Technical Report. Asymptotic Analysis and Variance Estimation for. Testing Quasi-independence under Truncation

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1 Teha Repor Asmpo Aass a Varae Esmao for Tesg Qas-epeee er Trao Taesh Emra a eg ag emra@pharm.asao-.a.p ag@sa..e. Dso of Bosass Shoo of Pharmaea Sees Kasao Uers 5-9- Sroae Mao- Too Japa se of Sass aoa hao-tg Uers s-h Taa R.O.. Absra A ass of eghe og-ra sass for esg qas-epeee for rae aa s propose. Ths masrp oas asmpo aass for he propose mehos a smao ses o eaae her fe-sampe performaes. B emporar gorg esorg e ere arge-sampe properes of he propose ess Seo a sss arae esmao Seo. Seo 3 osers he eee sao ha aos for rgh esorg. Seo 4 e ompare o mehos for esmag he arae of he propose sass. Oe esmaor has a aaa forma a he oher s he afe esmaor. Or smaos ae ha he afe arae esmaor hh s easer o mpeme has reabe performae. Seo 5 oas some og remars. Deae proofs are ge he Appe. Ke ors: Foa ea meho; epeee es; Jafe meho; og-ra sass; Mae-easze es; To-b-o abes; ea oergee. - -

2 . ARGE SAMPE PROPERTES ABSEE OF ESORG e oser he rao seg hh a par of feme arabes a be e he sampe o f. Mos mehos for aazg rae aa are base o he assmpo of qas-epeee Tsa 99 hh a be sae as : F S / here Pr > a F a a sra fos a s a osa sasfg S are rgh oos srbo F S. hs seo e oser obsere aa of he form K } sbe o b emporar gorg eera esorg.. The Propose Tes Sass For esg he propose eghe og-ra pe sass a be re as R here R a s a pre-spefe egh fo. a be sho ha er A sef spea ase of E R. 3 R has he form 4 R here > a [ s a arbrar osa. / he he obseraos hae o es sh ha he aes of K are a K - -

3 s e hae a a be epresse as A} sg } 5 R < here A } a a a sg s efe o be - or f < or > respee. f he form of / R s eerms he epresso of 5 s a U-sas. For eampe f / R he sass eas he ooa Kea s a sass K A }sg }. < Properes of U-sass hae bee ze b Mar a Bees 5 o ere e aa ress of K or s eee ersos. oeer f he fo / R s sbe o raom arao hs ehqe s o appabe.. arge Sampe Aass ere e oser a geera ass of hh a e febe egh fos. s epee ha a approprae egh fo s hepf for reasg he poer of he orrespog es. For arge-sampe aass e aop he foa ea meho a poerf oo ha a hae egh fos oag pgge- esmaors. parar e oser he foog o sass: } 6 R } 7 R here s a o fo hh s oos ffereabe a. R : < R - 3 -

4 oe ha a ffer heher he esmaor ĉ s oe. Aso oe ha hese o sass a be osere as appromaos of he base sass } R } R respee. The foa ea meho a hae he era esmao of he hpohesze egh fo a ssema a. To smpf he aass e assme ha he srbos of er he hpohess are absoe oos. The forma 6 a 7 a be re-epresse as he foog foa forms: } sg } g } sg } here g s a foa sasfg g efe Appe A.3. The proofs of he aboe heorea ress are proe he Appees A. a A.3. Bref speag hese foas a be sho o be aamar ffereabe fos of ge he ffereab of. B appg he foa ea meho Va Der Vaar 998 p. 97 e oba he foog asmpo epressos: / / U o / / U o P P here U a U are efe Appe A. a A.3. Theorem : Uer / oerges srbo o a mea-zero orma raom arabe h arae σ E[ U ] here U s efe Appe A

5 oroar : Uer / / a spea ase of h oerges srbo o a mea-zero orma raom arabe h arae E[ U ] here U / }. } sg sg } } A aa arae esmaor for he ass s presee Seo 4. To s he sass e ee o eame he proper of ĉ hh s ose reae o he marga esmaors of F a S. Asmpo orma of a be esabshe er he foog oo. efab Assmpo : There ess o pose mbers < sh ha U F > S F a S >. The aboe saeme s a efab oo for F S hh has bee roe U U se heorea aass of rao aa. For eampe he pper m U pas he same roe as he oao T ag e a Assmpo aso garaees he oo ha R / s aa from zero asmpoa haeb e a. 6 so ha he eomaor erms R a R for ĉ s aa from zero. Ths er a Assmpo ĉ s a reabe esmaor of. Theorem : Uer a Assmpo / oerges srbo o a mea-zero orma srbo h arae σ E[ U ] here U s efe Appe A.3.. JAKKFE ESTMATOR OF VARAE - 5 -

