Asymptotic normality of conditional distribution estimation in the single index model

Transcription

1 Acta Univ. Sapientiae, Mathematica, 9, DOI: 0.55/ausm Asymptotic normality of conditional distribution estimation in the single index model Diaa Eddine Hamdaoui Laboratory of Stochastic Models, Statistic and Applications, Dr. Tahar Moulay University, Algeria Abbes abhi Mathematics laboratory, Djillali Liabes University of Sidi Bel Abbes, Algeria rabhi Amina Angelika Bouchentouf Mathematics laboratory, Djillali Liabes University of Sidi Bel Abbes, Algeria bouchentouf Toufik Guendouzi Laboratory of Stochastic Models, Statistic and Applications, Dr. Tahar Moulay University, Algeria Abstract. This paper deals with the estimation of conditional distribution function based on the single-index model. The asymptotic normality of the conditional distribution estimator is established. Moreover, as an application, the asymptotic γ confidence interval of the conditional distribution function is given for 0 < γ <. Introduction The single functional index models have received a considerable attention because of their wide applications in many areas such as economics, medicine, financial econometric and so on. The study of these models has been developed rapidly, see Ait-Saidi et al. 2005, 2008a, 2008b. ecently, Attaoui et 200 Mathematics Subject Classification: 62G07, 62G5 62G20 Key words and phrases: asymptotic normality, single functional index model, conditional distribution 62

2 Asymptotic normality of conditional distribution estimation al. 20 investigated the kernel estimator of the conditional density of a scalar response variable Y, given a Hilbertian random variable X when the observations are from a single functional index model. The pointwise and the uniform almost complete convergence of the estimator with rates in this model were obtained for independent observations. Furthermore, Ling et al. 202 obtained the asymptotic normality of the conditional density estimator and the conditional mode estimator for the α-mixing dependence functional time series data. Ling et al. 204 investigated the pointwise almost complete consistency and the uniform almost complete convergence of the kernel estimation with rate for the conditional density in the setting of the α-mixing functional data, which extend the i.i.d case in Attaoui et al. 20 to the dependence setting, the convergence rate of the kernel estimation for the conditional mode was also obtained. The main contribution of this paper is to establish the asymptotic normality for the estimator of conditional distribution function in the i.i.d. case when the single functional index θ is fixed. As an application, the asymptotic γ confidence interval for the conditional density function Fθ, y, x is presented. The outline of the present paper is as follows. In section 2, we introduce the model as well as basic assumptions that are necessary in deriving the main result of this paper. In section 3, we state the main result of the paper; the asymptotic normality of the estimator for the conditional distribution function. As an application, the asymptotic γ confidence interval of the conditional distribution function is given for 0 < γ <. Finally, the technical proofs are related to section 4. 2 Model and some basic assumptions Let {X i, Y i, i n} be n random variables, identically distributed as the random pair X, Y with values in H, where H is a separable real Hilbert space with the norm. generated by an inner product <.,. >. Under such topological structure and for a fixed functional θ, we suppose that the conditional probability distribution of Y given < X, θ >=< x, θ > exists and is given by y, Fθ, y, x = PY y < X, θ >=< x, θ >. The nonparametric kernel estimator Fθ, y, x of Fθ, y, x is defined as follows,

3 64 D. E. Hamdaoui, A. A. Bouchentouf, A. abhi, T. Guendouzi Fθ, y, x = i= Kh K < x X i, θ >Hh H y Y i i= Kh K < x X i, θ >, 2 where K is a kernel, H is a cumulative distribution function cdf and h K = h K,n resp,h H = h H,n is a sequence of positive real numbers which goes to zero as n tends to infinity, and with the convention 0/0 = 0. Let, for any x H, i =,..., n and y K i θ, x := Kh K < x X i, θ >, and H i y := Hh H y Y i. We denote by B θ x, h = {X H/0 < < x X, θ > < h} the ball centered at x with radius h, let N x be a fixed neighborhood of x in H, S will be a fixed compact subset of. Now, we introduce the following basic assumptions that are necessary in deriving the main result of this paper. H PX B θ x, h K =: φ θ,x h > 0, φ θ,x h 0 as h 0. H2 The conditional cumulative distribution Fθ, y, x satisfies the Hölder condition, that is: y, y 2 S S, x, x 2 N x N x. Fθ, y, x Fθ, y 2, x 2 C θ,x x x 2 b + y y 2 b 2, b > 0, b 2 > 0. H3 For j = 0,, H j satisfies the lipschitz conditions and m := inf t [0,] KtH t > 0, with H tdt =, H 2 tdt < and t b 2 H tdt < H4 The kernel K is nonnegative, with compact support [0, ] of class C on [0, such that K > 0 and its derivative K exists on [0, and K t < 0. H5 For all u [0, ], lim h 0 φ θ,x uh φ θ,x h = lim h 0 ξθ,x h u = ξθ,x 0 u.

