Local linear estimation of the conditional cumulative distribution function: Censored functional data case

Abstract

In this paper, we estimate the conditional cumulative distribution function of a randomly censored scalar response variable given a functional random variable using the local linear approach. Under this structure, we state the asymptotic normality with explicit rates of the constructed estimator. Moreover, the usefulness of our results is illustrated through a simulated study.

Appendix

1.1 Proof of Lemma 1

We mention that the proof of this lemma is very close to the proof of Theorem 2.1 of Bouanani et al. (2019). Specifically, we have:

$$ \begin{array}{@{}rcl@{}} {}{}{}{}\lefteqn{\sqrt{n\phi_{x}(h_{\scriptscriptstyle K})}A_{n}(x,y)= \displaystyle\frac{\sqrt{n\phi_{x}(h_{\scriptscriptstyle K})}}{n\mathbb{E}({\Delta}_{1}K_{1})} \sum\limits_{j=1}^{n} {\Delta}_{j}K_{j}\left( \delta_{j} [\overline{G} (T_{j})]^{-1} J_{j} - F^{x}(y)\right)-}\\ {}{}{}{}& &\mathbb{E} \left( \sum\limits_{j=1}^{n} {\Delta}_{j}K_{j}\left( \delta_{j} [\overline{G} (T_{j})]^{-1} J_{j}-F^{x}(y)\right) \right). \end{array} $$

(3.5)

Next, by using Eqs. 2.4, 3.5 can be rewritten as follows:

$$ \begin{array}{@{}rcl@{}} &&\quad\lefteqn{ \sqrt{n\phi_{x}(h_{\scriptscriptstyle K})}(\widetilde{ F}^{x}_{N}(y)-\mathbb{E}(\widetilde{ F}^{x}_{N}(y))}\\ & &= \displaystyle\frac{1}{n\mathbb{E}({\beta_{1}^{2}}K_{1})}\displaystyle\sum\limits_{i=1}^{n} {\beta_{i}^{2}}K_{i} \displaystyle\frac{\sqrt{n\phi_{x}(h_{\scriptscriptstyle K})}\mathbb{E}({\beta_{1}^{2}}K_{1})}{\mathbb{E}({\Delta}_{1}K_{1})} \displaystyle\sum\limits_{j=1}^{n} K_{j} \left( \delta_{j} [\overline{G} (T_{j})]^{-1} J_{j}-F^{x}(y)\right)\\ & &- \displaystyle\frac{1}{n\mathbb{E}(\beta_{1}K_{1})}\displaystyle\sum\limits_{i=1}^{n}\beta_{i}K_{i} \displaystyle\frac{\sqrt{n\phi_{x}(h_{\scriptscriptstyle K})} \mathbb{E}(\beta_{1}K_{1}) }{\mathbb{E}({\Delta}_{1}K_{1})}\displaystyle\sum\limits_{j=1}^{n} \beta_{j}K_{j} \left( \delta_{j} [\overline{G} (T_{j})]^{-1} J_{j}-F^{x}(y)\right)\\ & & - \mathbb{E}\left( \displaystyle\frac{1}{n\mathbb{E}({\beta_{1}^{2}}K_{1})}\displaystyle\sum\limits_{i=1}^{n}{\beta_{i}^{2}}K_{i} \displaystyle\frac{\sqrt{n\phi_{x}(h_{\scriptscriptstyle K})}\mathbb{E}({\beta_{1}^{2}}K_{1})}{\mathbb{E}({\Delta}_{1}K_{1})}\sum\limits_{j=1}^{n} K_{j}\left( \delta_{j} [\overline{G} (T_{j})]^{-1} J_{j}-F^{x}(y)\right)\right)\\ & & + \mathbb{E}\left( \frac{1}{n\mathbb{E}(\beta_{1}K_{1})}\displaystyle\sum\limits_{i=1}^{n} \beta_{i}K_{i} \displaystyle\frac{\sqrt{n\phi_{x}(h_{\scriptscriptstyle K})}\mathbb{E}(\beta_{1}K_{1})}{\mathbb{E}({\Delta}_{1}K_{1})} \displaystyle\sum\limits_{j=1}^{n} \beta_{j}K_{j} \left( \delta_{j} [\overline{G} (T_{j})]^{-1} J_{j}-F^{x}(y)\right)\right). \end{array} $$

