Abdelkader Tatachak

Elias Ould-Saïd

Ould-Saïd, Elias

Tatachak, Abdelkader

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 344 (2007) 651–656http://france.elsevier.com/direct/CRASS1/Statistics/Probability TheoryAsymptotic properties of the kernel estimatorof the conditional mode for the left truncated modelElias Ould-Saïd a, Abdelkader Tatachak ba L.M.P.A. J. Liouville, Université du littoral côte d’opale, BP 699, 62228 Calais, Franceb Laboratoire M.S.T.D, U.S.T.H.B, faculté de mathématiques, BP 32, El-Alia, 16111, AlgeriaReceived 27 June 2006; accepted after revision 23 March 2007Available online 7 May 2007Presented by Paul DeheuvelsAbstractIn this Note we propose a non-parametric kernel estimator of the conditional mode function, when the variable of interest issubject to random left-truncation. We establish the rate of the strong uniform consistency of the estimate as well as its asymptoticnormality. To cite this article: E. Ould-Saïd, A. Tatachak, C. R. Acad. Sci. Paris, Ser. I 344 (2007).© 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméPropriétés asymptotiques de l’estimateur à noyau du mode conditionnel pour un modèle tronqué à gauche. Dans cetteNote nous proposons un estimateur non paramétrique de la fonction mode conditionnel, lorsque la variable d’intérêt est assujettieà une troncature aléatoire à gauche. Nous établissons dans un premier temps la convergence uniforme presque sûre de l’estimateurainsi que sa vitesse, puis nous en donnons la normalité asymptotique. Pour citer cet article : E. Ould-Saïd, A. Tatachak, C. R.Acad. Sci. Paris, Ser. I 344 (2007).© 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeDans le modèle de données tronquées aléatoirement à gauche (TAG), on considère deux suites (Yi)i1 et (Ti)i1mutuellement indépendantes de fonctions de répartition (f.d.r) F et G respectivement. On suppose que l’on disposed’un échantillon de taille N . La variable d’intérêt Y est dite tronquée aléatoirement à gauche par la variable T si lesseules variables aléatoires observables sont (Yi, Ti) vérifiant Yi  Ti , sinon rien n’est observé. Nous noterons par n lataille de l’échantillon observé qui est connue mais aléatoire, avec n  N . Sous ce modèle, soit à présent, X ∈ Rd unvecteur aléatoire de covariables de f.d.r L possédant une densité . Dans toute la suite, la variable T sera supposéeindépendante du vecteur (X, Y ).Il est clair que si α := P(Y  T ) = 1, le mode conditionnel Θ(x) de Y sachant X = x, supposé unique, est donnépar :E-mail addresses: ouldsaid@lmpa.univ-littoral.fr (E. Ould-Saïd), atatachak@usthb.dz (A. Tatachak).1631-073X/$ – see front matter © 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2007.03.023652 E. Ould-Saïd, A. Tatachak / C. R. Acad. Sci. Paris, Ser. I 344 (2007) 651–656Θ(x) = arg max−∞<y<∞g(y|x)où g(y|x) = f (x, y)/(x) est la densité conditionnelle de Y sachant X = x, et f (·, ·) est la densité jointe de (X, Y ).Sous le modèle de données TAG, nous construisons un estimateur non paramétrique du mode conditionnel Θ(x),basé sur l’observation de n triplets (Xi , Yi, Ti).Dans cette Note, en fixant d = 1, nous établissons la convergence uniforme presque sûre de Θ̂n(x), solution del’équationΘ̂n(x) = arg max−∞<y<∞ ĝn(y|x)où ĝn(·|x) est un estimateur convergent de g(·|x). Enfin, nous établissons la normalité asymptotique de Θ̂n(x) conve-nablement normalisé et affirmons que :(nh4nG(Θ(x))(f(0,2)(x,Θ(x)))2αf (x,Θ(x))∫R2[K1(r)Λ(1)(s)]2 dr ds)1/2(Θ̂n(x) − Θ(x)) D−→N (0,1)où K1(·) et Λ(·) sont des noyaux, hn est une suite telle que