4 Asymptotic normality of conditional distribution estimation H6 The bandwidth h H satisfies, i log n 0, as n. n φ θ,x h K ii nh 2 H φ2 θ,x h K, and nh3 H φ θ,xh K log 2 n iii nh 2 H φ3 θ,x h K 0, as n. as n. H7 i φ θ,xh n + φ x h = O n. ii nφ θ,x h 0 as n. Comments on the assumptions. Assumption H is the same as one given in Ferraty et al Assumption H2 is a regularity conditions which characterize the functional space of our model and is needed to evaluate the bias term of our asymptotic results. Assumptions H3 and H5 and H6 are technical conditions for the proofs. Assumptions H4 is classical in functional estimation for finite or infinite dimension spaces. emark Assumption H5 is known as for small h the concentration assumption acting on the distribution of X in infinite dimensional spaces. The function ξ x h intervening in assumption H9 is increasing for all fixed h. Its pointwise limit ξ x 0 plays a determinant role. It is possible to specify this function with ξ 0 u := ξ x 0 u in the above examples by:. ξ 0 u = u γ, 2. ξ 0 u = δ u, where δ is Dirac function, 3. ξ 0 u = ]0,] u. 3 Main results: Asymptotic normality of the estimator Fθ, y, x In this part of paper, we give the asymptotic normality of the conditional cumulative distribution function in the single functional index model. The main result is given in the following theorem. Theorem Under Assumptions H-H7 we have nφ θ,x h K σ 2 θ, y, x Fθ, y, x Fθ, y, x D N 0,.

5 66 D. E. Hamdaoui, A. A. Bouchentouf, A. abhi, T. Guendouzi Where σ 2 θ, y, x = C 2θ, xfθ, y, x Fθ, y, x C 2 θ, x, with C j θ, x = K j 0 sk sβ θ,x sds for j =, 2, " convergence in distribution. D " means the Proof. Consider, for i =,..., n, K i θ, x = Kh K < x X i, θ >, H i y = H h H y Y i, FN θ, y, x = K i θ, xh i y, n EK θ, x FD θ, x = n EK θ, x i= K i θ, x, i= i x, θ = Kh K < x X i, θ >. EK θ, x In order to establish the asymptotic normality of Fθ, t, x we have to consider the following decomposition FN θ, y, x Fθ, y, x Fθ, y, x = C θ, xfθ, y, x FD θ, x C θ, x = FN θ, y, x E F N θ, y, x FD θ, x FD θ, x ] C θ, x E [ FD θ, x = Fθ, y, x + FD θ, x C θ, xfθ, y, x E F N θ, y, x Fθ, y, x FD θ, x E F D θ, x FD θ, x FD θ, x A nθ, y, x + B n θ, y, x 3 where A n θ, y, x = nek x, θ i= { H i y Fθ, y, x K i θ, x

6 Asymptotic normality of conditional distribution estimation and } E [H i y Fθ, y, x K i θ, x] = nek x, θ N i θ, y, x, i= N i θ, y, x = H i y Fθ, y, x K i θ, x E [H i y Fθ, y, x K i θ, x]. It follows that, nφ θ,x h K Var A n θ, t, x = φ θ,xh K E 2 K x, θ VarN + φ θ,xh K ne 2 K x, θ n i j >0 CovN i, N j = V n θ, t, x 4 Then, the rest of the proof is based on the following Lemmas Lemma Under hypotheses H-H3, H5 and H7, as n we have nφ θ,x h K Var A n θ, y, x Vθ, y, x, where Vθ, y, x = C 2θ, x Fθ, y, x Fθ, y, x. C θ, x 2 Lemma 2 Under hypotheses H-H3 and H5-H7, as n we have where nφθ,x h K /2 A n θ, y, x D N 0,, Vθ, y, x D denotes the convergence in distribution. Lemma 3 Under assumptions H-H3 and H5-H7; as n we have nφθ,x h K B n θ, y, x 0 in Probabilty. Now, because the unknown functions C j θ, x and Fθ, y, x intervening in the expression of the variance, we need to estimate the quantities C θ, x, C 2 θ, x and Fθ, y, x, respectively.