Let

$$ \begin{array}{@{}rcl@{}} {\Upsilon}_{n}&=&\underbrace{\left( \displaystyle\frac{1}{n\mathbb{E}({\beta_{1}^{2}}K_{1})}\displaystyle\sum\limits_{i=1}^{n}{\beta_{i}^{2}}K_{i}-1 \right)\displaystyle\frac{\sqrt{n\phi_{x}(h_{\scriptscriptstyle K})} \mathbb{E}({\beta_{1}^{2}}K_{1})}{\mathbb{E}({\Delta}_{1}K_{1})}\displaystyle\sum\limits_{j=1}^{n} K_{j} \left( \delta_{j} [\overline{G} (T_{j})]^{-1} J_{j}-F^{x}(y)\right)}_{S_{1.n}}\\ &-&\mathbb{E}\left( S_{1.n} \right), \end{array} $$

$$ {\Theta}_{n}=\underbrace{\displaystyle\frac{\sqrt{n\phi_{x}(h_{\scriptscriptstyle K})} \mathbb{E}({\beta_{1}^{2}}K_{1})}{\mathbb{E}({\Delta}_{1}K_{1})}\displaystyle\sum\limits_{j=1}^{n} K_{j}\left( \delta_{j} [\overline{G} (T_{j})]^{-1} J_{j}-F^{x}(y)\right)}_{S_{2.n}}-\mathbb{E}\left( S_{2.n}\right),$$

and

$$ \begin{array}{@{}rcl@{}} D_{n} &=& \underbrace{\displaystyle\frac{1}{n\mathbb{E}(\beta_{1}K_{1})}\displaystyle\sum\limits_{i=1}^{n}\beta_{i}K_{i} \displaystyle\frac{\sqrt{n\phi_{x}(h_{\scriptscriptstyle K})} \mathbb{E}(\beta_{1}K_{1}) }{\mathbb{E}({\Delta}_{1}K_{1})}\displaystyle\sum\limits_{j=1}^{n} \beta_{j}K_{j} \left( \delta_{j} [\overline{G} (T_{j})]^{-1} J_{j}-F^{x}(y)\right)}_{S_{3.n}}\\ &-&\mathbb{E}\left( S_{3.n} \right). \end{array} $$

Finally, the decomposition (3.5) becomes:

$$\sqrt{n\phi_{x}(h_{\scriptscriptstyle K})} A_{n}(x,y)= {\Upsilon}_{n}+ {\Theta}_{n}-D_{n}.$$

So, it suffices to apply Slutsky’s Theorem to prove the asymptotic normality of Eq. 3.5. This last is reached by proving the following asymptotic results:

$$ {\Theta}_{n}\xrightarrow{\enskip d\enskip}\mathcal{N}(0, V_{H K}(x,y)). $$

(3.6)

$$ {\Upsilon}_{n}\xrightarrow{\enskip P\enskip}0. $$

(3.7)

$$ D_{n}\xrightarrow{\enskip P\enskip}0. $$

(3.8)

1.2 Proof of Lemma 3

Proof of (i)

Using the fact that $\delta _{1} T_{1} \overline {G}^{-1}(T_{1}) \leq \tau _{L} \overline {G}^{-1}(\tau _{L})$ (see, Kohler et al. (2002) for more details) and by assumptions ((H.2)-(i)), (H.3) and (H.5) we have:

$$ \begin{array}{@{}rcl@{}} h^{-l}_{K}\delta_{1} [ \overline{G}(T_{1})]^{-1} {K^{s}_{1}} \mid\beta_{1}\mid^{l}&\!\!\!\! \leq\!\!\!\!& h^{-l}_{K} \overline{G}^{-1}(T_{F}) {K^{s}_{1}} \mid\rho(X_{1},x)\mid^{l}, \\ &\!\!\!\! \leq\!\!\!\!& C h^{-l}_{K} \overline{G}^{-1}(T_{F})\!\! \mid\rho(X_{1},x)\mid^{l} \displaystyle{1\!\!1_{[-1,1]}(h^{-1}_{K} \rho(X_{1},x))},\\ &\!\!\!\! =\!\!\!\!&C^{\prime}\displaystyle{1\!\!1_{[-1,1]}(h^{-1}_{K} \rho(X_{1},x))}. \end{array} $$

Therefore, we obtain:

$$ \begin{array}{@{}rcl@{}} \mathbb{E}\left( \delta_{1} [ \overline{G}(T_{1})]^{-1} {K^{s}_{1}}\mid\beta_{1}\mid^{l}\right)& \leq& C^{\prime} {h^{l}_{K}} \mathbb{E}\left( \displaystyle{1\!\!1_{[-1,1]}(h^{-1}_{K} \rho(X_{1},x))}\right), \\ & \leq& C^{\prime} {h^{l}_{K}} \phi_{x}(h_{K}). \end{array} $$

Proof of (ii)

