Inference for L-Estimators of Location Using a Bootstrap Warping Approach

Transcription

1 Statstcs Research Letters (SRL Volume 5 6 do.4355/srl Iferece for L-Estmators of Locato Usg a Bootstrap Warpg Approach Ala D. Hutso Roswell Park Cacer Isttute Departmet of Bostatstcs & Boformatcs Elm& Carlto Streets Buffalo NY 463 USA Abstract I ths ote we propose a ew sem-parametrc bootstrap procedure for hy-pothess tests about a statstcal fucto ad termed bootstrap warpg. Ths procedure was motvated by emprcal lkelhood ad bootstrap tltg techques. The procedure s computatoally effcet ad has a fxed umber of parameters. We show that the warpg procedure has good type I error cotrol ad has moo-toe power as a fucto of sample sze ad shft alteratves. Keywords L-estmators; L-momets Itroducto Let deote a..d. sample draw from a absolutely cotuous populato wth cumulatve dstrbuto fucto (c.d.f. deoted as F ad correspodg quatle fucto deoted as Q(u = F (u. For the applcato descrbed ths ote we are terested makg fereces about a oe-dmesoal parameter of the form θ= T(F where our methodology T(F deotes a specfc smooth statstcal fuctoal for measurg expectato ad havg the form g( j( F( g( j( F( df T ( F E g( Q( u j( u du. (. I our applcatos we restrct j(f( to be a smooth absolutely cotuous weghtg fucto such that ( F( x df j( u du j s essetally a p.d.f. for j(u> u ( ad we assume g (x df s bouded e.g. suppose the parameter of terest s the populato mea the the statstcal fuctoal has the well-kow form T( F E( g( j( F( xdf Q( u du wth j ( F( x or alteratvely j(u= ad g(x = x. The classc bootstrap estmator of T(F s gve by replacg the c.d.f. F wth ts emprcal couterpart ( x I (. or alteratvely replacg Q wth ts emprcal couterpart Q ˆ( u x u where x x / ( [ ] deotes the floor fucto ad deotes the th order statstc. Substtutg to (. for F yelds the emprcal estmator of T(F whch has the well-kow form / T ( g( j( u du. (. ( / Some classc examples for T ( clude kerel desty ad quatle estmators sample momet estmators ad A Lestmators e.g. see Serflg (98 for a techcal overvew of estmators havg ths form relatve to ther asymptotc propertes. Now suppose we are terested testg a hypothess about the gve statstcal fuctoal T(F havg the form H F F or wthout loss of geeralty H F a semparametrc fasho. Note that we wll touch o two-sded tests later ths ote. Popular oparametrc approaches for testg hypotheses of ths type are gve by the well-kow emprcal lkelhood (EL method due to Owe (988 ad

2 Statstcs Research Letters (SRL Volume bootstrap tltg methods such as expoetal tltg or other multomal based resamplg schemes e.g. see Davso ad Hkley (997. The oparametrc EL ad bootstrap approaches provde the motvato for our ew semparametrc testg methodology. The key dea behd the EL ad bootstrap tltg approaches s to fd the oparametrc maxmum lkelhood estmator for the probablty desty fucto (p.d.f. f ad c.d.f F gve the costrat T ( F prescrbed uder H as estmated by ts emprcal couterpart T ( where u I ( xx ad the v parameters correspodg to f sum to ad are bouded betwee ad. The commo defto for the v parameters the cotuous case for the dscretzed model s gve as v F( x F( x where F( x P( x ad F( x P( x respectvely e.g. see Owe (988 for the techcal argumet pertag to ths defto. I the most commo scearo the lkelhood uder the ucostraed alteratve hypothess yelds the classc estmates of v / for smple statstcs such as the sample mea. Other weghts may occur for fuctoals correspodg to trmmed estmators e.g. see Q ad Taso ( wth respect to the weghts for trmmed mea. The weghts v uder the ull hypothess are geerally determed by mmzg a gve dstace measure such as the Kullback-Lebler dstace D( v v v log (v where the vectors v= (/ / / ad v=(v v v. Alteratvely oe may use costraed maxmum lkelhood approaches for determg the vector v=(v v v uder H e.g. see Vexler ad Gurevch ( a typcal model scearo.we use ths dea of a dscretzed model as a startg pot for developg a alteratvebased o smooth statstcal fuctoal feretal procedure usg what we term statstcalwarpg as defed the ext secto. I ths Secto we outle the bootstrap warpg procedure ad follow ths wthsmulato study Secto 3. Bootstrap Warpg ad Hypothess Testg The key features of bootstrap warpg as cotrasted wth the EL ad bootstrap tltg approaches s that t s a sem-parametrc approach ad that the umber of parameters the model s reduced from to codtoal o the observed data.e. we warp the observed e.d.f. ( x I versus treatg each dscretzed x x / ( segmet o the cotuum as a parameter. I addto our resamplg scheme follows the classc bootstrap multomal resamplg scheme wth cell probabltes of / versus bootstrap tltg whch requres the weghts to be determed codtoal o the dataset uder vestgato thus addg a layer of complexty to the computatoal compoets to these problems. The drect beeft of ths parameterzato s computatoal ease wthout sufferg the curse of dmesoalty assocated wth bg data scearos. Ths wll be descrbed detal below. Addtoally terms of future work covarate adjustmets may be made through the warpg model parameters thus extedg the utlty of ths approach to more complex settgs. I terms of testg H F F we eed to frst defe T ( F ˆ relatve to obtag a emprcal verso of the costrat T ( F. Towards ths ed we defe the warped emprcal estmator of T ( F based o formulato (. as ˆ T( F g( K J( K J( (. g( f ( (. 3

