Nonparametric local linear regression estimation for censored data and functional regressors

Abstract

In this work, we introduce a local linear nonparametric estimation of the regression function of a censored scalar response random variable, given a functional random covariate. Under standard conditions, we establish the pointwise and the uniform almost-complete convergences, with rates, of the proposed estimator. Then, we carry out a simulation study and a real data analysis in order to compare the performances of our methodology with those of the kernel method.

Appendix

In what follows, let C be some strictly positive generic constant and for any $x\in {\mathcal F}$, and for all $i=1,\ldots ,n$:

$$\begin{aligned} K_i(x):=K(h^{-1} d(X_i, x)) \;\; \text{ and }\;\;\beta _i(x):=\beta (X_i,x). \end{aligned}$$

• 1 To treat the pointwise almost-complete convergence of $\widehat{m}(x)$ we need Lemma A.1 introduced in Barrientos-Marin et al. (2010)

Proof of Lemma 2

As $(X_i,Z_i,\delta _i)$ are i.i.d., we get

$$\begin{aligned} E \widetilde{m}_{1}(x)-{m}(x)=\frac{1}{E[W_{12}(x)]} E \left\{ W_{12}(x) \left[ E \left( Z_2 S^{-1}(Z_{2}) \delta _2| X_2\right) -{m}(x) \right] \right\} . \end{aligned}$$

Hypothesis (H4), combining with the fact that $E(\delta _2|X_{2},Y_2)=S(Y_{2})$, give that

$$\begin{aligned} E \left[ Z_2 S^{-1}(Z_{2}) \delta _2| X_2\right] =E \left[ Y_2 S^{-1}(Y_{2}) E\left( \delta _2 |X_{2},Y_2 \right) | X_2\right] =m(X_2). \end{aligned}$$

Then, we get

$$\begin{aligned} E \widetilde{m}_{1}(x)-{m}(x)=\frac{1}{E[W_{12}(x)]} E \left[ W_{12}(x) \left( m(X_2)-{m}(x) \right) \right] . \end{aligned}$$

(9)

The claimed result is obtained by using the last relation and the condition (H2). $\square$

Proof of Lemma 3

We need to show that

$$\begin{aligned} \sum _{n}{P}\left( |\widetilde{m}_{1}(x)-E(\widetilde{m}_{1}(x)|>\epsilon \sqrt{\frac{\ln n}{n\varPhi _{x}(h)}}\;\right) <\infty . \end{aligned}$$

By following the same decomposition idea as in the proof of Lemma 4.4 in Barrientos-Marin et al. (2010), we can write

$$\begin{aligned} \widetilde{m}_{1}(x)= & {} \frac{1}{n(n-1) E\left[ W_{12}(x)\right] }\sum _{{i,j=1}}^n W_{ij}(x) \delta _j Z_j S^{-1}(Z_j) \nonumber \\= & {} \frac{n^2h^{2}\varPhi _{x}^{2}(h)}{n(n-1) E \left[ W_{12}(x)\right] } \Big[ \left( \frac{1}{n\varPhi _{x}(h)}\sum _{j=1}^n K_{j}(x) Z_j \delta _j S^{-1}(Z_j) \right) \left( \frac{1}{n\varPhi _{x}(h)}\sum _{j=1}^n \frac{K_{j}(x)\beta ^{2}_j(x)}{h^{2}}\right) \nonumber \\&- \left( \frac{1}{n\varPhi _{x}(h)}\sum _{j=1}^n \frac{K_{j}(x)\beta _j(x) Z_j \delta _j S^{-1}(Z_j)}{h} \right) \left( \frac{1}{n\varPhi _{x}(h)}\sum _{j=1}^n \frac{K_{j}(x)\beta _j(x) }{h} \right) \Big] \nonumber \\= & {} Q(x)[M_{2,1}(x)M_{4,0}(x)-M_{3,1}(x)M_{3,0}(x)], \end{aligned}$$

