Adaptive Kernel Estimation of the Conditional Quantiles

Transcription

1 Internatonal Journal of Statstcs and Probablty; Vol. 5, No. ; 206 ISSN E-ISSN Publsed by Canadan Center of Scence and Educaton Adaptve Kernel Estmaton of te Condtonal Quantles Rad B. Sala, Hazem I. El Sek Amed 2, Hossam O. EL-Sayed 3 Te Islamc Unversty of Gaza, Department of Matematcs, Palestne 2 Al Quds Open Unversty, Department of Matematcs, Palestne 3 UNRWA, Department of Educaton, Palestne Correspondence: Hazem I. El Sek Amed, Al Quds Open Unversty, Department of Matematcs, Palestne. E-mal: sakamad@qou.edu Receved: November, 205 Accepted: December 5, 205 Onlne Publsed: December 23, 205 do:0.5539/jsp.v5np79 Abstract URL: ttp://dx.do.org/0.5539/jsp.v5np79 In ts paper, we defne te adaptve kernel estmaton of te condtonal dstrbuton functon (cdf for ndependent and dentcally dstrbuted (d data usng varyng bandwdt. Te bas, varance and te mean squared error of te proposed estmator are nvestgated. Moreover, te asymptotc normalty of te proposed estmator s nvestgated. Te results of te smulaton study sow tat te adaptve kernel estmaton of te condtonal quantles wt varyng bandwdt ave better performance tan te kernel estmatons wt fxed bandwdt. Keywords: Quantle regresson, condtonal dstrbuton, kernel estmaton, asymptotc normalty. Introducton Suppose {(X, Y } n are R R random varables and we assume tat te regresson model defned by Y m(x + ϵ,, 2,... n. We assume tat te response varable Y depends on an ndependent random varable X wt common probablty densty functon f and m(x s te condtonal mean curve m(x E(Y X x y f (x, y dy, f (x and a nonparametrc kernel estmaton of te regresson functon can be obtaned as ˆ f (x n. Slverman (986 gave an algortm wt tree steps for adaptve kernel estmaton of densty functon. At te frst, a pror kernel estmator wt a fxed bandwdt s obtaned. Te second step, te local bandwdt factor λ s defned by λ { ˆ f (X g} β, 0 β, were g ( assumng g 0 s te geometrc mean of f ˆ(X. Fnally, for one varable te condtonal kernel estmaton s gven by f n (x n. Sala (2009 dscussed te same metod for te adaptve estmaton of te azard rate functon. Sala and El Sek Aamed (2009 also dscussed te same metod for te adaptve estmaton of te condtonal mode. Te adaptve kernel estmaton s equvalent to te kernel estmaton wt fxed bandwdt wen β s equal to 0. Wen β, ten te adaptve kernel estmaton s equvalent to te nearest negbor estmaton. Demr, Toktams (200 gave an algortm to adaptve kernel estmaton of densty functon by settng g f ˆ(X n. Te condtonal dstrbuton functon as an mportant role for quantle regresson. Yu and Jones (998, Hall et al. (999, Sala (2006 and Dette (20 ave recently consdered metods for estmatng condtonal dstrbutons. Yu (2003 dscussed some problems for estmaton of te condtonal quantles. 79

2 Internatonal Journal of Statstcs and Probablty Vol. 5, No. ; 206 Abberger (997 defned a kernel estmaton of te condtonal dstrbuton functon as ˆF(y x I {Y y} K( x X, and te condtonal α quantle s gven by ˆq α (x nf{y R ˆF(y x α}, 0 < α <. Abberger (997 dscussed te bas, varance and te mean squared error of ts estmator, and proved tat te asymptotc normalty of te proposed estmator. From te prevous, we ave notced a lot of statstcans use te kernel estmaton of te densty and condtonal dstrbuton functon, but a queston s rased: Is tere any adjustments or mprovements to ts estmated wose gves better results? In ts paper, we wll use te adaptve kernel estmaton of te condtonal dstrbuton functon to estmate te condtonal quantles. We proposed te followng estmator of te condtonal dstrbuton functon as F n (y x I {Y y}, were we defne te local bandwdt factor λ by [ ] β fn (X λ, 0 β, g were g s te artmetc mean of f n (X. Now, we defned te adaptve of te condtonal α quantle s gven by 2. Prelmnares q n,α (x nf{y R F n (y x α}, 0 < α <. In ts secton, we wll consder an assumptons of te kernel functon, te bandwdt and te condtonal dstrbuton functon. Tese assumptons are: (A n s sequence of postve number satsfes te followng: ( 0, for n ; ( n, for n ; (A2 Te kernel K s a Borel functon and satsfes te followng: ( ( ( (v K as compact support; K s symmetrc; K s Lpsctz-contnuous; K(udu ; 80