3 Statstcs Research Letters (SRL Volume 5 6 where J( v j( u du f s a weghtg fucto defed more formally below at (.5 j(. s defed at v ( (. ad we defe K such that K ( t t E(g( the T uder H. For example f we were terested testg about K K (.3 ( g( ad uder H true we would have T( g(. / The compoets of the weghtg fucto (. deoted as K ( are defed as the c.d.f. of a Kumaraswamy dstrbuto ad gve as where u ad K ( u ( u (.4. The choce of the Kumaraswamy dstrbuto terms of a weghtg fucto s due to ts umercal tractablty ad flexblty terms of the relatve shapes t cotas.e. our test wll be sestve to a umber of alteratves gve H va the choce of ths weghtg fucto. Our sem-parametrc desty utlzed wth (. s ow defed as dscretzed type model smlar to what s used the EL methodology ad bootstrap tltg ad gve as ( ( ( f F F K J K J (.5 where F( x P( x ad F( x P( x ad f represets a pot mass correspodg to the th order statstc. The Kumaraswamy dstrbuto was chose over other caddate dstrbutos e.g. the beta dstrbuto due to ts well-behaved umercal propertes ad relatvely straghtforward parameterzato. See Joes (9 for a detaled descrpto of the Kumaraswamy dstrbuto ad a descrpto of ts close relatoshp to the beta dstrbuto. I essece f serves as a stadard weghtg fucto such that whe ad the T ( equates to F ˆ T. ( ( The test of terest ths ote s gve as H t( F t( F. As EL methods ad bootstrap ( tltg the frst step s to maxmze the costraed pseudo-lkelhood L f ( (.6 wth respect to ad ad uder the costrat H F where f ( x ad gve H s true. The bootstrap resamplg scheme for our feretal method the s as follows Calculate the observed test statstc t (. Obta ad ˆ from (.6. s defed at (.5. Clearly 3 Geerate B oparametrc bootstrap samples of sze.e. geerate uform ( radom varables ad apply Q ˆ( u x u to those radomly geerated uform varates. * 4 Calculate ˆ ˆ T ( from (. replacg wth ˆ ad wth ˆ ( for = B. 4

