Strong Convergence Rates of Wavelet Estimators in Semiparametric Regression Models with Censored Data*

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1 8 The Ope ppled Maheacs Joural Srog Covergece Raes of Wavele Esaors Separaerc Regresso Models wh Cesored Daa Hogchag Hu School of Maheacs ad Sascs Hube Noral Uversy Huagsh Cha bsrac: The paper sudes a separaerc regresso odel Y X + g( T + e L. where Y s cesored o he rgh by aoher rado varable C wh kow or ukow dsrbuo G. Frsly he wavele esaors of paraeer ad oparaeer g ( are gve by wavele soohg ad he syhec daa ehods. Secodly uder geeral codos he laws of he eraed logarh for he wavele esaors of paraeer ad srog ufor covergece raes for he wavele esaors of oparaeer are vesgaed. Lasly he valdy of ehod s lluaed by he sulao exaple. Keywords: Separaerc regresso odel Wavele esae Cesored daa Law of he eraed logarh Srog ufor covergece rae.. INTRODUCTION Cosder he separaerc regresso odel Y X + g( T + e L. ( where Y s are scalar respose varables X s ad T s are explaaory varables s a oe-desoal ukow paraeer g( s a ukow regresso fuco o [0] ad e s are depede ad decally dsrbued rado errors wh zero ea ad fe varace. Followg Specka (See [] deoe X f( T +. ( f s soe ukow sooh fuco o [ ] where ( 0 s are depede ad decally dsrbued rado errors wh zero ea ad fe varace ad depede of Y s ad T s. Sce he roducory work by Egle e al. (See [] he odel ( has bee wdely suded (See [3-6] ad refereces here ad pu o use ay felds of appled sascs. paral ls of esao ehods for ad g ( cludes pealzed leas squares ehod (See [7] soohg sples ehod (See [8] pecewse polyoal ehod (See [3] ear eghbor ehod (See [9] ad wavele ehod (See [0-3] ec. I pracce parcularly edcal sudes Y ay be cesored radoly o he rgh by soe cesorg varable C L ad hece cao be observed co- X T Z L where pleely. Oe oly observes { } ddress correspodece o hs auhor a he School of Maheacs ad Sascs Hube Noral Uversy Huagsh Cha; E-al: reuoe@63.co Z ( Y C I( Y C C are assued o be depede of Y s ad s. For rgh cesored daa he odel ( has bee suded by Wag ad L [4] ad Wag ad Zheg [5] ad he odel (-( has bee vesgaed by Q ad Ca [6] Q ad Jg [7] Pa ad Fu [8] ad Lag ad Zhou [9]. Wh wavele ehod he paper dscusses he odel (- (. The orgazao of he paper s as follows. I seco he wavele esaors of paraeer ad oparaeer g ( are gve by wavele soohg ad he syhec daa ehods. Uder geeral codos he laws of he eraed logarh for he wavele esaors of paraeer ad srog ufor covergece raes for he wavele esaors of oparaeer are vesgaed seco 3. The a proofs are preseed seco 4 wh he sulao exaple seco 5.. ESTIMTION METHOD Le F ( ad G( deoe he dsrbuos for Y ad C respecvely. ssue ha G( s couous ad kow a he oe. Usg he syhec daa ehod (See [670] ad refereces here we rasfor he cesored daa {( Z L } o he followg syhec daa Y Z ; G + Z ; G (3 ( ( ( are couous fucos whch sasfy where ( ( ( ( ( ad do o deped o F. The class of all pars ( ( GY ( YG ; + Y GdG ; Y ( of such fucos wll be deoed by K. I ca be easly showed ha f( K he 874-4/ Beha Scece Publshers Ld.

