectus
etnetusetmalesuadafamesacturpiseges-
tas.Mauriscongueligulaegettortor.Nul-
lam laoreeturnased enim.Donecegeteros ut
erosvolutpatconvallis.Praesentturpis.Inte-
germaurisdiam,elementumquis,egestasac,
rutrumvel,orci.Nullafacilisi.Quisqueadip-
iscing,nullavitaeelementumporta,semurna
volutpatleo,sedportaenimrisussedmassa.
Integeracenimquisdiamsodalesluctus.Ut
eget eros a ligula commodo ultricies.Donec eu
urnaviverradolorhendreritfeugiat.Aliquam
acorcivelerosconguepharetra.Quisque
rhoncus,justoeuvolutpatfaucibus,augueleo
posuerelacus,arhoncuspuruspedevelest.
Proinultricesenim.
Aenean tincidunt laoreet dui.Vestibulum ante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Integeripsumlec-
tus,fermentumac,malesuadain,eleifendut,
lorem.Vivamusipsumturpis,elementumvel,
hendrerit ut,semper at,metus.Vivamus sapi-
entortor,eleifendid,dapibusin,egestaset,
pede.Pellentesquefaucibus.Praesentlorem
neque, dignissim in,facilisis nec, hendrerit vel,
odio.Namatdiamacnequealiquetviver-
ra.Morbidapibusligulasagittismagna.In
lobortis.Donecaliquetultricieslibero.Nunc
dictumvulputate purus.Morbi varius.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Intempor.Phaselluscommodoporttitor
magna.Curabiturvehiculaodioveldolor.
Praesentfacilisis,augueaadipiscingvene-
natis,liberorisusmolestieodio,pulvinarcon-
sectetuerfeliseratacmauris.Namvestibu-
lumrhoncusquam.Sedveliturna,pharetra
eu,eleifendeu,viverraat,wisi.Maecenasul-
tricesnibhatturpis.Aeneanquam.Nulla
ipsum.Aliquamposuereluctuserat.Cur-
abiturmagnafelis,laciniaet,tristiqueid,ul-
tricesut,mauris.Suspendissefeugiat.Cras
eleifendwisivitaetortor.Phasellusleopurus,
mattissitamet,auctorin,rutrumin,magna.
Inhachabitasseplateadictumst.Phasellus
imperdietmetusinsem.Vestibulumacenim
nonsemultriciessagittis.Sedveldiam.
Integervelenimsedturpisadipiscingbiben-
dum.Vestibulumpededolor,laoreetnec,po-
suerein,nonummyin,sem.Donecimperdiet
sapienplaceraterat.Donecviverra.Aliquam
eros.Nunc consequat massaid leo.Sed ullam-
corper,loreminsodalesdapibus,risusmetus
sagittislorem,non porttitorpurusodioneco-
dio.Sedtinciduntposuereelit.Quisqueeu
enim.Donecliberorisus,feugiatac,dapibus
eget, posuere a,felis.Quisque vellectusutme-
tustincidunteleifend.Duis utpede.Duis velit
erat,venenatisvitae,vulputate a,pharetra sit
amet,est.Etiamfringillafaucibusaugue.
Aeneanvelitsem,viverraeu,tempusid,
rutrumid, mi.Nullam necnibh.Proinullam-
corper,dolorincursustristique,erosaugue
tempornibh,atgravidadiamwisiatpurus.
Donec mattis ullamcorper tellus.Phasellus vel
nulla.Praesentinterdum,erosinsodalessol-
licitudin,nuncnulla pulvinarjusto, aeuismod
erossemnecnibh.Nullamsagittisdapibus
lectus.Nullamegetipsumeutortorlobortis
sodales.Etiampurusleo,pretiumnec,feu-
giatnon,ullamcorpervel,nibh.Sedvelelit
etquamaccumsanfacilisis.Nuncleo.Sus-
pendissefaucibuslacus.
Pellentesqueinterdumsapiensednulla.Proin
tincidunt.Aliquamvolutpatestvelmassa.
Seddolorlacus,imperdietnon,ornarenon,
commodoeu,neque.Integerpretiumsem-
perjusto.Proinrisus.Nullamidquam.
Namneque.Duisvitaewisiullamcorperdi-
amcongueultricies.Quisqueligula.Mauris
vehicula.
Curabiturnuncmagna,posuereeget,vene-
natiseu,vehiculaac,velit.Aeneanornare,
massaaaccumsanpulvinar,quamlorem
laoreetpurus, eusodalesmagna risusmolestie
lorem.Nunceratvelit,hendreritquis,male-
suadaut,aliquamvitae,wisi.Sedposuere.
Suspendisseipsumarcu,scelerisquenec,ali-
quameu,molestietincidunt,justo.Phasellus
iaculis.Sedposuereloremnonipsum.Pel-
lentesquedapibus.Suspendissequamlibero,
laoreeta,tincidunteget,consequatat,est.
Nullamutlectusnonenimconsequatfacili-
sis.Maurisleo.Quisquepedeligula,auctor
vel,pellentesquevel,posuereid,turpis.Cras
ipsumsem,cursuset,facilisisut,tempuseu-
ismod,quam.Suspendissetristiquedoloreu
orci.Maurismattis.Aeneansemper.Viva-
mustortormagna,facilisisid,variusmattis,
hendreritin,justo.Integerpurus.
Vivamusadipiscing.Curabitur imperdiet tem-
pusturpis.Vivamussapiendolor,congueve-
nenatis,euismodeget,portarhoncus,magna.
Proincondimentumpretiumenim.Fusce
fringilla,liberoetvenenatisfacilisis,erosen-
imcursus arcu,vitaefacilisisodioauguevitae
orci.Aliquamvariusnibhutodio.Sedcondi-
mentumcondimentumnunc.Pellentesque
egetmassa.Pellentesquequismauris.Donec
utligulaacpedepulvinarlobortis.Pellen-
tesqueeuismod.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentelit.Utlaoreet
ornareest.Phasellusgravidavulputatenulla.
Donecsitametarcuutsemtempormalesua-
da.Praesenthendreritaugueinurna.Proin
enimante,ornarevel,consequatut,blandit
in,justo.Donecfeliselit, dignissimsed,sagit-
tisut,ullamcorpera,nulla.Aeneanpharetra
vulputateodio.
Quisqueenim.Proinvelitneque,tristiqueeu,
eleifendeget,vestibulumnec,lacus.Vivamus
odio.Duisodiourna,vehiculain,elementum
aliquam,aliquetlaoreet,tellus.Sedvelit.Sed
velmiacelitaliquetinterdum.Etiamsapi-
enneque,convalliset,aliquetvel,auctornon,
arcu.Aliquamsuscipitaliquamlectus.Proin
tinciduntmagnasedwisi.Integerblanditla-
cusutlorem.Sedluctusjustosedenim.
Morbimalesuadahendreritdui.Nuncmauris
leo, dapibus sitamet, vestibulum et, commodo
id,est.Pellentesquepurus.Pellentesquetris-
tique,nuncacpulvinaradipiscing,justoeros
consequatlectus, sitamet posuerelectusneque
velaugue.Cras consectetuerlibero aceros.Ut
egetmassa.Fuscesitametenimeleifendsem
dictum auctor.In eget risusluctus wisi conva-
llis pulvinar.Vivamus sapien risus,temporin,
viverra in, aliquetpellentesque, eros.Aliquam
euismodliberoasem.
Nunc velit augue,scelerisquedignissim, lobor-
tiset,aliquamin,risus.Ineueros.Vestibu-
lumanteipsumprimisinfaucibusorciluctus
etultricesposuerecubiliaCurae;Curabitur
vulputateelitviverraaugue.Maurisfringilla,
tortorsitametmalesuadamollis,sapienmi
dapibusodio,acimperdietligulaenimeget
nisl.Quisquevitaepedeapedealiquetsus-
cipit.Phasellustelluspede,viverravestibu-
lum,gravidaid,laoreetin,justo.Cumsoci-
isnatoquepenatibusetmagnisdisparturient
montes,nasceturridiculusmus.Integercom-
modoluctuslectus.Maurisjusto.Duisvar-
iuseros.Sedquam.Craslacuseros,rutrum
eget,variusquis,convallisiaculis,velit.Mau-
ris imperdiet, metus at tristique venenatis, pu-
rusnequepellentesquemauris,aultriceselit
lacusnectortor.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentmalesuada.Nam
lacuslectus,auctorsitamet,malesuadavel,
elementumeget,metus.Duisnequepede,fa-
cilisiseget,egestaselementum,nonummyid,
neque.
