Uniformly Consistency of the Cauchy-Transformation Kernel Density Estimation Underlying Strong Mixing

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1 Appl. Mat. If. Sci. 7, No. L, 5-9 (203) 5 Applied Matematics & Iformatio Scieces A Iteratioal Joural c 203 NSP Natural Scieces Publisig Cor. Uiformly Cosistecy of te Caucy-Trasformatio Kerel Desity Estimatio Uderlyig Strog Miig Wa-Jyi Horg Departmet of Hospital ad Healt Care Admiistratio, Cia Na Uiversity of Parmacy & Sciece, No. 60, Er-Je RD., Sec., Je-Te, Taia, 770, Taiwa Received: 7 Sep. 202, Revised: 7 Oct. 202, Accepted: 28 Dec. 202 Publised olie: Feb. 203 Abstract: I tis paper, oe uses te idea of Caucy-trasformatio to costruct a Caucy-trasformatio kerel desity estimator uderlyig te coditio of strog miig. Te uiformly strog cosistecy ad covergece rates of te proposed estimator are obtaied uderlyig te papers of Cai ad Roussas [] ad Kim ad Lee [6]. Te proposed estimator ca improve te boudary effects of te empirical (or uiform)-trasformatio kerel desity estimator i te boudary area. Besides, te proposed estimator ca also be applied to estimate te azard fuctio. Keywords: kerel desity estimator, Caucy-trasformatio, empirical distributio, covergece rate, strog miig. Itroductio Let X,...,X be a strog miig (α-miig) sequece of radom variables. Suppose tat te X ave a distributio fuctio F() ad probability desity fuctio f (). Give te miig coefficietα() ad α() 0 as. Te α-miig coefficiets of te sequece {X k of radom variables is defied by α() = { P(A B) P(A)P(B) : A I k,bi +k k=,2,... werei m = σ (X t : m t < ), m <, ereσis teσ algebra. Te strog miig coditios ca also refer to te paper of Bradley[7]. I te literature, various desity estimators for f () uderlyig te coditio of strog miig, based o a radom sample X,...,X ave bee proposed ad teir properties are studied, for eamples, te papers of Scuster [3], Roussas [2], Cai ad Roussas [], Kim ad Co [4], Liebscer [7], Tae ad Co [5], Kim ad Co [5], Kim ad Lee [6] ad Hase [6]. A kerel desity estimator for f() is defied as follows: ˆf (;) = K { Xj, (.) were K is a (symmetric) kerel fuctio ad =() is called te badwidt. Estimator (.) is called te kerel desity estimator (KDE) of Parze [9] ad Roseblatt [0]. Ruppert ad Clie [] propose a empirical-trasformatio KDE wic it is defied as follows: ˆf RC (;) = F () F (X j ) K ˆf (;), (.2) weref ()is te empirical distributio fuctio of te data to estimate F() ad ˆf (;) is defied as above. We poit F() ears te boudary regios of [0,], te empirical-trasformatio kerel desity estimator (Ruppert ad Clie []) will be suffered to te problem of boudary effects, for eample, te epectatio of ˆf RC (;)is ot equal f(), as F() [0,c),0 c <. Some boudary modificatio metods ave bee proposed by, for eamples, Joes [3] ad Müller [8]. I tis paper, by te ideas of Caucy-trasformatio metod ad te estimator (.2), we propose a Caucy-trasformatio KDE wic it does ot ave te boudary effects problem of te empirical-trasformatio KDE. Te proposed estimator is defied as follows: f T (;) = T () T (Xj ) K T () (), (.3) weret () = ta[π(f() 0.5)], F() is te distributio fuctio of X adq( ) deotes te Caucy desity of Correspodig autor wj7902@mail.ca.edu.tw c 203 NSP Natural Scieces Publisig Cor.

