Research Article Gamma Kernel Estimators for Density and Hazard Rate of Right-Censored Data

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1 Joural of Probability ad Statistics Volume 211, Article ID , 16 pages doi:1.1155/211/ Research Article Gamma Kerel Estimators for Desity ad Hazard Rate of Right-Cesored Data T. Bouezmari, 1 A. El Ghouch, 2 ad M. Mesfioui 3 1 Départemet de Mathématiques, Uiversité de Sherbrooke, Québec, QC, Caada JIK 2RI 2 Istitut de Statistique, Biostatistique et Scieces Actuarielles, Uiversité Catholique de Louvai, 1348 Louvai-La-Neuve, Belgium 3 Départemet de Mathémathique et d Iformatique, Uiversité duquébec àtroisrivières, Trois Rivières, QC, Caada G9A 5H7 Correspodece should be addressed to T. Bouezmari, taoufik.bouezmari@usherbrooke.ca Received 15 September 21; Revised 29 March 211; Accepted 18 April 211 Academic Editor: Michael Lavie Copyright q 211 T. Bouezmari et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. The oparametric estimatio for the desity ad hazard rate fuctios for right-cesored data usig the kerel smoothig techiques is cosidered. The classical fixed symmetric kerel type estimator of these fuctios performs well i the iterior regio, but it suffers from the problem of bias i the boudary regio. Here, we propose ew estimators based o the gamma kerels for the desity ad the hazard rate fuctios. The estimators are free of bias ad achieve the optimal rate of covergece i terms of itegrated mea squared error. The mea itegrated squared error, the asymptotic ormality, ad the law of iterated logarithm are studied. A compariso of gamma estimators with the local liear estimator for the desity fuctio ad with hazard rate estimator proposed by Müller ad Wag 1994, which are free from boudary bias, is ivestigated by simulatios. 1. Itroductio Cesored data arise i may cotexts, for example, i medical follow-up studies i which the occurrece of the evet times called survival of idividuals may be preveted by the previous occurrece of aother competig evet called cesorig. I such studies, iterest focuses o estimatig the uderlyig desity ad/or hazard rate of the survival time. Noparametric estimatio usig kerel smoothig method has received cosiderable attetio i the statistical literature. A popular approach for estimatig the desity fuctio ad the hazard rate fuctio is doe usig a fixed symmetric kerel desity with bouded support ad a badwidthparameter. The kerel determies the shape of the local eighborhood while the badwidth cotrols the degree of smoothess. I order to get a reasoable estimator, these two parameters, the kerel ad the badwidth parameter, have to be chose carefully.