(10)

where, for $p \in \{2,3,4\}$ and $l \in \{0,1\}$,

$$\begin{aligned} M_{p,l}(x)=\frac{1}{n\varPhi _{x}(h)}\sum _{j=1}^n \frac{K_{j}(x)\beta ^{p-2}_j(x) Z_j^l \delta _j^l S^{-l}(Z_j)}{h^{p-2}} \; \; and \;\; Q(x)=\frac{n^2h^{2}\varPhi _{x}^{2}(h)}{n(n-1) E \left[ W_{12}(x)\right] }. \end{aligned}$$

So, we have

$$\begin{aligned} \widetilde{m}_{1}(x)-E\widetilde{m}_{1}(x)=Q(x)\left\{ \left[ M_{2,1}(x)M_{4,0}(x)-E(M_{2,1}(x)M_{4,0}(x))\right] -\left[ M_{3,1}(x)M_{3,0}(x)-E(M_{3,1}(x)M_{3,0}(x))\right] \right\} . \end{aligned}$$

Notice that, $Q(x)=O(1)$ [see the proof of Lemma 4.4 in Barrientos-Marin et al. (2010)], so, we have to show that, for $p \in \{2,3,4\}$ and $l \in \{0,1\}$

$$\begin{aligned} \sum _{n}{P}\left( |M_{p,l}(x)-E(M_{p,l}(x))|>\epsilon \sqrt{\frac{\ln n}{n\varPhi _{x}(h)}}\right) <\infty , \;\;\; E[M_{p,l}(x)]=O(1), \end{aligned}$$

and that almost-surely

$$\begin{aligned} E(M_{2,1}(x))E(M_{4,0}(x))-E(M_{2,1}(x)M_{4,0}(x))=O\left( \sqrt{\frac{\ln n}{n\varPhi _{x}(h)}}\right) \end{aligned}$$

and

$$\begin{aligned} E(M_{3,1}(x))E(M_{3,0}(x))-E(M_{3,1}(x)M_{3,0}(x))=O\left( \sqrt{\frac{\ln n}{n\varPhi _{x}(h)}}\right) . \end{aligned}$$

• Firstly we have

  $$\begin{aligned} M_{p,l}(x)-E(M_{p,l}(x))= & {} \frac{1}{nh^{p-2}\varPhi _{x}(h)} \sum _{i=1}^{n}\left[ K_{i}(x)\beta ^{p-2}_i(x) Z_i^l \delta _i^l S^{-l}(Z_i)-E \left( K_{i}(x)\beta ^{p-2}_i(x) Z_i^l \delta _i^l S^{-l}(Z_i)\right) \right] \\= & {} \frac{1}{n} \sum _{i=1}^{n}\frac{1}{h^{p-2}\varPhi _{x}(h)} \left[ K_{i}(x)\beta ^{p-2}_i(x) Z_i^l \delta _i^l S^{-l}(Z_i)-E \left( K_{i}(x)\beta ^{p-2}_i(x) Z_i^l \delta _i^l S^{-l}(Z_i)\right) \right] \\:= & {} \frac{1}{n}\sum _{i=1}^{n} \eta _i^{(p,l)}(x), \end{aligned}$$

  where

  $$\begin{aligned} \eta ^{(p,l)}_{i}(x) :=\frac{1}{h^{p-2}\varPhi _{x}(h)} \left[ K_{i}(x)\beta ^{p-2}_i(x) Z_i^l \delta _i^l S^{-l}(Z_i)-E \left( K_{i}(x)\beta ^{p-2}_i(x) Z_i^l \delta _i^l S^{-l}(Z_i)\right) \right] . \end{aligned}$$

  (11)

