A Direct Approach to Inference in Nonparametric and Semiparametric Quantile Models


 Percival Pope
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1 A Direct Approach to Iferece i Noparametric ad Semiparametric Quatile Models Yaqi Fa ad Ruixua Liu Uiversity of Washigto, Seattle Workig Paper o. 40 Ceter for Statistics ad the Social Scieces Uiversity of Washigto First versio: December 202 This versio: September 203 Abstract This paper makes two mai cotributios to iferece for coditioal quatiles. First, we costruct a geeric cofidece iterval for a coditioal quatile from ay give estimator of the coditioal quatile via the direct approach. Our geeric cofidece iterval makes use of two estimates of the coditioal quatile fuctio evaluated at two appropriately chose quatile levels. I cotrast to the stadard Wald type cofidece iterval, ours circumvets the eed to estimate the coditioal desity fuctio of the depedet variable give the covariate. We show that our ew cofidece iterval is asymptotically valid for ay quatile fuctio parametric, oparametric, or semiparametric, ay coditioal quatile estimator stadard kerel, local polyomial or sieve estimates, ad ay data structure radom samples, time series, or cesored data, provided that certai weak covergece of the coditioal quatile process holds for the prelimiary quatile estimator. I the same spirit, we also costruct a geeric cofidece bad for the coditioal quatile fuctio across a rage of covariate values. Secod, we use a specific estimator, the YagStute also kow as the symmetrized knn estimator for a oparametric quatile fuctio, ad two popular semiparametric quatile fuctios to demostrate that oftetimes by a judicious choice of the quatile estimator combied with the specific model structure, oe may further take advatage of the flexibility ad simplicity of the direct approach. For istace, by usig the YagStute estimator, we costruct cofidece itervals ad bads for a oparametric ad two semiparametric quatile fuctios that are free from additioal badwidth choices ivolved i estimatig ot oly the coditioal but also the margial desity fuctios ad that are very easy to implemet. The advatages of our ew cofidece itervals are bore out i a simulatio study. Keywords: Geeric Cofidece Iterval; Geeric Cofidece Bad; Partially Liear Quatile Regressio; SigleIdex Quatile Regressio; Rearraged Quatile Curve JEL Codes: C2; C4; C2 Departmet of Ecoomics, Uiversity of Washigto, Box , Seattle, WA We thak Matias Cattaeo, Xumig He, YuChi Hsu, Sag Soo Park, ad semiar participats at Northwester, Pekig Uiversity, Simo Fraser Uiversity, Texas A&M, Uiversity of British Clumbia, ad the Uiversity of Michiga, as well as participats at 203 Tsighua Iteratioal Coferece i Ecoometrics ad the 203 North America Ecoometric Society Meetigs for helpful commets ad discussios. We also thak Sog Sog for sharig his code with us ad Jo Weller for fidig the referece, Thompso 936, for us.
2 Itroductio I their semial paper, Koeker ad Bassett 978 propose to use liear quatile regressio to examie effects of a observable covariate o the distributio of a depedet variable other tha the mea. Sice the, liear quatile regressio has become a domiat approach i empirical work i ecoomics, see e.g., Buchisky 994 ad Koeker Followig Koeker ad Bassett 978, this approach has bee exteded to cesored data i Powell 986, Buchisky ad Hah 998, Hoore, Kha ad Powell 2002, ad to uit root quatile regressio models i Koeker ad Xiao 2004, further broadeig its scope of applicatios. Liearity adopted i Koeker ad Bassett 978 has bee relaxed to accommodate possibly oliear effects of the covariates o the coditioal quatile of the depedet variable i oparametric ad semiparametric quatile regressio models. The check fuctio approach of Koeker ad Bassett 978 has bee exteded to estimatig these models as well, see e.g., Truog 989, Chaudhuri 99, ad He, Ng, ad Portoy 998 for oparametric estimatio of coditioal quatiles; Chaudhuri, Doksum, ad Samarov 997 for oparametric average derivative quatile estimatio; Fa, Hu ad Truog 994, Yu ad Joes 998 ad Guerre ad Sabbah 202 for local polyomial estimatio of regressio quatiles; Lee 2003 ad Sog, Ritov, ad Hardle 202 for partial liear quatile regressio models; Wu, Yu, ad Yu 200 ad Kog ad Xia 202 for sigle idex quatile regressio models; ad Che ad Kha 200 for partially liear cesored regressio models. For oparametric quatile regressio models, a alterative estimatio approach to the check fuctio approach is take i Stute 986, Bhattacharya ad Gagopadhyay 990, Fa ad Liu 20, ad Li ad Racie 2008, amog others. I this approach, the coditioal distributio fuctio of the depedet variable Y give the covariate X is estimated first ad the geeralized iverse of this estimator at a give quatile level p 0, is take as a estimator of the pth coditioal quatile. Stute 986 ad Bhattacharya ad Gagopadhyay 990 focus o uivariate covariate ad estimate the coditioal distributio fuctio by knn method, while Fa ad Liu 20 ad Li ad Racie 2008 allow for multivariate covariate ad adopt respectively knn ad kerel estimators of the coditioal distributio fuctio. Uder regularity coditios, existig work establish asymptotic ormality of the coditioal quatile estimators which is the basis for the Waldtype iferece, i.e., usig the t statistic to test hypotheses or form cofidece itervals CI for the true coditioal quatiles. Regardless of the approach used to estimate the coditioal quatile i parametric, semiparametric, or oparametric quatile regressio models, oe commo feature of the asymptotic distributios of the coditioal Coditioal quatile fuctio also plays a importat role i the oseparable structural ecoometrics literature, see e.g., Chesher 2003, Holderlei ad Mamme 2007 ad i the estimatio of quatile treatmet effects, see e.g., Firpo 2007 ad Fa ad Park 20.