By applying Hypothesis (H.5) and using the fact that $\displaystyle {1\!\!1_{Y_{1}\leq C_{1}}} \varphi$ $(T_{1})=\displaystyle {1\!\!1_{Y_{1}\leq C_{1}}} \varphi (Y_{1})$, where φ(.) is a measurable function, we get:

$$ \begin{array}{@{}rcl@{}} \mathbb{E}\left( \delta_{1} T_{1} [ \overline{G}(T_{1})]^{-1} \mid X_{1}\right)&=&\mathbb{E}\left( \displaystyle{1\!\!1_{Y_{1}\leq C_{1}}} Y_{1} [\overline{G}(Y_{1})]^{-1} \mid X_{1}\right),\\ &=&\mathbb{E}\left( Y_{1} \overline{G}^{-1}(Y_{1}) \mathbb{E}\left( \displaystyle{1\!\!1_{Y_{1}\leq C_{1}}}\mid(X_{1},Y_{1})\mid\right)\mid X_{1}\right),\\ &=&\mathbb{E}\left( Y_{1}\mid X_{1}\right). \end{array} $$

Proof of (iii)

From the previous relationship (ii) and by assumption (H.5), we have:

$$ \begin{array}{@{}rcl@{}} \mathbb{E}\!\left( \displaystyle\!\!\frac{1}{n}\displaystyle\!\sum\limits_{i=1}^{n} \delta_{i} [\overline{G}(T_{i})]^{-1} U\!(\!X_{i},T_{i})\!\!\right)\!\!&\! = \!& \mathbb{E}\left( \displaystyle{1\!\!1_{Y_{1}\leq C_{1}}} [\overline{G}(Y_{1}]^{-1}) U(X_{1},Y_{1})\right),\\ &\! = \!&\mathbb{E}\left( [\overline{G}(Y_{1})]^{-1} \mathbb{E}\left( \displaystyle{1\!\!1_{Y_{1}\leq C_{1}}}\!\mid\!\!(\!X_{1},Y_{1})\!\right) \!U\!(X_{1},Y_{1})\!\right),\\ &\! = \!&\mathbb{E}(U(X_{1},Y_{1})). \end{array} $$

Proof of (iv)

This assertion is obtained by a straightforward application of the Glivenko-Cantelli Theorem for the censored case (see, for more details Kohler et al., 2002).

1.3 Proof of Lemma 4

By the definition of conditional variance we have:

$$ \begin{array}{@{}rcl@{}} Var \!\!\left( \!\!\delta_{1} [ \overline{G}(T_{1})]^{-1} J\!\!\left( \displaystyle\frac{y - T_{1}}{h_{J}}\!\right)\!\mid X_{1}\!\right)\!\!\!\!\! &=&\!\!\! \mathbb{E}\!\left( \left( \delta_{1} ([ \overline{G}(T_{1})]^{-1}) J\left( \displaystyle\frac{y-T_{1}}{h_{J}}\right)\right)^{2}\mid X_{1}\right) \\ \!\!&-&\!\!\! \mathbb{E}\left[\!\left( \delta_{1} [ \overline{G}(T_{1})]^{-1} J\!\left( \displaystyle\frac{y - T_{1}}{h_{J}}\!\right)\!\mid X_{1}\right)\right]^{2}. \\ \end{array} $$

For the second term of the right hand side of Eq. 3.9, by using our technical Lemma 3, we have:

$$ \begin{array}{@{}rcl@{}} \mathbb{E}\left( \!\delta_{1} [ \overline{G}(T_{1})]^{-1} J\!\left( \displaystyle\frac{y - T_{1}}{h_{J}}\!\right)\!\mid X_{1}\!\right)\!\!\!\!\!&=\mathbb{E}\left( \displaystyle{1\!\!1_{Y_{1}\leq C_{1}}} [\overline{G}(Y_{1})]^{-1} J\left( \displaystyle\frac{y - Y_{1}}{h_{J}}\right)\mid X_{1}\!\right)\\ &= \mathbb{E}\left( J\left( \displaystyle\frac{y-Y_{1}}{h_{J}}\right)\mid X_{1}\right). \end{array} $$

(3.9)

Then, using an integration par parts and a change of variables together with the notation (2.5) allow to get:

$$ \begin{array}{@{}rcl@{}} \mathbb{E}\left( \!J\!\left( \displaystyle\frac{y - Y_{1}}{h_{J}}\right)\!\mid X_{1}\!\right)\!\!\!\! &=& \displaystyle{\int}_{R} J(h_{\scriptscriptstyle J}^{-1}(y-z))f^{X_{1}}(z) dz. \\ \mathbb{E}(J_{1}\mid X_{1}) &=& \displaystyle{\int}_{R} J^{\prime}(t)F^{X_{1}}(y-h_{\scriptscriptstyle J}t) dt.\\ &=& \displaystyle{\int}_{R} J^{\prime}(t)\left( F^{X_{1}}(y-h_{\scriptscriptstyle J}t)-F^{x}(y)\right)+ \displaystyle{\int}_{R} J^{\prime}(t)F^{x}(y)dt\\ &=& F^{x}(y). \end{array} $$

This last result is obtained by using assumption ((H.1)-(iii)) and ((H.3)-(ii)).