hn ↘ 0 lorsque n ↗ ∞ et D−→ signifie la convergence enloi.Des simulations ont été effectuées pour illustrer la consistance de notre estimateur ainsi que la normalité asymp-totique à distance finie, qui ne figurent pas dans la Note par manque de place ; cependant, elles peuvent être obtenuesauprès des auteurs.1. IntroductionLet Y and T be independent random variables (rv) with distribution function (df) F and G respectively, bothassumed to be continuous. Let (Yi, Ti){i1} be a sequence of independent and identically distributed (iid) as (Y,T ),where the population size N is deterministic but unknown. In the random left-truncation model, the rv of interest Yis interfered by the truncation rv T , in such a way that Yi and Ti are observable only if Yi  Ti , whereas nothing isobserved if Yi < Ti . Without possible confusion, we still denote (Yi, Ti)1in (n  N), these observed iid pairs. Asa consequence of truncation, the size of the actually observed sample, n is a Bin(N,α) rv, with α := P(Y  T ) = 0.Let X be a Rd -valued random covariates vector, assumed to be absolutely continuous with df L having density ,and let g(·|x) = f (x, ·)/(x) denote the conditional probability density function (pdf) of Y given X = x, where f (·, ·)is the joint pdf of (X, Y ). Assuming that g(·|x) has a unique mode at the point Θ(x), the conditional mode functionof Y given X = x, is defined by the following equation:Θ(x) = arg max−∞<y<∞g(y|x). (1)Since N is unknown and n is known (although random), our results will not be stated with respect to the probabilitymeasure P (related to the N -sample) but will involve the conditional probability P (related to the n-sample). Note alsothat E and E denote the expectation operators related to P and P, respectively. In the sequel, we denote by a super-script (∗) any df that is associated with the truncated rv. Under the left-truncation sampling scheme, the conditionaljoint distribution (Stute [5]) of (Y,T ) becomes:V∗(y, t) = P{Y  y,T  t} = P{Y  y,T  t | Y  T }= α−1y∫−∞G(t ∧ v)dF(v).The marginal distributions and their empirical versions are defined byF ∗(y) := V∗(y,∞) = α−1y∫G(v)dF(v) and F ∗n (y) = n−1n∑i=11(Yiy),−∞E. Ould-Saïd, A. Tatachak / C. R. Acad. Sci. Paris, Ser. I 344 (2007) 651–656 653G∗(t) := V∗(∞, t) = α−1+∞∫−∞G(t ∧ v)dF(v) and G∗n(t) = n−1n∑i=11(Tit).Since N is unknown, in the sequel we will use a consistent estimator αn (He and Yang [1]) of α, defined byαn = Gn(y)(1 − Fn(y−))Rn(y), (2)for all y such that Rn(y) > 0, where Gn(·) and Fn(·) are the product-limit estimators (Lynden-Bell [2]) for F(·) andG(·) and Rn(y) = n−1 ∑ni=1 1(TiyYi).Suppose that one observes the n triplets (Xi , Yi, Ti) among the N ones and for any df W, define aW =inf{x: W(x) > 0} and bW = sup{x: W(x) < 1} as the endpoints of the W support. We point out that Woodroofe[6] proved that F and G can be estimated completely only ifaG  aF , bG  bF and∞∫aFdFG< ∞.Suppose now that T is independent of (X, Y ) and let F(· , ·) be the joint df of (X, Y ) related to the N -sample, wehave the following trivariate conditional distribution H ∗:H ∗(x, y, t) = P{X  x, Y  y,T  t} = α−1∫ux∫aGvyG(v ∧ t)F(du,dv).Taking t = +∞, the observed pair (X, Y ) then has the following df F∗(·, ·):F∗(x, y) = H ∗(x, y,∞) = α−1∫ux∫aGvyG(v)F(du,dv). (3)By differentiating in the statement (3) and integrating the result over y, we obtainF(dx,dy) = [α−1G(y)]−1F∗(dx,dy) for y  aG (4)andL(x) = α∫ux∫aGy1G(y)F∗(du,dy).A natural estimator of L(x) is then given byL̃n(x) = αnn∑i=11G(Yi)1(Xix). (5)Note that in (5) and in the