M-estimation of the regression function under random left truncation and functional time series model

Abstract

In this paper we study the M-estimation of the functional nonparametric regression when the response variable is subject to left-truncation by an other random variable. Under standard assumptions, we get the almost complete convergence rate of this robust estimate when the sample is an $\alpha$-mixing sequence. This approach can be applied in time series analysis to the prediction problem. Our asymptotic results are confronted by some simulations study.

Appendix

For the proofs of the Theorems 1 and 2 we use the fact that $\rho$ is a strictly convex function and continuously differentiable wrt the second component, then $\psi$ is strictly monotone and continuous wrt the second component. We give the proof for the case of a increasing $\psi (Y,\cdot )$, decreasing case being obtained by considering $-\psi (Y,\cdot )$. From this, it is clear that the optimization problems (1.3) and (2.1) correspond to vanishing the functions $\Psi (\cdot ,\cdot )$ and ${\widehat{\Psi }}(\cdot ,\cdot )$ respectively. Therefore, we can write, under this consideration, for all $\epsilon >0$

$$\begin{aligned} \Psi (\chi ,\theta _\chi -\epsilon )\le & {} \Psi (\chi ,\theta _\chi )=0\le \Psi (\chi ,\theta _\chi +\epsilon )\, \text{ and } \, {\widehat{\Psi }}(\chi ,\widehat{ \theta _\chi } -\epsilon )\le {\widehat{\Psi }}(\chi ,\widehat{ \theta _\chi })\\= & {} 0\le {\widehat{\Psi }}(\chi ,\widehat{ \theta _\chi } +\epsilon ). \end{aligned}$$

Hence, for all $\epsilon >0$, we have

$$\begin{aligned} \mathbf{P}\left( |\widehat{ \theta _\chi }-\theta _\chi |\ge \epsilon \right)\le & {} \mathbf{P}\left( |{\widehat{\Psi }}(\chi ,\theta _\chi +\epsilon )-\Psi (\chi ,\theta _\chi +\epsilon )| \ge \Psi (\chi ,\theta _\chi +\epsilon )\right) \\&+\,\mathbf{P}\left( |{\widehat{\Psi }}(\chi ,\theta _\chi -\epsilon )-\Psi (\chi ,\theta _\chi -\epsilon )|\ge -\Psi (\chi ,\theta _\chi -\epsilon )\right) . \end{aligned}$$

So, it suffices to show that

$$\begin{aligned} {\widehat{\Psi }}(\chi ,t)-\Psi (\chi ,t)\rightarrow \, 0\, \quad a.s. \text{ for } \quad t := \theta _\chi \pm \epsilon . \end{aligned}$$

(5.1)

Moreover, under ((H2) (i)), we get that

$$\begin{aligned} \widehat{ \theta _\chi }-\theta _\chi =\frac{\Psi (\chi ,\widehat{ \theta _\chi })-{\widehat{\Psi }}(\chi , \widehat{ \theta _\chi })}{\Psi ^{\prime }(\chi ,\xi _n)} \end{aligned}$$

where $\xi _n$ is between $\widehat{\theta _\chi }$ and $\theta _\chi$. As long as we could be able to check that

$$\begin{aligned} \exists \tau >0, \ \sum _{n=1}^{\infty } \mathbf{P}\left( \Psi ^{\prime }(\chi ,\xi _n)<\tau \right) \ < \ \infty , \end{aligned}$$

(5.2)

we would have

$$\begin{aligned} \widehat{ \theta _\chi }-\theta _\chi =O_{a.co.}\left( \sup _{t\in [\theta _\chi -\delta ,\, \theta _\chi +\delta ]} |\Psi (\chi ,t)-{\widehat{\Psi }}(\chi , t)|\right) . \end{aligned}$$

Therefore, all what is left to do, is to study the convergence rate of

$$\begin{aligned} \sup _{t\in [\theta _\chi -\delta ,\, \theta _\chi +\delta ]} |\Psi (\chi ,t)-{\widehat{\Psi }}(\chi , t)|. \end{aligned}$$

To do that, we write

$$\begin{aligned} {\widehat{\Psi }}(\chi ,t)=\frac{{\widehat{\Psi }}_N(\chi ,t)}{{\widehat{\Psi }}_D(\chi )} \end{aligned}$$

with

$$\begin{aligned} {\widehat{\Psi }}_N(\chi ,t)= & {} \frac{\tau _{n}}{n\mathbf{E}[K_{1}]}\displaystyle \sum _{i=1}^{n} \frac{1}{G_{n}(Y_{i})}K\left( \frac{d(\chi ,{\varvec{\chi }}_{i})}{h}\right) \psi (Y_{i},t),\\ {\widehat{\Psi }}_{D}(\chi )= & {} \frac{\tau _{n}}{n\mathbf{E}[K_{1}]}\displaystyle \sum _{i=1}^{n}\frac{1}{G_{n}(Y_{i})}K\left( \frac{d(\chi ,{\varvec{\chi }}_{i})}{h}\right) \end{aligned}$$

and we consider the following decomposition

$$\begin{aligned} {{\widehat{\Psi }}}(\chi , t)-\Psi (\chi ,t)= & {} \frac{1}{{{\widehat{\Psi }}}_D (\chi )} \Big [ \Big ({\widehat{\Psi }}_N (\chi ,t)- \mathbf{E}\left[ {\widehat{\Psi }}_N (\chi ,t)\right] \Big )\nonumber \\&-\,\Big (\Psi (\chi ,t)- \mathbf{E}\left[ {\widehat{\Psi }}_N (\chi ,t)\right] \Big ) \Big ]\nonumber \\&+\,\frac{\Psi (\chi ,t)}{{\widehat{\Psi }}_D (\chi )} \Big [ \mathbf{E}\left[ {\widehat{\Psi }}_D (\chi )\right] -{\widehat{\Psi }}_D(\chi ) \Big ]. \end{aligned}$$