3 quatile estimators is that their asymptotic variaces deped o the coditioal quatile desity fuctio of Y give X = x ad some eve deped o the desity fuctio of X, see e.g., Horowitz 998, Kha 200, Koeker ad Xiao 2002, Li ad Racie 2008, Hardle ad Sog 200, ad Sog, Ritov, ad Hardle 202, amog others. As a result, iferece procedures for the coditioal quatiles based o the asymptotic distributios of these estimators require cosistet estimators of the coditioal quatile desity fuctio of Y give X = x ad/or the desity of X both ivolvig badwidth choice. Numerical evidece preseted i De Agelis, Hall, ad Youg 993, Buchisky 995, Horowitz 998, ad Kochergisky, He, ad Mu 2005 shows that although asymptotically valid, these iferece procedures are sesitive i fiite samples to the choice of smoothig parameter used to estimate the coditioal quatile desity fuctio. Various alterative approaches have bee proposed i the curret literature to improve o the fiite sample performace of Waldtype ifereces. Most of these are developed for liear or parametric coditioal quatile regressio models. First, Goh ad Kight 2009 propose a differet scale statistic to stadardize the estimator of the model parameter i liear quatile regressio models resultig i a ostadard iferece procedure; Secod, Zhou ad Portoy 996 costruct cofidece itervals/bads directly from pairs of estimates of coditioal quatiles i the locatioscale forms of liear quatile regressio models extedig the direct or order statistics approach for sample quatiles i Thompso 936, see also Serflig 980, Csorgo 983, ad va der Vaart 998; Third, Gutebruer ad Jureckova 992 ad Gutebruer, Jureckova, Koeker, ad Portoy 993 employ rak scores to test a class of liear hypotheses; Fourth, Whag 2006 ad Otsu 2008 apply the empirical likelihood approach to parametric quatile regressio models; Lastly, MCMC related approaches have bee proposed to improve stadard resamplig or simulatio paradigms: He ad Hu 2002 resample estimators from the margial estimatig equatio alog the geerated Markov chai; ad Cherozhukov, Hase, ad Jasse 2009 develop fiite sample iferece procedures based o coditioal pivotal statistics i parametric quatile regressio models. A ice survey of various iferece procedures targeted at liear quatile regressio models could be foud i Kochergisky, He, ad Mu Compared with parametric quatile regressio models, iferece i oparametric ad semiparametric quatile regressio models is still i its ifacy. The oly alterative approach to the Waldtype ad bootstrap ifereces that is curretly available is the empirical likelihood procedure i Xu 202 for oparametric quatile regressio models. I semiparametric quatile regressio models icludig partial liear ad sigle idex models, oly Waldtype ad bootstrap ifereces are available. Although the empirical likelihood approach i Xu 202 avoids estimatio of the coditioal quatile desity fuctio ad performs better tha the Waldtype iferece procedures, it is kow to be computatioally costly. Amog existig approaches to iferece i parametric quatile regressio models, the direct approach is the simplest to implemet ad least costly 2
4 computatioally it oly requires computig pairs of the quatile estimate. I additio, it does ot rely o ay estimate of the coditioal quatile desity fuctio ad exhibits superior fiite sample performace compared with the Waldtype iferece, see Zhou ad Portoy 996. However, as discussed i Portoy 202, it appears that the direct approach i Zhou ad Portoy 996 has theoretical justificatio oly uder locatioscale forms of liear quatile regressio models. This paper aims at bridgig this gap. Specifically, it makes two mai cotributios to iferece o coditioal quatiles. First, we costruct a geeric cofidece iterval CI for a coditioal quatile from ay give estimator of the coditioal quatile via the direct approach. Our geeric cofidece iterval makes use of two estimates of the coditioal quatile fuctio evaluated at two appropriately chose quatile levels. If the origial quatile estimator is mootoe i the quatile level p 0,, the the two estimates are computed from this estimator; else the two estimates are computed from the mootoe rearraged versio of the origial quatile estimator as proposed i Cherozhukov, FeradezVal, ad Galicho 200. I cotrast to the stadard Wald type cofidece iterval, ours circumvets the eed to estimate the coditioal desity fuctio of the depedet variable give the covariate. We show that our ew cofidece iterval is asymptotically valid for ay quatile fuctio parametric, oparametric, or semiparametric, ay coditioal quatile estimator stadard kerel, local polyomial or sieve estimates, ad ay data structure radom samples, time series, or cesored data, provided that certai weak covergece of the coditioal quatile process holds for the prelimiary quatile estimator. I the same spirit, we also costruct a geeric cofidece bad CB for the coditioal quatile fuctio across a rage of covariate values focusig o the oparametric settig ad a class of quatile estimators obtaied from ivertig proper estimators of the coditioal distributio fuctio of Y give X. Sice members of this class of quatile estimators are mootoe by costructio, mootoe rearragemet is avoided. Secod, we use a specific estimator, the YagStute also kow as the symmetrized knn estimator for a oparametric quatile fuctio, ad two popular semiparametric quatile fuctios to demostrate that oftetimes by a judicious choice of the quatile estimator combied with the specific model structure, oe may further take advatage of the flexibility ad simplicity of the direct approach. For istace, by usig the YagStute estimator, we costruct cofidece itervals ad bads for a oparametric ad two semiparametric quatile fuctios that are free from additioal badwidth choices ivolved i estimatig ot oly the coditioal but also the margial desity fuctios ad that are very easy to compute. The reaso that we choose the Yag Stute estimator is its simplicity ad elegace; It iherits the socalled asymptotic distributioalfree property Stute, 984b ad avoids estimatig covariate s margial desity fuctio ulike stadard kerel estimators, so we are able to elimiate all uecessary tuig parameters. Besides, as we directly ivert coditioal distributio fuctios, the resultig coditioal quatile estimators are ideed mootoe, so there is o eed for mootoe rearragemet. Of course, practitioers 3