On the other hand, the first term on the right hand side of Eq. 3.9, can be written as follows:

$$ \begin{array}{@{}rcl@{}} \lefteqn{\mathbb{E}\left( \left( \delta_{1} ([ \overline{G}(T_{1})]^{-1}) J\left( \displaystyle\frac{y-T_{1}}{h_{J}}\right)\right)^{2}\mid X_{1}\right)=}\\ & & \mathbb{E}\left( \delta_{1} ([ \overline{G}(T_{1})]^{-1})^{2} \left( J\left( \displaystyle\frac{y-T_{1}}{h_{J}}\right)\right)^{2}\mid X_{1}\right),\\ & & =\mathbb{E}\left( ([\overline{G}(Y_{1})]^{-1})^{2} J\left( \left( \displaystyle\frac{y-Y_{1}}{h_{J}}\right)\right)^{2}\mathbb{E}\left( \displaystyle{1\!\!1_{Y_{1}\leq C_{1}}}\mid(X_{1},Y_{1})\right)\mid X_{1}\right),\\ & &=\mathbb{E}\left( [\overline{G}(Y_{1})]^{-1} J\left( \left( \displaystyle\frac{y-Y_{1}}{h_{J}}\right)\right)^{2}\mid X_{1}\right),\\ & &=\int J\left( \left( \displaystyle\frac{y-z}{h_{J}}\right)\right)^{2} [\overline{G}(z)]^{-1}dF^{X_{1}}(z). \end{array} $$

Again, by applying the same change of variables and using a Taylor expansion of order one of $\overline {G}(y-th_{J})$, we obtain:

$$ \begin{array}{@{}rcl@{}} \int \!J\!\!\left( \!\!\left( \displaystyle\frac{y - z}{h_{J}}\right)\!\!\right)^{\!\!2} \!\![\overline{G}(z)]^{-\!1}dF^{X_{1\!}}(\!z\!) \!\!\!\!\!&=&\!\!\!\! \int J^{2}(t) [\overline{G}(y-th_{J})]^{-1}dF^{X_{1}}(y-th_{J}), \\ &=&\!\!\!\! \int J^{2}(t) [\overline{G}(y)]^{-1}dF^{X_{1}}(y-th_{J}),\\ &+&\!\!\!\! \frac{h_{J} }{\overline{G}^{2}(y)} \!\int \!J^{2}(t) \!y \overline{G}^{\prime\!}(y^{\ast})dF^{X_{1\!}}(y - th_{J}) + o(h_{J}),\\ &=&\!\!\! \omega_{1}+\omega_{2}, \end{array} $$

where y^∗ is between y and y − th_J.

Now, under assumption ((H.3)-(ii)) and ((H.5)-(ii)), we have:

$$\omega_{2}\leq C \frac{{h_{J}^{2}} }{\overline{G}^{2}(\tau_{L})} \int J^{2}(t) y f^{X_{1}}(y-th_{J})dt+o(1)=O({h^{2}_{J}}).$$

Concerning the term w₁, with integration by parts we get:

$$ \begin{array}{@{}rcl@{}} w_{1}&=& {\int}_{\mathbb{R}}[\overline{G}(y)]^{-1}J^{2}(t)dF^{X_{1}}(y-th_{J}),\\ &=& [\overline{G}(y)]^{-1} {\int}_{\mathbb{R}}2 J(t)^{\prime}J(t)\left( F^{X_{1}}(y-th_{J})\right)-F^{x}(y) dt,\\ && +[\overline{G}(y)]^{-1} {\int}_{\mathbb{R}}2 J(t)^{\prime}{J(t)}F^{x}(y)dt. \end{array} $$

By the continuity of F^x and remarking that ${\int \limits }_{\mathbb {R}}2 J(t)^{\prime }{J(t)}F^{x}(y)dt=F^{x}(y)$, we deduce that: $w_{1}=\displaystyle \frac {F^{x}(y)}{\overline {G}(y)}$.

