On Testing Simple and Composite Goodness-of-Fit Hypotheses When Data are Censored

Transcription

1 ALT`8 Jue 9- Bordeux O Testg mple d Composte Goodess-of-Ft Hypotheses Whe Dt re Cesored 35 EV Chmtov BYu Lemesho Novosbrs tte Tehl Uversty Russ Abstrt Problems of pplto of the oprmetr olmogorov Crmer-vo Mses-mrov Aderso-Drlg goodess-of-ft tests for esored dt hve bee osdered ths pper The overgee of sttst dstrbutos to the orre s- podg lmtg dstrbuto lws hs bee vestgted uder some true ull hypothess by mes of sttstl smulto methods s well s the test power gst lose ompetg hypotheses The dstrbutos of test sttsts uder study hve bee vestgted for omposte hypotheses dex Terms goodess-of-ft tests esored dt the mrov trsformto the Rey test the olmogorov-mrov test the Crmer-vo Mses-mrov test the Aderso-Drlg test NTRO DUCTO N Les us osder pl of expermet whe we observe the frst r of order sttsts X () X () X from the smple X X X of depedet detlly dstrbuted rdom vrbles of the sze ( r ) The problem of testg smple d omposte goodess-offt hypotheses wth the oprmetr olmogorov-mrov Rey Crmer-vo Mses-mrov Aderso-Drlg rter hs bee osdered ths pper The lmtg dstrbutos of the bove sttsts were obted []-[3] for testg smple hypotheses of the d H : F( x) F( x ) where Fx (θ) s the probblty dstrbuto futo wth whh observed smple s tested for ft d θ s ow (slr or vetor) prmeter vlue se of omposte hypotheses of the d H : F( x) F( x θ) θ where estmte ˆθ s used sted of the uow prmeter θ the dstrbuto of oprmetr sttsts G( H ) essetlly dffers from the orrespodg dstrbuto whe smple hypothess s tested t depeds o the form of the lw Fx ( ) o whh the ull hypothess s bsed o estmto method for the prmeter d umber of other ftors [4]- [5] Note tht the estmte ˆθ s lulted from the sme smple tht the goodess-of-ft hypothess s tested by f ˆθ s obted from other smple the the hypothess uder test Ths reserh ws supported by the Russ Foudto for Bs Reserh projet o s smple spte of the ft tht these rter were obted log go ther propertes for lmted smple szes hve t bee properly studed Probbly for ths reso the usge of the tests for esored dt hs t bee relzed y progrm system of sttstl lyss we re fmlr wth Hee they re ot vlble for most spelsts M Nul [6] ttrted our tteto to the possblty of effetve pplto of oprmetr goodess-of-ft tests for the lyss of omplete dt by mes of the mrov trsformto d the rdomzto The dvtges of suh pproh re evdet s we move to the problem of tes tg goodess-of-ft of the emprl dstrbuto obted fter trsformtos to the otuous (uform) dstrbuto lw For solvg suh problem oe use the well studed tehque of testg goodess-of-ft hypotheses from omplete smples The pper s med t vestgtg prtl spets of oprmetr goodess-of-ft tests to lyze esored dt o the rght usg the mrov trsformto d rdomzto mog other thgs the pper the overgee of test sttst dstrbutos to the orrespodg lmtg lws s vestgted by mes of the Mote-Crlo method The power of bove tests s lso studed whe testg lose ompetg hypotheses The olmogorov-mrov test The olmogorov-mrov sttsts re defed s []: D ( ) sup F ( x) F( x ) F ( x) D ( ) f F ( x) F( x ) F( x) D ( ) sup F ( x) F( x ) F ( x) prte the olmogorov sttst s more oveet to use wth the Bolshev orreto [7]: 6D () 6 where D mx{ D D } D mx F( X () ) r D r () mx F( X ) [] the lmtg sttemet ws obted P

2 ALT`8 Jue 9- Bordeux ( ) exp( ) P X ( ) where X s the stdrd orml rdom vrble Whe the esorg degree the lmtg dstrbuto of the sttst odes wth the olmogorov dstrbuto for omplete smples: The Rey test ( ) ( ) exp( ) The Rey test sttst to test smple hypothess H re expressed s [] [7]: F ( ) ( ) ( ()) x F x R ( ) sup mx F( x) r F( X ) F ( x) () F( x) F ( ) ( ()) ( ) x R ( ) sup mx F( x) F( X ) F ( x) r () F ( x) F( x) R ( ) sup mx R R F ( x) Fx ( ) () The rdom vrbles R ( ) d R ( ) re dstrbuted detlly d s Rey showed [] the followg lmtg formuls re used: lm P R ( ) ( ) lm P R R( ) L( ) where ( ) s the