5 are free to choose their favorite prelimiary quatile estimators ad uder the mild high level assumptios below, our geeric CIs/CBs would apply. Like the empirical likelihood cofidece iterval for a oparametric quatile fuctio i Xu 202, our cofidece itervals/bads for oparametric quatiles based o the YagStute estimator iteralize the coditioal quatile desity estimatio of Y give X ad the covariate desity estimatio ad they are ot ecessarily symmetric. Compared with Xu 202, our procedure is much easier to implemet ad does ot require optimizatio. For coditioal quatiles i partial liear ad sigle idex quatile regressios, a direct applicatio of the geeric CI ad CB would require mootoe rearragemet, but by makig use of the model structures, we costruct CIs ad CBs that are easy to implemet avoidig mootoe rearragemet. A small scale simulatio study demostrates the advatages ad feasibility of our cofidece itervals/bads over existig oes i practically relevat model setups. Fially, we poit out that there is a iterestig coectio betwee our geeric CI ad the wellkow CI for ucoditioal quatiles based o order statistics origially proposed i Thompso 936, see also Serflig 980 ad va der Vaart 998. I fact by usig pairs of the stadard knn asymmetric quatile estimate, our geeric CI employes pairs of order statistics of the iduced order statistics of Y, so it shares the elegace ad simplicity of the cofidece iterval for ucoditioal quatiles based o order statistics. The rest of this paper is orgaized as follows. Sectio 2. presets our geeric cofidece iterval ad shows its asymptotic validity uder a high level assumptio o the prelimiary quatile estimator. The high level assumptio is verified i four examples icludig the asymmetric knn estimator, local polyomial quatile regressio, a oparametric quatile regressio with cesorig, ad the class of coditioal quatile estimators i Doald, Hsu, ad Barrett 202 which icludes parametric quatile estimators as well. A geeric CB is proposed i Sectio 2.2. Sectio 3. cosiders the oparametric quatile regressio with a uivariate covariate. It costructs a ew cofidece iterval ad a ew cofidece bad usig the YagStute estimator. Sectio 3.2 exteds the cofidece itervals/bads developed i Sectio 3. to two popular semiparametric models, partial liear ad sigle idex quatile regressio models. Sectio 4 provides a simulatio study comparig the fiite sample performace of our ew cofidece itervals with Waldtype cofidece itervals ad two bootstrap versios for oparametric ad partial liear quatile regressios. We coclude i the last sectio. All the techical proofs are collected i the Appedices. 2 Geeric Results o the Direct Approach to Quatile Iferece Cosider the radom vector X, Y with margial distributio fuctios F X x, F Y y respectively, where x X R d ad y Y R. Let F Y X x deote the coditioal distributio fuctio of Y give X = x with desity fuctio f Y X x. For 0 < p <, we are iterested i coductig iferece o the pth coditioal quatile of Y give X = x: ξ p x F Y X p x either 4
6 at a specific locatio x = x 0 X or for all x i a subset of the support of X. Let ξ p x deote a cosistet ad asymptotically ormally distributed estimator of ξ p x. To itroduce the direct approach to quatile iferece, cosider iferece for ξ p x 0 for a fixed x 0 X. Suppose ξ p x is mootoe i the quatile level p 0,. The the CI for ξ p x 0 based o the direct approach takes the form of a closed iterval with the ed poits give by ξ x 0 evaluated at two appropriately chose quatile levels, oe smaller ad oe larger tha p, see 3 below with ξ p x 0 beig replaced by ξ p x 0. I cotrast to Waldtype CIs, CIs based o the direct approach are ot depedet o ay estimate of the coditioal desity fuctio of Y give X = x 0. To esure the validity of the resultig CI, it is essetial that the quatile estimator beig used is mootoe i the quatile level p 0,. It is well kow that some commoly used quatile estimators icludig the liear quatile estimator of Koeker ad Bassett 978 ad local polyomial quatile estimators are ot mootoe i the quatile level. This is kow as the quatile crossig problem He, 997; Cherozhukov, FeradezVal, ad Galicho, 200. To rectify this issue, various methods of mootoizatio have bee proposed i the literature icludig Cherozhukov, FeradezVal, ad Galicho 200 who propose a mootoe rearraged quatile estimator from a give prelimiary quatile estimator ad Dette ad Volgushev 2008 who propose smooth mootoe quatile estimators from cosistet estimators of the coditioal distributio fuctio F Y X x. Of course quatile estimators costructed from ivertig mootoe estimators of the coditioal distributio fuctio F Y X x are also mootoe. I Sectio 2., we costruct a CI for ξ p x 0 from a prelimiary cosistet estimator ξ p x 0 usig the direct approach ad refer to it as the geeric CI. For omootoe ξ p x 0, our geeric CI makes use of the rearraged versio of ξ p x 0 i Cherozhukov, FeradezVal, ad Galicho I Sectio 2.2, we costruct a geeric CB from the direct approach which is valid for all x i a compact subset of X. 2. A Geeric Cofidece Iterval For ay cosistet estimator ξ p x of ξ p x, the mootoe versio of ξ p x i Cherozhukov, FeradezVal, ad Galicho 200 is based o the fact that F Y X y x = ξ y x = 0 I {ξ u x y} du, so we ca replace ξu x with ξu x i the expressio o the right had side of to get a mootoe estimator of F Y X y x. Sice the resultig estimator of F Y X y x is mootoe, its geeralized 2 We could use the smooth mootoe estimators proposed i Dette ad Volgushev 2008 as well. But sice they ivolve the choice of a additioal smoothig parameter, we fid the method i Cherozhukov, FeradezVal, ad Galicho 200 more suitable for our purpose. Marmer ad Sheyerov 202 offer a alterative method for costructig a mootoe quatile estimator from a prelimiary quatile estimator, but the asymptotic properties of their mootoe estimator are ukow. 5
7 iverse is a cosistet ad mootoe estimator of ξ p x. This is the mootoe rearraged versio of ξ p x 0 proposed by Cherozhukov, FeradezVal, ad Galicho 200: ξ p x = if y : 0 { ξ u x y}du p. 2 We ote that if the origial estimator ξ p x is mootoe i p 0,, the ξ p x = ξ p x for all p 0,. So we will use ξ p x i this sectio to itroduce our geeric CI for ξ p x 0. Below we first provide assumptios o the quatile fuctio ξ p x 0 ad a high level assumptio o the origial estimator ξ p x 0 uder which our geeric CI is asymptotically valid ad the verify the high level assumptio for four examples i Sectio 2... Assumptio GI i ξ p x 0 is a cotiuously differetiable fuctio i p 0, ad for fixed p 0,, ξ p x is cotiuously differetiable at x = x 0 ; ii Let q p x p ξ p x = /f Y X ξ p x x deote the coditioal quatile desity fuctio. The q p x 0 > 0 for p p, p 2 0, ; iii The quatile estimator ξ x 0 takes its values i the space of bouded measurable fuctios defied o p, p 2 0,, where p p, p 2, ad i l p, p 2, c ξ x0 ξ x 0 = q x 0 B x 0, as a stochastic process idexed by p p, p 2, where {B p x 0, p p, p 2 } is a Gaussia process with