Proof of Eq. 3.6

In order to apply the Lindeberg central limit Theorem, we need to compute the asymptotic term Θ_n. For this we have:

$$ \begin{array}{@{}rcl@{}} \lefteqn{ Var({\Theta}_{n})=\displaystyle\frac{n^{2}\phi_{x}(h_{\scriptscriptstyle K})\mathbb{E}^{2}({\beta^{2}_{1}}K_{1})}{\mathbb{E}^{2}({\Delta}_{1}K_{1})} \left( \mathbb{E}\left( {K^{2}_{1}}\left( \delta_{1} [\overline{G}(Y_{1})]^{-1} J_{1}-F^{x}(y)\right)^{2}\right)\right)}\\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!& &\quad\quad\quad~ -\displaystyle\frac{n^{2}\phi_{x}(h_{\scriptscriptstyle K})\mathbb{E}^{2}({\beta^{2}_{1}}K_{1})}{\mathbb{E}^{2}({\Delta}_{1}K_{1})} \mathbb{E}^{2}\left( K_{1}\!\left( \delta_{1} [\overline{G}(Y_{1})]^{-1} J_{1} - F^{x}(y)\right)\right). \end{array} $$

(3.10)

Concerning the second term on the right hand side of Eq. 3.10, we have:

$$ \begin{array}{@{}rcl@{}} \mathbb{E}^{2}(K_{1}(\delta_{1} [\overline{G}(Y_{1})]^{-1} J_{1} - F^{x}(y)))\!=\mathbb{E}^{2}\left( K_{1}(\mathbb{E}(\delta_{1} [\overline{G}(Y_{1})]^{-1} J_{1} \mid X_{1}) - F^{x}(y))\right). \end{array} $$

Moreover, by using equation of Eq. 3.9, we get:

$$ \mathbb{E}(\delta_{1} [\overline{G}(Y_{1})]^{-1} J_{1} \mid X_{1})-F^{x}(y)\longrightarrow0\quad\text{as}\quad n\longrightarrow \infty. $$

(3.11)

Now, for the first term on the right hand side of Eq. 3.10, we have:

$$ \begin{array}{@{}rcl@{}} &&\quad\lefteqn{\frac{n^{2}\phi_{x}(h_{\scriptscriptstyle K}) \mathbb{E}^{2}({\beta_{1}^{2}}K_{1})}{\mathbb{E}^{2}({\Delta}_{1}K_{1})}\mathbb{E}\left( (\delta_{1} [\overline{G}(Y_{1})]^{-1} J_{1}-F^{x}(y))^{2}{K_{1}^{2}}\right)}\\ && =\frac{n^{2}\phi_{x}(h)\ \mathbb{E}^{2}({\beta_{1}^{2}}K_{1})}{ \mathbb{E}^{2}({\Delta}_{1}K_{1})} \mathbb{E}\left( \mathbb{E}((\delta_{1} [\overline{G}(Y_{1})]^{-1} J_{1}-F^{x}(y))^{2}\mid X_{1}) {K_{1}^{2}}\right),\\ & & =\frac{n^{2}\phi_{x}(h_{\scriptscriptstyle K})\mathbb{E}^{2}({\beta_{1}^{2}}K_{1})}{\mathbb{E}^{2}({\Delta}_{1}K_{1})}\mathbb{E}\left( Var(\delta_{1} [\overline{G}(Y_{1})]^{-1} J_{1}\mid X_{1}){K_{1}^{2}}\right),\\ & &\quad+\frac{n^{2}\phi_{x}(h_{\scriptscriptstyle K})\mathbb{E}^{2}({\beta_{1}^{2}}K_{1})}{\mathbb{E}^{2}({\Delta}_{1}K_{1})} \mathbb{E}\left( (E(\delta_{1} [\overline{G}(Y_{1})]^{-1} J_{1}\mid X_{1})-F^{x}(y))^{2}{K_{1}^{2}}\right).\\ \end{array} $$

(3.12)

Combining Lemmas A.1 in Zhou and Lin (2016) and Eqs. 4 with Eq. 3.11, we get:

$$ \displaystyle\frac{n^{2}\phi_{x}(h_{\scriptscriptstyle K})\mathbb{E}^{2}({\beta_{1}^{2}}K_{1})}{\mathbb{E}^{2}({\Delta}_{1}K_{1})} \mathbb{E}\left( Var(\delta_{1} [\overline{G}(Y_{1})]^{-1} J_{1}\mid X_{1}){K_{1}^{2}}\right)\xrightarrow[n\to\infty]{} \displaystyle\frac{M_{2}}{{M^{2}_{1}}} F^{x}(y) \left( \displaystyle\frac{1}{\overline{G}(Y_{1})}-F^{x}(y)\right), $$