sequel, the sum is taken only for the i’s such that G(Yi) = 0. Finally (5) yields the estimatorof the density function of the vector X as̃n(x) := 1adn∫RdKd(x − uan)L̃n(du) = αnadnn∑i=11G(Yi)Kd(x − Xian), (6)where Kd is a fixed kernel with∫RdKd = 1, and (an)n1 ↘ 0 as n ↗ ∞. Here, we point out that the estimate (6) hasbeen already defined by Ould-Saïd and Lemdani [3] in the context of the estimation of the regression function. Whileadopting the same methodology, we get an estimator of F(·, ·) as followsF̃n(x, y) = αnn∑i=11G(Yi)1(Xix,Yiy). (7)Now, using (7) we define the kernel estimate of the joint pdf f (· , ·) by654 E. Ould-Saïd, A. Tatachak / C. R. Acad. Sci. Paris, Ser. I 344 (2007) 651–656f̃n(x, y) = 1adnbn∫Rd∫RKd(x − uan)Λ(y − vbn)F̃n(du,dv)= αnadnbnn∑i=11G(Yi)Kd(x − Xian)Λ(y − Yibn), (8)where Λ is a fixed kernel with∫RΛ = 1, and (bn)n1 ↘ 0 as n ↗ ∞. Finally, for all x ∈ Rd, we get a new estimatorof the conditional density g(y|x). Indeed, we haveg̃n(y|x) := f̃n(x, y)̃n(x)=∑ni=1(G(Yi))−1Kd((x − Xi )/an)Λ((y − Yi)/bn)bn∑ni=1(G(Yi))−1Kd((x − Xi )/an), (9)with the convention 00 = 0.The natural estimator of Θ(x) is defined as the random variable Θ̂n(x), satisfyingg̃n(Θ̂n(x)|x) = max−∞<y<∞ g̃n(y|x). (10)However, the computation of Θ̂n(x) from (10) is not possible since both α and G(·) are usually unknown. One wayto overcome this difficulty, is to replace G(·) by Gn(·) and α by αn. Therefore, the feasible estimator of Θ(x) is thengiven as the solution of the equationĝn(Θ̂n(x)|x) = max−∞<y<∞ ĝn(y|x) = max−∞<y<∞f̂n(x, y)̂n(x), (11)where f̂n(·, ·), ̂n(·) and the corresponding df’s F̂n(· , ·) and L̂n(·) are defined by the same expressions as in (8), (6),(7) and (5) respectively, in which α and G(·) are replaced by αn and Gn(·).2. Main resultsWith the aim of simplifying the results and the proofs, let us fix d = 1, an = bn = hn, and set Ξ0 = {x ∈ R | (x) >0}. Let also Ξ ⊂ Ξ0 be a compact set of R, and C() be the set of continuity points of . The assumptions are gatheredbelow for easy reference.(A) (a) f (· , ·) is differentiable up to order 4 and supx,y |f (i,j)(x, y)| < ∞, i + j  4;(b) f (0,2)(·, ·) is continuous and does not vanish in the neighborhood of Θ(·);(c) (·) satisfies the Lipschitz condition and (x)  γ0 > 0;(d) g(y|x) is twice differentiable, uniformly continuous in y and the second derivative g(2)(·|x) is continuous;(e) Θ(·) is such that for any ε > 0 and any function η(·), there exists a β > 0 such that supx∈Ξ |Θ(x) − η(x)|  εimplies supx∈Ξ |g(Θ(x)|x) − g(η(x)|x)|  β .(B) (a) The kernel K (K = K1 or K = Λ) is continuous and satisfies lim|u|→∞ K(u) = 0 and∫RK(u)du = 1;(b)∫RujK(u)du = 0; j = 1,2;(c)∫R|u|3K(u)du < ∞;(d) K1 is differentiable and of bounded variations;(e) Λ is three times differentiable and Λ, Λ(1) and Λ(2) are functions of bounded variations.(H) (a) The sequence hn is such that hn ↘ 0 and nh8nlogn → ∞ as n ↗ ∞;(b) nh9n → 0 as n ↗ ∞.Remark 1. Assumptions (A, a,b,e) are needed to establish Theorem 2.1. The conditions (B, a–c) are very usual inthe functional estimation; they allow us to study the asymptotic variance term of f̃ (0,1)n (· , ·) while (B, d,e) are usedto state the uniform convergence of f̂n(· , ·). The Assumptions (H, a) are needed to obtain the uniform convergenceof the second derivative f̂ (0,2)n . The hypothesis (H, b) allows to state the convergence of the bias term of f̃(0,1)n (· , ·).Note that these assumptions are usual and not very restrictive. We point out that we obtain the same bandwidth as theone used in the complete data case (see, Samanta