variace σ 2 p x 0 V ar B p x 0 which does ot deped o q p x 0 ad c is a sequece of positive costats such that c as. Assumptio GI i ad ii are take directly from Cherozhukov, FeradezVal, ad Galicho 200. Assumptio GI iii is a special case of Assumptio 2 i Cherozhukov, Feradez Val, ad Galicho 200. It imposes a specific structure o the asymptotic variace of the quatile estimator ξ p x 0 which esures the asymptotic validity of the followig geeric cofidece iterval obtaied from the direct approach: CIG α = ξ p z α/2 σ p x 0 x 0, c ξ p + z α/2 σ p x 0 x 0, 3 c where α 0,, σ p x 0 is a cosistet estimator of σ p x 0, ad z α/2 is the upper quatile of stadard ormal radom variable N, i.e. Pr { N > z α/2 } = α/2. For may quatile estimators ξ p x regardless of the model ad data structure, Assumptio GI iii is either established i existig work or ca be show usig results i existig work, see Sectio 2.. ad Sectio 3 for examples of parametric, oparametric ad semiparametric quatile estimators. Moreover, for may quatile estimators, σ 2 p x 0 takes the form of p p ϖ 2 x 0 for some positive costat ϖ x0 depedig o x 0, see the first three examples i Sectio 2.. ad Sectio 3. Example 2.4 presets 6
8 a example of σ 2 p x 0 that does ot take this form. 3 I additio to the examples i Sectio 2.. ad Sectio 3, aother parametric example of ξ p x is the quatile estimator of Koeker ad Bassett 978. Uder stadard regularity coditios, the quatile estimator of Koeker ad Bassett 978 satisfies Assumptio GI iii for the special class of locatio scale forms of liear quatile regressio models, see Zhou ad Portoy 996, Gutebruer ad Jureckova 992, Koeker ad Xiao 2005, ad Portoy 202, so our geeric cofidece iterval CIG α defied i 3 is asymptotically valid. I fact, for locatio scale forms of the liear quatile regressio models, our geeric cofidece iterval CIG α is just the cofidece iterval i Zhou ad Portoy 996 usig the origial estimator of Koeker ad Bassett 978, due to the absece of quatile crossig problem as demostrated by He 997. THEOREM 2. Suppose Assumptio GI holds. coverage probability equal to α. The CIG α is asymptotically valid with Proof. Resortig to Corollary 3 i Cherozhukov, FeradezVal, ad Galicho 200 which asserts that the rearraged estimator ξ p x 0 has the same first order asymptotic properties as ξ p x0. I particular, Assumptio GI iii implies that c ξ p x0 ξ p x 0 = q p x 0 B p x 0. Makig use of stochastic equicotiuity of the process { } c ξ p x0 ξ p x 0, p 0, ad c, we have c ξ p ± z α/2 σ p x 0 x 0 c ξ p x 0 = c ξ p ± z α/2 σ p x 0 x 0 ξ p x 0 + o p. c Now use the simple fact that uder Assumptio GI i ad cosistecy of σ p x 0 : c ξ p ± z α/2 σ p x 0 x 0 ξ p x 0 = ±q p x 0 z c α/2 σ p x 0 + o p. Hece the geeric cofidece iterval is asymptotically valid: lim Pr {ξ p x 0 CIG α } { = lim Pr ξ p z α/2 σ p x 0 x 0 c = lim Pr = α. ξ p x 0 ξ p + z } α/2 σ p x 0 x 0 c { ξ p x0 z α/2σ p x 0 c q p x 0 ξ p x 0 ξ p x 0 + z α/2σ p x 0 c q p x 0 3 We thak YuChi Hsu for suggestig this example. } Q.E.D 7
9 Remark 2.. Uder Assumptio GI, the stadard Wald type cofidece iterval is costructed by ceterig aroud ξ p x 0 with the stadard error multiplied by the ormal critical value as ξ p x0 z α/2 σ p x 0 f Y X ξ p x0 x 0 c, ξ p x 0 + z α/2 σ p x 0 f Y X ξ p x0 x 0 c, 4 where f Y X y x 0 is a cosistet estimator of f Y X y x 0 such as a kerel coditioal desity estimator. It is well kow that the fiite sample performace of the Wald CI i 4 is very sesitive to the choice of the smoothig parameter ivolved i the estimate f Y X y x 0. Distict from the Wald type iterval, our ew cofidece iterval i 3 avoids the estimatio of the coditioal desity fuctio f Y X y x 0 ad the two ed poits of the CI are ot ecessarily symmetric aroud ξ p x Examples of the Geeric Cofidece Iterval Let { X i, Y i } i= deote the sample iformatio o X, Y. It could be a radom sample or a time series. To demostrate the broad applicability of the cofidece iterval, CIG α, defied i 3, we preset four examples i this subsectio. They iclude a ovel cofidece iterval for oparametric coditioal quatiles based o order statistics/iduced order statistics i Example 2.; a ew cofidece iterval for oparametric quatile regressio based o local polyomial estimator i Example 2.2; a ew cofidece iterval for oparametric cesored quatile regressio i Example 2.3, ad fially ew cofidece itervals for the class of coditioal quatile models i Doald, Hsu, ad Barrett 202. For the first three examples, σ 2 p x 0 takes the form: p p ϖ 2 x 0 for some positive costat ϖ x0 depedig o x 0, while for the last example it does t have this specific form. Example 2.. A Novel Order Statistic Approach: 4 The geeric cofidece iterval i 3 whe applied to the stadard asymmetric knn estimator of the coditioal quatile leads to a ovel cofidece iterval for coditioal quatiles based o pairs of order statistics of a appropriately chose set of iduced order statistics of {Y i } i=. It exteds cofidece itervals for ucoditioal quatiles based o pairs of order statistics of {Y i } i=, see Thompso 936 or va der Vaart 998 for radom samples ad Wu 2005 for time series observatios. To itroduce it, let R i = X i x 0, for i =,,, where is the stadard Euclidea orm i R d, ad Y,i i= deote the collectio of iduced order statistics by rak R i i=, i.e., Y j = Y,i iff R j = R i ad R i is the ith order statistic of R i i=. For k, the stadard asymmetric 4 After fiishig the first versio of this paper, we came across Kapla 203, who also proposed similar iferece procedures to this example by further takig liear combiatios of those order statistics. Higher order properties have bee give i Kapla 203 usig Dirichlet process theory. 8