3 Internatonal Journal of Statstcs and Probablty Vol. 5, No. ; 206 (v K s bounded; (A3 For fxed y R tere exsts F (y x 2 F(y x x 2 n a negborood of x. Now, we state and prove some lemmas tat wll elp us to aceve te man result n ts paper: Lemma For any random varable X wt densty f, we ave Proof: Assume tat And we assume tat µ 2 (x K(udu, ten we ave E[ f n (x] f (x 2 2 µ 2 (x f (x + o( 2. ( u 2 K(udu <. uk(udu 0. E[ f n (x] E[ n ] f (ydy Let x y z, ten y x z, and so E[ f n (x] Now, usng Taylors expanson for f (x z to get n K(z f (x zdz K(z f (x zdz Now, f (x z f (x z f (x + (z2 2 f (x + o( 2 Tus, we ave te result. E[ f n (x] + K(z[ f (x z f (x + (z2 2 f (x + o( 2 ]dz K(z f (xdz f (x o( 2 K(zdz f (x µ 2 (x f (x + o( 2. Lemma 2 ( Integral approxmaton of te sum over te kernel functon zk(zdz f (x z 2 K(zdz Wt (A2(, Lpsctz-contnuty, (A2(, U (x x X, x (, and te mean value teorem of ntegraton, t follows:. lm 2. lm n K(U n K2 (U K(udu K 2 (udu 8

4 Internatonal Journal of Statstcs and Probablty Vol. 5, No. ; lm n U K(U Proof: See Abberger (997. uk(udu 3. Bas, Varance and Mean Squared Error In ts secton, te bas and varance wll be evaluated. For ndependent {Y } te expectaton and te varance of te estmator F n (y x are follows as: E[F n (y x] [ K( x X F(y X, (2 ] and Var[F n (y x] K 2 ( x X [ K( x X [F(y X F 2 (y X ]. (3 ] 2 3. Te Mean Squared Error In ts subsecton, we wll ntroduce te defnton of te mean squared error and gve te man teorem about te asymptotc mean squared error. Te mean squared error ( MSE s used to measure te error wen estmatng te densty functon at a sngle pont. It s defned by Tat s, equaton (4 can be wrtten as MS E{ f n (x} E{ f n (x f (x} 2. (4 MS E{ f n (x} (E f n (x f (x 2 + Var f n (x (5 Teorem Let {Y } be ndependent and let (A(,, (A2(,,, v and (A3 be satsfed. Ten t olds for n Proof: MS E(F n (y x [ ( 2 F (y x u 2 K(udu] n( (F(y x F2 (y x K 2 (udu. By equaton (5 and equaton ( te bas term s old. Tat s [E[F n (x] F(x] 2 [ ( 2 F (y x u 2 K(udu] 2 2 we only want to fnd Var[F n (y x]. Usng Taylors expanson for F(y X and F 2 (y X to get F( y x U F(y x U F (y x + ( 2 U 2 F (y x + o( 2, F 2 ( y x U F 2 (y x + ( U [F 2 (y x] + ( U 2 [F 2 (y x] + o( 2 F 2 (y x 2 U F(y xf (y x + 2( 2 U 2 F 2 (y x + 2( 2 U 2 F(y xf (y x + o( 2, 82

5 Internatonal Journal of Statstcs and Probablty Vol. 5, No. ; 206 Ten F( y x U F 2 ( y x U As n, and 0, we ave 2 0. F(y x U F (y x + ( 2 U 2 F (y x F (y x F 2 (y x + 2 U F(y xf (y x 2( 2 U 2 F 2 (y x 2( 2 U 2 F(y xf (y x Tat s F(y X F 2 (y X F(y x F 2 (y x, so we ave Tus, we ave te result. 4. Asymptotc Normalty [F(y x F 2 (y x] + U [ F (y x + 2F(y xf (y x] + ( 2 U 2 [( 2 U 2 2F(y xf (y x 2F 2 (y x] Var[F n (y x [ K 2 ( x X ] 2 (n (n [F(y x 2 F2 (y x] (n [F(y x F2 (y x] [F(y x F 2 (y x] K 2 (udu K 2 (udu In ts secton, te next teorems sow te asymptotc normalty (n 2 (F n (y x E[F n (y x] and (n 2 (F n (y x [F(y x]. Teorem 2 Let te condton of te last teorem be satsfed. Ten t olds for n, (n d 2 (Fn (y x E[F n (y x] N (0, [F(y x F 2 (y x] K 2 (udu. (6 Proof: To prove ts teorem, we use Lapunov s condton. So let Q n, (x K( x X [I {Y y} F(y X ] Var[Fn (y x] Terefor, Tat s Ts means tat Q n, (x Q n, (x K( x X [I {Y y} F(y X ] Var[Fn (y x] I {Y y} Var[Fn (y x] F(y X 83