Proinnonsem.Donecnecerat.Proinlibero.
Aliquamviverraarcu.Donecvitaepurus.
Donecfelismi,semperid,scelerisqueporta,
sollicitudinsed,turpis.Nullainurna.Integer
variuswisinonelit.Etiamnecsem.Mau-
risconsequat,risusnecconguecondimentum,
ligulaligula suscipiturna,vitae portaodioer-
at quissapien.Proinluctus leo iderat.Etiam
massametus,accumsanpellentesque,sagittis
sitamet,venenatisnec,mauris.Praesentur-
na eros, ornarenec, vulputate eget, cursussed,
justo.Phasellus nec lorem.Nullam ligula ligu-
la,mollissitamet,faucibusvel,eleifendac,
dui.Aliquameratvolutpat.
Fusce vehicula, tortoret gravida porttitor, me-
tusnibhconguelorem,ut tempuspurusmau-
risapede.Integertinciduntorcisitamet
turpis.Aeneanametus.Aliquamvestibulum
lobortisfelis.Donecgravida.Sedsedurna.
Maurisetorci.Integerultricesfeugiatligu-
la.Seddignissimnibhamassa.Donecorci
dui,temporsed,tinciduntnonummy,viverra
sit amet, turpis.Quisque lobortis.Proin vene-
natistortornecwisi.Vestibulumplacerat.In
hachabitasseplateadictumst.Aliquampor-
tamiquisrisus.Donecsagittisluctusdiam.
Nam ipsumelit, imperdiet vitae,faucibus nec,
fringillaeget,leo.Etiamquisdolorinsapien
porttitorimperdiet.
Craspretium.Nullamalesuadaipsumut
libero.Suspendissegravidahendrerittellus.
Maecenas quislacus.Morbifringilla.Vestibu-
lumodioturpis,temporvitae,scelerisquea,
dictumnon,massa.Praesenteratfelis,por-
tasitamet,condimentumsitamet,placer-
atet,turpis.Praesentplaceratlacusaen-
im.Vestibulumnoneros.Utcongue.Donec
tristiquevariustortor.Pellentesquehabitant
morbi tristique senectus et netus et malesuada
famesacturpisegestas.Namdictumdictum
urna.
Phasellusvestibulumorcivelmauris.Fusce
quamleo,adipiscingac,pulvinareget,mo-
lestiesitamet,erat.Seddiam.Suspendisse
erosleo,tempuseget,dapibussitamet,tem-
puseu,arcu.Vestibulumwisimetus,dapibus
vel,luctussitamet,condimentumquis,leo.
Suspendissemolestie.Duisinante.Utso-
dales sem sit amet mauris.Suspendisse ornare
pretiumorci.Fuscetristiqueenimegetmi.
Vestibulumeroselit,gravidaac,pharetrased,
lobortisin,massa.Proinatdolor.Duisac-
cumsanaccumsanpede.Nullamblanditelit
inmagnalaciniahendrerit.Utnonummyluc-
tuseros.Fusceegettortor.
Utsitametmagna.Crasaligulaeuurna
dignissimviverra.Nullamtemporleoporta
ipsum.Praesentpurus.Nullamconsequat.
Maurisdictumsagittisdui.Vestibulumsol-
licitudinconsectetuerwisi.Insitametdiam.
Nullammalesuadapharetrarisus.Proinlacus
arcu, eleifendsed, vehiculaat, conguesit amet,
sem.Sedsagittispedeanisl.Sedtincidunt o-
dioapede.Seddui.Nameuenim.Aliquam
sagittislacusegetlibero.Pellentesquediam
sem,sagittismolestie,tristiqueet,fermentum
ornare,nibh.Nulla ettellus nonfelis imperdi-
etmattis.Aliquameratvolutpat.
Vestibulumsodalesipsumidaugue.Integer
ipsumpede,convallissitamet,tristiquevi-
tae,temporut,nunc.Namnonligulanon
lorem convallis hendrerit.Maecenas hendrerit.
Sedmagnaodio,aliquamimperdiet,portaac,
aliqueteget,mi.Cumsociisnatoquepenati-
busetmagnisdisparturientmontes,nascetur
ridiculusmus.Vestibulumnislsem,dignissim
vel,euismodquis,egestasut,orci.Nuncvi-
taerisusvelmetuseuismodlaoreet.Crassit
ametnequeaturpislobortisauctor.Sedali-
quamsemacelit.Crasvelitlectus,facilisis
id,dictumsed,portarutrum,nisl.Namhen-
dreritipsumsedaugue.Nullamscelerisque
hendreritwisi.Vivamusegestasarcusedpu-
rus.Utornarelectussederos.Suspendisse
potenti.Maurissollicitudinpedevelvelit.In
hachabitasseplateadictumst.
Suspendisseeratmauris,nonummyeget,
pretiumeget,consequatvel,justo.Pellen-
tesqueconsectetuereratsedlacus.Nullam
egestasnullaacdui.Doneccursusrhoncus
ipsum.Nuncetsemeumagnaegestasmale-
suada.Vivamusdictummassaatdolor.Mor-
biestnulla,faucibusac,posuerein,interdum
ut,sapien.Proinconsectetuerpretiumurna.
Donecsit ametnibhnec purusdignissim mat-
tis.Phasellusvehiculaelitatlacus.Nullafa-
cilisi.Crasutarcu.Sedconsectetuer.Integer
tristiqueelitquisfelisconsectetuereleifend.
Crasetlectus.
Utcongue malesuadajusto.Curabiturcongue,
felisathendreritfaucibus,maurislacusport-
titor pede, nec aliquam turpisdiam feugiat ar-
cu.Nullamrhoncusipsumatrisus.Vestibu-
lumadolorseddolorfermentumvulputate.
Sednecipsumdapibusurnabibendumlobor-
tis.Vestibulum elit.Namligula arcu, volutpat
eget,laciniaeu,lobortisac,urna.Nammol-
lisultricesnulla.Crasvulputate.Suspendisse
atrisus atmetuspulvinar malesuada.Nullam
lacus.Aliquamtempusmagna.Aliquamut
purus.Prointellus.
Vestibulumante ipsumprimisin faucibusorci
luctus et ultrices posuere cubilia Curae; Donec
scelerisquemetus.Maecenasnonmiutmetus
portahendrerit.Nuncsemper.Crasquiswisi
utloremposueretristique.Nuncvestibulum
scelerisquenulla.Suspendissepharetrasollic-
itudinante.Praesentatauguesitametante
interdum porta.Nunc bibendumaugue luctus
diam.Etiam nec sem.Sed eros turpis, facilisis
nec,vehiculavitae,aliquamsed,nulla.Cur-
abiturjustoleo,vestibulumeget,tristiqueut,
tempusat,nisl.
Nullavenenatisloremidarcu.Morbicur-
susurnaaipsum.Donecporttitor.Integer
eleifend,estnonmattismalesuada,minulla
convallismi,etauctorlectussapienutpu-
rus.Aliquamnullaaugue,pharetrasitamet,
faucibussemper,molestievel,nibh.Pellen-
tesquevestibulummagnaetmi.Sedfringilla
dolorveltellus.Nuncliberonunc,venenatis
eget,convallishendrerit,iaculiselementum,
mi.Nullamaliquam,felisetaccumsanvehic-
ula,magnajusto vehiculadiam,eu condimen-
tumnislfelisetnunc.Quisquevolutpatmauris
avelit.Pellentesquemassa.Integeratlorem.
Nammetuserat,laciniaid,convallisut,pul-
vinarnon,wisi.Crasiaculismaurisutneque.
Crassodales,semvitaeimperdietconsequat,
pede purussollicitudin urna,ac aliquammetus
orciinleo.Utmolestieultricesmauris.Viva-
musvitaesem.Aliquameratvolutpat.Prae-
sentcommodo,nislacdapibusaliquet,tortor
orci sodales lorem, non ornare nulla lorem quis
nisl.
