Department of Statistics, College of Science, Persian Gulf University, Bushehr, Iran

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1 Appied Mahemaics Pubished Oie Juy 4 i SciRes hp://wwwscirporg/oura/am hp://dxdoiorg/436/am453 Wavee Desiy Esimaio of Cesorig Daa ad Evauae of Mea Iegra Square Error wih Covergece Raio ad Empirica Disribuio of Give Esimaor Mahmoud Afshari Deparme of Saisics Coege of Sciece Persia Guf Uiversiy Bushehr Ira Emai: afshar@pguacir Received 9 Apri 4; revised 9 May 4; acceped Jue 4 Copyrigh 4 by auhor ad Scieific Research Pubishig Ic This wor is icesed uder he Creaive Commos Aribuio Ieraioa Licese (CC BY hp://creaivecommosorg/iceses/by/4/ Absrac Wavee has rapid deveopme i he curre mahemaics ew areas I aso has a doube meaig of heory ad appicaio I siga ad image compressio siga aaysis egieerig echoogy has a wide rage of appicaios I his paper we use wavee mehod for esimaig he desiy fucio for cesorig daa We evauae he mea iegraed squared error covergece raio of give esimaor Aso we obai empirica disribuio of give esimaor ad verify he cocusio by wo simuaio exampes eywords Wavee Esimaio Cesorig Mea Iegra Error Covergece Iroducio Oe of daa ypes which researchers are exremey ieresed i is carig o he ime ierva i he occurrece of cerai eves such as deah ec Ay process waiig for a specific eve produces surviva daa Surviva fucio which is show by S( idicaes he raio of peope who survived sice he base ime which is he poi hey eer he experime Faiure i surviva aaysis meas he occurrece of he eve we were waiig for The ime where surviva is measured afer ha poi is caed he sar ime The faiure ime is he ime ha faiure occurs for each idividua which is deoed by T i for i = 3 The faiure ime is occurred from he base ime up o whe he faiure occurs ad i s ow as T i I s o aways possibe o observe he faiure ime for each idividua I such cases cesorship occurs The rae of occurreces of a eve (faiure i a spe- How o cie his paper: Afshari M (4 Wavee Desiy Esimaio of Cesorig Daa ad Evauae of Mea Iegra Square Error wih Covergece Raio ad Empirica Disribuio of Give Esimaor Appied Mahemaics hp://dxdoiorg/436/am453

2 M Afshari cific shor period of ime providig ha o faiure occurred before ha ime is he cocep which is discussed by he ame hazard fucio i surviva aaysis Hazard fucio for he faiure ime ie is as foows: h P T + T F S = im = = F S Wavees ca be used for rasie pheomea aaysis or fucios aaysis which someimes chages rapidy ad hey are symmerica ad have imied period uie rugged Sie waves hus he sigas wih radica chages are aayzed beer The cose reaioship bewee wavee coefficies ad some spaces wavee bases beig orhogoa ad aso usefu properies of hem i wavee issues simpify he compuaioa agorihms Wavees heory was proposed by Afred Harr [] for he firs ime i 9 He showed ha a coiuous fucio ca be approximaed as foows: Such ha φ φ f i = f x i x d x = φ φ + φ φ + + φ φ f x f x f x f x ( φ x = I I = = x< x< Aso for moher wavee ad faher wavees he foowig: φ ( x = φ( x ψ ( x = ψ ( x Defiiio -: Assume ha V = spa { ϕ : z} ; { z} Spaces { V Z} ϕ is a orhogoa ui base for V ad V coais a secioay cosa fucios ad heir exac egh is wice he ierva egh of V + are caed muiresoaio aaysis or scae fucio ϕ if i saisfies he foowig codiios: -V V+ Z - V = L R V = P 3- { } z z 4- z f ( x V f ( x V 5- z f ( x V f ( xh V 6- ϕ ( x V i codiio ha { ϕ ( x : z} is a orhogoa base for V If we cosider he scae fucio i he ierva [ ] he he image of f o he space V is defied as f = α ϕ ( x which is a fucio wih he resouio ad because of he fac ha V = L ( R V z f V hus P is a good approximaio of fucio f for arge amous of Le he esed sequece of cosed subspaces; V V V+ Z be a muiresouuo approximaio o L ( R Defie W Z o be orhogoa compeme of V i V + The erm wavees are used o refer o a se of basis fucios wih very specia srucure The specia of wave- es basis for fucio f L ( R as scaig fucio ϕ ad moher wavee ψ such ha { ϕ ( x } Z forms a orhogoa basis for V ad { ψ ( x } Z forms a orhoorma basis for W Oher wavees i he basis are he geeraed by rasaio of he scaig fucio ad diaios of he moher wavee by usig he reaioships: ( x m ( m x ( x ( x ϕ = ϕ ψ = ψ ( m Give above Wavee basis a fucio f ( R where = d = L ca be wrie a forma expasio: f = α ϕ + β ψ m m Z = m Z (3 α f x ϕ x x δ f x ψ dx As for geera orhogoa series esimaor Daubechies [] desiy esimaor ca be wrie as: fˆ = ˆ α φ x + ˆ β ψ x = P f + ˆ β ψ (4 m m m Z m Z m Z z 63

