Un test d'adéquation global pour la fonction de répartition conditionnelle
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(8) MFIyj Z K b jnI b IHG fchnIXQ^fchnILGkô Z MFG3B Z heC+Q^IHG+CFILôIXC b U^I_C -I V Z hijeC l U b Z ôyôyIHK b IXC B l CÊCFIHôyIXC b U^IXC : UWQWQgIHG h bFd l CFôI
(9) B Z heCýM Z hDM j9S lp]Z CXjRô[S l V Z UWC/BeC Z B Z GJE hnKyGJh YI_M
(10) jnIkM d GJI l hiGJG7U U^K\MFE_CFILGJG l K\M IkQ^IkCFILôyI_C b U^I l hnGJGYU]B Z hDC M Z heMFIkG Z K l U^jnI DGJILGK Z ô p CXILhnGJILG Q^I b MFheCFILGIXMCFI_QgI b MFheCFIHGB]IHKnj l K\MPQ^ILGjnIXCFKeU^IXCFGô Z U^GRjiI CFELj l b MJU Z K¼IXMB Z hDC b Im Z MB Z heCPQ l CFI ` bFd I_C bFd IPfch9S UWQ l GJh[ôI4M¢C l KnGJôyI_MJMJCFI5jnhDC l K\M b ILG l KnKiEHILG . .I V Z hijeC l UºB Z heCFGJheUWVcCFI5ILKCFIHôyIXC b U l K\M3VcUWV\IHôyILKNM+Q^ILGkjnILh C l BnB Z CJMFIHhDCXG
(11) jnI b IXMJMXI1M d GJInôyILG ` GYU^ILheCFG U bFd I_Q 25l CFIXQ3IXM l G b l Q * l CFj l IQ^IHGPCFILôyI_C b U^IILKB l CJMJU b hDQWU^IXCjiI M Z heM;QTS U^K\MFE_CXMfchºS UWQ^G Z K\MkB Z CJMFEPOô Z KyMJC l V l UWQ . 8 KÔmªC l KijÿôyIXC b U-O I l K `,kZ
(12) Q jnI1M d GJI<IXM5B Z heCkG l mqIHK\MJUWQWQgIHGJGJIROô. l b C Z B Z hDC l V Z UWC l bLb ILBeMFEjºSXMJCFIRQ^I CXELGYU^jnILK\M<jiIô Z K 7 hDC Z K?ELm l CFj . yI CFIHôyI_C b U^IELm l QgIHôyIHK\M EXCoôyI * l C l bHb Z jºS l V Z UWC l bLb ILBeMFEjnI l UWCFIB l CJMJU^IjnIô Z K YheCjiI M d GJIrô l U^G l hnGJGYU5B Z heCM Z heMFIÄQ l mqIHK\MJUWQWQ^ILGJGJIÿfch9S UWQ l ILh¾OÔô Z KELm l CFjjnhDC l K\M b ILG l KnKiEHILG Z XS l U+BnhÔQ^I b Z M Z \I_C O rZ K\MFB]I_QWQWU^IXC IQ^IyCFIHôyI_C b U^IyEHm l Q^ILôILK\MRB Z heCPM Z hnGPQ^ILGRBeCFE b U^ILh b Z KiG¢IXUWQ^G fchºS UWQ l BnhrôyIPj Z KnKiI_C5IXM.B Z heCkQ l b Z K
(13) ]l K b IPfchºS U Q ô[S l MFELô Z U^moKiEHIPILK}ôyIPBeC Z B Z G l K\MkjeUWV\I_CFG5ILK ` GJI_UgmoKnILôyIHK\MFG.jiheC l KNM5ô Z K l KnKnELIRjºS MJM l bFd EHI ILôyB Z C l UWCFI;jºS KiGJI_U^mqKnILôyIHK\M.IXM jiI I bFd I_C bFd IO [Z K\MFB]I_QWQWUgI_C .