Sedatsemvitaepurusultricesvestibulum.
Vestibulumtinciduntlacusetligula.Pel-
lentesquevitaeelit.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;Duisornare,erateget
laoreetvulputate,lacusipsumsuscipitturpis,
etbibendumnislorcinonlectus.Vestibu-
lumnecrisusnecliberofermentumfringilla.
Morbinonvelitinmagnagravidahendrerit.
Pellentesquequislectus.Vestibulumeleifend
lobortisleo.Vestibulumnonaugue.Vivamus
dictumtempordui.Maecenasatligulaidfe-
liscongueporttitor.Nullaleomagna,egestas
quis,vulputatesitamet,viverraid,velit.
Utlectuslectus,ultriciessitamet,semper
eget,laoreetnon,ante.Proinatmassaquis
nuncrhoncusmattis.Aliquamlorem.Cur-
abiturpharetraduiatneque.Aliquameutel-
lus.Aeneantempus,felisvitaevulputateia-
culis,estdolorfaucibusurna,inviverrawisi
nequenonrisus.Fusceveldolornecsapien
pretiumnonummy.Integerfaucibusmassaac
nulla ornarevenenatis.Nullaquissapien.Sed
tortor.Phasellusegetmi.Crasnunc.Crasa
enim.
Quisque nisl.In dignissim dapibus massa.Ae-
neansemmagna,scelerisquenec,ullamcorper
quis,porttitorut,lectus.Fuscedignissimfa-
cilisistortor.Vivamusgravidafelissitamet
nunc.Nampulvinarodiovelenim.Pellen-
tesquesitamet est.Vivamuspulvinar leonon
sapien.Aliquamerat volutpat.Utelementum
auctormetus.Maurisvestibulumnequevitae
eros.Pellentesquealiquamquam.Donecve-
nenatistristiquepurus.Innisl.Nullavelit
libero,fermentumat,portaa,feugiatvitae,
urna.Etiamaliquetornareipsum.Proinnon
dolor.Aeneannuncligula,venenatissuscipit,
porttitorsit amet, mattis suscipit,magna.Vi-
vamusegestasviverraest.Morbiatrisussed
sapiensodalespretium.
Morbicongueconguemetus.Aeneansedpu-
rus.Nampedemagna,tristiquenec,porta
id,sollicitudinquis,sapien.Vestibulumblan-
dit.Suspendisse ut augue ac nibhullamcorper
posuere.Integereuismod,nequeateleifend
fringilla,augueelitornaredolor,veltincidunt
purusest idlacus.Vivamusloremdui, commo-
doquis,scelerisqueeu,tinciduntnon,magna.
Cras sodales.Quisque vestibulum pulvinar di-
am.Phasellustincidunt, leovitae tristiquefa-
cilisis,ipsum wisiinterdum sem,dapibus sem-
pernulla velit vel lectus.Crasdapibusmauris
etaugue.Quisquecursusnullainlibero.Sus-
pendisseetloremsitametmaurismalesuada
mollis.Nullamidjusto.Maecenasvenenatis.
Doneclacusarcu,egestasac,fermentumcon-
sectetuer,tempuseu,metus.Proinsodales,
seminpretiumfermentum,arcusapiencom-
modomauris,venenatisconsequataugueur-
nainwisi.Quisquesapiennunc,variuseget,
condimentumquis,laciniain,est.Fuscefacil-
isis.Praesentnecipsum.
Suspendisseadolor.Namerateros,congue
eget,sagittisa,laciniain,pede.Maecenas
inelit.Proinmolestievariusnibh.Vivamus
tristiquepurussedaugue.Proinegestassem-
pertortor.Vestibulumanteipsumprimisin
faucibusorciluctusetultricesposuerecubil-
iaCurae;Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Vestibulumorcienim,sagittis
ornare,eleifendut,mattisat,ligula.Nulla
molestieconvallisarcu.Uterostellus,condi-
mentum at,sodalesin,ultricesvel,nulla.
Duismagnaante,bibendumeget,eleifend
eget,suscipitsed,neque.Vestibuluminmi
sedmassacursuscursus.Pellentesquepulv-
inarmollisneque.Fusceutenimvitaemauris
malesuadatincidunt.Vivamusaneque.Mau-
rispulvinar,sapienidcondimentumdictum,
quamarcurhoncusdui,idtemporlacusjusto
etjusto.Proinsitametorcieudiameleifend
blandit.Nunceratmassa,luctusac,fermen-
tumlacinia,tinciduntultrices,sapien.Prae-
sentsed orci vitae dolorsollicitudin adipiscing.
Crasaneque.Utrisusdui,interdumat,plac-
eratid,tristiqueeu,enim.Vestibulumante
ipsum primisin faucibus orciluctus et ultrices
posuerecubiliaCurae;Etiamadipiscingeros
vestibulumdolor.Pellentesquealiquam,diam
eget eleifendposuere,augue erosporttitorlec-
tus,acdignissimduimetusnecfelis.Quisque
lacinia.Vestibulumtellus.Suspendissenec
wisi.Aeneanacfelis.Aliquamultricesmetus
etnulla.
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ametneque.Sedcommodomalesuadalectus.
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aliquameu,quam.Aeneanquisquamquis
wisigravidavehicula.Pellentesqueamassa
aleopretiumrhoncus.Suspendisseultrices.
Doneclaciniamalesuadamassa.Classapten-
ttacitisociosquadlitoratorquentperconu-
bianostra,perinceptoshymenaeos.Donec
pretiumornaremauris.Phasellusauctorer-
ategetenim.Integerscelerisque,feliseucon-
sequatfringilla,loremwisiultriciesvelit,id
vehiculapurusnullaegetodio.Nullammat-
tis,diamarutrumfermentum,odiosapien
tristiquequam,idmollistellusquaminodi-
o.Mauriseusapien.Donecaliquamloremsit
ametlorempharetralobortis.
Donecacvelit.Sedconvallisvestibulumsapi-
en.Vivamustemporlacussedlacus.Nunc
utlorem.Utettortor.Nullamvariuswisiat
diam.Etiamultricies,dolorsitametfermen-
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elit.Quisquevenenatislobortisaugue.Sed
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pededuimalesuadamauris,velultricesurna
maurisacnibh.Etiamnibhodio,ultriciesve-
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lis.Vivamuspulvinar.Aliquam eratvolutpat.
Nullaegestasvenenatismetus.Namfeugiat
nuncquiselitegestassagittis.Sedvitaefe-
lis.Inliberoarcu,rhoncusin,commodoeget,
auctorin,enim.Vivamussuscipitest.Nul-
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massahendreritlacus,acrutrumtellustellus
sitametfelis.Crasviverra.
Suspendisseeununc.Aliquamdignissimurna
sit ametmauris.Cras commodo, urna ut port-
titor venenatis, arcu metussodales risus,vitae
gravidasapienligulainest.Donecvulputate
sollicitudinwisi.Donecvehicula,estidinter-
dumornare,nibhtellusconsectetuerjusto,a
ultrices felis erat at lectus.In est massa, male-
suadanon,suscipitat,ullamcorpereu,elit.
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tissed,temporeget,libero.Praesentligula.
Suspendissenulla.Etiamdiam.Nullaante
diam,vestibulumet,aliquetac,imperdietvi-
tae,urna.Fuscetinciduntlacusvelelit.Mae-
cenasdictum,tortornoneuismodbibendum,
pedenibh pretiumtellus,at dignissimleo eros
egetpede.Nullavenenatiseleifenderos.Ae-
nean ut odio dignissim augue rutrum faucibus.
Fusceposuere,tellusegetviverramattis,er-
attellusportami,atfacilisissemnibhnon
urna.Phasellusquisturpisquismaurissus-
cipitvulputate.Sedinterdumlacusnonvelit.
Vestibulumante ipsumprimisin faucibusorci
luctusetultricesposuerecubiliaCurae;
Vivamus vehicula leoa justo.Quisque necau-
gue.Morbimauriswisi,aliquetvitae,dignis-
simeget,sollicitudinmolestie,ligula.Indic-
tumenim sitametrisus.Curabiturvitae velit
eudiamrhoncushendrerit.Vivamusutelit.