3 M Afshari where he obvious coefficie esimaor ca be wrie: ˆ α E X X = ϕ = ϕ β = E ψ X = ψ X ˆ (5 m m m i i i= i= We divide ime axis io wo pars he iervas ad he umber of eves i each ierva We deermie umber of eves ad hazard fucio accordig o he observaios The we fae hem separaey via iear wavee desiy esimaio o he whoe ime ad he we cacuae he fucio esimaor ad evauae he asympoic disribuio I his paper we obai esimaor desiy for cesorig daa by usig wavee mehod ad evauae mea iegra square error wih covergece raio ad empirica disribuio of give esimaor Esimaor of Desiy by Usig Wavee Mehod Wavees ca be used for rasie pheomea aaysis or fucios aaysis which someimes chages rapidy ad hey are symmerica ad have imied period uie rugged Sie waves hus he sigas wih radica chages are aayzed beer The cose reaioship bewee wavee coefficies ad some spaces wavee bases beig orhogoa ad aso usefu properies of hem i wavee issues simpify he compuaioa agorihms As a resu umerous arices have bee pubished abou desiy fucio esimaio The mahemaica heorem of wavees ad heir appicaio i saisics have bee sudied as a echique for oparameric curve esimaors by Aoiadys [3] Afshari [4]-[6] have doe some researches abou desiy fucio esimaor he desiy fucioa derivaive ad he oparameric regressio fucio for he mixig radom variabes Doohu [7] yacharya Picard [8] Maa [9] Meyer [] ad some arices have bee pubished i his fied Ha ad Pai [] have foud a formua for he Mea Iegraed Squared Error of oiear Wavee based o desiy esimaors Aoiadys e a [] achieved he desiy fucio esimaor ad he hazard fucio for righ-cesored daa wih he wavees I his secio we obai esimaor of desiy fucio for cesorig daa by usig wavee mehod Suppose X X X3 X are faiure ime of ess ha are sudied They are o-egaive idepede ideicay disribued wih he desiy fucio f ad disribuio fucio F ad C C C3 C are correspodig o cesored imes o-egaive idepede ideicay disribued wih he desiy fucio g ad disribuio fucio G Assumig idepedecy of faiure imes ad cesored ime of he observed radom variabe fucio δ ad Hazard fucio are show as beow: i F( f Z= mi ( X C δ = I h = F < i i i i X i C i Such ha I( A is idicaor fucio of A For daa cesorig if G( < f ( ( G( h = F < ( F( ( G( Aso we defiie as foows: Z i ad he he we have as he foowig: L( = P( Zi = P( Zi > = P( Xi > Ci > = ( F( ( G( f ( f ( = f( ( G h = L < L To esimae f ( we divide he ime axis io wo pars of sma iervas ad he amous of eves ( or i each ierva ad he we divide hese vaues o he egh of iervas Esimaio procedures of f ( ca be summarized as he foowig: Seec > ad coec he observed faiures i + iervas wih he egh ad usig wavee esimaio o he coeced daa We fid a esimae of sub desiy This meas ha we cacuae he coeced wavee coefficies daa o he scae of ( by choosig he decomposiio eve ( ad he we esi- f I is ecessary o sae he foowig symbos o show he deais: mae 64