stdrd orml dstrbuto L ( ) s the Rey dstrbuto futo: 4 ( ) ( ) L ( ) exp 8 The Crmer-vo Mses-mrov test The Crmer-vo Mses-mrov sttst s lulted by: r () (3) [3] the upper peretge pots of the Crmer-vo Mses- mrov sttst dstrbuto re gve for vrous esorg degrees se of testg smple hypothess H se of omplete smple ( r ) ths sttst hs the dstrbuto of the form ( j / ) 4 j (4 j ) ( ) exp s (/ ) ( j ) 6 j 4 4 (4 j ) (4 j ) 6 6 where ( ) ( ) re the modfed Bessel futos 4 4 z ( z) ( ) ( ) z rg z The Aderso-Drlg test The Aderso-Drlg sttst from the esored smple o the rght s lulted ordg to the formul: r l () ( ) ( ) ( ) ( ) l l () () () 36 [3] the upper peretge pots of the Aderso-Drlg sttst dstrbuto re gve for vrous esorg degrees whe testg smple hypothess H se of omplete smple ( r ) ths sttst hs the dstrbuto of the form j (4 j ) j (4 j ) ( ) ( ) exp j 8 ( j ) exp (4 j ) y 8( y ) 8 dy THE NVETGATO N O F TATTC DTRBUTO N WHEN TETNG MPLE HYPO THEE We hve studed the Rey sttst dstrbutos for vrous dstrbutos lws wth whh observed smple s tested for ft vrous smple szes d esorg degrees by sttstl modelg methods t hs bee show tht the Rey sttst dstrbutos essetlly deped o the esorg degree For exmple fgure the emprl dstrbutos of the Rey sttst re represeted whe testg goodess-of-ft to the expoetl lw wth the desty futo x f ( x) e se of the esorg degree 9 Fg The dstrbutos of the Rey sttst R whe the smple sze 3 d 9

3 ALT`8 Jue 9- Bordeux As t s see from the fgure for the smple sze d 9 the Rey sttst dstrbutos essetlly dffer from the lmtg lw L ( ) The vestgto of the Rey sttst dstrbutos depedg o the esorg degree hs show tht the best goodess-of-ft to the lmtg lw L ( ) s rehed for 5 d for smll or o the otrry hgh esorg degrees the emprl sttst dstrbutos essetlly dffer from L ( ) otrst to the Rey test the vestgtos of the olmogorov sttst dstrbutos hve show ther good o - vergee to the orrespodg lmtg dstrbuto futos For exmple fgure the olmogorov emprl dstrbutos G( H ) d the orrespodg lmtg lws re represeted whle testg smple hypothess of goodess-of-ft to the expoetl lw Fg The dstrbutos of the olmogorov sttst for vrous esorg degrees 5 As s t see from the fgure the emprl dstrbutos of the olmogorov sttst re very lose the orrespodg lmtg lws whe the smple sze 5 d the esorg degree 5 but for 7 d 9 the emprl dstrbutos of the sttst () osderbly dffer from ther lmtg lws Thus s result of vestgtos wth omputer smulto methods suffet goodess-of-ft of the emprl dstrbutos ( G H ) to the lmtg lw ( ) hs bee show for the smple sze begg from 3 whe the esorg degree less th 5 Whle resg esorg degree up to 95 suffet goodess-of-ft of G( H ) to ( ) hs bee observed oly for 5 mlr regulrtes for the sttst dstrbutos hve bee observed for the Crmer-vo Mses-mrov test d the Aderso-Drlg test For these rter we hve ompred smple qutles obted from the emprl dstrbutos of the test sttsts wth the upper peretge pots gve [3] depedg o the smple sze d the esorg degree THE MRNO V TRANFO RMATO N AND RANDO MZATO N FO R CENO RED DATA mrov trsformto s used rther ofte sttstl lyss t s possble to move from esored smple to the smple of rdom vrbles U U U uformly dstrbuted o [] se of rght esorg we hve U F( X () ) U F( X () ) Ur F( X ) d the vlues Ur Ur U re smulted uformly o the tervl F( X ) The rdomzto s tehque of overso esored dt to smple of omplete observtos s relly pplble oly omputer lyss The lssl olmogorov Crmer-vo Mses-mrov d Aderso-Drlg tests (for omplete smples) be ppled to lyze trsformed smple We hve vestgted the sttst dstrbutos of these rter by sttstl smulto for vrous dstrbutos o whh the ull goodess-of-ft hypothess s bsed o d vrous smple szes As exmple let us osder the problem of testg smple hypothess of goodess-of-ft to the Webull dstrbuto wth the desty futo x x f x ( ) exp by esored smple of sze d r 4 The emprl dstrbutos of the olmogorov Crmer-vo Mses-mrov d Aderso-Drlg sttsts obted whe testg goodess-of-ft of the smples U U U obted fter mrov trsformto d rdomzto wth the uform o [] dstrbuto re represeted the fg 3 Fg 3 The dstrbutos of the olmogorov Crmer-vo Mses-mrov d Aderso-Drlg sttsts whe r 4 37 As t s see from the fgure 3 the emprl dstrbutos of the oprmetr olmogorov Crmer-vo Mses-mrov d Aderso-Drlg sttsts lulted by trsformed smple U U U re ord wth the orrespodg lmtg lws lredy whe depedetly o the esorg degree Ths oluso s ofrmed by the hgh sgfe levels heved whle testg goodess-of-ft of the