2 2 Joural of Probability ad Statistics For a review about kerel smoothig approaches, we refer the reader to Silverma 1 ad Izema 2 for ucesored data ad Sigpurwalla ad Wog 3, Taer ad Wog 4, Padgett ad McNichols 5, Mieliczuk 6, ad Lo et al. 7 i the case of right cesorig. It is well kow from the literature that the kerel has less impact tha the badwidth o the resultig estimate. However, whe the desity fuctio of the data have a bouded support, usig the classical kerel leads to a estimator with a large bias ear the edpoits. The problem of bias, called also the boudary effect,becomesaseriousdrawbackwhe a large portio of the sampled data are preset i the boudary regio. I fact, whe we estimate the uderlyig fuctio ear the edpoits, as the support of the kerel exceeds the available rage of the data, the bias of the resultig estimator becomes larger. This is especially the case i survival aalysis, sice the survival time is assumed to be oegative variable. So, ear zero, the symmetric kerel estimator of the desity ad the hazard fuctios uderestimates the true oes. For ucesored data, several methods are available i the literature to overcome this problem, for example, the reflectio method of Schuster 8, the smooth optimum kerel of Müller 9, the local liear estimator of Lejeue ad Sarda 1, the trasformatio approach of Marro ad Ruppert 11, ad the boudary kerel of Joes 12 ad Joes ad Foster 13. The local liear estimator is a special case of boudary kerel method. The mai idea behid the boudary kerel method is to use a adaptive kerels i the boudary regio ad to use a fixed symmetric kerel i the iterior regio. For oegative data ad i order to overcome the boudary bias problem, Che 14 cosiders the gamma kerel estimator. Simulatio results of Joes 12 ad Che 14 show that the local liear estimator outperforms the boudary kerel estimator of Müller 9. For right-cesored data ad to resolve this problem, Müller ad Wag 15 propose a ew class of polyomial boudary kerel estimator for hazard rate fuctio, where the kerel ad the badwidth parameter deped o the poit where the estimate is to be evaluated. Hess et al. 16 show umerically its performace via a extesive simulatio study. I this paper, we adapt the gamma kerel smoothig procedure to estimate the margial desity ad the hazard fuctio of positive idepedet survival data that are subject to right cesorig. We show that both estimators are robust agaist boudary problems. Also, we establish the mea itegrated square error, the asymptotic ormality, ad the law of iterated logarithm of the two estimators. Furthermore, via a Mote Carlo study, the fiite sample performace of the estimators is ivestigated uder various scearios. The paper is orgaized as follows. Sectio 2 itroduces the gamma kerel estimators for the desity ad the hazardratefuctios for right-cesoreddata.i Sectio3, we establish the asymptotic properties of the gamma kerel estimators. I Sectio 4, we ivestigate the fiite sample properties of the gamma kerel estimators. Sectio 5 provides a applicatio to the classical boe marrow trasplat data. The last sectio is a appedix that gathers the proofs. 2. Methodology Let T 1,...,T survival times ad C 1,...,C cesorig times be two i.i.d. oegative idepedet radom sequeces with distributio fuctios F ad G, respectively. Uder the cesorig model, istead of observig T i, we observe the pair X i,δ i,wherex i mi T i,c i ad δ i I Y i C i with I beig the idicator fuctio. We deote by f the desity fuctio

3 Joural of Probability ad Statistics 3 of F ad by h f/ 1 F the correspodig hazard fuctio. The oparametric maximum likelihood approach proposed by Kapla ad Meier 17 leads to the estimator of F give by i δ i 1 if x<x F x i:1 i,x i i 1 x 1, otherwise, 2.1 where X 1 X ad δ 1,...,δ are the correspodig δ i s. This estimator was studied by may authors. For referece, we cite Breslow ad Crowley 18,Wag 19, ad Stute ad Wag 2 amog may authors. Lo ad Sigh 21 expressed the KM estimator as a i.i.d. mea process with a remaider of egligible order. This result was improved by Lo et al. 7 ad will be useful i this paper to make the coectio betwee the ucesored ad cesored case. From ow o, we will deote the right edpoit of a give sub distributio L by T L,thatis,T L sup{t,l t < 1}. WewillalsousetheotatioL for the correspodig survival fuctio, that is, L 1 L. LetH be the distributio fuctio of X, thatis, H F G. We suppose that T G T F or equivaletly T H T F. I the remaider of this paper, except if metioed otherwise, all the itegratios are take over the iterval,t, where T T F. Based o the smooth kerel techique, we propose to estimate the desity by the gamma kerel estimator defied as follows: f b x K x, b t d F t K x, b X i W i, 2.2 i 1 where the kerel K is give by K x, b t tρ b x 1 exp t/b b ρ b x Γ ρ b x, x if x 2b ρ b b x 1 x if x, 2b, 4 b the weights W i s are the jumps of F at X i for i 2,...,,ad<b b is the badwidth parameter. Naturally, the gamma kerel estimator that we propose for the hazard rate h is f b x h b x F x As we will see later, those estimators are free of boudary bias; this is due to the fact that the gamma kerel is defied o the positivereal, ad so o weight is assiged outside the