  In order to apply an exponential inequality, we focus on the absolute moments of the r.r.v. $\eta ^{(p,l)}_{i}(x)$. By Lemma A.1(i) in Barrientos-Marin et al. (2010), we can write

  $$\begin{aligned} E \vert \eta ^{(p,l)}_{i}(x) \vert ^m = O \left( [\varPhi _{x}(h)]^{-m+1} \right) . \end{aligned}$$

  Finally, it suffices to apply Corollary A.8-(ii) in Ferraty and Vieu (2006) with $a^2_n =[\varPhi _{x}(h)]^{-1}$ to get, for $p \in \{2,3,4\}$ and $l \in \{0,1\}$

  $$\begin{aligned} M_{p,l}(x)-E M_{p,l}(x)=O_{a.co.}\left( \sqrt{\frac{\ln n}{n\varPhi _{x}(h)}} \; \right) . \end{aligned}$$

  (12)

• It is easy to see that under (H1), (H3), (H4) and (A1), we get, for $p \in \{2,3,4\}$ and $l \in \{0,1\}$,

  $$\begin{aligned} E[M_{p,l}(x)]=h^{2-p}\varPhi _{x}(h)^{-1} E\left[ K_{1}(x)\beta ^{p-2}_1(x) Z_1^l \delta _1^l S^{-l}(Z_1)\right] \le C, \end{aligned}$$

  (13)

  the last inequality is obtained by using the Lemma A.1(i) in Barrientos-Marin et al. (2010).

• Treatment of the term $E(M_{2,1}(x))E(M_{4,0}(x))-E(M_{2,1}(x)M_{4,0}(x))$

  We can write

  $$\begin{aligned} E(M_{2,1}(x))E(M_{4,0}(x))-E(M_{2,1}(x)M_{4,0}(x)) = \frac{1}{nh^{2} \varPhi _{x}(h)^{2}} E[ K_1(x) \beta _1^{2}(x)] E[ K_1(x) m(X_1)] +O\left( (n \varPhi _{x}(h))^{-1} \right) . \end{aligned}$$

  By using Lemma A.1(i) in Barrientos-Marin et al. (2010), it is easy to see that

  $$\begin{aligned} E(M_{2,1}(x))E(M_{4,0}(x))-E(M_{2,1}(x)M_{4,0}(x))=O\left( (n \varPhi _{x}(h))^{-1} \right) , \end{aligned}$$

  (14)

  which is negligible with respect to $O\left( \sqrt{\frac{\ln n}{n\varPhi _{x}(h)}} \; \right)$, under (H5).

• By similar arguments, one can state

  $$\begin{aligned} E(M_{3,1}(x))E(M_{3,0}(x))-E(M_{3,1}(x)M_{3,0}(x))=O\left( \sqrt{\frac{\ln n}{n\varPhi _{x}(h)}} \; \right) . \end{aligned}$$

  (15)

$\square$

Proof of Lemma 5

Because the assumption (A1) and the definitions of $\widehat{m}_{1}(x)$ and $\widetilde{m}_{1}(x)$ in (6) and (7), we can write

$$\begin{aligned} \left| \widehat{m}_{1}(x)-\widetilde{m}_{1}(x)\right| \nonumber= & {} \left| \frac{1}{n(n-1) E\left[ W_{12}(x)\right] } \sum _{i\not =j}W_{ij}(x)\delta _j Z_j\left( \frac{1}{S_n(Z_j)} - \frac{1}{S(Z_j)} \right) \right| \nonumber \\\le & {} \frac{ \vert T \vert \sup \limits _{t \le T} \vert S_n(t)-S(t)\vert }{S_n (T)S(T)} \left| \frac{1}{n(n-1) E\left[ W_{12}(x)\right] } \sum _{i\not =j}W_{ij}(x)\right| \nonumber \\\le & {} \frac{ \vert T \vert \sup \limits _{t \le T} \vert S_n(t)-S(t)\vert }{S_n (T)S(T)} \vert \widehat{m}_0(x) \vert , \end{aligned}$$

(16)

where $\widehat{m}_0(x)$ is defined in (6).