4 ALT`8 Jue 9- Bordeux emprl sttst dstrbutos to the orrespodg lmtg lws As t see from the fgure 4 the emprl sttst dstrbuto s lose to the pproxmto of the lmtg lw for omplete smples whe the esorg degree s smll ( r 95) But wth deresg the umber of observtos r from 6 d lower the devto to the rght from the lmtg lw osderbly reses mlr regulrtes hve bee observed whe testg omposte hypotheses of goodess-of-ft to other dstrbuto lws The devto to the rght of the emprl dstrbutos from the lmtg lw for hgh esorg degrees be expled wth the ft tht for lmted smple szes d hgh esorg degree the prmeter MLE turs out to be bsed [9] 38 V THE NVETGATO N O F TATTC DTRBUTO N WHEN TETNG CO MPO TE HYPO THEE Whe testg omposte hypotheses the estmte ˆθ whh s lulted from the sme smple tht the goodess-of-ft hypothess s tested by s used s the uow prmeter Estmtes of the dstrbuto prmeters be obted by the mxmum lelhood method whh s uversl oerg the form of dt regstrto Whe lultg mxmum lelhood estmtes (MLE) from esored smple o the rght the followg system of lelhood equtos s solved r l f( X( j) ) l P ( ) ( r) m j where m s the dmeso of the prmeter vetor ( ) f( x ) s the desty futo of the m rdom vrble P ( ) f ( x ) dx X ( r ) B Lemesho hs ppers obted the dstrbuto models pproxmtg the lmtg dstrbuto lws of the oprmetr sttsts for umber dstrbuto lws wth whh observed smple s tested for ft usg MLE The pproxmtos for vrous dstrbutos re represeted [8] Let us vestgte the oprmetr sttst dstrbutos whe usg the mrov trsformto d rdomzto MLE lulted from the esored smples re used s the uow prmeter The we ompre the emprl sttst dstrbutos obted wth the orrespodg pproxmtos gve [8] t hs bee show tht the dstrbutos of the oprmetr sttsts uder osderto whe testg omposte h y- potheses essetlly deped o the esorg degree For exmple the emprl dstrbutos of the olmogorov sttst () obted whe testg omposte hypothess of goodess-of-ft to the Webull dstrbuto re represeted the fgure 4 for the smple sze d vrous esorg degrees The pproxmto of the lmtg dstrbuto lw ( ) for the olmogorov sttst whe tes t- g goodess-of-ft to the Webull dstrbuto d estmtg the sle d form prmeters by mxmum lelhood method s lso represeted the fgure Fg 4 The olmogorov sttsts dstrbutos for V THE TET PO WER NVETGATO N The power of the oprmetr tests hs bee vestgted for vrous prs of lose ompetg hypotheses d e- pedg o the smple sze d the esorg degree the pper The test power reses wth the smple sze growth for y pr of ompetg hypotheses The test power behvor reltve to the esorg degree growth essetlly depeds o the d of hypotheses H d H uder test Let us osder s exmple the problem of testg smple hypothess H : expoetl dstrbuto wth the sle prmeter equl to gst H : the Webull dstrbuto wth the form prmeter d the sle prmeter The power s estmtes for the olmogorov-mrov test re gve se of the smple sze d the sgfe level the tble Tble Whe lultg the sttst from the orgl (wthout trsformto) esored smple r 95 r 8 r 65 r 5 r 4 r Whe lultg the sttst usg the mrov trsformto d rdomzto r 95 r 8 r 65 r 5 r 4 r 8 As t s see from the tble the test power the frst se (wthout trsformto) hges ot stedly for ths pr of ompetg hypotheses The power s estmtes for the olmogorov-mrov test se of testg smple hypothess H : expoetl dstrbuto wth the sle prmeter 5 gst H : expoetl dstrbuto wth the sle prmeter 7 re gve the tble As the prevous exmple here the smple sze d the sgft level