4 4 Joural of Probability ad Statistics support of the uderlyig desity ad/or hazard rate fuctios. The shape of the gamma kerel ad the amout of smoothig are ot oly cotrolled by the badwidth parameter, but also vary accordig to the positio where the fuctio is estimated. For ucesored data, Che 14 shows that the gamma kerel desity estimator achieves the optimal rate of covergece for the mea itegrated squared error withi the class of oegative kerel desity estimators. Bouezmari ad Scaillet 22 state the uiform weak cosistecy for the gamma kerel estimator o ay compact set ad also the weak covergece i terms of mea itegrated absolute error. I the ext sectio, we will prove that eve whe the data are cesored, the gamma kerel estimators perform i the iterior ad the boudary regios. 3. Asymptotic Properties I this sectio, we state the asymptotic properties of the gamma kerel estimator of the desity ad the hazard rate fuctios. We start with the followig theorem which will play a importat role for the remaider of this sectio. To be cocise, i the followig, we will deote by Z either the desity or the hazard rate fuctio ad Ẑ b either the gamma kerel estimator for the desity or the hazard fuctio. Theorem 3.1. Assume that f is twice cotiuously differetiable. If i log 2 / b 3/2, the the itegrated mea squared error of Ẑ b is IMSE Ẑb b 2 B 2 x dx 1 b 1/2 V x dx o b 2 o 1 b 1/2, 3.1 where for the desity fuctio B ad V are give by B f x 1 2 xf x, V f x 1 2 π x 1/2 f x, 3.2 G x ad for the hazard rate fuctio B ad V are give by B h x B f x F x, V h x V f x. F x The optimal badwidth parameter which miimizes the asymptotic IMSE Ẑ b is give by b 2/5 1 V x dx 2/5, 4 B x dx 3.4

5 Joural of Probability ad Statistics 5 ad the correspodig optimal asymptotic itegrated mea squared error is IMSE Ẑb 5 4/5 V x dx 4 4/5 1/5 4/ B x dx Theorem 3.1 states that the gamma kerel estimators of the desity ad hazard rate fuctios are free of boudary bias ad provides the theoretical formula of the optimal badwidth. However, i real data aalysis, to choose a appropriate badwidth, oe eeds to use a data drive procedure, for example, the cross-validatio, the bootstrap, or the method proposed by Bouezmari ad Scaillet 22. Of course, all those methods eed to be carefully adapted to the cesorig case. Also, from A.25 i the appedix, the asymptotic variaces of the gamma kerel estimator are of a larger order O 1 b 1 ear the boudaries tha those O 1 b 1/2 i the iterior. However, Theorem 3.1 shows that the impact of the icreased variace ear the boudary o the mea itegrated squared error is egligible ad the optimal rate of covergece i term of itegrated mea squared error is achieved by the gamma kerel estimators. The followig propositio deals with the asymptotic ormality of the gamma kerel estimators. Propositio 3.2. Uder the same coditios i Theorem 3.1. if ii b 5/2 o 1, the for ay x such that f x > ad x T, oehas 1/2 b 1/4 Ẑb x Z x N, 1, i distributio, 3.6 V x where V x if x b V x Γ 2κ 1 b 2 1 2κ Γ 2 κ 1 2 πx 1/2 V x if x κ. b 3.7 We establish i the ext propositio the law of iterated logarithm of the gamma kerel estimators. Let Φ x, b 2 1 b 1/2 V x loglog, wherev is defied i Propositio 3.2. Propositio 3.3. Uder the same coditios i Theorem 3.1. if, iii b /b m 1, as, m such that /m 1, ad 2log 4 / b 2 Φ x, b. iv b o log log / 2/5, the lim sup Φ x, b 1/2 Ẑ b x Z x 1, a.s. 3.8