2 Srog Covergece Raes of Wavele Esaors The Ope ppled Maheacs Joural 008 Volue 9 EY ( X T EY ( X T ad EY ( EY ( (4 where X { X L X} T { T L T}. The class K cludes ay eresg cases (See [9-0]. If G( s ukow we use a odfed verso G% ( of he Kapla-Meer esaor of G( defed by Where G% ( ( f Z( or ( 0 G( Z( f > Z( ad ( ( ( 0 I Z + + ( ( + ( G N Z N Z ( { Z } ( ( > + N I Z Z ax ad ( s he correspodg. I s easly showed ha G% + < for all. Subsug G ( ( ( % for G( (3 we oba he followg syhec daa: ( ( ( Y Z G % Z G % (5 ; + ; Suppose ha here exss a scalg fuco ( x he Schwarz space S l ad a ulresoluo aalyss { V } he cocoa Hlber space L ( R wh s reproducg kerel E ( s gve by E ( s E 0 s ( ( k ( s k where Z deoes he colleco of egers. Le [ s s ] be ervals ha paro [0] wh ad. The esae ehod wll be roduced as followg: Frsly suppose ha s kow we defe esaor of g ( by k Z ( ( ( g Y X E s ds I successo we defe wavele esaor by zg arg Y X g ( XY X where. % % %. ( ( X % X E s dsx Y % Y E s dsy. Fally we defe lear wavele esaor of g ( by ( ( ( g ( g Y X E s ds Usg he slar procedure f ( we defe esaors of ad g ( by XY X g ( Y X E s ds G s ukow he % % % ( ( where %Ŷ Ŷ Ŷ E ( s. ds To oba our resuls he followg codos are suffce. g( ad f ( belog o he Sobolev space wh order >. g ( ad f ( sasfy he Lpschz codo wh order > 0. s he Schwarz space wh order l sasfes he Lpschz codo wh order ad has a copac suppor. Furheore ( O( as 0 where s he Fourer rasfor of. ( 4 ax( s s O ( ad O ( 3. ( 5 F ( ad G( are couous fucos. I he case 3 ( G ( F < assue ha F ( F ( s dg( s < { } where F f : F(. I he case G( F assue ha G( u q O F u as u F for soe q (0. ( ( ( I he followg dscusso we shall cofe ourselves o soe ore resrced classes of ( ha he class K. I parcular we defe K {( K : for each s < here exss a cosa G 0 < C < such ha ad {( ( G C } sup ; s % here exs cosas 0 L L ( s K K : > 0 such ha ( % ( %( ( < < ad sup G ; G ; Lsup G G s s for all d.f. G % wh sup G% ( G( <. } s For ease of exposo we shall roduce he followg oaos whch wll be used laer he paper. Defe

3 0 The Ope ppled Maheacs Joural 008 Volue Hogchag Hu % ( E X Y X g T X % Y X g T E ( ( XT ( ( ( ( ( e Y E Y Y X g T ( ( ( ( f% T f T f T E s ds g% T g T g T E s ds ( ( E s ds. e% e e E s ds 3. STTEMENT OF THE RESULTS Theore 3.. Suppose ha codos hold. Ee < E < ad ( K. The for every 3 > 3 ( ( ( 3 sup g ( g( O( log as.. (6 lsup log log E e as.. (7 0 Theore 3.. Suppose ha codos hold. Ee < E < ad ( K %. The for every 3 > 3 ( ( ( 3 sup g ( g( O( log as.. (8 lsup log log E e as.. ( PROOFS OF THEOREMS Throughou hs paper le C deoe a geerc posve cosa whch could ake dffere value a each occurrece. Before he proofs of he heores we roduce soe prelary resuls. Lea 4.. (oads e al. []. If codo ( 3 holds he ( E0 ( s C ( k k + s ad k E( s Ck (+ s for k N where C k s a real cosa depedg o k oly. sup E s O. (II ( ( 0s (III E ( s ds C sup. 0 ( Lea 4.. (Hu ad Hu []. Le whe < < 3 whe 3 whe > 3. If codos ( ( 4 hold he ( ( ( k ( + ( sup f ( E s ds f T O O k k ( ( ( k ( + ( sup g ( E s ds g T O O k k Lea 4.3. (Qa ad Ca [3]. If codos ( + ( 4 hold ad < for soe > 0 he 3 Ee l X % as.. Lea 4.4. (Qa ad Ca [3]. If codos ( ( 4 hold