In order hands, by adapt Theorem 1 of Bitouzé et al. (1999), we get

$$\begin{aligned} \sup \limits _{t \le T} \vert S_n(t)-S(t)\vert =O_{a.co.}\left( \sqrt{\frac{\ln n}{n}} \; \right) , \end{aligned}$$

(17)

which is equals to $O_{a.co.}\left( \sqrt{\frac{\ln n}{n\varPhi _x(h)}}\right)$. The proof is completed by using Lemma 4. $\square$

• 2. To treat the uniform convergence of $\widehat{m}(x)$ we need to Lemma 4.1 introduced in Messaci et al. (2015)

Proof of Lemma 7

It is a direct proof, by combining Eq. (9) and hypothesis (U2). $\square$

Proof of Lemma 8

We use again the decomposition (10) and by following the same steps as in (13), (14) and (15), with using Lemma 4.1 in Messaci et al. (2015) instead of lemma A.1 in Barrientos-Marin et al. (2010), we obtain under the assumptions (U1), (U3), (U4), (U6) and (A1), for $p=2,3,4$ and $l=0,1$,

$$\begin{aligned}&\sup _{x\in S_{\mathcal F}}E(M_{p,l}(x))=O(1),\;\;\sup _{x\in S_{\mathcal F}}Q(x)=O(1),\nonumber \\&\quad \sup _{x\in S_{\mathcal F}}| E(M_{2,1}(x)) E(M_{4,0}(x))-E(M_{2,1}(x)M_{4,0}(x))|=O\left( \frac{1}{n\varPhi (h)}\right) , \end{aligned}$$

(18)

and

$$\begin{aligned} \sup _{x\in S_{\mathcal F}}| E(M_{3,1}(x)) E(M_{3,0}(x))-E(M_{3,1}(x)M_{3,0}(x))|=O\left( \frac{1}{n\varPhi (h)}\right) , \end{aligned}$$

which is, in view of hypothesis (U5), equals to $O\left( \sqrt{\frac{\ln d_n}{n\varPhi (h)}}\right)$.

So, we need to check that for $p=2,3,4$ and $l=0,1$,

$$\begin{aligned} \sup _{x\in S_{\mathcal F}}\left| M_{p,l}(x) - E(M_{p,l}(x))\right| =O_{a.co.}\left( \sqrt{\frac{\ln d_n}{n\varPhi (h)}}\right) . \end{aligned}$$

Now, we consider the following decomposition

$$\begin{aligned} \sup _{x\in S_{\mathcal {F}}}\left| M_{p,l}(x)-E(M_{p,l}(x)) \right|\le & {} \sup _{x\in S_{\mathcal {F}}}\left| M_{p,l}(x)-M_{p,l}(x_{j(x)}) \right| \\&+\sup _{x\in S_{\mathcal {F}}}\left| E(M_{p,l}(x_{j(x)}))-E(M_{p,l}(x_{j(x)})) \right| \\&+\sup _{x\in S_{\mathcal {F}}}\left| E(M_{p,l}(x_{j(x)}))- E(M_{p,l}(x)) \right| \\:= & {} \sum _{i=1}^{3}F_{i}^{p,l}. \end{aligned}$$

Study of the terms $F_1^{p,l}$ and $F_3^{p,l }$.