7 68 D. E. Hamdaoui, A. A. Bouchentouf, A. abhi, T. Guendouzi By assumptions H-H4 we know that a j θ, x can be estimated by Ĉjθ, x which is defined as Ĉ j θ, x = K j i θ, x, j =, 2 n φ θ,x h K where φ θ,x h K = n i= I { <x Xi,θ> <h k }. i= By applying the kernel estimator of σ 2 θ, x can be estimated finally by: σ 2 θ, x = Ĉ2θ, x Fθ, y, x Ĉ 2 θ, x Next, we can derive the following corollary: Fθ, y, x given above, the quantity H 2 tdt. Corollary Under assumptions of Theorem, we have n φ θ,x h K σ 2 θ, y, x Fθ, y, x Fθ, y, x D N 0,. Thus, following this Corollary we can approximate γ confidence interval of Fθ, y, x by σθ, x Fθ, y, x ± tγ/2, where t γ/2 is the upper γ/2 quantile of n φ θ,x h K standard Normal N 0,. 4 Proofs of technical lemmas Proof. [Proof of Lemma ] Let V n θ, y, x = φ θ,xh K [ E 2 K θ, x E K 2 θ, x H y Fθ, y, x 2] = φ θ,xh K E 2 K θ, x E [ K 2 θ, xe H y Fθ, y, x 2 < θ, X > Using the definition of conditional variance, we have [ ] 2 E Hh H y Y Fθ, y, x < θ, X > = J n + J 2n, where ] 5

8 Asymptotic normality of conditional distribution estimation Concerning J n. Let J n = Var Hh H y Y < θ, X >, and [ ] 2 J 2n = E Hh H y Y < θ, X > Fθ, y, x [ ] [ y J n = E H 2 Y < θ, x > E H = J + J 2 h H y Y h H ] 2 < θ, X > By the property of double conditional expectation, we get that [ ] y J = E H 2 Y < θ, X > h H y v = H 2 dfθ, v, X h H = H 2 tdfθ, y h H t, X. 6 On the other hand, by integrating by part and under assumption H3, we have J = 2HtH tfθ, y h H t, X du = 2HtH t Fθ, y h H t, X Fθ, y, x du + 2HtH tfθ, y, xdu. Clearly, we have 2HtH tfθ, y, xdu = [ ] + H 2 tfθ, y, x = Fθ, y, x, 7 thus H 2 tdfθ, y h H t, X = Fθ, y, x + Oh b K + hb 2 H. 8

9 70 D. E. Hamdaoui, A. A. Bouchentouf, A. abhi, T. Guendouzi Concerning J 2. Let I = E H i y < X, θ > y Y y u E H < X, θ > = H fθ, y, X du, h H h H y u = H dfθ, y, X, h H y u = H Fθ, u, X du, h H = H t Fθ, y h H t, X Fθ, y, x dt +Fθ, y, x H tdt. Because H is a probability density and by hypotheses H2 and H3, we can write: I C x,θ H t h b K + t b 2 h b 2 H dt + Fθ, y, x = O h b K + hb 2 H Finally, by hypothesis H3 we get + Fθ, y, x. J 2 F 2 θ, y, x, as n. 9 The last equality is due to the fact that H is a probability density, thus we have by hypothesis H3 H t Fθ, y h H t, X Fθ, y, x dt Concerning J 2n. We have by integration by parts and changing variables J 2n = E H y < θ, X > y Y = E H < θ, X > h H y v = H fθ, v, X dv h H H t t b 2 h b 2 H + hb K dt 0. n