5 ALT`8 Jue 9- Bordeux Tble Whe lultg the sttst from the orgl (wthout trsformto) esored smple r 95 r 8 r 65 r 5 r 4 r Whe lultg the sttst usg the mrov trsformto d rdomzto r 95 r 8 r 65 r 5 r 4 r Here the test power stedly dereses wth the esorg degree growth both ses We lso hve studed the test power for umber of other prs of ompetg hypotheses Alyzg the results gve the tbles d s well s the results of other smultos the followg regulrtes hve bee observed: whe testg smple hypotheses for smll esorg degrees the power of the olmogorov-mrov test sgftly hgher whe lultg sttst from the orgl esored smples X () X () X th se of usg the mrov trsformto d rdomzto But whle resg the esorg degree the dvtge of the test by the orgl esored smple beomes more osderble omprg the test whh the olmogorov sttst s lulted from the trsformed smple U U U se of omposte hypothess testg the test power s hgher th for smple hypotheses for the sme d of ompetg hypotheses wht s ofrmed by sttstl modelg As exmple we osder testg omposte hypothess H : expoetl dstrbuto gst H : the Webull dstrbuto wth the form prmeter d the sle prmeter by smples of sze The power s estmtes re gve tble 3 for the olmogorov-mrov test the sgfe level Tble 3 Whe lultg the sttst from the orgl (wthout trsformto) esored smple r 95 r 8 r 65 r 5 r 4 r Whe lultg the sttst usg the mrov trsformto d rdomzto r 95 r 8 r 65 r 5 r 4 r As t s see from the tble 3 the test power stedly dereses wth esorg degree growth mlrly to the se of smple hypotheses the test power s hgher whe the sttst s lulted from the orgl esored smples X () X () X th se of usg the mrov trsformto d rdomzto d dfferee the power reses wth the esorg degree growth V CO NCLUO N The Rey sttst dstrbutos hve bee vestgted for vrous smple szes d esorg degrees t hs bee show tht the sttst dstrbutos overge to the lmtg lw L ( ) very slowly espelly for hgh or o the otrry 39 very low esorg degrees Ths result does t llow reommedg usg the Rey test prte The results of vestgtg the dstrbutos the olmogorov sttst Crmer-vo Mses-mrov sttst d the Aderso-Drlg sttst eble to olude good possblty to use the pproh osdered (mrov trsformto wth rdomzto) for orret pplto of the lssl op - rmetr goodess-of-ft tests for esored dt se of smple hypothess testg the dstrbutos of the bove sttsts overge to the orrespodg lmtg dstrbutos very quly For the smple sze oe use the lmtg lws wthout rs of mg gret mste for y esorg degree The pplto of mrov trsformto wth rdomzto s qute effet for relzto softwre systems of sttstl lyss The oprmetr tests usg the mrov trsformto d rdomzto re t dsdvtge by power omprg wth the orrespodg tests wthout dt trsformto espelly for hgh esorg degrees Whe testg omposte hypotheses from esored dt d estmtg uow prmeters by mxmum lelhood method the dstrbutos of oprmetr test sttsts osderbly deped o the esorg degree REFERENCE [] Brr DM Dvdso T A olmogorov-mrov test for esored smples Tehometrs 973 Vol 5 N 4 [] Rey A O the theory of order sttsts // At Mthem Ad Hug 953 Vol 4 P 9-3 [3] Petttt AN tephes MA Modfed Crmer vo Mses sttsts for esored dt // Bometr 976 Vol 63 N [4] Lemesho B Yu Errors whe usg oprmetr fttg rter // Mesuremet Tehques 4Vol 47 N P34-4 [5] Lemesho BYu Postovlov N Frtsuzov AV Applto of the oprmetr goodess-of-ft tests to testg oprmetr model dequy // Optoeletros strumetto d Dt Proessg N P 3- [6] Greewood PE Nul M A Gude to Ch-qured Testg Joh Wley & os p [7] Bolshev LN mrov NV Tbles of Mthemtl ttsts Mosow ee 983 ( Russ) [8] R 537- Reommedtos for stdrdzto Appled sttsts Rules of he of expermetl d theoretl dstrbuto of the oset Prt Noprmetr goodess of ft test Mosow Publshg house of the stdrds - 64 p ( Russ) [9] Lemesho BYu Chmtov EV vestgto of the estmtes propertes d goodess-of-ft test sttsts from esored smples wth omputer modelg tehque // Proeedgs of the eveth tertol Coferee Computer Dt Alyss d Modelg: Robustess d Computer tesve Methods eptember 6-4 Ms Vol P 43-46