(14) IPCFIHôyIXC b U^I l hnGJGYU Q l U^K
(15) IXC¢QWUgKnIXM<B Z hDC5QTS U^K\MFE_CXM fch9S UWQ l B Z CJMFEOô Z KrMJC l V l UWQ ILK l bLb ILBeM l K\M nj I l UWCFI<B l CJMJU^I;jnI;ô Z K YheCjnI.M d GJI . IG Z h d l UWMFI_C l UiO4BeCFELGJIHK\MýCXILôyI_C b U^I_C U^IXCJCFI \l b Zop jeUWCFI b MFILheCÊjnI-QS fcheU^B]I C Zop¶lªp UWQWUWMFELGaIXM * M l¿` MJU^GYMJU^fchnI I3QgI3CFILôyI_C b U^I3ILKyB l CJMJU b heQWU^IXC-jnI+ô[S l V Z UWC b Z K
(16) E3jiIHG-ILKnGJIXU^moKiIHôyILK\MXGÊILK U Z GYM l MJU^GYMJU^fchnILG jnhDC l K\M5ô Z K l KnKiEHIRj9S MJM l bFd ELI IHôyB Z C l U CXI1jºS KiGJI_U^mqKnILôILK\M.IXM.jnI I bFd IXC bFd IO [Z K\MFB]I_QWQWUgI_C . I.Q^I1CFIHôyIXC b U^I.ELm l Q^ILôILK\MkB Z heC
(17) Q^IHGkK Z ô p CFIHh b Z KnGJIXUWQ^G5fchºS UWQ l BnhôyI.j Z KnKiI_C+I_M5B Z hDCM Z heMFI5G l moILK\MJUWQWQ^ILGJG¢I . I V Z hijeC l U¶O.BDCFEHGJILK\M CFILôIXC b UgI_C/Q^ILG l hDM¢CFILG/ôyIHô p CFIHGýjnI
(18) QTS fcheUgB9I C Zqp¶lp UWQWUWMFELG/I_M * M l MJU^GYMJU^fchnI fcheU-OyQ^ILheCRô l KDU CFIRô[S Z K\M p I l h b Z hiB l BnB Z C¢MFERjiheC l K\M b ILG l KiKnELIHG I b Z ôyôyIHK b IXC l UÊB l C.CFI ` ôyI_C b U^I_C ILK Z M +l jeCFIkB Z hDC-M Z hDMXI3QTS l U^jiI.fchºS UWQ]ô[S lRl BiB Z CJMFEHIkB Z heC b I_MJMFI5M d GJI5ô l U^G l hiG¢GYUºB Z hDC G Z KG Z hDMJU^IHKô Z C l Q-IXMB Z heCRô[S l V Z UWCRô Z MJUWVNELIyj l KnG;Q^IHGô Z ôyIHK\MFG<jDU b UWQ^ILG ICFIHôyI_C b U^I l hnGJGYU hij Z VcU b IHKiKnIXMFI l hB Z heC-G Z K l U^jnI4I_M+G Z KyG Z heMJU^ILKyG¢hDCJM Z hDM Q Z CFGjiI5ôyIHG-BDCFIHôU CXILG qZ hDCFKnELIHG-jiI * M l MJU^GYMJU^fchnI O -CFh IXQWQ^ILG ILK µ qµ a I_C b U l hiGJGYU
(19) O rZ d l ôôyILj IXQWQ Z h I_M QWU :;l KiK Z hnKÔB Z hDC M Z hDMFI Q^ILheCmoILK\M¢U QWQ^IHGJGJIrIXM ôyIXC b UjiIÄô[S l V Z UWC b Z K
(20) E bFd l b hnKQ^ILheCM Z heC ajiIHGILKnGJIXU^moKiIHôyILKNMFGRjiI * M l MJU^GYMJU^fchnIPIHK ILhnm IPCFIHôyI_C b U^I l hiGJG7U : EXC l CFj U l hÚB Z heC5GJILG.BeCFE b U^IHh b Z KnGJIXUWQ^G l U^KnGYUÊfchnI KnjDCFE Úl GkB Z heC+M Z heMFI4Q l G eôyB l M d U^I5fch9S UWQaô[S l MFEHô Z U^moKiEHI;jnILBnheU^G+G Z K l CJCJUWV\EHIPO [Z K\MFB]IXQWQWU^I_C . . . . IPV Z hnjDC l U(ô l U^K\MFILK l KNMkCFILôIXC b U^I_C5M Z heMFIHG3Q^IHG;B]I_CFG Z KiKnILG+MJC l V l UWQWQ l K\M OQTS ,8k * 5 jiI +l C ` b l GJG Z KiKnI-fchDUeôrS l bHb hnIXUWQWQ^ILK\M ILKPM l K\MÊfchºS MJM l bFd EHI ILôyB Z C l UWCFI/j9S KnGJIXU^moKiIHôyIHK\MýI_MÊjnI I bFd I_C bFd I B Z hDC b I_MJMFI l KnKiEHIhnKeU VNIXCXGYUWM l UWCFI µ µ ICFILôyI_C b U^I ILKB l CJMJU b hDQWU^I_CPG Z KÔjDUWCFI b MFIHhDC ILK eG d l l CFj Z K l U^KnGYUcfchnI * cQWVcU^I 5U^mqheU^IXC `, Q l B Z heC(M Z heMFI/G l moILK\MJUWQWQ^IHGJGJI
(21) jnILBnheU^Gýô Z K l CJCJUWV\EHI+OQS ,8k I_C b U l hnGJGYU O U bFd IXQ ILGYMFI [Z hnK l <l ôIXQ d U^I_CJC * BeU^K Z G l VNIXQ eKnI CFôyGYMJC Z Knm :;lp CJU^I_Q. . .