(3.13)

and

$$\displaystyle\frac{n^{2}\phi_{x}(h_{\scriptscriptstyle K})\mathbb{E}^{2}({\beta_{1}^{2}}K_{1})}{\mathbb{E}^{2}({\Delta}_{1}K_{1})} \mathbb{E}\left( (\mathbb{E}(\delta_{1} [\overline{G}Y_{1})]^{-1} ( J_{1}\mid X_{1})-F^{x}(y))^{2}{K_{1}^{2}}\right))\longrightarrow 0, \quad\text{as}\quad n\longrightarrow \infty. $$

To complete the proof of the claim (3.6), it suffices to use the Lindeberg’s central limit theorem on R_jn which satisfies the following condition:

$$ \begin{array}{@{}rcl@{}} &&\frac{1}{Var({\Theta}_{n})}\displaystyle\sum\limits_{j=1}^{n}\mathbb{E}\left( R_{jn}^{2}\displaystyle{1\!\!1_{{\mid R_{jn}\mid>\varepsilon \sqrt{Var({\Theta}_{n})}} }}\right)\\&=&\displaystyle\frac{1}{Var({\Theta}_{n})}\displaystyle\mathbb{E}\left( (\sqrt{n} R_{1n})^{2}\displaystyle{1\!\!1_{{\mid \sqrt{n}R_{1n}\mid>\varepsilon \sqrt{n Var({\Theta}_{n})}} }}\right) \!\to\! 0, \text{ for all } \varepsilon>0, \end{array} $$

where

$$ R_{1n}=\displaystyle\frac{\sqrt{n\phi_{x}(h_{\scriptscriptstyle K})} \mathbb{E}({\beta_{1}^{2}}K_{1})}{\mathbb{E}({\Delta}_{1}K_{1})}\left( K_{1}\left( \delta_{1} [\overline{G} (T_{1})]^{-1} J_{1}-F^{x}(y)\right)\right)-\mu_{n}.$$

μ_n is the mean of the first term on the right side.

On the other hand, we have:

$$\mathbb{E}\left( (\sqrt{n} R_{1n})^{2}\right)=Var({\Theta}_{n})\longrightarrow V_{J K}(x,y) \text{ as } n\longrightarrow \infty. $$

From the technical lemma A.1 of Zhou and Lin (2016) and by using the fact that $\mid J_{1} \delta _{1} [\overline {G}(Y_{1})]^{-1} -F^{x}(y)\mid \leq \overline {G}^{-1} (\tau _{L})+1$, (see Ould Saïd et al. 2011) and under assumption (H.3)(i), (H.3)(ii) and (H.4)(iii), we obtain

$$ \begin{array}{@{}rcl@{}} \left( \displaystyle\frac{n}{ n Var({\Theta}_{n})}\right)^{\frac{1}{2}}\mid R_{1n}\mid\!\!\! &\leq & \!\!\!C \displaystyle \!\left( \!\! \frac{ n}{(n - 1)^{2} \phi_{x}(h_{K}) Var({\Theta}_{n})}\!\!\right)^{\frac{1}{2}} \!\longrightarrow\! 0,\! \text{ as } n\!\longrightarrow\! \infty. \end{array} $$

Therefore, for n large enough, we deduce that $\left \lbrace \sqrt {n}\mid R_{1n}\mid >\epsilon \sqrt {n Var({\Theta }_{n})}\right \rbrace$ is an empty set. The proof of the claim Eq. 3.6 is therefore complete.

proof of Eq. 3.7

By using Cauchy-Schwarz’s inequality, we obtain:

$$ \begin{array}{@{}rcl@{}} \!\!\!\!\!\!\lefteqn{\mathbb{E}\mid(S_{1.n} - \mathbb{E}(S_{1.n})\mid\leq 2 \sqrt{\mathbb{E} \left( \displaystyle\frac{1}{n\mathbb{E}({\beta_{1}^{2}}K_{1})}\displaystyle\sum\limits_{i=1}^{n}{\beta_{i}^{2}}K_{i}-1 \right)^{2} }\times }\\ \!\!\!\!\!\!\!\!& & \sqrt{\mathbb{\!E}\!\left( \underbrace{\displaystyle\!\frac{\sqrt{n\phi_{x}(h_{\scriptscriptstyle K})} \mathbb{E}({\beta_{1}^{2}}K_{1})}{\mathbb{E}({\Delta}_{1}K_{1})}\displaystyle\sum\limits_{j=1}^{n} K_{j} \left( \delta_{j} [\overline{G} (T_{j})]^{-1} J_{j} - F^{x}(y)\right)}_{L_{n}}\!\right)^{2}}. \end{array} $$