4 Statstcs Research Letters (SRL Volume j B * 5 Calculate the approxmate oe-sded bootstrap p-value pboot I( T ( ˆ ˆ t( / B. For a test of H F F smply reverse the equalty step (5 above. For the test H F F there s a added assumpto of the symmetry of the dstrbuto of * ˆ ˆ uder H. Uder ths assumpto the two-sded p-value s gve as T ( j B * pboot I( T ( t( ˆ ˆ t( / B. I geeral for most of the tests of terest the test statstc wll have a asymptotc ormal dstrbuto thus most two-sded tests should satsfy approxmately a symmetry assumpto.e. the statstcs are based o smooth fuctos whch the tur led themselves to well-behaved ad symmetrc bootstrap resamplg dstrbutos. As wth smlar bootstrappg methodologes for ferece. e.g. see Davso ad Hkley (997 the key s that T s a cosstet estmator of F uder H whch the methodology preseted above holds gve ad ( uder H coverges to F ad T ( coverges to (F T gve the smoothess codtos outled earler whch s by defto the statstcal fucto of terest e.g. see va der Vaart (998. The large sample proof of ths cocept s gve by the followg theorem Theorem. Uder H F ad as ( ˆ ˆ has a cetered bvarate ormal dstrbuto wth varace-covarace matrx B B assocated wth the Kumaraswamy desty dmesoal radom vector whose compoets are gve by where where B s the stadard maxmum lkelhood based formato matrx k dk ad s the varace-covarace matrx of a - log f ( / Walpha( (.7 log f ( / Wbeta ( (8 (.9 Proof. W W I( F ( log( f ( u df ( u u (. u ( I( F ( log( f ( u df ( u. u (. u ( The techcal detals have bee worked out a elegat fasho for the case of a sem-parametrc copula model wth margal dstrbuto fuctos estmated by the emprcal dstrbuto fucto estmator. The result Theorem follows drectly from the theoretcal developmets used the copula approach Secto 4 of the copula paper Geest et al. (995 by smply replacg the multvarate copula fucto wth the uvarate beta desty whch s essetally a specal case of the hgher dmeso copula model. Estmates of the varacecovarace matrx B B are ot as straghtforward to obta ad we recommed bootstrap resamplg for ths purpose. SIMULATION RESULTS For our smulato study we focused o the trmmed mea wth kow statstcal fuctoal gve as 5

5 Statstcs Research Letters (SRL Volume 5 6 T ( F Q Q ( ( xdf ( / Q u du. Smlar results terms of behavor hold for momet estmators ad kerel estmators ad are ot preseted here. We cetered our smulato study for the trmmed mea o the hypothess test H T( F T( F at type I error rate.5 for symmetrc dstrbutos wth trmmg proportos =.. ad samples of sze = 5. For each smulato result we utlzed teratos wth the umber of bootstrap resamples set to B= 5. For the expoetal dstrbuto we tested H T( F T ( T( F T ( wth shfted expoetal alteratves. For the F trmmed mea we ca smplfy (.3 such that F T( g( k k K( K( k k where k. For power examatos we used shft alteratves of δ=.5 ad δ = dfferet from H. (3. The results of our smulato study are preseted Table. We see that the type I error s cotrolled at the omal level ad that fluctuatos about that level are prmarly due to smulato error. The power s mootoe crease δ ad. As compared wth a optmal scearo such as a t-test uder ormalty ad δ =.5 the oesample t-test has power of ad.967 as compared to the warpg powers of ad.966 for samples of sze of ad 5 respectvely. Table type error ad power δ= Normal Logstc Expoetal δ=.5 Normal Logstc Expoetal δ= Normal Logstc Expoetal ACKNOWLEDGMENTS We wsh to thak the referee ad assocate edtor for ther postve commets. Ths work was supported by Roswell Park Cacer Isttute ad Natoal Cacer Isttute (NCI grat P3CA656. REFERENCES [] Davso A. C. ad Hkley D. V. (997 Bootstrap Methods ad ther Applcatos. Cambrdge Seres Statstcal ad Probablstc Mathematcs Cambrdge Uversty Press New York NY. 6

6 Statstcs Research Letters (SRL Volume [] Geest C. Ghoud K. ad Rvest L. P. (995 A semparametrc estmato procedure of depedece parameters multvarate famles of dstrbutos. Bometrka [3] Joes M.C. (9 Kumaraswamy s dstrbuto A beta-type dstrbuto wth some tractablty advatages. Statstcal Methodology [4] Owe A. B. (988 Emprcal lkelhood rato cofdece tervals for a sgle fuctoal. Bometrka [5] Q G. ad Tsao M. ( Emprcal lkelhood rato cofdece tervals for the trmmed mea. Commucatos Statstcs Theory ad Methods [6] Serflg R. J. (98 Approxmato Theorems of Mathematcal Statstcs. Joh Wley & Sos New York. [7] va der Vaart (998 Asymptotc Statstcs. Cambrdge Seres Statstcal ad Probablstc Mathematcs Cambrdge Uted Kgdom. [8] Vexler A. ad Gurevch G. (. Emprcal Lkelhood Ratos Appled to Goodess-of-Ft Tests Based o Sample Etropy. Computatoal Statstcs ad Data Aalyss