2 6 Wa-Jyi Horg: Uiformly Cosistecy of te Caucy-Trasformatio... T(). T() is cotiuous ad differetiable at te poit. Ad q(t ( )) is cotiuously differetiable. Te estimator (.3) is called te ideal Caucy-trasformatio KDE. Here researcer uses te estimator of f T (; ) to estimate f(). We will study te uiformly cosistecy ad covergece rates of estimator (.3). We te fuctio F() (or f ()) is ukow, te estimator (.3) will be ot a practical estimator, terefore, we eed furter to estimate F(). Te corrected estimator of (.3) is give by ˆf T (;) = K ˆT () ˆT (X j ) ˆT () (), (.4) were ˆT () = ta[π(f () 0.5)], F ()is te empirical distributio fuctio of te data. Tis paper is orgaized as follows: I sectio 2, we state te uiformly cosistecy of estimator (.4) ad gives te eplicit formula for te covergece rate. I sectio 3, we give te proofs of te asymptotic results. 2 Uiformly Cosistecy I tis sectio, oe states te mai results of estimator (.4). Before oe state te mai results oe will give te followig assumptios: (A) X,...,X is α-miig sequece wit te probability desity fuctio f (), f () is bouded o its domai, ad cotiuous at te poit ad f (m) () eists, for m 2. (A2) K(u) is bouded variatio. K( ) is te kerel fuctio ad satisfies (i) lim u k+2 K(u) <, for eac k 0. u (ii) K () (u) du <. (A3) = log (loglog)+r α() coverges for some r>0, te coditio of Cai ad Roussas []. Let us ow give te mai results of te estimator (.4). I te followig Teorem 2., we give te uiformly strog covergece of te estimator (.4) at te poit (-, ). Teorem 2.. Assume tat coditios (A)-(A3) are satisfied. Ad pose tat 0 ad, as, te we ave ˆf T (;) f () 0, (2.) (, ) almost surely, as. Remark : Te estimator (.4) as te strog covergece property as tat of te traditioal kerel desity estimator. Te proposed estimator does ot ave te boudary effects problem of te empirical-trasformatio KDE, te kerel fuctio does ot eed to use te boudary kerel fuctio, te boudary problem ca also refer to te paper of Müller [8]. Remark 2: As te estimator of F() is ˆF(;) = ˆf (t;)dt, ere ˆf (;)is te KDE of Parze (962), te asymptotic result of Teorem 2. is also eld. If f (2) ()is uiformly cotiuous fuctios o (-, ), te, we ave te followig teorem about te covergece rate of te estimator ˆf T (;). Teorem 2.2. Uder te assumptios of Teorem 2. ad assume tat f (2) ()is uiformly cotiuous, te we ave ˆf T (;) f () loglog = O( 2 + ) (, ) (2.2) Remark 3: From Teorem 2.2, te covergece rate is costructed for te estimator (.4). Te coice of te smootig parameter ca refer to, for eample, te book of Hrdle [2]. A applicatio of Teorem 2.-2, oe ca use te proposed estimator to estimate te azard fuctio. Te azard fuctio is defied as follows: H() = f () F() = f () F(), (2.3) were F() = F(). Te estimator of H() is defied as follows: Ĥ(;) = ˆf T (;) F () = ˆf T (;) F (), (2.4) were ˆf T (;)adf ()are defied as above. Let us ow give te asymptotic results of te estimator (2.4). I te followig Teorem 2.3-4, oe also provides te uiformly strog cosistecy, as (, ). Teorem 2.3. Assume tat coditios (A)-(A3) are satisfied. Ad pose tat 0 ad, as, te we ave Ĥ (;) H() 0, (2.5) (, ) almost surely, as. Teorem 2.4. Uder te assumptios of Teorem 2.3 ad assume tat f (2) ()is uiformly cotiuous, te we ave Ĥ(;) H() loglog = O( 2 + ) (, ) (2.6) c 203 NSP Natural Scieces Publisig Cor.