(22). . .
(23) C l U^GJGJIIXM I l K `,}l C b 2 him Z hnKiKnI_MÊB Z hDCaM Z heMFI/Q l moILK\MJUWQWQ^IHGJGJI+I_MÊQ^ILG/IHK b Z heC l moILôILK\MFGafchºS UWQ^G/ôrS Z K\M MFEHô Z U^moKiEHG+B Z heC b I_MJMFI1M d GJI . IV Z hijeC l U-B Z heCFGJheUWVcCFIILKÔCFILôIXC b U l K\M l BeMJU^GYMFI d l BnhDU^G l MPB Z heC;M Z hDMFI G Z K l U^jiI ILKU^K Z C ` ô l MJU^fchnIô l U^GRGJheCJM Z heMRB Z hDCG l B l MJU^IHK b IIXM G l moILK\MJUWQWQ^IHGJGJI}O[ô Z KEHm l CFjÙB]IHKnj l K\MPM Z hDMFIHG b IHG l KiKnELIHG IkCFIHôyI_C b U^I l hiGJG7U bFd l Q^ILheCFILhnGJIHôyILK\M U^IXCJCXIXMJMFI CFK l hnjyI_M IXCFK l jnIXM¢MFI (l b l KB Z heC/Q^ILheC jeU^GJB Z KeU p UWQWUWMFE cQ^ILheC+G Z heMJU^ILKj l KnG/Q^ILG3ô Z ôILK\MFG/jeU b UWQ^ILGkI_M
(24) M Z hDMXI3Q^IHhDC3moILK\MJUWQWQ^ILGJG¢I ¶ I_C b U l hnGJGYUºO U b Z Q^I : C l bFd IXM : ILKnI_VcU V\I U l CXj UWCFI_UWQWQ^I U^fchiI_M<IXM C¢U b 2 him Z hnKiIHK º I_C b U ILK
(25) KÿO * l C l d h p¶l fcheU^IXC1IXM +l M d C l GJBeUgK?B Z heC+Q^ILheC5moILK\MJUWQWQ^ILGJG¢IRIXMkQ^ILheC5jeU^GJB Z KDU p UWQWU MXE . . . I V Z hijeC l UnB Z heCFGJhDUWVcCFI-IHKCFIHôyI_C b U l K\M-ôyILG l ôU^G/j Z b M Z C l K\MFGýI_M-j Z b MFIHhDCFG/fcheU Z K\M/CFILKnjih b IXMJMFI M d GJI BDQ^hnG l mªCFE lªp Q^I ICFILôIXC b U^IyK Z M l ôyôyIHK\M Úl CJU^I ` qZ GJE }l CJMJU^KnI B Z heCPG Z K l ô UWMJU^EjnILBnhDU^G jnE ¢OjnILh Ôl KnG IQ l CFIHôyI_C b U^I l hnGJGYU/jºS l V Z UWCPB l C¢M l moERjiIHG;ô Z ôyIHK\MFG5jiE_MFILK\MXILG.GJB Z C¢MJU G Z I_QWQgI l GJh?ôyI4CXILô Z K\MFIXCQ^I;ô Z C l Q]j l KnG
(26) Q^IHGkô Z ôyILK\MXG
(27) jeU b UWQgIHG I1CFIHôyIXC b U^I l hnGJGYU l heCFILK\M kZ heVcU CFI B Z hDCM Z hDMFI.G l mqIHK\MJUWQWQ^ILGJGJI ]8 KnI;B9IXMJUWMFI.B]ILKnGJEHI l hiGJGYUB Z heC rZ d l KnK l j 25l G¢G l K d Z ô l G -IXCJC Z K ô l U^G l hnGJGYUaB Z heCkQ^ILG l K b U^IHKiG -U Z GYM l MJU^GYMJU b UgIHKiG QWU VDUgI_C : U^ôyILKnI U^I_CJCFI (l l \IRjnI U bFd l h
(28) IHK Z UWM C¢U bFd Z MIXM }l CJU^IHô U^K\M `, Q `,rZ heVcU^j I K9S Z h p QWU^IyB l G l hiG¢GYU-jiI CFILôyI_C b U^I_C : heUWQWQ l hnôyI U b Z MJM l B Z heC
(29) M Z heMFI1QTS l ôUWMJU^E;fchºS UWQ(ôyI1MXELô Z U^mqKnI.jiIHBiheU^GkfchnIXQ^fchnILG l KiKnELIHGkjnE JO . . . . .