6 Internatonal Journal of Statstcs and Probablty Vol. 5, No. ; 206 Now, te Lapunov s condton Q n, (x F n(y x E[F n (y x] Var[Fn (y x]. Te numerator s E lm E Q n, (x 3 lm E [I {Y y} F(y X ] 3 (Var[F n (y x] 3 2 [I {Y y} F(y X ] 3 E{ { { n [n 3 [I {Y y} F(y X ] 3 } 3 E [I {Y y} F(y X ] 3 } 3 } K 3 ( x X du du] 3 [ K 3 ( x X ] 3 o(n o(n 3 λ 3 o( n 2 λ 2 But, te varance of F n (y x follows as Var[F n (y x] o( n Tat s (Var[F n (y x] 3 2 o( n 3 2 λ 3 2 Ten O( lm E Q n, (x 3 n 2 λ 2 O( o(. O( n 2 λ 2 n 3 2 λ 3 2 Tat s te Lapunov s condton olds. Ten F n (y x E[F n (y x] Var[Fn (y x] N(0, Tus (n 2 (Fn (y x E[F n (y x] d N (0, [F(y x F 2 (y x] K 2 (udu. 84

7 Internatonal Journal of Statstcs and Probablty Vol. 5, No. ; 206 Corollary Let te condton of Teorem be satsfed, and n( 5 0. Ten t olds for n, (n d 2 (Fn (y x F(y x N (0, [F n (y x F 2 (y x] K 2 (udu. Proof: Snce (n 2 and (F n (y x E[F n (y x] d N (0, [F(y x F 2 (y x] K 2 (udu. E[F n (y x] F(y x 2 ( 2 u 2 K(uduF (x + o( 2 O( 2 Ten (n 2 E[Fn (y x] F(y x (n 2.O(λ 2 O(n[ ] 5 2 Wt n[λ] 5 0, we ave E[F n (y x] F(y x Tat s (n d 2 (Fn (y x F(y x N (0, [F n (y x F 2 (y x] K 2 (udu. Now, let H n,α (θ(x. [α I {Y θ(x}] Usng te central lmt teorem, we ave H n,α (θ(x E[H n,α (θ(x] Var[Hn,α (θ(x] d N ( 0,, n. Wt H n,α (θ(x te mean squared error of q n,α (x can be calculated. Teorem 3 Let te condtons of Teorem be satsfed and let F n (q n,α (x x F(q α (x x α be unque. Ten t olds MS E[q n,α (x] [ F (q α (x x 2 (λ2 u 2 K(udu] 2 f (q α (x x α( α + K 2 (udu n f 2 (q α (x Proof: By te Taylor expanson of te condtonal dstrbuton functons of teorem (3.. and θ(x q n,α (x follows and wt ntegral approxmaton olds E[H n,α (q n,α (x] f (q α (x x [q n,α (x q α (x] U 2 K(U + 2 ( 2 nf (q α (x K(U E[H n,α (q n,α (x] f (q α (x x [q n,α (x q α (x] + 2 (λ2 nf (q α (x u 2 K(udu. 85

8 Internatonal Journal of Statstcs and Probablty Vol. 5, No. ; 206 Now, Ts mples for n Var[H n,α (q n,α (x] [ K(U ] 2 K 2 (U [ F(q n,α (x x U F 2 ((q n,α (x x U ] [ [ α( α] K(U ] 2 α( α n f (q α (x x[q n,α (x q α (x] + 2 λ2 F (q α (x α( α K nλ 2 (udu K 2 (udu. K 2 (U u 2 K(udu Corollary 2 Let te condton of Teorem be satsfed and let n 5 n 0, for n. Ten t olds 5. Smulaton Studes (n 2 (qn,α (x q α (x d N ( 2 (nλ5 2 d N( 0, α( α f 2 (q α (x x F (q α (x x f (q α (x x α( α f 2 (q α (x x N( 0,. K 2 (udu ; K 2 (udu. u 2 K(udu, In ts secton, two smulaton studes were conducted to compare te performance of te KRE estmaton of te condtonal dstrbuton functon and te AKRE estmaton of te condtonal dstrbuton functon. In ts studes, 400 data ponts are smulated from R program. 5. Case Study We generated samples usng R program of sze 25, 50, 75, 50, 250, 400, 700 from te regresson functon y cos x + ex, were te x were drawn from a unform dstrbuton based on te nterval [0, ], and te e ave a normal dstrbuton N(0,. Te fxed bandwdt was computed by Slverman (986 and Wand and Jones (995 as opt.06 sd(t n 5. Now, we defne a plot estmate f n (X tat satsfes f n (X 0, were {X } s a famly of d. In ts paper we select te Epanecnkov kernel K(t 3 4 ( 5 t2 / 5 for t < 5. We defne te local bandwdt factor λ by [ ] fn 2 (X λ, g were g s te artmetc mean of f n (X. We ave consdered tat te equaton I {Y y} λ F n (y x K( x X, 86