(3.14)

Again, by applying the technical lemma A.1 of Zhou and Lin (2016) for the first term on the right-hand side of Eq. 3.14, we get:

$$ \begin{array}{@{}rcl@{}} \mathbb{E} \left( \frac{1}{n\mathbb{E}({\beta_{1}^{2}}K_{1})}\sum\limits_{i=1}^{n}{\beta_{i}^{2}}K_{i}-1 \right)^{2} &=& Var \left( \frac{1}{n\mathbb{E}({\beta_{1}^{2}}K_{1})}\sum\limits_{i=1}^{n}{\beta_{i}^{2}}K_{i} \right),\\ &=&\frac{1}{n^{2}\mathbb{E}^{2}({\beta_{1}^{2}}K_{1})} n Var({\beta_{1}^{2}}K_{1}),\\ &\leq & \frac{1}{n(O(h^{4}{\phi^{2}_{x}}(h_{K})))}\mathbb{E}({\beta_{1}^{4}}{K_{1}^{2}}), \\ &\leq & \left( \frac{1}{n\phi_{x}(h_{K})}\right). \end{array} $$

(3.15)

Concerning the second term on the right-hand side of Eq. 3.14, we have:

$$ \begin{array}{@{}rcl@{}} \mathbb{E}({L_{n}^{2}})&=&\displaystyle\frac{ n\phi_{x}(h_{\scriptscriptstyle K})\mathbb{E}^{2}({\beta_{1}^{2}}K_{1})}{\mathbb{E}^{2}({\Delta}_{1}K_{1})}\left( n\mathbb{E}(K_{1} (\delta_{1} [\overline{G} (T_{1})]^{-1} J_{1}-F^{x}(y)))^{2}\right) \\ &+&\displaystyle\frac{ n\phi_{x}(h_{\scriptscriptstyle K})\mathbb{E}^{2}({\beta_{1}^{2}}K_{1})}{\mathbb{E}^{2}({\Delta}_{1}K_{1})} \left( n(n-1)\mathbb{E}^{2}(K_{1} (\delta_{1} [\overline{G} (T_{1})]^{-1} J_{1}-F^{x}(y)))\right). \\ \end{array} $$

By using the fact that $\mid J_{i} \delta _{j} [\overline {G}Y_{i})]^{-1} -F^{x}(y)\mid \leq [\overline {G}(\tau _{L})]^{-1} +1$, (see, Ould Saïd et al. 2011) and by the application of the same technical lemma A.1 of Zhou and Lin (2016), we obtain under assumption (H3) and Eq. 3.11:

$$ \begin{array}{@{}rcl@{}} \mathbb{E}({L_{n}^{2}})&=&\displaystyle\frac{n \phi_{x}(h_{K})O({h_{K}^{2}}{\phi_{x}^{2}}(h_{K})}{(n-1)^{2,}O({h_{K}^{4}}{\phi_{x}^{4}}(h_{K})}\left( nO(\phi_{x}(h_{K})) +n(n-1)o({\phi_{x}^{2}}(h_{K})) \right)\\ &=& O(1)+o(n\phi_{x}(h_{K})).\\ \end{array} $$

(3.16)

In addition, by combining (3.15) and (3.16), we deduce that

$$\mathbb{E}\mid(S_{1.n}-\mathbb{E}(S_{1.n})\mid=o(1).$$

Finally to obtain the convergence in probability it suffices to use Bienaymé-Tchebychev’s inequality. Indeed, we obtain for all ε > 0

$$\mathbb{P}(\mid S_{1.n}-\mathbb{E}(S_{1.n})\mid >\varepsilon) \leqslant\frac{\mathbb{E}(\mid(S_{1.n}-\mathbb{E}(S_{1.n})\mid) }{\varepsilon}\xrightarrow{\enskip \enskip}0.$$

Proof of Eq. 3.8

By following the same idea as in the proof of claim (3.7), we have

$$\mathbb{E}\mid(S_{3.n}-\mathbb{E}(S_{3.n})\mid=o(1).$$

In addition, by applying Bienaymé-Tchebychev’s inequality, we deduced that:

$$ S_{3.n}-\mathbb{E}(S_{3.n})\longrightarrow0, as n\longrightarrow\infty.$$

1.4 Proof of Lemma 2

To prove the convergence in probability of R_n(x, y) to 0, it suffices to show the two following expressions:

$$ \mathbb{P}(\mid \widehat{F}^{x}_{N}(y)-\widetilde{F}^{x}_{N}(y)\mid>\varepsilon)\leq\displaystyle\frac{\mathbb{E}(\mid \widehat{F}^{x}_{N}(y)-\widetilde{F}^{x}_{N}(y)\mid)}{\varepsilon}, $$