First, let us analyze the term $F_1^{p,l}$. Since K is supported in [0, 1] and according to (U1), we can write

$$\begin{aligned} F_1^{p,l}\le \frac{Cr_n}{nh\varPhi (h)}\sup _{x\in S_{\mathcal F}}\sum _{i=1}^{n} \vert Z_i^l \vert \delta _i^l S^{-l}(Z_i)1_{B(x,h)\cup {B(x_{j(x)},h)}}(X_i). \end{aligned}$$

Let

$$\begin{aligned} \xi _{i}=\frac{C r_n \vert Z_i^l \vert \delta _i^l S^{-l}(Z_i)}{h\varPhi (h)}\sup _{x\in S_{\mathcal F}}1_{B(x,h)\cup {B(x_{j(x)},h)}}(X_i). \end{aligned}$$

The assumptions (A1) and (U7), implie that

$$\begin{aligned} E|\xi _{1}^{m}|\le \frac{C r_{n}^{m}}{h^{m}\varPhi (h)^{m-1}}, \end{aligned}$$

(19)

so, by applying corollary A.8-(ii) in Ferraty and Vieu (2006), with $a_{n}^{2}=\frac{r_n}{h\varPhi (h)}$,

$$\begin{aligned} \frac{1}{n}\sum _{i=1}^{n}\xi _{i}=E (\xi _{1})+O_{a.co.}\left( \sqrt{\frac{r_n\ln n}{nh\varPhi (h)}}\right) . \end{aligned}$$

Applying (19) again (for $m=1$), one gets

$$\begin{aligned} F_{1}^{p,l}=O\left( \frac{r_{n}}{h}\right) +O_{a.co.}\left( \sqrt{\frac{r_n\ln n}{nh\varPhi (h)}}\right) . \end{aligned}$$

Combining this equation with assumption (U5) and the second part of the assumption (U1), we obtain

$$\begin{aligned} F_{1}^{p,l}=O_{a.co.}\left( \sqrt{\frac{\ln d_n}{n\varPhi (h)}}\right) . \end{aligned}$$

(20)

Second, since

$$\begin{aligned} F_{3}^{p,l}\le E\left( \sup _{x\in S_{\mathcal F}}\left| M_{p,l}(x) - M_{p,l}(x_{j(x)})\right| \right) , \end{aligned}$$

we deduce that

$$\begin{aligned} F_{3}^{p,l}=O_{a.co.}\left( \sqrt{\frac{\ln d_n}{n\varPhi (h)}}\right) . \end{aligned}$$

(21)

Study of the term $F_{2}^{p,l}.$

For all $\zeta >0$, we have that

$$\begin{aligned} {P}\left( F_{2}^{p,l}>\zeta \sqrt{\frac{\ln d_n}{n\varPhi (h)}}\;\right)= & {} {P}\left( \sup _{x\in S_{\mathcal {F}}}\left| M_{p,l} (x_{j(x)})- E(M_{p,l}(x_{j(x)})\right|> \zeta \sqrt{\frac{\ln d_n}{n\varPhi (h)}}\;\right) \\\le & d_{n} \max _{x_{j(x)}\in \{x_1,\ldots ,x_{d_n}\}}{P}\left( \frac{1}{n}\left| \sum _{i=1}^n \eta ^{(p,l)}_{i}(x_{j(x)})\right| >\zeta \sqrt{\frac{\ln d_n}{n\varPhi (h)}}\;\right) , \end{aligned}$$

where $\eta ^{(p,l)}_{i}$ is defined in (11). By using again Corollary A.8-(ii) in Ferraty and Vieu (2006) and the assumption (U5), we obtain

$$\begin{aligned} F_{2}^{p,l}=O_{a.co.}\left( \sqrt{\frac{\ln d_n}{n\varPhi (h)}}\;\right) . \end{aligned}$$

(22)

Finally, the result of Lemma 8 follows from the relations (20), (22) and (21). $\square$

Proof of Lemma 10

By the relation (16), we can write

$$\begin{aligned} \sup _{x\in S_{\mathcal {F}}} \left| \widehat{m}_{1}(x)-\widetilde{m}_{1}(x)\right| \le C \sup \limits _{t \le T} \vert S_n(t)-S(t) \vert \sup _{x\in S_{\mathcal {F}}} \vert \widehat{m}_{0}(x) \vert . \end{aligned}$$

The proof is completed by the relation (17) and Lemma 9. $\square$