(30). . . . 5U^ILKnKiIHK\MyIHK
(31) KQ^IHGCFIHôyIXC b U^ILôyIHK\MFGRjnh b Z IHhDC IV Z hnjDC l U b Z ôyôyIHK b IXC B l CCFILôIXC b UgI_C ôyILG B l CFIHK\MFG/B Z hDC/M Z hDMXI+Q^ILheC
(32) moILK\M¢UWQ Q^IHGJGJI5IXMQ^ILheC b Z ôyBeCFE d ILKnGYU Z KM Z heM l hQ Z KimRjnIkôyIHG/EXMFhnjiIHG D IXC b U jnI5ô[S l V Z UWC+GJhnBnB Z CJMXELI5ô HôyIkj l KnG Q^IHG+ô Z ôyIHK\MFG/jDU b UWQ^IHGkIXM3ILK b Z CFI1ôIXC b U]jºS l V Z UWC+M Z h Z heCFG-EXMFE Q OB Z heCkô Z U]M Z hDM3GYU^ôyBeQ^ILôILK\M n8 KrmªC l KnjôyIXC b UaORô Z K?B9IXMJUWM C CFI E_CFELôU^I.fcheU l M Z h Z heCFGEXMFE hnKC l Z K jiI3G Z Q^I_UWQ]j l KnG/ô l VcU^I IkG Z h d l UWMFI
(33) Q^I+CFIHôyIXC b U^IXC-B Z heC/M Z heMÊQ^IkG Z hDM¢UgIHKfchºS UWQ¶ô[S l BnB Z CJMFI
(34). .
(35) jnILBnhDU^G+M Z h Z hDCFG3IXM5B Z hDC3G l BDCFEHGJILK b I GYU(BDCFE b U^IHhiGJIO rZ K\MFB]I_QWQWU^IXC5BeC GkjnI;ô Z U kI V Z hnjDC l U l hiG¢GYU CFIHôyIXC b U^IXC bFd l Q^ILheCFILhnGJIHôyILK\Mô Z KyB l BeU]I_M+ô l ô l ôU^IkB Z hDC/Q^IHhDC b Z hiBjiI
(36) Q fch Z MJU^jeU^ILK?fcheU(ô[S l G Z heV\IHK\M+CFIHô Z K\MXE4Q^IPô Z C l Q]I_M5B Z hDC+Q^IHhDC1G Z heMJU^ILKrB]IXCXô l KnILKNM e I_C b UajºS l V Z UWC *cb Z h p U+BeC GjnIyV Z hnG ýUWQ l -O}G l ô l KDU CFI ILô p I_QWQ U4ôyIHG+IHI NILKnjiGO * MFI
(37) I_C b U l hnGJGYUkO}ô l mC l Knj ` ô CFI5B Z hDC3M Z heMFIHG+GJILG5B9ILKnGJELIHGkIXM5GJIHGkBDC¢U CFILG4B Z heCkô Z U . . I;CFIHôyI_C b U^IPEHK Z CFôyEHôyILKNM heQWU^ILK ¶ô Z KrG Z hDMJU^IHK[B9IXCFô l KnILK\M efcheU l VNE b h b I_MJMFI;M d GJI l h Z hDC Q^I Z hDC3OPôILG b MFELG IkKnI3Q^IkCFIHôyI_C b U^IXC l U l ô l U^G l GJGJI .B Z hDC l V Z UWC3EXMFE5GYU^ôyBeQ^ILôILK\M-BeC G-jnI5ô Z U GJheCJM Z heM-Q l jiI_CFKDU CFI l KnKiEHI5fchDU l E_MFE4Q l BeQ^hiGjDU b UWQ^I<B Z heC+ô Z U9ô Z C l Q^ILôILK\M D Q l GJh?ô[S E b Z heMFIXC ôyI5C l GJGJhDCXIXCkI_M l M Z h Z hDCFG b CFhrILKrô Z U b I fchDU l l UWM5ô l Z C b I . . . I.MFIXCFô UgKnI_C l UºIHKÄCFIHôyI_C b U l K\MkjºS lªp]Z CFjô l l ôUWQWQ^I VNI cC Z KiK l U^GJI<IXM * l V Z l CFjiI.jºSXMJCFI.V\IHKchiI Oô l G Z hDMXILK l K b IPjnI;M d GJI IXM.jnIPô[S l V Z UWC5IHK b Z heC l moELI jiheC l K\M4M Z heMFILGkôILG5E_MFhijnILG I G Z h d l UWMFI Q lRp U^IHK\V\IHKchiI<OPô l B]IXMJUWMFI b Z hnGYU^KiI 5U b cU^IPOPfcheU YI5jnELjeU^I b IXMJMFI5M d GJI.I_M 7I4CFIHôyIXC b U^I5jih Z Kijjnh b Z IHheC5G l ô l ô l K d CJU^GYMFIXQaB Z hDC3M Z heM b I fch9S I_QWQgI l l UWM.B Z hDC5ô Z U(jnILBnhDU^G+M Z h Z hDCFG ¶ IXC b U l hnGJGYU Oô l ô l C¢C l U^KnI QWU^G lp IXM d I_M.Oô Z K b Z hnGYU^K hDQWU^ILKrB Z hDC3M Z heMFI5Q^ILheC5mqIHK\MJUWQWQ^ILGJGJI .