6 6 Joural of Probability ad Statistics 4. Simulatios Results The fiite sample performace of the proposed methodology is studied i this sectio. Two models are cosidered. i Model A: the survival time follows a expoetial distributio, ad the cesorig times were geerated from a uiform distributio,c. c was chose to be the solutio of the followig equatio pc exp c 1, where p is the desired percetage of cesorig. ii Model B: the survival times follow a Weibull distributio with scale parameter b 2 ad shape parameter a 1.2, ad the cesorig times are also geerated from a Weibull distributio with shape parameter a ad a scale parameter give by b 1 p /p 1/a. This esures that the degree of cesorig is equal to p. First, we show the results for the desity estimator. To evaluate the performace of the gamma kerel estimator for the desity fuctio, we compare this later with the local liear estimator of Joes 12 adapted for the right cesorig case. The local liear method is kow to be a robust techique agaist boudary bias problems. We cosider differet sample sizes, 125, 25, 5, 1. As a measure of errors of the estimators, we aalyze the mea ad the stadard deviatio of the L 2 -orm. Tables 1 ad 2 provide the results obtaied with 1 replicatios for the mea ad the stadard deviatio, respectively. Firstly ad as expected, whe the sample size icreases, the mea itegrated square error for the two estimators decreases; see Table 1.Thisistruefor both models ad all degrees of cesorig. For example, i model A, we ca see that with 1% of cesorig ad usig the gamma kerel estimator, the mea error decreases from.117 to.11 whe the sample size goes from 125 to 25. Note that, for the 5% of cesorig, the rate at which the mea error decreases is much smaller. Secodly, except the 5% cesorig case with model A, the gamma kerel estimator outperforms strogly the local liear kerel estimator. Thirdly, for model B, whe the degree of cesorig icreases the mea itegrated square error icreases as expected. I fact, for model B, the cumulative distributio of the cesorig times icreases whe the degree of cesorig p icreases. This implies that the mea itegrated squared error of the gamma kerel estimators icreases. From Table 2, oe ca see that the variace of both models ad both estimators decreases with the sample size, for all the degree of cesorig. For model B, the variace of gamma kerel estimator icreases with the degree of cesorig. Aother poit to remark is that, for model B, the local liear estimator is domiated i terms of variace by gamma kerel estimator i all situatios. For model A, the variace of the gamma kerel estimator is smaller tha the variace of the local liear estimator for p.1 ad.25, but ot for p.5. For the hazard fuctio, we compare our gamma kerel estimator with the boudarycorrected hazard estimator of Müller ad Wag 15.Table3 shows the mea ad the variace of L 2 errors based o 1 replicatios for both 125 ad 25. The same datageeratig procedure, as for desity fuctio, was used. This results clearly demostrate that our method is far more efficiet tha the method proposed by Müller ad Wag Applicatio To illustrate our approach with real data, we cosider the classical boe marrow trasplat study. The variable of iterest is the disease-free survival time, that is, time to relapse or

7 Joural of Probability ad Statistics 7 Table 1: Mea 1 2 of L 2 error for the desity fuctio estimators Model % ces G LL G LL G LL G LL A B G: gamma estimator; LL: local liear estimator; Model A: expoetial survival times with uiform cesorig times; Model B: Weibull survival times with Weibull cesorig times. The results are based o 1 replicatios. Table 2: Stadard deviatio 1 2 of L 2 error for the desity fuctio estimators Model % ces G LL G LL G LL G LL A B G: gamma estimator; LL: local liear estimator; Model A: expoetial survival times with uiform cesorig times; Model B: Weibull survival times with Weibull cesorig times. The results are based o 1 replicatios. Table 3: Mea ad Stadard deviatio 1 2 of L 2 error for the hazard rate fuctio estimators Mea Variace Mea Variace Model % ces G MW G MW G MW G MW A B G: gamma estimator. MW: Müller ad Wag estimator. Model A: expoetial survival times with uiform cesorig times. Model B: Weibull survival times with Weibull cesorig times. The results are based o 1 replicatios. death. Amog a total of 137 observatios, there are 83 cesored times all of them are caused by the ed of the follow-up period. The data together with a detailed descriptio of the study ca be foud i 23,Sectio1.3. Figure 1 shows the estimated hazard fuctio usig both our method G adthatproposedbymüller ad Wag 15 MW with boudary correctio. For the Muller estimator, we use the data-drive global optimal badwidth as proposed by the authors. To calculate the MW estimator ad its badwidth parameter, we make use of the R package muhaz; see Hess ad Getlema 24. The estimator of Müller ad Wag 15 show i Figure 1 is based o the badwidth b 24.5 while for the gamma

8 8 Joural of Probability ad Statistics.2 Hazard Time MW G Figure 1: Gamma G ad Müller ad Wag MW estimators of the hazard fuctio for the boe marrow trasplat data. kerel estimator b This is a data-drive badwidth based o the least squared crossvalidatio method of Marro ad Padgett 25 adapted to our case. Note that this is just a practical choice ad may be far from the optimal oe. From Figure 1, oe caot decide which estimator is better but, give the results of our simulatio study, we suspect that the MW method has a global tedecy of uderestimatig the real hazard fuctio ad so the real risks after a boe marrow trasplat. Appedix We start this sectio with some otatios. Let g t t 1 H s 2 dh 1 s, A.1 where H 1 t : P X i t, δ i 1 t 1 G s df s deote the subdistributio of the ucesored observatios. For positive real umbers z ad x, adδ or1,let ξ z, δ, t g z t 1 H 1 I z t, δ 1. A.2 We also set ξ i x ξ X i,δ i,x. First ote that E ξ i x, Cov ξ i t,ξ i s g s t. A.3 The followig two lemmas will play a importat role i the demostratios. The first lemma is a due to Lo et al. 7 which expressed the KM estimator as a i.i.d. mea process with a remaider of egligible order.