and Thavaneswaran [4]).E. Ould-Saïd, A. Tatachak / C. R. Acad. Sci. Paris, Ser. I 344 (2007) 651–656 655Our first result deals with the uniform almost sure (a.s.) convergence of the estimator (11):Theorem 2.1. Suppose that Assumptions (A: a,c), (B: c,d,e) and (H: a) hold. Then,limn→∞ supy∣∣ĝn(y|x) − g(y|x)∣∣ = 0, a.s.Moreover, we have for any compact set Ξ ⊂ Ξ0,supx∈Ξsupy∣∣ĝn(y|x) − g(y|x)∣∣ = O[max{((nh4n)−1 logn)1/2, hn}], P-a.s.From this Theorem we deduce the following corollary:Corollary 2.2. Under Assumptions (A: a,b,d,e), (B: d,e), (H:a), we havelimn→∞ supx∈Ξ∣∣Θ̂n(x) − Θ(x)∣∣ = 0, a.s.In addition, for all n large enough, we havesupx∈Ξ∣∣Θ̂n(x) − Θ(x)∣∣ = O[max{((nh4n)−1 logn)1/4, h1/2n }], P-a.s.Remark 2. The proof of Theorem 2.1 is based mainly on the following classical decomposition, which holds withprobability one and the convergence of each of the termssupx∈Ξsupy∣∣ĝn(y|x) − g(y|x)∣∣  1infx∈Ξ ̂n(x){supx∈Ξsupy∣∣f̂n(x, y) − f (x, y)∣∣ + κ. supx∈Ξ∣∣̂n(x) − (x)∣∣}, (12)where κ is the upper bound of g(y|x).The following result states the asymptotic normality of the kernel conditional mode estimator:Theorem 2.3. Suppose that Assumptions (A: b), (B: a,b,d,e) and (H: a,b) hold. Then, we have for any x ∈ A,(nh4n(f(0,2)(x,Θ(x)))2Var(x,Θ(x)))1/2(Θ̂n(x) − Θ(x)) D−→ N (0,1),where A ={x: x ∈ C(f ),Var(x,Θ(x)) = 0},Var(x,Θ(x)) = αf (x,Θ(x))G(Θ(x))∫R2[K1(r)Λ(1)(s)]2dr dsand D−→ denotes the convergence in distribution.Remark 3. A plug-in estimate for the asymptotic standard deviation σ(x,Θ(x)) := (Var(x,Θ(x)))1/2, can be easilyobtained using the estimators Gn(·), (2), (11) and f̂n(· , ·) of G(·), α, Θ(x) and f (· , ·) respectively, which is givenby σ̂ (x, Θ̂n(x)) = (αnf̂n(x,Θ̂n(x))Gn(Θ̂n(x))∫R2[K1(r)Λ(1)(s)]2 dr ds)1/2. Then σ̂ (x, Θ̂n(x)) can be used to get the followingapproximate (1 − ζ )% confidence interval for Θ(x).Θ̂n(x) ± t1−ζ/2 ×(σ̂ 2(x, Θ̂n(x))nh4n(f̂(0,2)n (x, Θ̂n(x)))2)1/2.A simulation study has been performed to illustrate the behavior of our estimator and how good is the asymptoticnormality in the finite sample. This study has not been included in this Note for lack of place, however, it can beobtained from the authors.656 E. Ould-Saïd, A. Tatachak / C. R. Acad. Sci. Paris, Ser. I 344 (2007) 651–656AcknowledgementsThe authors wish to thank the two anonymous referees for their remarks and comments that permitted them toimprove this Note.References[1] S. He, G.L. Yang, Estimation of the truncation probability in the random truncation model, Ann. Statist. 26 (1998) 1011–1027.[2] D. Lynden-Bell, A method of allowing for known observational selection in small samples applied to 3CR quasars, Monthly Notices Roy.Astronom. Soc. 155 (1971) 95–118.[3] E. Ould-Saïd, M. Lemdani, Asymptotic properties of a nonparametric regression function estimator with randomly truncated data, Ann. Inst.Statist. Math. 58 (2006) 357–378.[4] M. Samanta, A. Thavaneswaran, Nonparametric estimation of the conditional mode, Comm. Statist. Theory Methods 19 (1990) 4515–4524.[5] W. Stute, Almost sure representations of the product-limit estimator for truncated data, Ann. Statist. 21 (1993) 146–156.[6] M. Woodroofe, Estimating a distribution function with truncated data, Ann. Statist. 13 (1985) 163–177.

Asymptotic properties of the kernel estimator of the conditional mode for the left truncated model

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Asymptotic properties of the kernel estimator of the conditional mode for the left truncated model

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