3 Appl. Mat. If. Sci. 7, No. L, 5-9 (203) / Proofs I tis sectio our mai purpose is to prove Teorem Te four lemmas below will be used to justify te teorems. Lemma 3.. Let X,...,X be a statioary α-miig sequece of real-valued radom variables wit distributio fuctio F() ad miig coefficiet α() satisfyig (A3), ad let F () be te empirical distributio fuctio based o te segmet X,...,X. Te F () F() 0, (3.) almost surely, as. Proof: Te proof follows from te Teorem i te book of Tucker [4] ad te Corollary 2. i te paper of Cai ad Roussas []. Lemma 3.2. Let X,...,X be a statioary α-miig sequece of real-valued radom variables wit distributio fuctio F() ad miig coefficiet α() =O( δ ),for some δ > 3, ad let F () be te empirical distributio fuctio based o te segmet X,...,X. Ad let F() satisfies te Lipscitz coditio F() F(y) b 0 y, for some costat b 0. Te P{lim[( 2loglog )/2 F () F() ] = 2 =, (3.2) almost surely, as. Proof: Te proof follows from te Teorem 3.2 i te paper of Cai ad Roussas []. Lemma 3.3. Assume tat te coditios (A)-(A2) are satisfied. Te we ave E f T (;) f () = O( 2 ), (3.3) for (-, ). Proof: By te calculatio of epectatio, as (-, ), we ave E f T (;) = { K T () T (y) f (y)t () ()dy = { K T () T (y) = K T () u q(u)dut () () q(t (y))t () (y)dyt () () = K {wq(t () w)dwt () () = q(t ())T () () + µ q (2) (u) u=t () T () () + o( 2 ) = f () + O( 2 ), from teciques of Calculus uderlyig te coditios (A)-(A2) ad q(t ( )) is cotiuously differetiable. Terefore, te proof of Lemma 3.3 is proved. Lemma 3.4. Uder te coditios (A)-(A2), te we ave f T (;) E f T (;) loglog, = O( ) (3.4) almost surely, for all o real lie. Proof: We kow tat f T (;) E f T (;) = K { T () T (y) T () ()df (y) F (y) F(y) dk T () T (y) K T () T (y) T () ()df(y) T () () v F () F(), (3.5) by te teciques of calculus ad te coditios (A)ad A(2), were ν is te variatio of K{ T () (). From Lemma 3.2 ad (3.5), we ave f T (;) E f T (;) = O( almost surely, as. + Proof of Teorem 2.2. We kow tat ˆf T (;) f () ˆf T (;) f T (;) + f T (;) f () =I()+I(2), From I(), we ave ˆf T (;) f T (;) K K { T () T (Xj ) ˆT () ˆT (X j ) K ˆT () () T () () +C 0 loglog ) (3.6) ( ˆT () () T () ()) T () T (Xj ) ˆT () () ˆT () () T () () T () T K (Xj ) T () T (Xj ) K() ( X j ) ˆT () () T () T K (Xj ) C F () F() +C 2 F () F() K () T () T (Xj ) ( X j ) ˆT () () =I(3)+I(4), for some costat C 0,C ad C 2. By I(3), we ave q(t () T () T K (Xj ) K(w) dw < (3.7) c 203 NSP Natural Scieces Publisig Cor.