4 M Afshari { } { } { } { } T = sup : F < T = sup : G < T = sup : L < = mi T T F G L F G We figure esimaors o he fiie ierva [ ] isic i of he sequece i Suppose ha Suppose ha τ i which τ < TL oe ha if Z ( i is he ordia order sa Z he TL = Z( TL I fac we suppose τ = Z( is a ieger ha coud be depede o ad he esimaed pois are as foows: τ = = = = τ ad we divide he ierva [ τ ] of ime axis o + iervas wih og τ = τ = = τ+ = τ The -h ierva is mared by J τ τ + = J = τ τ ow we defie he foowig idicaor fucio ha idicaes he umber of ucesored faiures i he ime ierva J : Yi = IJ ( Z i δi i = = We assume ha U he observed faiures raio i he ierva J oher words: U = Y = J so: = [ for [ ] i i = Theorem -: Suppose ha he sub desiy f is a coiuous fucio o [ ] τ ad i s m imes differeiabe he if v or we have: Proof: see [3] We smooh he daa We ca wrie where τ ( U f U Var = + O E = f + O U U Cov = f ( f ( + O U by a appropriae wavee smooher o fid he esimaio of f = φ φ + ψ ψ f f f = = f g f g d The compex srucura poymorphism aaysis causes a efficie ree cosrucio agorihm for aaysis of fucios i V wih heoreic scae wavee coefficies f ϕ However he iegra scae f ϕ is o we avaiabe ad we eed a iiia vaue for a fas wavee rasform Aoyadys [4] suggesed he foowig iiia amou: m f φ = f + O As a resu a reasoabe esimae for image of f wih cariy is: U f ( = ϕ ( (7 = If we assume ha he coeced vauesu which are equa o he esimaors of f ( are i Soboev space m * W ([ τ ] ad ϕ is reguar of degree m We esimae he uow fucio f as foows o eve he < : daa wih a beer rae for he sampe size ad he sequece Tha i is he orhogoa image of f ( fˆ = P f * V ( o he eveer approximaio space V ( (6 (8 65

5 M Afshari Theorem -: Suppose ha he sub desiy f is a coiuous fucio o [ ] τ ad i s m imes differe- iabe he if for we have: Proof: by usig heorem (- we ca wrie: Sice sup φ = M { } f ( E f = PV + O Var f = O + O U f E f O = = = φ = + ( φ he ϕ ( ϕ( So Equaios (9 ca be wrie as foows: By usig Equaio ( we have: ( φ φ = f + O( = = = ad we ca wrie as he foowig: sup φ = M = φ E f f + O M ( m+ f O M f O O = = ( φ + ( = φ + φ + ( m+ = f φ φ ( + O + O( = By usig Equaios ( ad ( we have: f ( E f ( = PV + O( U U U Var f = Var + Cov = = = φ φ φ By usig heorem (- we ca wrihe as foows: ( f Var f ( = + O( φ ( = + f f + O = = Usig his fac ha f is uiformy bouded o [ ] φ φ τ ad O( = we have: Var f C + O + C = = φ φ φ φ + O φ Sice ϕ is reguar i order m we ca wrie: φ φ φ φ = ( = O( ( ( = O( (9 ( ( (3 66

6 M Afshari Accordig Equaio (3 we ca wrie: { } Var f = O + O compee he proof 3 Evauae of Mea Iegra Square Error wih Covergece Raio I his secio we evauae mea iegra square error ad covergece raio is ivesigaed Defiiio 3-: The mea iegraed square error (MISE of ere esimaor of a desiy fucio f is give r MISE C h + Ch I his formua deoes he righ ad ef covergece whe deoes he sampe size h deoes he esimaor badwidh core r deoes core eve ad C C deoe ere depede quaiies wih uow desiy Theorem 3-: Suppose ha he sub desiy f is a coiuous fucio o [ τ ] ad i s m imes differeiabe he if for ad ( he ( ˆ τ { ˆ m } MISE f = E f f d O + O + O (4 Proof: ˆ f ( f ( f ( f ( V V V V E = f f = P f = P P + P f = S + A By usig Equaio (5 ad heorem (- for m we ca wrie as he foowig: V f ( f ( V E E = E S + A = P + O P + A (5 Because V V we ca wrie as he foowig: So by usig Equaios (6 ad (7 we ca wrie: For evauae Var MISE ( f ˆ we ca wrie: = ( + E E O A (6 f ( m( V P f = O (7 ( ˆ m( f O O( MISE = + (8 Aso we ca wrie: f ( U V = φ ϕ = ϕ ϕ ϕ = = = P f U EU f E f V V = φ φ φ P P he U EU f E f PV P d V = φ φ By usig heorem (- ad expecaio of Equaio (9 we ca wrie as he foowig: By usig heorem (- we have: ( U EU ( Var = E ϕ ϕ U Uh + Cov ϕ ϕ ϕ ϕ h h (9 ( 67