(38). . PI CFIHôyIXC b U^IRILK
(39) KÃôyILG l ôU^G * E_M Z U^G.IXM rZ K\MFB]I_QWQWU^EXC l U^KiG l U^KnGYU/fchnIPM Z heMFIHG3Q^IHG;B]I_CFG Z KiKnILG.fcheU G Z K\MkV\IHKchiG l GJGYU^GYMFI_C<Oô l G Z heMFILK l K b I<jiI<M d GJI 9 I_Q l ô[S lp I l h b Z hnBM Z h bFd ELI ] IXC b UÊIHK[B l CJM¢U ` b heQWU^IXC<O UgI_CJCFI * Z B d U^ I 5l M d l QWU^I Úl m l QWUýI_M * Z B d U^I .
(40)
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(186) ýü ú ? A CFEHG ) q : t+ K[GJhnBnB Z GJI q t I_M Z KrGJhnBiB Z GJI.jiI<BDQ^hnGkfchnI $ B . nQ Z CFGJfchnI $ . . õþ . q µ Pt K GJhnBiB Z GJIq t _I Mq : t XI M Z KGJhnBnB Z GJI jnIBeQ^hiG fchiI $Q,) 4 B o$ % (Q Z CFGJfchnI. ). $ . . . q =< t KÙGJhnBnB Z GJI q t )q : t IXM@q µ t IXM Z KÙGJhnBiB Z GJIjnIBDQ^hnGfchiI $ B o G % Q Z CFG¢fchiI $ ILG d eB Z M d GJILGkE_M l K\M5ô U^GJILGkIHK[BeQ l b I Z K[B]IHhDM5ô l U^K\MFIHK l K\MkGHS U^K\MXEXCFILG¢GJIXC.OQS l BnBDC Z U^ô l MJU Z K jnhÄK Z l h? A C E G " $
(187) KCFIXVcU^IHK\M/ô l U^K\MFILK l K\M/O.K Z MJCFIBDCFIHôU^IXC Zqp 7I b M¢U 9fcheU b Z KnGYU^GYMFI4O l BiBeC Z U^ôyI_CýQ^I+K Z l h ? A CFEHG B l C.hnKiI Z K b MJU Z KÃBeQ^hiGRv GYM lp Q^Ixw * hDUWMFIOyG Z KÃI BeCFILGJGYU Z KÃj Z KnKiEHIPILK q : : · t Z K b Z KnGYM l MFIRfchºS IHK U^K\MJC Z jnheUgG l K\MQ l ô l MJCJU b I 3P ) C ! E E E *B G-jiI<jDU^ôILKnGYU Z K C
(188) ! G C
(189) ! G l Q Z CFG @ n # ! *. B. B ? ACG C E E E C G G C G. @ n ! ! $ B $ B $ C!. EEE G C G. @ ! n $ B C eG C !#. E E E G C 2G. q : :{r t A Z [Z K l B Z GJE ! A C eG n $ B $ n E q : µ t ). . . . . . . . . . . KrjºS l BnBDC Z U^ôIXC b I_MJMFI;I @BDCFIHGJGYU Z K[jnI CG Z rIHGYMýQ l ô l MJCJU b I-jnI jeU^ôyIHKiGYU Z K C
(190) !@ G jnE
(191) KDU^G?ILKYq : : p t * Z hnGQTS d eB Z M d G¢Iq . . . . . . . .
(192). . . . ? A CFEHG njnE
(193) KDU^GJG Z @ KiG4ô l U^K\MXILK l K\M n C !#. E E E uG C 2G. q : µ : t C
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