4 8 Wa-Jyi Horg: Uiformly Cosistecy of te Caucy-Trasformatio... By I(4), we ave T () T (Xj ) K() ( X j ) ˆT () () wk f () () (w) dw <, (3.8) were f()=q(t())t () (). By (3.7) ad (3.8), we ave ˆf T (;) f T (;) C 3 F () F() (3.9) for some costat C 3. From I(2), we ave f T (;) f () f T (;) E f T (;) + E f T (;) f () (3.0) From (3.9), (3.0) ad Lemma 3.3-4, te proof of Teorem 2.2 is completed. Proof of Teorem 2.. By (3.5) ad Lemma 3., we ave f T (;) E f T (;) 0, (3.) almost surely, as 0,. From (3.0), (3.) ad Lemma 3.3, te proof of Teorem 2. is proved. Proof of Teorem 2.3. By (2.3) ad (2.4), we ave Ĥ(;) H() = F() F (). { F()[ ˆf T (;) f ()] + f ()[F () F()] (3.2) From (3.2), we obtai Ĥ(;) H() C 4 ˆf T (;) f () +C 5 F () F(), (3.3) for some costats C 4 ad C 5. By (3.3), Lemma 3. ad Teorem 2., te Teorem 2.3 ca be proved. Proof of Teorem 2.4. From Teorem 2.3 ad Lemma 3.2, te Teorem 2.4 ca be proved, te details are omitted. 4 Coclusios I tis paper, oe uses te tecique of Caucy-trasformatio metod to costruct a Caucy-trasformatio kerel desity estimator uderlyig te coditio of strog miig. Te uiformly strog cosistecy ad covergece rates of te proposed estimator are obtaied uderlyig te papers of Cai ad Roussas [] ad Kim ad Lee [6]. Te empirical result sows tat te proposed estimator ca improve te boudary effects of te uiform-trasformatio kerel desity estimator i te boudary area (Ruppert ad Clie []). Besides, te proposed estimator ca also be applied to estimate te azard fuctio ad oters desity fuctio. Refereces [] Z. Cai ad G.G. Roussas, Uiform strog estimatio uderαmiig, wit rates, Statistics & Probability Letters 5, (992) [2] W. Hrdle, Smootig Teciques wit Implemetatio i S, Spriger-Verlag, New York (99) [3] M.C. Joes, Simple boudary correctio for kerel desity estimatio, Statistics Computig 3, (993) [4] T.Y. Kim ad D.D. Co, Uiform strog cosistecy of kerel desity estimators uder depedece, Statistics & Probability Letters 26, (996) [5] T.Y. Kim ad D.D. Co, Covergece rates for average square errors i kerel smootig estimators, Joural Noparametric Statistics 3, (200) [6] T.Y. Kim ad S. Lee, Kerel desity estimator for strog miig processes, Workig paper (2004) [7] E. Liebscer, Strog covergece of sums of α-miig radom variables wit applicatios to desity estimatio, Stocastic Process. Appl. 65, (996) [8] H.G. Müller, O te boudary kerel metod for oparametric curve estimatio ear edpoits, Scadiavia Joural of Statistics 20, (993) [9] E. Parze, O estimatio of a probability desity fuctio ad mode, Aals Matematical Statistics 33, (962) [0] M. Roseblatt, Remarks o some o-parametric estimates of a desity fuctio, Aals Matematical Statistics 27, (956) [] D. Ruppert ad D.B.H. Clie, Trasformatio kerel desity estimatio: bias reductio by empirical trasformatio, Aals Statistics 22, (994) [2] G.G. Roussas, Noparametric estimatio i miig sequeces of radom variables, J. Statist. Pla. Iferece 8, (988) [3] E.F. Scuster, Estimatio of a probability desity fuctio ad its derivatives, Aals matematical Statistics 40, 87-95(969) [4] H.G. Tucker, A Graduate Course i Probability, Academic Press, StateplaceNew York (967) [5] Y.K. Tae ad D.D. Co, Uiform strog cosistecy of kerel desity estimator uder depedece, Statist. & Probab. Lett. 26, (996) [6] B.E. Hase, Uiform covergece rates for kerel estimatio wit depedece data, Ecoometric Teory 24, (2008) [7] R.C. Bradley, Basic properties of strog miig coditios: A Survey ad some ope questios. Probab. Surveys 2, (2005) c 203 NSP Natural Scieces Publisig Cor.

5 Appl. Mat. If. Sci. 7, No. L, 5-9 (203) / 9 Wa-Jyi Horg received te MS degree i Applied Matematics from Natioal Ceg Kug Uiversity i 989, ad te PD degree i Statistics sciece, from te departmet of Istitute Statistics, Natioal Tsig Hua Uiversity i 995. He is curretly a associate professor i Cia Na Uiversity. His researc iterests are i te areas of applied statistics, applied matematics, ad fiacial data aalysis. c 203 NSP Natural Scieces Publisig Cor.