7 M Afshari ( U EU ( f ( E = + O ( f = + O ϕ V φ φ ϕ ϕ d ϕ ϕ = P = ϕ = ϕ = φ d = ( By usig Equaio ( ad his fac ha f is uiformy bouded we ca wrie as he foowig: ( U EU ( E φ φ = O O O + = The secod par of Equaio ( ca be wrie as he foowig: U Uh Cov ϕ ϕ ϕ h ϕ h = f f + O h = h ϕ ϕ ϕ ϕ h ( ( h ϕ ϕ ϕ h ϕ ϕ ϕ ϕ h ϕ = ϕ ϕ h φ φ = φ = ( By usig O + O = O he proof is compee 4 Empirica Disribuio of Purpose Esimaor I his secio we ivesigae empirica disribuio of esimaor uder some codiio Theorem 4- Suppose ha he sub desiy f is a coiuous fucio o [ τ ] ad i s m imes differeiabe for τ we 3 ( m have: Proof: he for ierva [ ] ( ˆ ϕ f f f = ( ( ( fˆ ˆ ˆ ˆ f = f E f + E f f By usig heorems (- ad (- we ca wrie as he foowig: ˆ f ( E f ( f ( = O( + PV f m m f f sup PV f O P = m V f = O f W m ˆ E f ( f ( O O = + (3 (4 68

8 M Afshari So by usig equaio of (3 ad (4 we ca wrie as he foowig: ˆ ( ˆ ( ˆ ( ( ( ( ( ( ˆ f E f = f f + f E f + E f E f = Ι + ΙΙ + ΙΙΙ We prove ha II has asympoicay orma disribuio ad aso I III ed o zero whe Firs we show ha I III ed o zero whe Accordig o Equaio (4 we have: m ˆ f f( f ( = PV f = O m E( f ( ˆ ( ( ˆ ˆ E f E f E f E f f( = O (5 By usig Equaio (3 we have: f ( V m P f So by usig Equaio (4 ad (5 he phrase I III ed o zero whe ad fiay we have: So we have: U f = = U = = = Uφ( = Yiφ( = i= = φ( φ( ˆ ( ( ˆ ( i φ ( i (6 f E f = Y p = Z i= = i= Such ha for each fixed whie i = Yi radom sampe wih he mea as foows: By usig cushy Schwarz iequaiy: is defied as a idepede ad ideicay disribued p f ( E( f ( φ ( ( Yi p = i= M ( φ ( i ( i = = i= = i= f E f Y p Y p (7 So we ca wrie as he foowig: M E sup f E f Var Yi τ = Usig his fac ha f is uiformy bouded ad Var ( Yi f ( O wrie: ( i Var Y = f + O M ( c E sup f ( E f ( f + c Thus he Equaio (6 sae is coverge i Aso by usig Theorem (- we have: τ = = + = we ca L ad hus i he disribuio Var [ Zi ] = f ( φ ( O + = 69