10 knn estimator of the distributio fuctio of Y give X = x 0 is defied as F,k y x 0 = k k I Y,i y i= ad the asymmetric knn estimator of ξ p x 0 is give by { ξ p x0 = if y : F,k y x 0 kp } k = the kp th order statistic of Y,, Y,2,, Y,k, 5 where k k is a sequece of costats such that k ad k = o 4 4+d. Assumig GI i ad ii, the asymptotic validity of CIG α based o the asymmetric knn estimator relies o Assumptio GI iii. For a radom sample {X i, Y i } i=, Dabrowska 987 provides primitive coditios uder which the stadard knn estimator of the coditioal distributio fuctio coverges weakly to a Gaussia process which ca be used to show that Assumptio GI iii holds for ξ p x 0 i 5 with c = k ad ϖ 2 x 0 = π d/2 /Γ d/2 +. We refer the reader to Sectio 3.3 ad the proof of Propositio 3.4 i Dabrowska 987 for further details icludig the primitive coditios. Sice by defiitio ξ p x 0 i 5 is mootoe i p, the cofidece iterval 3 reduces to: CIO α = = ξ p zα/2 σ k x 0, ξ p + zα/2 σ k x 0 Y,kp zα/2 σ k, Y,kp+z α/2 σ k, 6 where σ k = p pϖ 2 x 0 /k ad Y,i deotes the ith order statistic of {Y,i } k i=. Notice that σ k ivolves o covariates desity ad the costat factor ϖ 2 x 0 is the volume of the uit ball i R d, which appears i the asymptotic variace of the stadard asymmetric knn estimator. The ew cofidece iterval CIO α defied i 6 for coditioal quatiles shares the elegace ad simplicity of the cofidece iterval for ucoditioal quatiles based o order statistics. Example 2.2. Local Polyomial Quatile Regressio: A local polyomial estimator defied as the miimizer of a weighted check fuctio is the atural oparametric aalog of the liear quatile estimator origiated by Koeker ad Bassett 978. Recall that the check fuctio is of the form: ρ p t = pt + + p t, where subscripts +, stad for the positive ad egative parts respectively. The local polyomial estimator of order s usig kerel fuctio K is defied by ξ p x 0 = e θ p x 0, with e =, 0,..., 0 ad θ p x0 = arg mi ρ p Y i θ T x0 X i P s K θ where P s z T i= x0 X i = z v v!, v s for v = v,, v d with v = v + + v d, v! = d vectors of v N d beig ordered lexicographically, z v = 9 d i= z v i i, i= v i! ad the see Chaudhuri, 99; Fa, Hu
11 ad Truog, 994; Guerre ad Sabbah, 202. With i.i.d. observatios, uder primitive coditios Guerre ad Sabbah 202 establish a Bahadur represetatio for ξ p x 0 valid uiformly over p 0,, where the liear represetatio is proportioal to q p x 0. is satisfied uder their coditios, where ϖ x0 So Assumptio GI iii = K 2 / f X x 0 ad c = h d with the badwidth. I the time series settig, Assumptio GI iii is satisfied uder the coditios i either Poloik ad Yao 2002 or Su ad White 202 uder proper mixig coditios. It is kow that the local polyomial estimator ξ p x 0 is ot guarateed to be mootoe i p, so our geeric CI employes the rearraged versio of ξ p x 0. Example 2.3. Noparametric Quatile Regressio With Cesorig: Cosider a oparametric cesored quatile regressio model where the depedet variable Y i is subject to coditioal radom cesorig by C i. So istead of observig {X i, Y i } i=, we observe a radom sample mi Y i, C i, δ i, X i i=, where δ i = {Y i C i }, Y i ad C i are idepedet of each other coditioal o X i. Dabrowska 987 exteds various oparametric quatile regressio estimators for radom samples icludig the kerel estimator, the symmetrized knn estimator, ad the stadard knn estimator to the above cesored case. Uder primitive coditios, she establishes weak covergece of the associated quatile processes which esures Assumptio GI iii with stadard oparametric covergece rate ad the factor ϖ x0 ow also ivolvig the coditioal cumulative hazard fuctio ad coditioal subsurvival fuctio see Corollary 2.2 i Dabrowska, 987. Thus our geeric cofidece iterval defied i 3 is asymptotically valid. Also oe could use local polyomial type estimators give the work of Kog, Lito ad Xia 203. Example 2.4. Ivertig Estimators of the Coditioal Distributio Fuctio: Doald, Hsu, ad Barrett 202 cosider various models for the coditioal distributio fuctio icludig fully parametric models of the form: F Y X y x F y x, θ 0 for a kow fuctio F ad a ukow parameter θ 0 ad semiparametric models of the form: F Y X y x F log y x θ 0 for a ukow F, θ 0. For each model, they provide primitive coditios uder which Assumptio GI iii is satisfied for radom samples, but σ 2 p x 0 is ot of the form: p p ϖ 2 x 0. We refer iterested readers to Sectio 3 i their paper for details. 2.2 A Geeric Cofidece Bad Ofte it is of iterest to coduct iferece simultaeously o ξ p x for all x i a subset of the support of X. I this sectio, we costruct a geeric CB for a oparametric quatile fuctio 5 by the direct approach agai avoidig the coditioal desity estimatio, whe x varies i some compact set J cotaied i the iterior of the support of X. Ulike the geeric CI i 3 which relies o the ormal limitig distributio, the geeric CB developed i this sectio will rely o the 5 For the locatioscale forms of liear quatile models, Zhou ad Portoy 996 costruct Scheffe type cofidece bad usig the direct approach see their Propositio 3. based o chisquare asymptotics. 0
12 extreme type limitig distributio of a statioary Gaussia process Bickel ad Roseblatt, 973; Leadbetter, Lidgre ad Rootze, 983 obtaied from characterizig the maximal deviatio of the origial oparametric estimator from the true coditioal quatile. Although the first order asymptotic properties of the mootoe quatile estimator i Cherozhukov, FeradezVal ad Galicho 200 are the same as the origial quatile estimator, this is ot suffi ciet for the purpose of costructig a geeric CB, see the discussio o Assumptio GB iii below for a more detailed explaatio. Because of this, we will adopt a special class of mootoe quatile estimators obtaied from ivertig mootoe estimators of the coditioal distributio fuctio of Y give X. Sice these estimators are mootoe by costructio, the geeric CI i 3 is applicable to them. For both practical ad techical reasos, we focus o the case whe X is uivariate. For multivariate covariate, semiparametric models icludig the partially liear ad sigle idex quatile regressio models are itroduced i the literature to alleviate the curse of dimesioality associated with fully oparametric models. I Sectio 3, we illustrate how CBs usig the direct approach ca be costructed for semiparametric quatile models. Let { X i, Y i } i= deote the sample iformatio o X, Y, radom sample or time series data. Cosider ay firststep estimator F y x of the coditioal distributio fuctio F Y X x takig the followig liear form: F y x = W i x I Y i y, 7 i= where {W i x} i= is a sequece of oegative weights summig up to oe. Let F p x deote the geeralized iverse of F x. The liear form give i 7 ests almost all commoly used kerel type local smoothers 6 such as NadarayaWatso estimator, YagStute estimator Yag, 98; Stute, 984b, Local Partitioed estimator the local costat versio i Chaudhuri, 99, ad Adjusted NadarayaWatso estimator Hall, Wolff ad Yao, 999: 7 W NW i x = W Y S i x = W P i x = K h x X i j= K x X j, 8 K h F x F X i j= K F x F X j, 9 I X i P x j= X, ad 0 j P x 6 Theorem 2.2 ad the high level assumptios are writte focusig o local smoothers. Upo chage of otatios, sieve type estimators could be icorporated as well, see FigueroafLopez As show by Hall, Wolff ad Yao 999, this Adjusted NadarayaWatso estimator is asymptoticall the same as local liear estimator upto the first order, hece it iherits the smaller bias property of local liear estimator, especially at the boudary. Meawhile it is suprior to local liear estimator as the weights are all positive ad summig up to, thus the estimated CDF is a proper distributio fuctio.