Praesentmattisipsumquisturpis.Curabitur
rhoncusnequeeudui.Etiamvitaemagna.
Namullamcorper.Praesentinterdumbiben-
dummagna.Quisqueauctoraliquamdolor.
Morbieuloremetestporttitorfermentum.
Nuncegestasarcuattortorvariusviverra.
Fusceeu nulla utnullainterdum consectetuer.
Vestibulumgravida.Morbimattisliberosed
est.
Abstract
In thispaperwe have demonstratedthe ability ofthe newBayesian
measureofevidenceofYin(2012,ComputationalStatistics,27:
237-249)tosolveboththeBehrens-FisherproblemandLindley’s
paradox.We haveprovided a general proof that for any prior which
yieldsalinearcombinationoftwoindependenttrandomvariablesas
posteriordistributionofthedifferenceofmeans,thenewBayesian
measureofevidencegiventhatpriorwillsolveLindleys’paradox
thereby serving asa general proof for the works of Yin andLi (2014,
JournalofAppliedMathematics,2014(978691))andGoltong
andDoguwa(2018,OpenJournalofStatistics,8:902-914).
UsingtheParetopriorasanexample,wehaveshown bytheuseof
simulation results that the new Bayesian measure of evidence solves
Lindley’sparadox.
Keywords
Behrens-FisherProblem,Lindley’sParadox,Metropolis-Hastings
Algorithm,ParetoPrior,tDistribution
1.Introduction
TheBehrens-Fisherproblemsimplyreferstoahypothesistesting
problem about the differenceoftwo means based ontwo random sam-
plesthatareassumedtobeindependentandnormallydistributed
withoutassumingthatthetwovariancesareequal.
[1]notedthatconclusionsfromahypothesistestbasedona
Bayesianframeworkcouldcontradictthatfromafrequentistframe-
work.Heshowedthatasn→∞,whileafrequentistwillrejectH
0
,a
Bayesianusingthe posteriorprobabilityunderH
0
willfailto rejectH
0
.
Eventhough[2]arguedthatthesetwoschoolsofstatisticalinference
neednotagreesincethetwoevaluatedifferentobjectsusingdifferent
measures amongotherpoints, healsonoted thatit isnot meaningfulif
aprocedurewouldalwaysconcludebypickingthenullwhateverbe the
valueofthe teststatistic.Thispoints tothe factthat even if onemay
DOI: 10.4236/ojs.2019.91001Jan.21,20191OpenJournalofStatistics
N.E.Goltong,S.I.Doguwa
notbeinterestedinhavingtheBayesianandFrequentistapproaches
reconciled,thereisneedtolookoutfortestproceduresthatwillnot
alwaysfavourthenullhypothesis.Morediscussionsandarguments
concerningLindley’sparadox,canbefoundin[3],[4]and[2].[5]de-
veloped a Bayesian hypothesis testing framework that solves Lindley’s
paradoxinaonesamplecase.UsingJeffreys’independentprior,[6]
extendedthemethodologyof[5]tosolvetheBehrens-FisherProblem
soastoavoidLindleys’Paradox.Also,[7]madeanattempttosolve
theParadoxin[1]byassigningGammapriorstotheprecisionsusing
themethodologyof[5].Itisnoteworthythattheposteriordistribu-
tionofthedifferenceofmeansderivedby[6]andthatderivedby[7]
aresimilarinthesensethatbotharealinearcombinationoftwot
distributedrandomvariables.Alsothesetwoposteriordistributions
differinthesensethat,thecoefficientsoftheselinearcombinations
arenotthesameandalso,thedistributionsofthetrandomvariables
differintheirdegreesoffreedom.Inthispaper,weproposetogive
ageneralresultofthemethodologyof[5]insolvingsimultaneously,
theBehrens-FisherproblemandLindley’sparadoxwhenanygiven
priorisassignedtotheunknownvarianceswhichyieldsaposterior
distribution thatisa linear combination of two independent trandom
variables.
2.LiteratureReview
InanattempttosolvetheBehrens-FisherProblem,[8]derivedgeneral-
izedpivotal quantitiesandgeneralizedp-valuesforhypothesistesting
inthepresenceofnuisanceparameters.Thegeneralizedp-valueis
givenby
p(x) = Pr
"
F
1,n
1
+n
2
2
x
1
¯x
2
)
2
(n
1
+n
2
2)
˜s
2
1
B
+
˜s
2
2
1B
1
#
(1)
wherePr[.]meanstheprobabilityof [.],s
2
i
isthevarianceofsampleiof
size n
i
s
2
i
= (1n
1
i
)s
2
i
,i= 1,2 andBBeta((n
1
1)/2,(n
2
1)/2).
[9]derivedageneralrecipeforconstructinggeneralizedpivotalquan-
tities and generalizedconfidence intervals using a systematicapproach
whichwasnotprovidedby [8].[10] usedageneralizedp-value approach
to derive tests and confidenceintervals for thecommon mean basedon
severalwellknownestimatesofthecommonmean.[11]usedStein’s
two-stagesamplingschemetosolvetheBehrens-Fisherproblemsoas
tocontroltheType-IIerrorandthelengthoftheconfidenceinterval
givenaspecifiedType-Ierror.[12]conductedacomparativestudyof
thedifferentmethodsofsolvingtheBehrens-Fisherproblem.
Theideaofusinginformativepriorsisadvocatedforby[13],who
disagreesthatdiffusenessofapriorreflectsignoranceaboutthedis-
tributionofsuchaparameter.[14],[15],and[16]investigatedthe
reconcilability of theBayesian and frequentist approaches to hypothe-
sis testing.[17]developed theposteriorpredictive p-value which takes
careofnuisanceparametersinhypothesistesting,givenas
ppp(x) =P{F
1,n
1
+n
2
x
1
¯x
2
)
2
(n
1
+n
2
)
s
2
1
+(¯x
1
µ)
2
]B
1
n
1
,n
2
+[˜s
2
2
+(¯x
2
µ)
2
](1B
n
1
,n
2
)
1
(2)
DOI: 10.4236/ojs.2019.910012OpenJournalofStatistics
N.E.Goltong,S.I.Doguwa
whereP{.}istheprobabilityof{.},s
2
i
isthevarianceofsampleiof
sizen
i
s
2
i
= (1n
1
i
)s
2
i
,i= 1,2andB
n
1
,n
2
Beta(n
1
/2,n
2
/2).
[5]developedaBayesiantestingprocedurefortestingaprecise
nullhypothesisintheonesamplecasethatavoidsthedichotomyof
theparameterspace,therebysolvingLindley’sparadox.Thisnew
Bayesianmeasureofevidenceisgivenby
P
B
(x) = P(|θE(θ|x)|≥|θ
0
E(θ|x)||x)(3)
where P(.) is the probability of (.), E(θ|x) is the posterior expectation
of θunder thepriorπ(θ),where asmallervalueof P
B
(x) corresponds
toabiggerdistancebetweenθ
0
andthetrueθandtherefore,sug-
gestingstrongerevidenceagainstthenullhypothesisH
0
.[18]showed
bothanalyticallyandbysimulationresultsthattheinfimumofthe
posterior probability of H
0
, Pr(H
0
|x) underthe conjugate classof pri-
orsisstrictlylessthanthethep-value andtheGeneralizedp-value in
testinganormalmeanandintestingtwonormalmeansrespectively.
InordertosolvetheBehrens-Fisherproblem,[6]extendedthe
methodologyof[5]assigningJeffreys’objectivepriortotheunknown
variancesgivenas
π(µ
1
2
2
1
2
2
)
1
σ
2
1
σ
2
2
(4)
and obtained the posterior distribution of the difference of means, θ|x,
as
θ|x¯x
1
¯x
2
S
1
T
n
1
1
n
1
S
2
T
n
2
1
n
2
(5)
therebyobtaining underJeffreysindependentprior, theBayesian mea-
sureofevidenceas
P
BF
J
(x) = P
S
1
T
n
1
1
n
1
S
2
T
n
2
1
n
2
≥|¯x
1
¯x
2
|
!
(6)
whereT
n
i
1
isatrandomvariablewithn
i
1degreesoffreedom.