9 Joural of Probability ad Statistics 9 Lemma A.1 see 7. For t T, F t F t 1 F t ξ i t r t, i 1 A.4 where log sup r x O x T a.s., A.5 ad, for ay α 1, [ ] log α sup E r x α O. A.6 x T The ext lemma gives a strog approximatio of the gamma kerel estimator of the desity. This lemma permits us to derive the asymptotic properties of the gamma kerel estimator for desity ad hazard rate fuctio. Lemma A.2. The gamma kerel desity estimator f b admits the strog approximatio o the iterval, T : f b x f x β x σ x e x, A.7 where β x f t K x, b t dt f x, σ x 1 ξ i t dk x, b t, A.8 log sup e x O x T b a.s. A.9 Also, for ay α 1, [log ] α sup E e x α O. A.1 x T b Proof of Lemma A.2. Let us first show that K x, b T ca be made arbitrary small for ay x< T. From Γ w 1 2π exp w w w 1/2, as w A.11

10 1 Joural of Probability ad Statistics it follows that for s x b, K x, b T T s/b exp T/b b s/b 1 Γ s/b 1 bs 1/2 2π { s exp b 1 T } T s log. s A.12 Observe that, for ay w>1, 1 w log w <. So, K x, b T o b α for ay α 1. Now, usig this result, the itegratio by part, ad Lemma A.1, weobtai f b x K x, b t d F t F t dk x, b t [ ] F t 1 ξ i t r t dk x, b t A.13 f t K x, b t dt 1 ξ i t dk x, b t r t dk x, b t f x β x σ x e x. We deduce the result of Lemma A.2 by usig the followig iequality: dk x, b t b 1 K x, b t K x b, b t dt 2b 1. A.14 Proof of Theorem 3.1. We start with the gamma kerel desity estimator. From the asymptotic bias of the gamma kerel estimator for ucesored data see Che 14 ad the fact that for the iterior regio log / b 1/2 b 1/4 log, 1/2 b 3/4 A.15 ad for the boudary regio log / b log, 1/2 b 1/2 1/2 b A.16 1/2

11 Joural of Probability ad Statistics 11 it ca be easily verified that, i our case, the bias is give by 1 E fb x f x 2 xf x b o b o 1/2 b 1/4 if x 2b, ξ b x bf x o b o 1/2 b 1/2 if x, 2b. A.17 where ξ b x 1 x ρ b x x/b / 1 bρ b x x. To calculate the asymptotic variace of the gamma kerel estimator, we oly eed to evaluate the variace of σ x sice the other terms are egligible. We start with the fact that Var σ x 1 F t F s g t s dk t dk s, A.18 where k k x, b. Usig itegratio by parts, the first itegral becomes F t g t s dk t s F t g t dk t F t g s dk t t s [ ] s F t g t k t s F t g t k t dt [ ] g s F t k t g s F t k t dt t s t s s k t dt F t g t g s F t k t dt. s A.19 So that, Var σ x F s s k t dt F t g t dk s F s g s F t k t dk s dt t s t k t F t g t F s dk s s t dt F t k t F t g t k t F t g t k t F s dk s dt s t F s g s dk s dt F t k t t F s g s dk s dt. A.2