13 W ANW i x = p i x K h x X i j= p j x K h x X j, where h 0 is a badwidth ad K h = K /h /h with stadard kerel desity fuctio K. Moreover i the YagStute estimator, F x is the empirical distributio fuctio of {X i } i= ; i the Adjusted NadarayaWatso weights, we maximize j= p j x subject to the followig costraits: p j x 0, j= p j x = ad j= K X j x X j x p j x = 0; ad i the local partitioed versio, P = {P, P 2,...} stads for a partitio of X with max j Leb P j of order, ad P x P, is the set cotaiig poit x. The high level assumptios below are writte for the local smoothers with tuig parameter defied above. Implicitly below we select a udersmoothed to kill the bias term. Assumptio GB i. A cotiuous ad positive coditioal desity fuctio f Y X y x exists uiformly over the iterval F Y X p x ɛ, F Y X p 2 x + ɛ for some ɛ > 0, where p, p 2 cotais p ad belogs to 0, ad for x J. ii. The first step estimator for the coditioal distributio fuctio i 7 has the followig covergece rate: sup x J sup C F y x FY X y x = O p. y Y Moreover the uiform local oscillatio could be bouded as follows: sup x J sup y y 2 OC C F y x F y 2 x F Y X y x + F Y X y 2 x = o p, ad for all x J the local weights satisfy: max i W i x = o C almost surely. iii. Give p 0,, we could fid A, D ad a determiistic fuctio Ψ x such that { lim Pr A Ψ x } F ξ p x x FY X ξ p x x D z = exp 2 exp z, sup x J with /2 D A = O C ad Ψ x beig uiformly bouded away from zero ad ifiity. For those local smoothers the uiform covergece rate is C = log uder stadard regularity coditios ad the magitude of local weights are all of order O. The local oscillatio could be readily hadled give the results i Stute 984a or Eimahl ad Maso Apropos of the covergece to the Gumbel distributio i GB iii, we shall rely o the bivariate Gaussia approximatio ad deduce that the limitig distributio is the maximum of a statioary Gaussia process, say h /2 K u v h db v for a Gaussia process B ad kerel fuctio K as demostrated i Hardle 989, Hardle ad Sog 200 for radom sample, Liu ad Wu 200 for time series data. Give the limitig distributio i GBiii ad coditio i 2
14 GBii for Bahadur represetatio of the coditioal quatile estimator by directly ivertig the coditioal CDF estimator, oe could obtai the limitig distributio for the maximal deviatio of this coditioal quatile estimator, also see Lemma 3.4 ad its followig remark. The diffi culty with the estimator ξ p x i Cherozhukov, FeradezVal ad Galicho 200 by rearragig a arbitrary ξ p x may ot be mootoe i p is due to the lack of uiform asymptotic property of ξ p x. To the best of our kowledge, the characterizatio of the maximal deviatio of ξ p x remais to be a ope questio. where Our geeric CB takes the followig form: CBG α = F τ σ x; α x, F τ + σ x; α x σ x; α = D + h Ψ x log 2 log log α A ad Ψ x deote a uiformly cosistet estimator of Ψ x. 2 THEOREM 2.2 Suppose Assumptio GB holds. desired asymptotic size: The the cofidece bad i 2 has the lim Pr {ξ p x CBG α, x J } = α. Proof. I order to prove the validity of the geeric cofidece bad, we first eed the followig type of Bahadur represetatio for ay ε = O C aroud p, uiformly over x J : F p + ε x ξ p x = p + ε f Y X ξ p x x F ξ p x x + o p C. 3 To see why the above result holds, set η = F p + ε x ξ p x. The followig strig of equalities hold i view of the order of its local oscillatio i Assumptio GB ii: F ξ p x + η x F ξ p x x 4 = F Y X ξ p x + η x F Y X ξ p x x + o p C = fy X ξ p x x η + o p C. Also due to the liear structure of F y x, ad the egligibility of the idividual weights i Assumptio GB ii, we have F ξ p x + η x = F F p + ε x x + o p C = p + ε + o p C. Thereafter replace F ξ p x + η x with p + ε + o p C at the LHS i 4, we have show the first claim. Similar represetatios as i 3 ad their proofs could be foud i Serflig 980 for margial quatile ad Zhou ad Portoy 996 for a liear quatile regressio. 3
15 Now with 3 i had, the ext strig of derivatios eed o further explaatio oce we set ε = σ x; α, ad for otatioal simplicity we suppress the smaller order term o p C alog the lies: { } Pr F p + σ x; α x ξ p x 0, x J { = Pr p + ε F ξ p x x 0, x J = Pr f Y X ξ p x x { F ξ p x x p D h Ψ x } } log 2 log log α, x J h Ψ x A { = Pr Ψ x } F ξ p x x FY X ξ p x x D log 2 log log α, x J. Similarly by omittig the smaller order terms, we get { } Pr F p σ x; α x ξ p x 0, x J { = Pr Ψ x F ξ p x x FY X ξ p x x + D } log log α log 2, x J. Thus applyig the extreme value type asymptotics i Assumptio GB iii, we get lim Pr {ξ p x CBG α, x J } = α. 3 Iferece Based O YagStute Estimator Q.E.D Let { X i, Y i } i= deote a radom sample o X, Y for a uivariate X. I this sectio, we give a detailed illustratio of the direct approach usig the coditioal quatile estimator defied as the geeralized iverse of particular estimator of F Y X x proposed by Yag 98 ad Stute 984b, 986. I the sequel we will proceed with the iterchageable otatio F Y X p x = ξ p x to highlight this geeralized iverse ature. We cosider three quatile models: oparametric, partially liear, ad sigle idex models. For each model, we costruct a geeric CI ad a geeric CB for the correspodig coditioal quatile fuctio ad provide primitive coditios uder which we show their asymptotic validity. We use partially liear ad sigle idex models here to demostrate how CIs/CBs may be costructed for semiparametric quatile models whe the covariate is multivariate. It is worth repeatig here that o mootoe rearragemet is eeded i this sectio, because the quatile estimator we adopt is mootoe by costructio. Compared with the other local smoothers i the previous sectio, icludig the Adjusted NadarayaWatso estimator, ad Local Partitioed estimator, we show that the advatage of YagStute estimator is that it allows us to costruct CIs 4