By theuse ofsimulation results,this approach was shown tosolve the
twoproblemsandalso,itwasshownmathematicallytoyieldcredible
intervalsthatactuallypossess1αcoverageprobability.
[7]alsoextendedthemethodologyof[5]byassigningGammapri-
orstotheprecisionsandshowedmathematicallythattheparadox
in[1]isavoidedwhilesolvingtheBehrens-Fisherproblem.Theyal-
soshowedthatresultsobtainedbyassigningGammapriorstothe
precisionscouldcoincidewithresultsobtainedbyassigningJeffrey’s
independentpriortothevariancesasisdonein[6].[19]provedthat
Lindley’s Paradox could be reversed given that theprobabilitydensity
ispersistentlyunboundedorwheresufficientregularityoflikelihood
orpriorisabsent.
3.MainResult
Inthissection,weproposeamoregeneralresultabouttheability
oftheBayesianmeasureofevidenceof[5]tosolveLindley’sparadox
givenanypriordistribution.Beforeproceedingtothisgeneralresult,
letusconsideradefinitionthatiscrucialtoitspostulation.
Definition1.Arationalfunction,B,issaidtobeasymptotically
independentofnifBcanbeexpressedinthefollowingmanner:
B=
P(n)
Q(n)
B
(7)
DOI: 10.4236/ojs.2019.910013OpenJournalofStatistics
N.E.Goltong,S.I.Doguwa
where P(n)and Q(n) are polynomials in n,andQ(n) is a polynomial of
degreeatleastequaltothatofP(n),andB
iscompletelyindependent
ofn.
Lemma1.Lettheposteriordistributionofθgivenanypriordistri-
bution,sayπ,begivenby
θ|x¯x
1
¯x
2
(A
1
T
1
A
2
T
2
)(8)
wherethecoefficientsA
1
andA
2
arefunctionsofn
p
1
andn
q
2
respec-
tively,suchthatthesecoefficientsareoftheformA
1
=B
1
n
p
1
and
A
2
= B
2
n
q
2
foranyB
1
eitherindependentorasymptoticallyindepen-
dentofn
1
,anyB
2
eitherindependentorasymptoticallyindependent
ofn
2
andforanyp,q>0.Then,whereT
1
andT
2
areindependent
trandomvariables,ifthedegreesoffreedomofT
1
andT
2
arelinear
functionsofn
1
andn
2
respectively,theBayesianmeasureofevidence
basedon(8)solvesLindley’sparadox,thatis
P
BF
π
(x) 0as(n
1
,n
2
) (,)
Proof.Recall that the methodology of [5] yields the Bayesian measure
ofevidenceas
P
B
(x) = P(|θE(θ|x)|≥|θ
0
E(θ|x)||x)(9)
Sincethedegreesoffreedomofthetworandomvariablesarelinear
formsofthecorrespondingsamplesizes,letk
1
n
1
+c
1
bethedegrees
offreedomofT
1
,wherek
1
isanyconstantandc
1
isindependentof
n
1
.Also,letk
2
n
2
+ c
2
bethedegreesoffreedomofT
2
,wherek
2
is
anyconstantandc
2
isindependentofn
2
.
ThenwhereB
1
andB
2
arecompletelyindependentofn
1
andn
2
respectively,wehavethat
Var(θ|x)
=
B
2
1
n
2p
1
(k
1
n
1
+c
1
)
(k
1
n
1
+c
1
2)
+
B
2
2
n
2q
2
(k
2
n
2
+c
2
)
(k
2
n
2
+c
2
2)
=
B
2
1
n
2q
2
(k
1
n
1
+c
1
)(k
2
n
2
+c
2
)+B
2
2
n
2p
1
(k
1
n
1
+c
1
)(k
2
n
2
+c
2
)
n
2p
1
n
2q
2
(k
1
n
1
+c
1
)(k
2
n
2
+c
2
)
(10)
wherec
1
= c
1
2andc
2
= c
2
2.Notethat(9) canbeexpressedas
P
BF
π
(x) = 2
1P
χ
2
1
<Z
2
0

(11)
andwealsohavethat
Z
2
0
=
[θ
0
E(θ|x)]
2
Var(θ|x)
(12)
Now,leta=[θ
0
E(θ|x)]
2
,independentofn
1
andn
2
,thenwehave
that
Z
2
0
=
an
2p
1
n
2q
2
(k
1
n
1
+c
1
)(k
2
n
2
+c
2
)
B
2
1
n
2q
2
(k
1
n
1
+c
1
)(k
2
n
2
+c
2
)+B
2
2
n
2p
1
(k
1
n
1
+c
1
)(k
2
n
2
+c
2
)
(13)
DOI: 10.4236/ojs.2019.910014OpenJournalofStatistics
N.E.Goltong,S.I.Doguwa
lim
n
1
→∞
Z
2
0
=lim
n
1
→∞
(
an
2p
1
n
2q
2
(k
1
n
1
+c
1
)(k
2
n
2
+c
2
)
B
2
1
n
2q
2
(k
1
n
1
+c
1
)(k
2
n
2
+c
2
)+B
2
2
n
2p
1
(k
1
n
1
+c
1
)(k
2
n
2
+c
2
)
)
=lim
n
1
→∞
an
2q
2
k
1
+
c
1
n
1
(k
2
n
2
+c
2
)
B
2
1
n
2p
1
n
2q
2
k
1
+
c
1
n
1
(k
2
n
2
+c
2
)+B
2
2
k
1
+
c
1
n
1
(k
2
n
2
+c
2
)
=
an
2q
2
(k
2
n
2
+c
2
)
B
2
2
(k
2
n
2
+c
2
)
(14)
Thenwehavethat
lim
n
2
→∞
lim
n
1
→∞
Z
2
0
=lim
n
2
→∞
an
2q
2
k
2
+
c
2
n
2
B
2
2
k
2
+
c
2
n
2
=(15)
Similarly,wehavethat
lim
n
2
→∞
Z
2
0
=
an
2p
1
(k
1
n
1
+c
1
)
B
2
1
(k
1
n
1
+c
1
)
(16)
Thenwehavethat
lim
n
1
→∞
lim
n
2
→∞
Z
2
0
=lim
n
1
→∞
an
2p
1
k
1
+
c
1
n
1
B
2
1
k
1
+
c
1
n
1
=(17)
Sincewehavefrom(15)and(17)that
lim
n
2
→∞
lim
n
1
→∞
Z
2
0
=lim
n
1
→∞
lim
n
2
→∞
Z
2
0
= (18)
Itimpliesthat
lim
(n
1
,n
2
)(,)
Z
2
0
= ,=lim
(n
1
,n
2
)(,)
P
BF
π
(x) = 0(19)
ToshowthatthisresultstillholdswhereB
1
andB
2
areonlyasymp-
toticallyindependentofn
1
andn
2
respectively,itsufficestoshow
thatundermildassumptions,B
1
andB
2
cansafelybeassumedtobe
completelyindependentofn
1
andn
2
respectively.Thisresultwillbe
shownintwosteps.First,letusconsiderthecasewhereQ(n)and
P(n)arepolynomialsofequaldegreeandrecallthat
B
i
=B
i
P(n
i
)
Q(n
i
)
,i= 1,2.
=B
i
P
(n
i
)+α
Q
(n
i
)+β
,i= 1,2.
(20)
whereP
(n
i
)isthatpartofthepolynomialP(n
i
)consistingofterms
thatcontainn
i
onlyexcludingthe constantterm, Q
(n
i
)isthat partof
thepolynomialQ(n
i
)consistingoftermsthatcontainn
i
onlyexclud-
ingtheconstantterm,whileαandβarethecorrespondingconstant
terms.
DOI: 10.4236/ojs.2019.910015OpenJournalofStatistics
N.E.Goltong,S.I.Doguwa
Withoutanylossofgenerality,assumeeachofthecoefficientsin
P
(n
i
)andQ
(n
i
)tobeequalto1,andα=β=0.Thenitis
obviousthatP
(n
i
)andQ
(n
i
)areequalandhence
B
i
=B
i
,i= 1,2.
(21)
Thus, B
1
andB
2
arecompletelyindependentof n
1
andn
2
respectively.