10 Asymptotic normality of conditional distribution estimation... 7 = y v H dfθ, v, X h H = H tfθ, y h H t, X dt = Fθ, y, x H tdt + H t Fθ, y h H t, x Fθ, y, x dt, the last equality is due to the fact that H is a probability density. Thus, we have: J 2n = Fθ, y, x + O h b K + hb 2 H 0 Finally, we obtain that J 2n n 0. Meanwhile, by H, H2, H4 and H5, it follows that: φ θ,x h K EK 2 θ, x E 2 K θ, x n Then, by combining equations 5-0, it leads to C 2 θ, x C θ, x 2, C 2 θ, x V n θ, y, x Fθ, y, x Fθ, y, x. n C θ, x 2 Proof. [Proof of Lemma 2] We will establish the asymptotic normality of A n θ, t, x suitably normalized. We have nφθ,x h K nφθ,x h K A n θ, y, x = N i θ, y, x nek θ, x i= φθ,x h K = N i θ, y, x nek θ, x = n n i= i= Ξ i θ, y, x = n S n 2 Now, we can write, Ξ i = φθ,x h K EK θ, x N i,

11 72 D. E. Hamdaoui, A. A. Bouchentouf, A. abhi, T. Guendouzi Thus VarΞ i = φ θ,xh K E 2 K θ, x VarN i = V n θ, y, x. Note that by, we have VarΞ i Vθ, y, x as n goes to infinity. Obviously, we have nφ θ,x h K Vθ, y, x A nθ, y, x = nvθ, y, x /2 S n. Thus, the asymptotic normality of nvθ, y, x /2 S n, is deduced from the following results { } E exp izn /2 S n n j=0 n j=0 n j=0 { } E exp izn /2 Ξ j 0, 3 E Ξ 2 j Vθ, y, x, 4 E Ξ 2 j 0, for every ε > 0. 5 { Ξ j >ε nvθ,y,x} While equations 3 and 4 show that the Υ j are asymptotically independent, verifying that the sum of their variances tends to Vθ, y, x. Expression 5 is the Lindeberg-Feller s condition for a sum of independent terms. Asymptotic normality of S n is a consequence of equations 3-5. Proof of 3 We make use of Volkonskii and ozanov s lemma see the appendix in Masry 2005 and the fact that the process X i is i.i.d. Note that using that V j = exp izn /2 S n, we have { } E exp izn /2 S n as n goes to infinity. n j=0 { } E exp izn /2 Ξ j 0 Proof of 4 Note that VarS n Vθ, y, x by equation and 2 by the definition of the Ξ i. Then because E S n 2 = Var S n = Var Ξ j, j=0

12 Asymptotic normality of conditional distribution estimation and, using the same arguments as those previously used in the proof of first term of equation 5, we obtain n as Var Ξ Vθ, y, x. j= Proof of 5 ecall that E Ξ 2 j = Var Ξ, Ξ j = Υ i. i=0 Finally, to establish 5 it suffices to show that the set is negligible for n large enough. { Ξ j > ε nvθ, y, x} By using assumptions H4 and H5, we have Υ i C φθ,x h K /2, therefore Ξj Cn φθ,x h K /2, which goes to zero as n goes to infinity. Since H i y Fθ, y, x. { Then for n large enough, the set Ξ j > ε nvθ, y, x /2} becomes empty, this completes the proof and therefore that of the asymptotic normality of nvθ, y, x /2 S n and the Lemma 2. Proof. [Proof of Lemma 3] We have nφθ,x h K B n θ, y, x = nφθ,x h K FD θ, x { E F N θ, y, x C θ, xfθ, y, x } +Fθ, y, x C θ, x E F D θ, x.

13 74 D. E. Hamdaoui, A. A. Bouchentouf, A. abhi, T. Guendouzi Firstly, observe that as n [ ] < x φ θ,x h K E K l Xi, θ > C l θ, x, for l =, 2 6 h K ] E [ FD θ, x C θ, x, 7 and ] E [ FN θ, y, x C θ, xfθ, y, x, 8 can be proved in the same way as in Ezzahrioui and Ould Said 2008 corresponding to their Lemmas 5. and 5.2. Then the proofs of 6-8 are omitted. Secondly, making use of 6, 7 and 8, we have as n { } E F N θ, y, x C θ, xfθ, y, x + Fθ, y, x C θ, x E F D θ, x 0. On other hand nφθ,x h K FD θ, x = nφθ,x h K Fθ, y, x FD θ, x Fθ, y, x = nφθ,x h K Fθ, y, x FN θ, y, x. 9 Because K H is continuous with support on [0, ], then by hypotheses H3 and H4 m = inf t [0,] KtH t such that FN θ, y, x m h H φ θ,x h K which gives nφ θ,x h K FN θ, y, x nh 2 H φ θ,xh K 3 m Finally, using H6, the proof of Lemma 3 is completed.

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