6 The asmpo arae of a be esmae b he afe esmaor: here has he form of ho he h obserao a /. Asmpo properes of he afe arae esmaor are ose reae o smoohess of he orrespog foa epresso. Spefa he es sass he ass 6 a be epresse as Φ here s a empra proess a Φ s a foa efe o a spae of fo Pr > see Appe A.. prog he asmpo orma e hae sho ha Φ s aamar ffereabe h respe o he argme. oeer o sho osse of he afe arae esmaor e ee a more sr smoohess oo o Φ ae oos Gaea ffereab Shao 993. Theorem 3: Uer he asmpo araes σ for a be osse esmae b he afe meho. Theorem 4: Uer a Assmpo he asmpo araes σ for a be osse esmae b he afe meho. The proofs of he aboe heorems are ge Appe A ARGE SAMPE PROPERTES UDER ESORG 3. The Propose Tes Sass he s frher sbe o esorg b obsere aa beome Z... } sbe o Z here Z a. Assme ha s epee of. A a esore fare po h e se he same oaos for he e a marga os h he foog mofe efos: Z Z - 6 -

7 Z a R Z. Aorg he propose og-ra sass beomes R hh a be epresse as here he ee B Z B} sg Z Z } 8 R Z < Z Z Z > & & Z Z > & & } mpes ha he par s omparabe a orerabe Mar a Bees 5. Uer he qas-epeee assmpo a be sho ha E [sg Z Z } B ]. For a osa [ he sass beomes 9 R here Z > / S } s a esmaor of a S s he e-be s 97 esmaor for Pr > S base o aa Z... }. 3. Asmpo Aass er esorg o e sss he asmpo orma of he o sass presee of esorg. For a oos ffereabe fo e oser asses of sass: } R - 7 -

8 - 8 - R }. here m a < R R :. Asmpo orma of he mofe sass s o bref sehe Appe A.5 se he reae forma er esorg are oo ompae a proe o e sgh. Defe / > > hh s he empra esmaor of Pr Z > >. foos ha } sg } ; < ϕ } sg } ; g < ϕ here ; ϕ a g are fos sh ha ; ϕ ν a g eah of hh s efe Appe A.5. Asmpo orma of a a be esabshe b appg he foa ea meho base o he fas ha boh of hem are aamar ffereabe fos of Ĥ a he proess / oerges ea o a Gassa proess. Smar o he esore ase osse of he afe arae esmaor s b base o he oos Gaea ffereab of he orrespog foa epresso. The proof s smar as ha for Theorem 3 a 4 a hee s ome. 4. EMPRA VARAE AD JAKKFE VARAE ESTMATOR absee of esorg asmpo arae of he es efe 4 has a

9 raabe form. Base o he meho of mome a appg he pg- prpe e oba he foog aa arae esmaor of : V < A A } } sg sg } }. The forma s ere Appe A.. presee of esorg hoeer mofao of he aboe forma oes ompae mahemaa eraos. To eame heher he era effor s orh or o e ompare V h he afe arae esmaor a smaos. oe ha he aer s ompaoa oee ee er he esorg sao a s heorea sfe Theorem 3 a 4. The frs se of smaos eaaes he es saarze b o arae forma ame he aa a afe esmaors absee of esorg. The arabes are geerae epee from epoea srbos h hazars λ λ respee. Fe ases h λ a.5.5 λ Pr a.5 respee are eame. A 5% ee of sgfae he hpohess s reee f / σ s greaer ha.96 here σ s eher he aa esmaor or afe esmaor. Tabe repors he ress of he es saarze b o arae esmaors er 5 a base o 5 repaos. The mea sqare error MSE for esmag he sampe arae ereases as he sampe sze ges arge for boh he aa a afe arae esmaor. oeer he aa arae esmaor has smaer MSE ha he afe esmaor. O he oher has he aa meho es o ssemaa eresmae he arae so ha he pe error rae of he orrespog es s fae. s ear ha he es saarze b he afe arae has pe error rae oser o he - 9 -