Onageneralnote,relaxingtheassumptionsonP(n
i
)andQ(n
i
)
excepttheassumptionthattheyarepolynomialsofthesamedegree,
thenitiseasilyseenthat
lim
n
i
→∞
B
i
=B
i
lim
n
i
→∞
P(n
i
)
Q(n
i
)
,i= 1,2.
=B
i
κ
i
,i= 1,2.
(22)
whereκ
i
isaconstantindependentofn
i
.Thusingeneral,where
B
i
isonlyasymptoticallyindependentofn
i
,andQ(n)andP(n)are
polynomialsofthesamedegree,thenB
i
cansafelybeassumedtobe
independentofn
i
.
Secondly,whereQ(n
i
)isapolynomialinn
i
ofdegreer
1
andP(n
i
)
ofdegreer
2
,andwehavethatr
1
r
2
= r>0,thenwehavethat
B
i
=B
i
P(n
i
)
Q(n
i
)
,i= 1,2.
=B
i
P(n
i
)
n
r
i
˜
Q(n
i
)
,i= 1,2.
=
˜
B
i
n
r
i
,i= 1,2.
(23)
where
˜
B
i
= B
i
P(n
i
)
˜
Q(n
i
)
.ThenitiseasilyseenthatP(n
i
)and
˜
Q(n
i
)are
polynomialsofthesamedegreeinn
i
.Tomakesenseoftheproof,
recallthat
A
i
=
B
i
n
p
i
1
(24)
wherep
1
= pandp
2
= qandsinceB
1
=
˜
B
1
n
r
1
,wehavethat
A
i
=
1
n
p
i
i
˜
B
i
n
r
i
=
˜
B
i
n
p
i
+r
i
(25)
Now since
˜
B
i
is arationalfunction ofn
i
where thenumerator and the
denominatorarepolynomialsofthesamedegree,itimpliesthat
˜
B
i
cansafelybeassumedtobeindependentofn
i
.Andthiscompletes
theproof.
Corollary1.Lettheposteriordistributionofθgivenanypriordis-
tributionbealinearcombinationoftwotdistributedrandomvariables
suchthatthecoefficients,say,A
1
andA
2
arefunctionsofn
p
1
and
n
q
2
respectively,andforanyp,q>0.Then,ifthedegreesoffree-
domofthevariablesarepolynomialsinn
1
andn
2
respectively,not
necessarilyofthesamedegree,thecorrespondingBayesianmeasureof
evidencesolvesLindley’sparadox.
Proof.RecallthatA
1
= B
1
n
p
1
andA
2
= B
2
n
q
2
.Forthepurposeof
this proof, itwill surficetouse any polynomial functionas anexample.
DOI: 10.4236/ojs.2019.910016OpenJournalofStatistics
N.E.Goltong,S.I.Doguwa
BecausetheproofoftheLemmadealswithpolynomialsofthesame
degree, we shalluse polynomials of different degrees in thiscase.Also,
itwillsurficetoassumethatB
1
and B
2
areindependent ofn
1
and n
2
respectively.
LetthetwotrandomvariablesbedenotedbyT
1
andT
2
,with
k
1
n
2
1
+c
1
n
1
+γ
1
andk
2
n
2
+γ
2
degreesoffreedomrespectively.Then
wehavethat
Var(θ|x) =
B
2
1
(k
1
n
2
1
+c
1
n
1
+γ
1
)
n
2p
1
(k
1
n
2
1
+c
1
n
1
+γ
1
2)
+
B
2
2
(k
2
n
2
+γ
2
)
n
2q
2
(k
2
n
2
+γ
2
2)
=
B
2
1
n
2q
2
(k
1
n
2
1
+c
1
n
1
+γ
1
)(k
2
n
2
+γ
2
)
n
2p
1
n
2q
2
(k
1
n
2
1
+c
1
n
1
+γ
1
)(k
2
n
2
+γ
2
)
+
B
2
2
n
2p
1
(k
1
n
2
1
+c
1
n
1
+γ
1
)(k
2
n
2
+γ
2
)
n
2p
1
n
2q
2
(k
1
n
2
1
+c
1
n
1
+γ
1
)(k
2
n
2
+γ
2
)
(26)
NowrecallthatZ
2
0
=
[θ
0
E(θ|x)]
2
Var(θ|x)
.Then,lettingA=[θ
0
E(θ|x)]
2
,
α= k
2
n
2
+γ
2
,andα
= k
2
n
2
+γ
2
,wehavethat
Z
2
0
=
A
Var(θ|x)
=
An
2p
1
n
2q
2
(k
1
n
2
1
+c
1
n
1
+γ
1
)α
B
2
1
n
2q
2
(k
1
n
2
1
+c
1
n
1
+γ
1
)α
+B
2
2
n
2p
1
(k
1
n
2
1
+c
1
n
1
+γ
1
)α
=
An
2q
2
(k
1
+c
1
n
1
1
+γ
1
n
2
1
)α
B
2
1
n
2p
1
n
2q
2
(k
1
+c
1
n
1
1
+γ
1
n
2
1
)α
+B
2
2
(k
1
+c
1
n
1
1
+γ
1
n
2
1
)α
(27)
lim
n
1
→∞
Z
2
0
=
An
2q
2
(k
2
n
2
+γ
2
)
B
2
2
(k
2
n
2
+γ
2
)
(28)
Andsowehavethat
lim
n
2
→∞
lim
n
1
→∞
Z
2
0
=(29)
Similarly, it can easily be shown thatlim
n
1
→∞
lim
n
2
→∞
Z
2
0
= .And the
restoftheargumentsaresameasthosefoundintheproofofLemma
1.
ItfollowsfromtheCorollarythatwhentheposteriordistribution
ofthedifferenceinmeansisalinearcombinationoftwotrandom
variables,itisnotnecessaryforthedegreesoffreedomstobelinear
functionsofthecorrespondingsamplesizes,butthatLindley’spara-
doxwillbesolvedevenwhenthedegreesoffreedomsarepolynomial
functionsofdifferentdegreesinthecorrespondingsamplesizes.
Itcaneasilybeseenthattheposteriordistributionsofthediffer-
enceofmeansderived by [6]and[7]underthedifferentcircumstances
satisfytheconditionsforLemma1tohold.Assuch,Lemma1then
provesmathematicallythatusingeithertheBayesianmeasureofev-
idenceunderJeffreys’independentpriorasobtainedby[6]orunder
theGammapriorasobtainedby[7],bothofwhichareextensionsof
the methodologyof [5],willsolve Lindley’sparadox.Notethat[6]did
notprovideamathematicalproofoftheabilityofthenewBayesian
measureofevidenceof[5]tosolveboththeBehrens-Fisherproblem
andLindley’sparadoxunderJeffreys’independentprior.