12 12 Joural of Probability ad Statistics Agai, usig the itegratio by parts we obtai, F t g t k t F s dk s dt F 2 t g t k 2 t dt O 1, s t F t g t k t F s dk s dt F t F t g t k 2 t dt O 1 s t A.21 F t k t t F s g s dk s dt F t F t g t k 2 t dt O 1. Therefore, from g t f t / G t F t 2,weget Var σ x F 2 t g t k 2 t dt [ ] f t / G t k 2 t dt A.22 B b x IE f 1 η x G ηx, where η x is a gamma 2x/b 1,b/2 radom variable ad B b x b 1 Γ 2x/b 1 2 2x/b 1 Γ 2 x/b 1. A.23 From Che 14, we have that for a small value of b, 1 2 π b 1/2 x 1/2, if x b B b x Γ 2κ κ Γ 2 κ 1 b 1, if x b, κ. A.24 So after some easy developmet, we get Var fb x 1 2 π b 1/2 x Γ 2κ κ Γ 2 κ 1 1/2 f x G x, if x, b f x b 1 G x, if x κ. b A.25 Fially, we derive the itegrated mea squared error from A.25 ad A.17. Now, for the gamma kerel estimator of the hazard fuctio, we start by the followig decompositio: MSE h b x E h b x h x 2 E I II III 2, A.26

13 Joural of Probability ad Statistics 13 where I f b x F x, F x F x [ fb x E fb x ] II F x, [ ] E fb x f x III F x. A.27 Observe that E II III III E II, because the term III is determiistic. Usig Schwartz iequity ad the boudedess of the gamma kerel desity estimator we foud that E I O 1/2 ad E I 2 O 1. Now, from the bias formula of the gamma kerel estimator ad the coditios o the badwidth parameter, we get O 1/2 O b o E I III III E I O 1/2 O b o 1/2 b 1/4 1/2 b 1/2 o o 1 b 1 x, if, b x, if κ. b 1 b 1/2 A.28 Agai, usig the Schwartz s iequality, we obtai E I II [ ] 1/2 [ E I 2] 1/2 Var fb x F x O 1/2 O O 1/2 O 1/2 b 1/4 1/2 b 1/2 o 1 b 1/2 o 1 b 1 if x b, if x b κ. A.29 Combiig these formulas, oe ca see that MSE h b x E II 2 E III 2 O 1 o b 1 E II 2 E III 2 o b 1. A.3 Fially, usig the expressio of the bias ad the variace of the gamma kerel desity estimator, we derive the desired result of the gamma kerel estimator for the failure rate fuctio. We will oly give the proof of the asymptotic ormality ad the iterated logarithm for the gamma kerel estimator. Thereafter, the result is straightforward for the gamma kerel

14 14 Joural of Probability ad Statistics hazard estimator. The proofs are based o Lemma A.2. Ideed, we oly eed to prove the asymptotic ormality ad the iterated logarithm for σ defied i A.8 sice the other terms are egligible. Proof of Propositio 3.2. By i, see Theorem 3.1, wehave 1/2 b 1/4 e x o 1 ad uder the coditios o the badwidth parameter, we have 1/2 b 1/4 β x o 1. Therefore, we eed to state that 1/2 b 1/4 σ x N, 1, i distributio. A.31 V x But sice σ x W i x, b, where W i x, b 1 ξ i t dk x, b t, A.32 it suffices to prove that i 1 E W i 3. A.33 3/2 var W 1 I fact, usig iequality A.14, 1 sup x W i x, b O. A.34 b Therefore, from the variace of σ x, i 1 W E i 3 1 var W 1 O O b 3/2 if x 2b, var W 3/2 1 1/2 b O 3/2 4/5 if x, 2b. o 1 sice b o 2/5. A.35 Proof of Propositio 3.3. Coditio iv esures that 1/2 b 1/4 β x o 1 ad 1/2 b 1/4 e x o 1. O the other had, we apply 26, Theorem1 to S i 1 W i,wherew i is defied as i the proof of Propositio 3.2; we get uder coditio iii o the badwidth parameter lim sup Φ x, b 1/2 σ 1, a.s. A.36 which cocludes the proof of the theorem.