9 Internatonal Journal of Statstcs and Probablty Vol. 5, No. ; 206 Te grap of te real regresson functon of te estmaton of te regresson functon and te adaptve regresson functon. 0 2 True Regresson WNW AWNW Fgure : Improved Kernel Quantle Regresson Estmator at n75 Fgure. sows te plot of te data togeter wc te true regresson curve, te kernel quantle regresson estmator and te mproved kernel quantle regresson estmator (Ts grap at n 75. Te contnuous lne ndcates tat te kernel regresson, te dotted lne ndcates tat te kernel regresson estmator and te ntermttent lne ndcates tat te mprovement kernel regresson estmator. Te mean square error was computed as Table. MSE and MSE Adaptve for samples n MSE MSE Adaptve * * * * * * * * Mnmum MSE n eac row. As seen from Table, for all sample szes, we ave te kernel estmators usng varyng bandwdts ave smaller MSE values tan te kernel estmator wt fxed bandwdt. In eac case, t s seen tat MSE as te best performance. 5.2 Case Study 2 Anoter study was conducted to compare te performance of te KRE estmaton of te condtonal dstrbuton functon and te AKRE estmaton of te conatonal dstrbuton functon. We generated samples usng R program of sze 25, 50, 75, 50, 250, 400, 700 from te regresson functon y x + 87

10 Internatonal Journal of Statstcs and Probablty Vol. 5, No. ; 206 3e 200x + ex, were te x were drawn from a unform dstrbuton based on te nterval [0, ], and te e ave a normal dstrbuton N(0, and, λ computed as te last smulaton. 0 2 True Regresson WNW AWNW Fgure 2: Improved Kernel Quantle Regresson Estmator at n75 Fgure 2. sows te plot of te data togeter wc te true regresson curve, te kernel quantle regresson estmator and te mproved kernel quantle regresson estmator (Ts grap at n 75. Te contnuous lne ndcates tat te kernel regresson, te dotted lne ndcates tat te kernel regresson estmator and te ntermttent lne ndcates tat te mprovement kernel regresson estmator. Te mean square error was computed as Table 2. MSE and MSE Adaptve for samples 6. Concluson n MSE MSE Adaptve * * * * * * * * Mnmum MSE n eac row. In ts paper, we ave studed te adaptve kernel estmaton of te condtonal quantles and we concluded te expectaton, varance, mean square error and asymptotcally dstrbuton of te proposed estmator. Te results of te smulaton studes sowed te adaptve kernel estmaton of te condtonal quantle wt varyng bandwdts provde better estmates tan te condtonal quantle estmate wt fxed bandwdt. 88

11 Internatonal Journal of Statstcs and Probablty Vol. 5, No. ; 206 References Abberger, K. (997. Quantle smootng n fnancal tme seres. Statstcal Papers, 38, ttp://dx.do.org/0.007/bf Demr, S., & Toktams, Ö. (200. On te adaptve Nadaraya-Watson kernel regresson estmaton. Hacettepe Journal of Matmatcs and Statstc, 39(3, Dette, H., & Sceder, R. (20. Estmaton of addtve quantle regresson. Ann Inst Stat Mat, 63, ttp://dx.do.org/0.007/s Sala, R. (2006. Kernel estmaton of te condtonal quantles and mode for tme seres. Unversty Of Macedona. Sala, R. (2009. Adaptve kernel estmaton of te azard rate functon. Te Islamc Unversty - Gaza Journal, 7(, 7-8. Sala, Rad B., & El Sek Amed, H. (2009. Local varable kernel estmaton of te condtonal mode. Rajastan Academy of Pyscal Scences, 8(3, Slverman, B. W. (986. Densty estmaton for statstcs and data analyss. Scool of Matematcs Unversty of Bat, UK. ttp://dx.do.org/0.007/ Wand, M. P., & Jones, M. C. (995. Kernel smootng. Unversty of New Sout Wales Asutrala. ttp://dx.do.org/0.007/ Yu, K. (2003. Quantle regresson: Applcaton and current reseac areas. Unversty of Plymout. UK. ttp://dx.do.org/0./ Yu, K., & Jones M. C. (998. Local lnear quantle regresson. Journal of te Amercan Statstcal Assocaton, 93(44, ttp://dx.do.org/0.080/ Copyrgts Copyrgt for ts artcle s retaned by te autor(s, wt frst publcaton rgts granted to te journal. Ts s an open-access artcle dstrbuted under te terms and condtons of te Creatve Commons Attrbuton lcense (ttp://creatvecommons.org/lcenses/by/3.0/. 89