10 oma 5% ee a he ases. Tabe smmarzes he ress of he es. Boh arae esmaors e o oeresmae he re arae b he afe esmaor has sgh smaer MSE a s proes more arae pe error probab ha he aa esmaor. The ress ae ha he es saarze b he afe arae esmaor has more reabe performae mos saos. The seo se of smaos eaaes he es saarze b he afe arae esmaor presee of esorg hh he aa forma s o aaabe. Three epee arabes are geerae from epoea srbos h hazar raes λ λ λ.5..5 respee eg P Z.5. Tabe 3 smmarzes he ress for hree ess a og er 5 a base o 5 repaos. The afe arae meho o he aerage s qe arae. The MSE for esmag he arae ges sma as he mber of sampe reases. Frhermore he pe error raes for he hree sass are a ose o he oma ee. 5. OUSO To esabsh asmpo orma e app he foa ea meho hh a hae more geera forms of sass ha he approahes base o U-sass or ra sass. The epresso of he propose sass as a sasa ffereabe foa aos s o ere a aa arae esmaor a sf he se of he afe meho arae esmao. Smao aass aes ha he afe meho s a beer aerae bease s oeee a reabe ress. eerheess oe ma r o mof he aa arae forma b g more ome erms he Taor epaso o see f he bas a be ree. oeer he eraos be er eos a ma o be orh from a praa po of e. - -

11 APPED: ASMPTOT AASS e D [ } a D [ } be he oeo of a rgh-oos fos h ef-se m efe o [ a [ respee er hh he orms are efe b f sp f for f D[ } a f sp f for f D[ }. e assme ha he fo F S / s absoe oos. ereafer epeao smbos represe he ooa epeao ge. The oao o P s shor of raom arabes ha oerge o zero probab er he ooa probab e b F S /. The empra proess o he pae s efe as: >. a be sho ha / oerges ea o a mea Gassa proess V o D [ } h he oarae ge b o V V } for a. [ A. Proof of Theorem B some agebra mapaos he sas 6 a be epresse erms of he eghe sm of sgs sh ha } R } A} sg } < R } A} sg } < } A} sg } - -

12 here he as eqao se he fas ha eah erm s smmer for es a a ha sg }. Usg he proper ha / for some oherse he aboe epresso a be re as Φ } sg } here he efo of he foa Φ : D [ } R s ge b Φ } sg }. The argme D[ } he preeg eqao oes o ee o be F S / b f so he aboe egra a be erpree as a epeao er qas-epeee. a be sho ha Φ se Φ E A} E A} } sg } } Esg } } a Esg } } for. ere he as eqao foos from haeb e a. 6. The bas ea of he foa ea meho s o f he asmpo behaor of Φ hrogh a fferea aass of Φ a eghborhoo of F S /. A frs orer epaso o hae he form: Φ Φ Φ Φ. here Φ be rgoros efe aer. The aass r he ea oergee of Φ o he ea oergee of hh e be frher esgae a - -

13 rgoros fasho. B re aaos e a proe he aamar ffereab of Φ. The ear ffereabe map of Φ a arbrar argme D[ } h reo h D[ } s: Φ h } h sg } } h sg } } sg } h here /. B appg he foa ea meho Va Der Vaar 998 p. 97 e oba he foog asmpo ear epresso: / / / / / Φ Φ Φ } Φ } o Φ P o here >. s eas o see ha he seqees P U Φ for K are raom arabes h E [ U ] a F S / se E[ Φ ] Φ E[ Φ ]. Base o he era m heorem / oerges srbo o a mea orma raom arabe h arae σ E[ U ]. A. Aa Varae Esmaor for he Tes The sass forms a e sbse of sh ha σ a be epresse b - 3 -

14 - 4 - aa forma. The foa epresso has he form Φ here Φ } sg / for } [ D a Φ for / S F. Spefa he orrespog erae map s ge b Φ. } sg } sg / h h h The arae ] [ E U σ a be esmae b / Φ here } } sg } }sg / Φ. Trg he egra o he obe smmao e oba }. sg } } sg } } sg } } sg } < < < Φ A A A A The seo a forh erms ombe o form /. Ths e ge. } sg } } sg } < Φ A A

15 Base o he aboe epressos e a esmae AVar σ b he foog empra esmaor: σ < A A } } sg sg } }. A.3 Proof of Theorem The sas oes ĉ hh s he esmaor of he ormazg osa. From he res of e a ag 998 ĉ a be re as here F < a S F S are he pro m esmaors of Pr a Pr > respee e-be 97. Defe g a e sho ha he map g : a s he omposo of o aamar ffereabe maps: a F S a S F. A. s e-o for rgh-esore aa ha he pro m esmaor s aamar ffereabe fo of he orrespog empra proesses. For rao aa e app he argmes of Eampe.5 Va Der Vaar 998 o sho he aamar ffereab of he maps from D [ } o D [ } : a F a S. To proe he former saeme e eompose eah map o hree ffereabe maps. For eampe oe a re - 5 -