15 Joural of Probability ad Statistics 15 Ackowledgmet M. Mesfioui ackowledges the fiacial support of the Natural Scieces ad Egieerig Research Coucil of Caada. Refereces 1 B. W. Silverma, Desity Estimatio for Statistics ad Data Aalysis, Moographs o Statistics ad Applied Probability, Chapma & Hall, New York, NY, USA, A. J. Izema, Recet developmets i oparametric desity estimatio, Joural of the America Statistical Associatio, vol. 86, o. 413, pp , N. D. Sigpurwalla ad M. Y. Wog, Kerel estimators of the failure-rate fuctio ad desity estimatio: a aalogy, Joural of the America Statistical Associatio, vol. 78, o. 382, pp , M. A. Taer ad W. H. Wog, The estimatio of the hazard fuctio from radomly cesored data by the kerel method, The Aals of Statistics, vol. 11, o. 3, pp , W. J. Padgett ad D. T. McNichols, Noparametric desity estimatio from cesored data, Commuicatios i Statistics Theory ad Methods, vol. 13, o. 13, pp , J. Mieliczuk, Some asymptotic properties of kerel estimators of a desity fuctio i case of cesored data, The Aals of Statistics, vol. 14, o. 2, pp , S. H. Lo, Y. P. Mack, ad J. L. Wag, Desity ad hazard rate estimatio for cesored data via strog represetatio of the Kapla-Meier estimator, Probability Theory ad Related Fields, vol.8,o.3,pp , E. F. Schuster, Icorporatig support costraits ito oparametric estimators of desities, Commuicatios i Statistics Theory ad Methods, vol. 14, o. 5, pp , H. G. Müller, Smooth optimum kerel estimators ear edpoits, Biometrika, vol.78,o.3,pp , M. Lejeue ad P. Sarda, Smooth estimators of distributio ad desity fuctios, Computatioal Statistics & Data Aalysis, vol. 14, o. 4, pp , J. S. Marro ad D. Ruppert, Trasformatios to reduce boudary bias i kerel desity estimatio, Joural of the Royal Statistical Society. Series B, vol. 56, o. 4, pp , M. C. Joes, Simple boudary correctio for kerel desity estimatio, Statistics ad Computig,vol. 3, o. 3, pp , M. C. Joes ad P. J. Foster, A simple oegative boudary correctio method for kerel desity estimatio, Statistica Siica, vol. 6, o. 4, pp , S. X. Che, Probability desity fuctio estimatio usig gamma kerels, Aals of the Istitute of Statistical Mathematics, vol. 52, o. 3, pp , H. G. Müller ad J. L. Wag, Hazard rate estimatio uder radom cesorig with varyig kerels ad badwidths, Biometrics, vol. 5, o. 1, pp , K. R. Hess, D. M. Serachitopol, ad B. W. Brow, Hazard fuctio estimators: a simulatio study, Statistics i Medicie, vol. 18, o. 22, pp , E. L. Kapla ad P. Meier, Noparametric estimatio from icomplete observatios, Joural of the America Statistical Associatio, vol. 53, pp , N. Breslow ad J. Crowley, A large sample study of the life table ad product limit estimates uder radom cesorship, The Aals of Statistics, vol. 2, pp , J. G. Wag, A ote o the uiform cosistecy of the Kapla-Meier estimator, The Aals of Statistics, vol. 15, o. 3, pp , W. Stute ad J. L. Wag, The strog law uder radom cesorship, The Aals of Statistics, vol. 21, o. 3, pp , S. H. Lo ad K. Sigh, The product-limit estimator ad the bootstrap: some asymptotic represetatios, Probability Theory ad Related Fields, vol. 71, o. 3, pp , T. Bouezmari ad O. Scaillet, Cosistecy of asymmetric kerel desity estimators ad smoothed histograms with applicatio to icome data, Ecoometric Theory, vol. 21, o. 2, pp , J. P. Klei ad M. L. Moeschberger, Survival aalysis. Techiques for Cesored ad Trucated Data,vol.15 of Statistics for Biology ad Health, Spriger, New York, NY, USA, 2d editio, 23.

16 16 Joural of Probability ad Statistics 24 K. Hess ad R. Getlema, Muhaz: hazard fuctio estimatio i survival aalysis, R package versio 1.2.5, J. S. Marro ad W. J. Padgett, Asymptotically optimal badwidth selectio for kerel desity estimators from radomly right-cesored samples, The Aals of Statistics, vol. 15, o. 4, pp , P. Hall, Laws of the iterated logarithm for oparametric desity estimators, Zeitschrift für Wahrscheilichkeitstheorie ud Verwadte Gebiete, vol. 56, o. 1, pp , 1981.

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