9 M Afshari Thus we have: ( ( φ Var f E f f We coro he Lidberg codiio i order o prove ha II is asympoicay orma For his purpose we Zi se: Ui = ad we show ha E{ Ui Ι } Ui > ε VarZ i By usig cushy Schwarz iequaiy: ( i = ( i ( i = { i Ι } { ( i U ε } ( ε > EU O EZ EZ O E U i EU foowig: EU { i Ι } = O Ui >ε ad compee he proof 5 Simuaio ad umerica Compuaio for Targe Esimaor I his secio we simuae = So we ca wrie as he fˆ o he daa of size by usig Semay s wavee We cosider covergece raio of give esimaor by compuig of average mea square error of give esimaors We use R sofware ad wavee pacage for simuaio Exampe : We geerae X X X3 X ~ Γ( 5 ad C C C3 C ~ E( 6 from he Sampes of size = 4 ad = 6 wih = 8 = 6 = 3 ad = 5 for opima surface = The resus i Tabe dispays he average mea square errors of subdesiy fucio esimaor for sampe sizes = 4 ad = 6 The pae i Figure dispays he wavee esimaor of subdesiy fˆ ( of observed faiures for a radiioa cesorig daa The soid ie is he desiy esimaor ad he doed ie is he rue desiy Exampe : Suppose ha X X X3 X ~ f = 6Y + 4W where Y ~ L ( ad W ~ ( 34 We geerae C C C3 C ~ E( 3 from sampe size of = 4 ad = 6 wih = 8 = 6 = 3 ad = 5 The resus i Tabe dispays he average mea square errors of subdesiy fucio esimaor for sampe sizes = 4 ad = 6 The pae i Figure dispays he wavee esimaor of subdesiy of observed faiures for a radiioa cesorig daa The soid ie dispays he subdesiy esimaes based acua daa ad he doed ie is he rue desiy Tabe The average mea square errors of subdesiy fucio esimaor by wavee mehod AMSE ˆ = ( f = f( f ( = = Tabe The average mea square errors of subdesiy fucio esimaor by wavee mehod AMSE = ˆ = ( f = f( f ( =

10 M Afshari Figure The wavee subdesiy ad rue desiy esimaor 6 Cocusio Figure The wavee subdesiy ad rue desiy esimaor I his paper we obai desiy esimaio for cesorig daa by usig wavee mehod ad evauae mea iegra square error We show ha covergece raio is accepabe ad empirica disribuio of give esimaor uder some codiio is orma Acowedgemes The suppor of Research Commiee of Persia Guf Uiversiy is greay acowedged Refereces [] Harr A (9 Zur Theorie der Orhogoae Fuioe Mahemaische Aae [] Daubechies I (988 Orhogoa Bases of Compacy Suppored Wavees Commuicaio i Pure ad Appied Mahemaics [3] Aoiadis A (996 Smoohig oisy Daa wih Tapered Coifes Series Scadiavia Joura of Saisics [4] Afshari M (3 A Fas Wavee Agorihm for Aayzig of Siga Processig ad Empirica Disribuio of Wavee Coefficies wih umerica Exampe ad Simuaio Commuicaio of Saisics-Theory ad Mehods [5] Afshari M (4 Esimaio of Hazard Fucio for Cesorig Radom Variabe by Usig Wavee Decomposiio ad Evauae of MISE AMSE Wih Simuaio Joura of Daa Aaysis ad Iformaio Processig -5 hp://dxdoiorg/436/daip4 [6] Afshari M (8 Wavee-ere Esimaio of Regressio Fucio for Uiformy Mixig Process Word Appied Scieces Joura [7] Dooha DL ad Johsoe IM (994 Idea Spaia Adapaio by Wavee Shriage Biomeria Joura hp://dxdoiorg/93/biome/8345 7

11 M Afshari [8] eryacharia G ad Picard D (993 Desiy Esimaio Byere ad Probabiiy McGraw-Hi Sciece ew Yor [9] Maa SG (989 A Theory for Muiresouio Siga Decomposiio: The Wavee Represeaio Trasformaios o Paer Aaysis ad Machie Ieigece [] Meyer Y (99 O de ees e operaeurs Herma Paris [] Ha P ad Pai P (995 Formua for Mea Iegraed Squarederror of o-liear Wavee Based Desiy Esimaors Aas of Saisics hp://dxdoiorg/4/aos/ [] Aoiadis A Gregoire G ad aso P (999 Desiy ad Hazard Rae Esimaio for Righ Cesored Daa Usig Wavee Mehods Joura of Roya Saisica Sociey Series B [3] Vidaovi B (999 Saisica Modeig by Wavees Wiey ew Yor hp://dxdoiorg//

12 Scieific Research Pubishig (SCIRP is oe of he arges Ope Access oura pubishers I is currey pubishig more ha ope access oie peer-reviewed ouras coverig a wide rage of academic discipies SCIRP serves he wordwide academic commuiies ad coribues o he progress ad appicaio of sciece wih is pubicaio Oher seeced ouras from SCIRP are ised as beow Submi your mauscrip o us via eiher submi@scirporg or Oie Submissio Pora