16 a / a a F < Λ } Λ here he aamar ffereab of he seo map foos from emma. Va Der Vaar 998 a he as map foos from he aamar ffereab of he pro egra. The aamar ffereab of he map a S a be esabshe b appg smar argmes. The aamar ffereab of he seo map A. a be fo emma. Va Der Vaar 998. Usg he ha re Va Der Vaar 998 Theorem.9 he map g s sho o be aamar ffereabe. e g h R be he fferea map of g a D[ } h reo h D[ } sh ha / / / / g g g o g P o P. The sass a be epresse as g } sg } Ψ. Appg smar argmes Seo A. e a sho Ψ. o e sho he aamar ffereab of he map ffereab of g foos ha Ψ : D [ } R. From he aamar g h g g h o form h ompa sbses of D [ }. Ths eas o he foog Taor epaso g h h } } h g h } o. a be sho ha he erae map of Ψ a D[ } h reo - 6 -

17 h D[ } a be re as Ψ h g } h sg } g h } sg } g } h sg } } sg } h. B appg he foa ea meho e oba he foog asmpo ear epressos: / / Ψ / Ψ Ψ } here he seqees / Ψ o P U Ψ K are mea... raom arabes. From he era m heorem / oerges ea o a mea orma srbo h he arae σ E[ U ]. A.4 osse of he Jafe Esmaor e hae sho ha / s asmpoa orma h a fe arae. o e sho osse of he afe arae esmaor of. Aorg o Theorem 3. of Shao 993 e ee o sho he oos Gaea ffereab of Φ a D[ }. oe ha he aamar ffereab s sroger ha he Gaea ffereab a hee he Gaea erae map s ge b Φ h aaabe from Seo A.. e o ee o sho he oos reqreme of he erae map. For seqee D[ } sasfg a for K e ee - 7 -

18 - 8 - o sho } o A Φ Φ Φ here > a o sas for / o form. The foog argmes for prog he oos Gaea ffereab are smar o hose Eampe.6 of Shao 993. The oos ffereab of a he assmpo esre he foog epaso } } } } } }] [ }] [ O form. ee sraghforar b eos aaos sho ha O D B A here } } sg } } } sg } }] [ B } } } } sg } sg } a

19 } sg } } } sg } } D Uer he assmpo ha a be see ha B a D hae orer o. Ths proes he heorem 3. To sho he osse of he afe arae esmaor for sae heorem 4 e o ee o he heher he o of he Gaea fferea map of Ψ hh s aaabe Seo A.3. The o reqreme for he Gaea ffereabe map h Ψ Seo A.3 a be erfe afer eos agebra operaos smar o he aboe argmes. A.5 Asmpo Aass Presee of esorg Base o he pro egra form of he e-be s esmaor S e oba he epresso ; } ν ϕ. A. B some agebra or he ee B a be re as } B <. The e oba he foog foa epresso: < < Z R Z B R } sg } ; } } ; } } ϕ ϕ

20 - - } sg } ; < ϕ. ere he as eqao foos from he proper oherse some for /. Base o smar argmes h Seo A3 e a epress he esmaor ĉ as a fo of Ĥ sh ha g. Smar agebra operaos a be appe o oba he foa epresso of.

21 REFEREES haeb. Res.-P. a Abos B. 6 Esmag Sra Uer a Depee Trao Bomera arrgo D. P. a Femg T. R. 98 A ass of Ra Tes Proeres for esore Sra Daa Bomera e S. a ag G Esmao of he Trao Probab he Raom Trao Moe Aas of Sass e-be D. 97 A Meho of Aog for Ko Obseraoa Seeo Sma Sampes Appe o 3R Qasars Mo. a. R. Asr. So Mar E.. a Bees R. A. 5 Tesg Qas-epeee of Fare a Trao a ooa Kea s Ta Jora of he Amera Sasa Assoao Shao J. 993 Dffereab of Sasa Foas a osse of he Jafe The Aas of Sass Tsa Tesg he Assoao of epeee of Trao Tme a Fare Tme Bomera Va Der Vaar. A Asmpo Sass ambrge Seres Sass a Probabs Mahemas ambrge: ambrge Uers Press. ag M.. Jee. P. a Tsa Asmpo Properes of he Pro-m Esmae a Rgh esore Daa Aas of Sas

22 Tabe. omparso of o afe s. aa arae esmaors of absee of esorg he 5 repaos for pars / a are geerae from epee epoea srbos Pr Varae of / Aerage MSE of Tpe error 5% he arae esmaors Jafe Aa Jafe Aa

23 / Tabe. omparso of o afe s. aa arae esmaors of absee of esorg he 5 repaos for pars of a are geerae from epee epoea srbos Pr Varae of / Aerage MSE of he arae esmaors Tpe error 5% Jafe Aa Jafe Aa

24 Tabe 3. Performae of / saarze b he afe arae esmaor absee of esorg he 5 repaos for pars a are geerae from epee epoea srbos Tes sass Varae of / Aerage MSE of he afe arae esmaor Tpe error 5% / / / og