DOI: 10.4236/ojs.2019.910017OpenJournalofStatistics
N.E.Goltong,S.I.Doguwa
TheParetoPrior
AsanexampletodemonstratetheimplicationofLemma1,weshall
assignParetopriorstotheprecisions.Considersamplesofsizesn
1
and
n
2
fromX
1
N(µ
1
2
1
)andX
2
N(µ
2
2
2
)respectively.Underthe
assumptionofindependence,andlettingx=(x
1
,x
2
),thelikelihood
functionisgivenby
f(x|µ
1
2
2
1
2
2
) σ
n
1
1
σ
n
2
2
exp
"
1
2
2
X
i=1
S
i
(µ
i
)
σ
2
i
#
(30)
where
S
i
(µ
i
) =
n
i
X
j=1
(x
ij
µ
i
)
2
(31)
Letτ
i
Pareto(α
i
,k
i
),i= 1,2thenwehavethat
π(τ
i
) = α
i
k
α
i
i
τ
(α
i
+1)
i
τ
(α
i
+1)
i
,i= 1,2.(32)
Letπ
Pa
(µ
1
2
2
1
2
2
|x)denotethePosteriordistributionof
(µ
1
2
2
1
2
2
)undertheParetoprior,thenwehavethat
π
Pa
(µ
1
2
2
1
2
2
|x) f(x|µ
1
2
2
1
2
2
)(µ
1
2
2
1
2
2
)
f(x|µ
1
2
2
1
2
2
)(µ
1
2
|σ
2
1
2
2
)(τ
1
)(τ
2
)
σ
n
1
1
σ
n
2
2
e
1
2
P
2
i=1
S
i
(µ
i
)
σ
2
i
τ
(α
1
+1)
1
τ
(α
2
+1)
2
(33)
ThenthecorrespondingPosteriordistributionof(µ
1
2
)isobtained
byintegrating(33)withrespecttoσ
2
1
andσ
2
2
toobtain
π
Pa
(µ
1
2
|x)
1+
n
1
x
1
µ
1
)
2
(n
1
1)S
2
1
(n
1
2α
1
4)/2
1+
n
2
x
2
µ
2
)
2
(n
2
1)S
2
2
(n
2
2α
2
4)/2
(34)
Let
t
i
=
p
n
i
(n
i
2α
i
5)(¯x
i
µ
i
)
p
(n
i
1)S
i
,i= 1,2.(35)
thenwehavethat
π
Pa
(µ
1
2
|x)
1+
t
2
1
(n
1
2α
1
5)
(n
1
2α
1
4)/2
1+
t
2
2
(n
2
2α
2
5)
(n
2
2α
2
4)/2
(36)
where(36)isthekernel ofthejoint distributionoftwoindependentt
random variables with n
1
2α
1
5 andn
2
2α
2
5 degreesof freedom
respectively andwe havethe posteriordistribution ofthedifference of
meansas
θ|x¯x
1
¯x
2
S
1
p
(n
1
1)
p
n
1
(n
1
2α
1
5)
T
1
S
2
p
(n
2
1)
p
n
2
(n
2
2α
2
5)
T
2
!
(37)
DOI: 10.4236/ojs.2019.910018OpenJournalofStatistics
N.E.Goltong,S.I.Doguwa
whereT
i
isarandomvariablethatfollowstdistributionwithn
i
2α
i
5degreesoffreedom.TheBayesianmeasureofevidenceunder
theParetoprioristhengivenby
P
BF
Pa
(x)
= P
S
1
p
(n
1
1)
p
n
1
(n
1
2α
1
5)
T
1
S
2
p
(n
2
1)
p
n
2
(n
2
2α
2
5)
T
2
≥|¯x
1
¯x
2
|
!
(38)
NotethatforthenewBayesianmeasureofevidenceunderthePareto
prior to be valid, sample sizes have to be at least 6 each since thereare
no negative degrees offreedom.In fact,even when sample sizesare as
smallassix,thechoice ofα
i
hastoensurethatn
i
2α
i
5 resultsto
apositivenumber.ItcaneasilybeseenthattheBayesian measureof
evidence of[5] solves the paradox in [1]when aParetoprior isassigned
tothenuisance parametersbynotingthat equation(37) fulfilsallthe
conditionsforLemma1tohold.
Also,sinceitcanbeeasilyseenfrom(37)thattheposteriordistri-
butionofθissymmetricaboutitsexpectedvalue,E(θ|x)=¯x
1
¯x
2
,
then Theorem2of [6]applies herewhichimplies thatunder thePareto
Prior, theBayesian measure ofevidenceof [5] yieldsthe1αcredible
intervalsforθ= µ
1
µ
2
centeredat¯x
1
¯x
2
.
4.SimulationResultsandDiscussion
For the simulation exercise, we used the Metropolis-Hastings algorith-
m with a thinningstepof12,while specifying the Normaldistribution
astheproposaldistribution.ThevaluesinTable1wereobtained
byfixingthefollowingvalues:s
2
1
=15,ands
2
2
=12.Thesevalues
arechosentodemonstratethecaseofmoderatetolargevariances.It
canbeseenfromtheresultsherethatthenewBayesianmeasureof
evidenceundertheParetopriorgivesconsistently,thehighestprob-
abilityvaluewhichcorrespondstotheweakestevidenceagainstthe
nullhypothesis.Infact,forsmallsamplesizes,conclusionbasedon
thenewBayesianmeasureofevidenceundertheParetopriorhasthe
tendencyoffailingtorejectthenullhypothesiswhiletheotherthree
measuresofevidencecanleadtotherejectionofthenullhypothesis.
However,asthesamplesizesincrease,theconclusionsfromthefour
measuresofevidencetendtomovetowardsthesamedirection.
ThevaluesinTable2wereobtainedbyfixingthefollowingvalues:
s
2
1
= 2,ands
2
2
= 3.Hereontheotherhand,thevalueschosenreflect
thecaseofsmallvariances.Inasimilarmanner,theresultsinthis
tableshowclearlythatthenewBayesianmeasureofevidenceunder
theParetopriorgivesaconsistentlyweakevidenceagainstthenull
hypothesiscomparedtotheotherthreemeasuresofevidence.While
forsmallsamplesizes,conclusionbasedonthenewBayesianmeasure
of evidence under the Pareto prior has the tendency of failing to reject
thenullhypothesisincontrasttotheotherthree,asthesamplesizes
increase,theconclusionsfromthefourmeasuresofevidencetendto
movetowardsthesamedirection.
ItisthenclearfromtheresultspresentedinbothTable1andTable2
thatthenewBayesianmeasureofevidenceundertheParetoprior
solvesLindley’sparadoxeventhoughtheevidenceagainstthenull
providedundertheParetopriorisnotasstrongasthatprovidedby
theotherthreemeasuresofevidencenomatterthecircumstances.
DOI: 10.4236/ojs.2019.910019OpenJournalofStatistics
N.E.Goltong,S.I.Doguwa
Table1.Thefourdifferentprobabilityvaluesfordifferentvaluesofα
1
2
1
andβ
2
.
n
1
= n
2
= 6
1
= 0.05
2
= 0.02,s
2
1
= 15,s
2
2
= 12
x
1
¯x
2
)2.5002.2001.9001.6001.3001.0000.7000.4000.100
p(x)0.26590.32410.39150.46810.55370.64750.74820.85420.9633
ppp(x)0.25000.30190.36400.43740.52240.61880.72550.84010.9597
P
BF
J
(x)0.31480.37260.43840.51160.59210.67900.77100.86760.9668
P
BF
Pa
(x)0.78780.81160.83630.86120.88650.91220.93820.96450.9912
n
1
= n
2
= 20
1
= 0.05
2
= 0.02,s
2
1
= 15,s
2
2
= 12
x
1
¯x
2
)2.5002.2001.9001.6001.3001.0000.7000.4000.100
p(x)0.03780.06590.11030.17660.27020.39480.55040.73250.9319
ppp(x)0.04370.07080.11290.17570.26550.38710.54190.72620.9301
P
BF
J
(x)0.04330.07320.11970.18800.28250.40690.56030.73910.9338
P
BF
Pa
(x)0.07590.11590.17220.24800.34560.46630.60840.76950.9415
n
1
= n
2
= 50
1
= 0.05
2
= 0.02,s
2
1
= 15,s
2
2
= 12
x
1
¯x
2
)2.5002.2001.9001.6001.3001.0000.7000.4000.100
p(x)0.00100.00350.01120.03190.08000.17670.34310.58750.8920
ppp(x)0.00160.00470.01310.03420.08170.17640.34050.58420.8910
P
BF
J
(x)0.00110.00380.01220.03380.08300.18050.34820.59060.8929
P
BF
Pa
(x)0.00170.00560.01640.04220.09690.20020.36860.60730.8975
n
1
= 7,n
2
= 10
1
= 0.003
2
= 0.005s
2
1
= 15,s
2
2
= 12
x
1
¯x
2
)2.5002.2001.9001.6001.3001.0000.7000.4000.100
p(x)0.19170.24750.31520.39530.48800.59250.70720.82980.9571
ppp(x)0.19070.23990.30120.37620.46590.57050.68880.81800.9540
P
BF
J
(x)0.22580.28290.35050.42880.51870.61800.72650.84160.9602
P
BF
Pa
(x)0.50570.55450.60640.66240.72080.78200.84630.91170.9779
n
1
= 25,n
2
= 20
1
= 0.003
2
= 0.005,s
2
1
= 15,s
2
2
= 12
x
1
¯x
2
)2.5002.2001.9001.6001.3001.0000.7000.4000.100
p(x)0.02750.05090.09000.15140.24180.36640.52620.71680.9277
ppp(x)0.03270.05570.09320.15170.23850.35990.51850.71090.9260
P
BF
J
(x)0.03170.05680.09800.16130.25290.37760.53620.72320.9294
P
BF
Pa
(x)0.05480.08880.13920.21000.30680.42980.57920.75090.9368
n
1
= 55,n
2
= 60
1
= 0.003
2
= 0.005,s
2
1
= 15,s
2
2
= 12
x
1
¯x
2
)2.5002.2001.9001.6001.3001.0000.7000.4000.100
p(x)0.00040.00180.00670.02170.06120.14860.31080.56190.8846
ppp(x)0.00070.00250.00810.02380.06310.14890.30880.55900.8836
P
BF
J
(x)0.00040.00200.00730.02310.06370.15220.31510.56500.8853
P
BF
Pa
(x)0.00080.00290.00960.02840.07360.16730.33260.57910.8897
n
1
= 55,n
2
= 60
1
= 5.3
2
= 3.9,s
2
1
= 15,s
2
2
= 12
x
1
¯x
2
)2.5002.2001.9001.6001.3001.0000.7000.4000.100
p(x)0.00040.00180.00670.02170.06120.14860.31080.56190.8846
ppp(x)0.00070.00250.00810.02380.06310.14890.30880.55900.8836
P
BF
J
(x)0.00040.00200.00730.02310.06370.15220.31510.56500.8853
P
BF
Pa
(x)0.00080.00290.00960.02840.07360.16730.33260.57910.8897
DOI: 10.4236/ojs.2019.9100110OpenJournalofStatistics
N.E.Goltong,S.I.Doguwa
Table2.Thefourdifferentprobabilityvaluesfordifferentvaluesofα
1
2
1
andβ
2
.