16 ad CBs for all three quatile models that do ot require estimatio of the covariate s margial desity fuctio i fact the method does ot eve require the existece of covariate s desity fuctio, thus achievig the socalled asymptotic distributiofreeess i Stute 984b. 3. Uivariate Noparametric Quatile Fuctio We ow itroduce our estimator of F Y X p x at a fixed value x 0 X. Let F y x 0 deote the estimator of F Y X y x 0 itroduced i Yag 98 ad further studied i Stute 984b, 986. It is of the form i 7 with weights defied i 9, so F y x 0 = i= {Y i y}k Fx0 F X i, 5 i= K Fx0 F X i where K is a kerel fuctio ad 0 is a badwidth. Notice that the local eighborhood aroud the poit of iterest is calibrated accordig to raks istead of Euclidea distace, so F y x 0 is also kow as the symmetrized knn estimator 8. A more ituitive view of YagStute estimator is the followig kerel estimator replacig F with F X : i= F {Y i y}k FX x 0 F X X i y x 0 =. i= K FX x 0 F X X i I Appedix A, we show that the differece betwee F ad F is egligible for the iferetial purposes cosidered i our paper, thus F could be viewed as a feasible versio of F where the probability itegral trasformatio F X o the covariate makes its margial desity equal to alog its whole support. The estimator of ξ p x 0 based o F y x 0 is defied as the geeralized iverse of F x 0 : ξ p x0 = F p x 0. 6 I the rest of this sectio, we provide primitive coditios uder which Assumptios GI ad GB hold for F y x 0 or F p x 0 so the geeric CI i 3 ad the geeric CB i 2 are applicable to F p x A New Cofidece Iterval Our ew level αcofidece iterval for F Y X p x 0 takes the followig form: CIN α = F p zα/2 σ p K x 0, F p + zα/2 σ p K x 0, 7 8 To illustrate its symmetry, suppose there are three observatios, X =.5, X 2 = 2, ad X 3 = 5, ad we are iterested i estimatig the coditioal fuctioal at x 0 = 3 usig effectively two observatios. The the stadard knn estimator would choose X, X 2 accordig to Euclidea distace, whereas the symmetrized knn estimator would pick X 2, X 3 based o rak. 5
17 where σ p K = R K p p 8 i which R K = K 2 u du. Recall that q p x = /f Y X F Y X p x x is the coditioal quatile desity fuctio of Y give X. It is obvious from 7 that our ew cofidece iterval CI, CIN α, has several advatages over existig CIs. First, compared with Waldtype cofidece itervals, our ew cofidece iterval, CIN α, does ot require either a cosistet estimator of the desity fuctio of X or the coditioal quatile desity fuctio of Y give X = x 0, q p x 0. Secod, compared with the CI based o the empirical likelihood approach i Xu 202, our CI is much easier to implemet; there is o optimizatio ivolved ad it oly requires evaluatig our coditioal quatile estimator ξ p x0 at two specific quatile levels, p z α/2 σ p K ad p + z α/2 σ p K. Below we provide a list of suffi ciet coditios for the asymptotic validity of CIN α. Assumptio S. Let H y u = F Y X y F X u. i Assume that sup t s τ H F Y t u H F Y s u = o l τ as τ 0 uiformly i a eighborhood of u 0 = F X x 0 ; ii Uiformly i y, H y belogs to the secod order Holder class at u 0 0,, i.e., for ay y, H y u is differetiable w.r.t u at u 0 ad there exists a eighborhood of u 0 such that for ay u, u 2 i this eighborhood, we have that H y u H y u 2 L u u 2 holds uiformly i y, where H y u = H y u / u ad L <. Assumptio H. The badwidth satisfies satisfies: h 5 0 ad h 3 as. = O δ for some δ /5, /3, i.e., it Assumptio K. The kerel fuctio K is a twice cotiuously differetiable desity fuctio with zero mea, compact support ad bouded secod order derivative. Assumptio X. The coditioal desity fuctio f Y X x 0 exists ad is cotiuous ad positive o the iterval F Y X p x 0 ɛ, F Y X p 2 x 0 + ɛ for some ɛ > 0, where p, p 2 cotais p ad belogs to 0,, also X has cotiuous distributio fuctio F X x. Assumptio S is chose i accordace with Assumptios A, B i Stute 986. For Si 9 is used by Stute 986 to show the tightess of the coditioal empirical process. It is pretty 9 To clarify some otatio, we added the correspodig quatile trasformatio sice Stute 986 directly works with X, Y with uiform margial distributios. 6
18 weak; if the term i absolute values could be bouded by ay polyomial order i τ, it would imply Si. I Fa ad Liu 20, we show that existece of copula desity fuctio of Y ad X would imply this result. Sii is writte slightly differetly from Assumptio B i Stute 986 as it does ot require secod order differetiability of H y u, but achieves the same purpose i cotrollig the bias term. The socalled uiform Holder class is adapted from Tsybakov 2008, see also Guerre ad Sabbah 202. Assumptio X spells out this asymptotic distributio freeess advocated by Stute 984b, as by the elemetary fact F X X i U 0,. The requiremet o the badwidth is stadard, with oe added coditio h 3, which is ecessary i dealig with the asymptotic variace term as demostrated i Stute 984b. Assumptio K esures that our quatile estimator F p x 0 is mootoe i p 0, so CIN α is oempty. Kerel fuctios satisfyig Assumptio K iclude Bisquare ad Triweight kerels. THEOREM 3. Suppose Assumptios X, S, K, ad H hold ad x 0 is a iterior poit ot o the flat part of F X. For 0 < α <, we get: Pr F Y X p x 0 CIN α α as. Give our geeric result i Sectio 2, the proof