ad E < he 6 ( ( sup E s ds O log as.. Lea 4.5. (Xue []. If codos ( ( 4 hold he ( ( ax E s ds O as.. Lea 4.6. (Gu ad La [3]. ssue codo ( 5 holds. If G( F < le ( sup { : F( } where 3< < he lsup ( log log sup G ( G( sup S( ( as.. F If G( F he lsup ( log log sup G ( G( sup S( ( as.. G G where S( G( ( ( G( s ( F( s dg( s. Lea 4.7. Uder he codos of heore we have ( o( as.. The proof s slar o ha of lea [7] ad hece oed here. Lea 4.8. (Hardle Lag ad Gao [4] Le{. L } be depede rado varables wh r E 0 ad fe varaces ad sup E C r. L s a sequece of real u- for soe 0< p < ad ssue ha { a } bers such ha p sup a O( p a O( for p ax{ 0 r p} < (. The

4 Srog Covergece Raes of Wavele Esaors The Ope ppled Maheacs Joural 008 Volue ax a O( s log for s ( p p as.. Lea 4.9. Le{. L } be depede rado varables wh E 0 ad fe varaces r ad sup E C r 3. ssue ha codos ( ( 4 hold. The ax < ( ( E s ds O( 3 log a.s. ( Proof Le r 3 p 3 p 0 ad a E s ds. The lea follows fro lea 4. lea 4.5 ad lea 4.8. Proof of Theore 3. Noe ha where X % ( I I (0 + ( ( XT I X% Y% E Y% I e E( s ds e f% ( T f% ( T E( s dse E( s ds e + I + I + I3 + I ( 4 X % g% ( T % ( %( + %( %( ( f T g T g T g T E s ds I + I ( Le ( e e I e < e e e h h % ( E e % XT h ( V a e h % E( e ad ( ( ( XT a f% T f% T E s ds. The for ay > 0 ( % I V + a e% h L + L (3 3 By E( e < (See lea [6] ad hree seres heore we ca easly oba ha e % < as.. By Berse s equaly ad Bore-Caell lea h log % as.. Hece fro lea 4. ad 4.5 we oba ha ax a sup f% ( ax + E ( s ds Csup f% O + O ( ( ( (4 (% % ax % % ( ( ( ( ( L a e + h a e + h O + O O log o as.. By Berse s equaly ad Bore-Caell lea V.. Fro (3 (5 ad (6 we have ha I3 ( (5 as (6 o as.. (7 Usg he slar argue as above by lea ad 4.5 we oba ha I ( as ( I 4 o o.. Fro lea 4. follows ha as.. I o (.. ( ( ( ( ( sup as (8 I f% g% O + O o (9 Fro (0-( ad (7-(9 ( + ( X% e o as.. (0 Hece he (6 follows edaely fro lea 4.3 ad LIL of Hara-Wer. Noe ha 0 ( ( % sup g g 0 sup ( ( X E s ds ( ( ( ( + sup g T E s ds g + sup e E s ds 0 0 I+ I + I ( 3 By lea 4. lea 4. ad lea 4.4 we have ha I C ( log log sup + ( log log sup 0 0 C ( log log ax + C ( log log sup 0 %f T ( E s % E ( s ds %f ( T sup 0 E 0 E ( sd ( ds ( s ds + ( C log log sup ax ( ( 0 E s ds E s ds ( ( + O ( ( log log ( 3 O( log log log O log log ( ( + O 3 log log log By lea 4. we have ha ( ( ( I O + O (3

5 The Ope ppled Maheacs Joural 008 Volue Hogchag Hu 3 Fro Ee < ad lea [6] we have 3 ha Ee <. Hece by lea ( I3 O log as.. (4 Thus (7 follows fro (-(4. We herefore coplee he proof of Theore 3.. Proof of Theore 3. Noe ha ( ( + ( The (8 follows edaely fro lea 4.7 ad heore 3.. Wre g ( g( g ( g ( + g ( g( I+ I (5 ( ( ( ( ( ( I E sdsy Y X E sds I I E s ds Y Y I Z I ( ( ( ( ( ( + E s ds Y Y I Z > (6 R + R (7 By lea 4. ad 4.6 we have ha R E s 0 ( dssup Ĝ (G ( C( log log o ( 3 log I he case G( ax F G as (8 Z as... By lea 4. ad ( ( ( sup R sup E s ds ax Z G Z G ( ( ( E 0 s ds G G G ( ( 3 sup sup C log log o log as.. (9 I he case G( F <. Le I( Z E( I( Z > > s are depede rado varables wh zero ea. Sce( K % (3 ad (5 ax Y Y < C as... Hece fro lea 4.9 we have ha sup sup R C sup ax E Ŷ Y ( ( ( sds I( Z > PZ ( > + PZ > E ( sds C 3 ( log + + C sup ( O 3 log E ( sdspz > ( ( a.s. (30 Therefore he desred cocluso (9 follows fro (5- (30 ad heore NUMERICL EXMPLE We wll sulae a sple separaerc regresso odel ( Y X + cos T + e L 64 where T 64 X 5T ( 0 + N ( e N. The rgh-cesored rado varables C are depede ad decally dsrbuo fuco G u exp.4u ( u 0. ( ( Choose ( ug ; u ( ug ; u G( u G( u + ad Daubeches scalg `fuco (. By calculao we have.058. I closely approxaes he rue value of paraeer. I ca be see ha our ehod s successful especal esag he paraeer. However a furher dscus- s eeded so of he choces he scalg fuco ad ( so ha we ca fd a good ehod o use praccal applcaos. CKNOWLEDGMENTS Suppored by Scefc Reasearch Ie of Educao Offce Hube (No.Q REFERENCES [] Specka P. Kerel soohg paral lear odels. J R Sas Soc 988; 50: [] Egle RF Grager WJ Rce J Wess. Separaerc esaes of he relao bewee weaher ad elecrcy sales. J Sa ssoc 986; 80: [3] Che H. Covergece raes for paraerc copoes a parly lear odel. Sa 988; 6(: [4] Baco Boee G. Robus esaors separarc parly lear regresso odels. J Sa Pla If 004; : 9-5. [5] Ive P Kg ML Zhag X. Soohg sple based ess for oleary a parally lear odel. J Sa Pla If 006; 36: [6] Hu H. Rdge esao of a separaerc regresso odel. J. Copu ppl Mah 005; 76: 5-. [7] Fscher B Heglad M. Collocao flerg ad oparaerc regresso par. ZfV 999; : [8] Gree PJ Slvera BW. Noparaerc regresso ad geeralzed lear odels. Lodo: Chapa & Hall 994. [9] Hog SY. The esae heory of a separaerc regresso odel. Sc Cha (Ser 99; : [0] Chag XW Qu LM. Wavele esao of parally lear odels. Copu Sa Daa al 004; 47: [] Xue LG. Raes of rado weghg approxao of wavele esaes separaerc regresso odel. Ch ca Mah ppl Sca 003; 6(: -5. [] Su H Zhao XM. Covergece raes of esaors for paraerc copoes ad oparaerc copoes paral lear odel. J Eg Mah 999; 6(3: -8. [3] Qa WM Ca GX. Srog approxably of wavele esae separarc regresso odel. Sc Cha (Ser 999; 9: [4] Wag QH L G. Eprcal lkelhood separaerc regresso aalyss uder rado cesorshp. J Mulvar al 00; 83: [5] Wag QH Zheg ZG. sypoc properes for he separaerc regresso odel wh radoly cesored daa. Sc Cha (Ser 999; 40: [6] Q GS Ca L. Esao for he asypoc varace of paraerc esaes paral lear odel wh cesored daa. ca Mah Sc 996; 6(: [7] Q GS Jg BY. sypoc properes for esao of paral lear odels wh cesored daa. J Sa Pla If 000; 84: 95-0.

6 Srog Covergece Raes of Wavele Esaors The Ope ppled Maheacs Joural 008 Volue 3 [8] Pa X Fu ZT. The asypoc properes of wavele esao a separaerc regresso odel uder rado cesorshp. ca Mah ppl Sca 006; 9(: [9] Lag H Zhou Y. sypoc oraly a separaerc parally lear odel wh rgh-cesored daa. Cou Sas Theory Meh 998; 7(: [0] Zheg ZK. class of esaors of he paraeers lear regresso wh cesored daa. ca Mah ppl Sca 987; 3(3: 3-4. [] oads Gregore G Mckeague IW. Wavele ehods for curve esao. J. Sa ssoc 994; 89: [] Hu HC Hu DH. Srog cossecy of wavele esao separaerc regresso odels. ca Mah Sca (Chese Seres 006; 49(6: [3] Gu MG La TL. Fucoal law of he eraed logarh for he produc-l esaor of a fuco uder rado cesorshp or rucao. Probab 990; 8: [4] Hardle WG Lag H Gao JT. Parally lear odels. Hedelberg: Physca-Verlag 000. Receved: Noveber Revsed: Deceber cceped: March