n
1
= n
2
= 7
1
= 0.04
2
= 0.07,s
2
1
= 2,s
2
2
= 3
x
1
¯x
2
)2.5002.2001.9001.6001.3001.0000.7000.4000.100
p(x)0.01200.02310.04410.08270.14990.25960.42370.64450.9078
ppp(x)0.03030.04250.06240.09570.15240.24750.39940.62050.9002
P
BF
J
(x)0.02260.03870.06600.11200.18690.30070.46200.67150.9149
P
BF
Pa
(x)0.29910.34700.40420.47140.54950.63880.73850.84720.9611
n
1
= n
2
= 25
1
= 0.04
2
= 0.07,s
2
1
= 2,s
2
2
= 3
x
1
¯x
2
)2.5002.2001.9001.6001.3001.0000.7000.4000.100
p(x)0.00000.00000.00010.00080.00550.03000.12410.37560.8240
ppp(x)0.00010.00020.00060.00220.00880.03500.12580.36980.8205
P
BF
J
(x)0.00000.00000.00010.00100.00670.03380.13200.38490.8276
P
BF
Pa
(x)0.00000.00010.00060.00300.01410.05480.17240.43090.8433
n
1
= n
2
= 50
1
= 0.04
2
= 0.07,s
2
1
= 2,s
2
2
= 3
x
1
¯x
2
)2.5002.2001.9001.6001.3001.0000.7000.4000.100
p(x)0.00000.00000.00000.00000.00010.00210.02920.20890.7525
ppp(x)0.00000.00000.00000.00000.00020.00300.03160.20810.7502
P
BF
J
(x)0.00000.00000.00000.00000.00010.00230.03120.21310.7545
P
BF
Pa
(x)0.00000.00000.00000.00000.00020.00360.03910.23280.7647
n
1
= 10,n
2
= 8
1
= 0.5
2
= 0.7,s
2
1
= 2,s
2
2
= 3
x
1
¯x
2
)2.5002.2001.9001.6001.3001.0000.7000.4000.100
p(x)0.00450.01040.02340.05100.10580.20580.36970.60510.8967
ppp(x)0.01760.02610.04090.06730.11560.20320.35350.58540.8901
P
BF
J
(x)0.00970.01870.03630.06950.13100.23640.39950.62670.9025
P
BF
Pa
(x)0.25950.30450.35970.42670.50720.60120.70930.82890.9567
n
1
= 20,n
2
= 25
1
= 0.5
2
= 0.7,s
2
1
= 2,s
2
2
= 3
x
1
¯x
2
)2.5002.2001.9001.6001.3001.0000.7000.4000.100
p(x)0.00000.00000.00020.00140.00820.03880.14290.39850.8322
ppp(x)0.00010.00030.00100.00340.01220.04380.14350.39150.8284
P
BF
J
(x)0.00000.00000.00030.00190.01000.04360.15230.40880.8361
P
BF
Pa
(x)0.00010.00050.00190.00740.02650.08200.21750.47620.8574
n
1
= 55,n
2
= 60
1
= 0.5
2
= 0.7,s
2
1
= 2,s
2
2
= 3
x
1
¯x
2
)2.5002.2001.9001.6001.3001.0000.7000.4000.100
p(x)0.00000.00000.00000.00000.00000.00090.01890.17620.7343
ppp(x)0.00000.00000.00000.00000.00010.00140.02080.17590.7322
P
BF
J
(x)0.00000.00000.00000.00000.00000.00100.02010.17990.7358
P
BF
Pa
(x)0.00000.00000.00000.00000.00010.00170.02680.20100.7481
n
1
= 55,n
2
= 60
1
= 2.5
2
= 3.7,s
2
1
= 2,s
2
2
= 3
x
1
¯x
2
)2.5002.2001.9001.6001.3001.0000.7000.4000.100
p(x)0.00000.00000.00000.00000.00000.00090.01890.17620.7343
ppp(x)0.00000.00000.00000.00000.00010.00140.02080.17590.7322
P
BF
J
(x)0.00000.00000.00000.00000.00000.00100.02010.17990.7358
P
BF
Pa
(x)0.00000.00000.00000.00000.00010.00300.03600.22500.7608
DOI: 10.4236/ojs.2019.9100111OpenJournalofStatistics
N.E.Goltong,S.I.Doguwa
5.Conclusion
Inthispaper,wehavebeenabletoinvestigatetheabilityofthenew
Bayesianmeasureofevidenceof[5]tosolveboththeBehrens-Fisher
problem and Lindley’s paradox when a prior is assigned to theparam-
eters which yields aposteriordistribution thatis alinearcombination
oftwoindependenttrandomvariablesprovidedthatthedegreesof
freedomoftheserandomvariablesarelinearfunctionsofthecorre-
spondingsamplesizes.Furthermore,wehaveshownthattheability
ofthemethodologyof[5]tosolveLindley’sparadoxisnotrestricted
to thecase wherethe degreesof freedomsof theposterior distribution
arelinearfunctionsofthesamplesizesbutextendtothecasewhere
thedegreesoffreedomarepolynomialfunctionsinthecorresponding
samplesizes,notnecessarilyofthesamedegree.Lemma1providesa
mathematical proofthatthe newBayesianmeasureof evidenceunder
Jeffreys’independentpriorsolvesLindley’sparadox.Moreinterest-
ingly,theresultswehaveobtainedshowsthatthemethodologyof[5]
doesnotonly solve Lindley’sparadoxforthe purposereconciling thet-
wo approaches (i.e.Bayesian and Frequentist), but that this Bayesian
procedurewillnotalwaysfavourthenullhypothesis.
UsingtheParetopriorasanexample,wehavedemonstratedusing
simulationresultsthatthenewBayesianmeasureofevidencesolves
Lindley’sparadoxeven thoughthenewBayesianmeasureof evidence
undertheParetopriorprovidesaconsistentlyweakevidenceagainst
thenullhypothesiscomparedtoeithertheGeneralizedp-value,the
Posterior Predictive p-value, orthe newBayesian measure ofevidence
underJeffreys’independentprior.Wewerealsoabletoproofthat
undertheParetoprior,a1 αcredibleintervalforthedifferencein
means centered aroundthe posterior expectation could be constructed
byusingtheα/2and1(α/2)quantilesofthe posteriordistribution
ofθ.
ConflictsofInterest
Theauthorsdeclarenoconflictsofinterestregardingthepublication
ofthispaper.
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