of this theorem follows immediately after Lemma 3.2 below ad as F y x 0 is a proper distributio fuctio by Assumptio K, the rearragig step could be skipped. The Lemma below demostrates the critical role played by the symmetrized knn estimator F y x 0 i our ew cofidece iterval whicot oly avoids the estimatio of the coditioal quatile desity fuctio of Y give X = x 0 but also the estimatio of the desity fuctio of X. Lemma 3.2 Suppose the coditios of Theorem 2. hold. The i F x 0 F Y X x 0 = B 0, where B 0 is the Browia Bridge with the followig covariace structure: CovB 0 y, B 0 y 2 = R K F Y X y y 2 x 0 F Y X y x 0 F Y X y 2 x 0 ; ii Moreover, the coditioal desity fuctio of Y give X is strictly positive o the iterval: F Y X p x 0 ɛ, F Y X p 2 x 0 + ɛ for some ɛ > 0. The { } h F p x 0 F Y X p x 0 : p p, p 2 = q p x 0 B 0 F Y X p x 0. Lemma 3.2 i is restated from Stute 986. It makes clear that i cotrast to the commoly used NadarayaWatso estimator or the local polyomial estimator of the coditioal distributio fuctio, the asymptotic variace of F y x 0 does ot deped o the desity of the covariate X. I fact, Lemma 3.2 does ot eve require that X has a desity. It is this desityfree feature of F y x 0 that eables us to dispese with the desity of X i our ew cofidece iterval. 7
19 Lemma 3.2 ii follows from Lemma 2.3 i, Lemma 2.3 i va der Vaart 998, ad the fuctioal Delta method. It implies that for a fixed p p, p 2, h F p x 0 F Y X p x 0 = N 0, σ 2 with σ 2 = R K p p qp 2 x 0. So eve though the use of F y x 0 frees us from estimatig the desity of X, the asymptotic variace of F p x 0 still depeds o the coditioal quatile desity q p x 0. As a result, Waldtype iferece procedures based o the asymptotic ormality of F p x 0 would still require a cosistet estimator of q p x 0 or f Y X F Y X p x 0 x 0 which our ew cofidece iterval avoids as well A New Cofidece Bad I may applicatios, uiformly valid cofidece bads over a rage of covariate values may be desirable, see Hardle ad Sog 200, Sog, Ritov, ad Hardle 202 for iterestig empirical applicatios i labor ecoomics. Below we exted our cofidece iterval CIN α to cofidece bads over a rage of covariate values. Let CBN α = F p c δ α, K σ p K x, where σ p K is defied i 8 ad F p + c δ α, K σ p K x, 9 c δ α, K = c α 2δ log /2 + d 20 i which c α = log 2 log log α ad 2 d = 2δ log /2 + 2δ log /2 K u du log. 2 4πRK Note that like our cofidece iterval CIN α, our cofidece bad, CBN α, is easy to compute ad shares the remarkable desityfree feature. Below we provide additioal coditios uder which we show the uiform asymptotic validity of our cofidece bad. Let J X deote a ier compact subset of X. Assumptio S. Assumptio S holds uiformly for x J. Assumptio X. The coditioal desity fuctio f Y X y x has bouded derivative with respect to y uiformly for x J. Notice Assumptio X plus the compactess of J would give uiform cotiuity of F X. We list this rather redudat assumptio for easy referece. Assumptio B. i h 3 log y >a f Y ydy = O, where f Y y is the margial desity of Y ad a = is a sequece of costats tedig to ifiity as ; ii if x J f Y X F Y X p x x > 8
20 0; iii sup y sup x J f Y X y x < ; iv Y has Lipschitz cotiuous distributio fuctio F Y ad X, Y has uiformly bouded copula desity fuctio c x, y. Assumptio B iii are added i accordace with the strog approximatio result i Hardle ad Sog 200. Sice we base our aalysis o the covariate X after empirical probability itegral trasform, some of the assumptios i Hardle ad Sog 200 will be satisfied automatically here such as their A5 ad A6. Also otice that our Assumptio H o the badwidth implies Assumptio A2 i Hardle ad Sog 200 ad our Assumptio K implies their assumptio A. Assumptio B iii, iv will be eeded to establish the uiform Bahadur represetatio. Specifically Assumptio Biii aims to cotrol the bias term i the local oscillatio uiformly, ad with the help of Biv we could utilize certai ice maximal iequality i Stute 984a to boud the local oscillatio of copula process withi a shrikig rectagle. Details could be foud i our Lemmas A7 ad A8. THEOREM 3.3 Suppose Assumptios B, S, X, K, ad H hold. The the cofidece bad CBN α is asymptotically valid with coverage probability α uiformly over x J. Compared with our cofidece iterval, our cofidece bad replaces z α/2 with c δ α, K. The Lemma below explais why. Lemma 3.4 Uder Assumptios B, S, X, K, ad H, it holds that } Pr 2δ log /2 σ p K sup {f Y X F Y X p x x F p x F Y X p x d z x J exp 2 exp z as. Remark 3.. The above lemma follows from Theorem 2.2 i Hardle ad Sog 200 whe the covariate is uiformly distributed betwee 0,. The detailed proof cosists of a characterizatio of maximal deviatio of the coditioal CDF estimator, as i our Assumptio GBiii ad a uiform Bahadur represetatio of the coditioal quatile estimator i terms of the coditioal CDF estimators evaluated at F Y X p x. 3.2 Semiparametric Quatile Models I most applicatios, the covariate is multivariate. Semiparametric quatile models are itroduced i the literature to alleviate the curse of dimesioality associated with fully oparametric models ad at the same time are more robust tha fully parametric models. Commoly used semiparametric quatile regressio models iclude partial liear ad sigle idex quatile regressio models. Although most work i the literature cocer root estimatio of the fiite dimesioal parameters, Sog, Ritov ad Hardle 202 have costructed uiform cofidece bads for partial 9
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