(3.17)

$$ B_{n}(x,y)=\mathbb{E}(\widetilde{F}^{x}_{N}(y))- F^{x}(y)\longrightarrow0, as n\longrightarrow \infty. $$

(3.18)

By the definition of $\widehat {F}^{x}_{N}(y)$ and $\widetilde {F}^{x}_{N}(y)$, we obtain:

$$ \begin{array}{@{}rcl@{}} \mid \widehat{F}^{x}_{N}(y)-\widetilde{F}^{x}_{N}(y)\mid\!\!\! &\leq&\!\!\!\! \displaystyle \frac{1}{n(n - 1)\mathbb{E}(W_{12})}\displaystyle \sum\limits_{i \neq j} \mid J_{i} \delta_{j} W_{ij} \left( [\overline{G}_{n}(T_{i})]^{-1} - [\overline{G}(T_{i})]^{-1} \right)\! \mid \\ &\leq&\!\!\!\! \displaystyle \frac{ C \sup\limits_{{t\leq \tau_{L}}}\mid \overline{G}_{n} (t)- \overline{G}(t) \mid}{ \overline{G}_{n} (\tau_{L}) \overline{G} (\tau_{L})} \mid\widehat{F}^{x}_{D}\mid. \end{array} $$

On the other side, using Lemma 3 in Leulmi (2019), we have: $\mid \widehat {F}^{x}_{D}\mid \leq \displaystyle \frac {log(n)}{n\phi _{x}(h_{K})}+1$ and under our technical Lemma 3 and by assumption (H.4)(ii) we get:

$$ \widehat{F}^{x}_{N}(y)-\widetilde{F}^{x}_{N}(y) \longrightarrow 0, as n\longrightarrow \infty.$$

Proof of Eq. 3.18

By the definition of $\widetilde { F}^{x}_{N}(y)$, we have:

$$ \begin{array}{@{}rcl@{}} \mathbb{E}(\widetilde{ F}^{x}_{N}(y))-F^{x}(y)&=&\displaystyle\frac{1}{\mathbb{E}({\Delta}_{1}K_{1})} \mathbb{E}\left( {\Delta}_{1}K_{1}\delta_{1} J_{1} [\overline{G}(T_{1})]^{-1} J_{1}\right)-F^{x}(y)\\ &=&\displaystyle\frac{1}{\mathbb{E}({\Delta}_{1}K_{1})} \mathbb{E}\left( {\Delta}_{1}K_{1}\mathbb{E}\left( \delta_{1} [\overline{G}(T_{1})]^{-1} J_{1}\mid X_{1}\right)\right) - F^{x}(y), \end{array} $$

where

$$ \begin{array}{@{}rcl@{}} \mathbb{E}\left( \delta_{1} [\overline{G}(T_{1})]^{-1} J_{1}\mid X_{1}\right) &=&\mathbb{E}\left( J_{1}[\overline{G}(Y_{1})]^{-1} \mathbb{E}\left( \displaystyle{1\!\!1_{Y_{1}\leq C_{1}}}\mid(X_{1},Y_{1})\mid \right)\mid X_{1}\right)\\ &=&\mathbb{E}\left( J(h^{-1}_{J}(y-Y_{1}))\mid X_{1}\right). \end{array} $$

Finally, by using Lemma 3.2 in Demongeot et al. (2014) and by assumption (H.4)(ii), we obtain

$$ B_{n}(x,y)\longrightarrow0, as n\longrightarrow\infty. $$

Conclusion

The theoretical study of this paper aims mainly at investigating relationships between scalar censored and functional variables throuth a local linear estimation of the CDF. The application part was devoted to a study of its performance on a finite sample. This study demonstrates more efficiency for the local linear method than the classical method in the presence of censored data. Accordingly, this paper displays three main statements. First, the focus has been on determining the leading terms, which are the bias and the asymptotic variance, and where results are obtained under standard hypotheses that allow a large flexibility in the selection of the parameters. Second, the proposed asymptotic study can be also considered as a preliminary investigation allowing various theoretical issues. Among them, optimal selection of the bi-functional operators β and δ, optimal choice of the smoothing parameters or the different types of censorship. Third, the estimation procedure used for this study can also be employed in other conditional models. Finally, this study contributes greatly in both theory and practice and which is assessed by the numerous open questions.