(5.3)

Unlike to Attouch et al. (2012) we must introduce the following pseudo-estimators

$$\begin{aligned} \widetilde{\Psi }_D(\chi )=\frac{\tau }{n\mathbf{E}[K_{1}]}\displaystyle \sum _{i=1}^{n}\frac{1}{G(Y_{i})}K\left( \frac{d(\chi ,{\varvec{\chi }}_{i})}{h}\right) \end{aligned}$$

and

$$\begin{aligned} \widetilde{\Psi }_N(\chi ,t)=\frac{\tau }{n\mathbf{E}[K_{1}]}\displaystyle \sum _{i=1}^{n}\frac{1}{G(Y_{i})}K\left( \frac{d(\chi ,{\varvec{\chi }}_{i})}{h}\right) \psi (Y_{i},t). \end{aligned}$$

Moreover, we put

$$\begin{aligned} \widehat{\Psi }_N(\chi ,t)=\frac{\tau _{n}}{n\mathbf{E}[K_{1}]}\displaystyle \sum _{i=1}^{n} \frac{1}{G_{n}(Y_{i})}K\left( \frac{d(\chi ,{\varvec{\chi }}_{i})}{h}\right) \psi (Y_{i},t) \end{aligned}$$

and

$$\begin{aligned} {\widehat{\Psi }}_{D}(\chi )=\frac{\tau _{n}}{n\mathbf{E}[K_{1}]}\displaystyle \sum _{i=1}^{n}\frac{1}{G_{n}(Y_{i})}K\left( \frac{d(\chi , {\varvec{\chi }}_{i})}{h}\right) . \end{aligned}$$

Now, we consider the following decomposition

$$\begin{aligned} {\widehat{\Psi }}(\chi ,t)-\Psi (\chi ,t)= & {} \frac{\widehat{\Psi }_N(\chi ,t)}{{\widehat{\Psi }}_{D}(\chi )}-\Psi (\chi ,t)\\= & {} \frac{{\widehat{\Psi }}_N(\chi ,t)}{{\widehat{\Psi }}_{D}(\chi )}-\frac{\Psi (\chi ,t)}{{\widehat{\Psi }}_{D}(\chi )}+\frac{\Psi (\chi ,t)}{{\widehat{\Psi }}_D(\chi )}-\Psi (\chi ,t)\\= & {} \frac{1}{{\widehat{\Psi }}_{D}(\chi )}\left( {\widehat{\Psi }}_N(\chi ,t)-\Psi (\chi ,t)\right) +\frac{\Psi (\chi ,t)}{{\widehat{\Psi }}_{D}(\chi )} \left( -{\widehat{\Psi }}_{D}(\chi )+1\right) \\= & {} \frac{1}{{\widehat{\Psi }}_{D}(\chi )}\left( {\widehat{\Psi }}_N(\chi ,t)-\widetilde{\Psi }_N(\chi ,t)\right) +\frac{1}{{\widehat{\Psi }}_{D}(\chi )} \left( \widetilde{\Psi }_N(\chi ,t)\right. \\&\left. -\;\mathbf{E}\left[ \widetilde{\Psi }_N(\chi ,t)\right] \right) +\frac{1}{{\widehat{\Psi }}_{D}(\chi )}\left( \mathbf{E}\left[ \widetilde{\Psi }_N(\chi ,t)\right] -\Psi (\chi ,t)\right) \\&+\;\frac{\Psi (\chi ,t)}{{\widehat{\Psi }}_{D}(\chi )}\left\{ \left( \widetilde{\Psi }_D(\chi )-{\widehat{\Psi }}_D(\chi )\right) \right. \\&\left. +\;\left( \mathbf{E}\left[ \widetilde{\Psi }_{D}(\chi )\right] -\widetilde{\Psi }_{D}(\chi )\right) + \left( -\mathbf{E}\left[ \widetilde{\Psi }_{D}(\chi )\right] +1\right) \right\} . \end{aligned}$$

Thus, both Theorems are a consequence of the following intermediates results. The first lemma is the main point of the proof.

Lemma 1

Under Hypotheses (H1) and (H3)–(H6), we have,

$$\begin{aligned} \sup _{t\in [\theta _\chi -\delta , \theta _\chi +\delta ]}\left| \widetilde{\Psi }_N (\chi ,t)- \mathbf{E}\left[ \widetilde{\Psi }_N (\chi ,t)\right] \right| = O\left( \sqrt{\frac{\log n}{n\phi _\chi (h)}}\right) \quad a.co. \end{aligned}$$

Proof of Lemma 1

The compactness of $[\theta _\chi -\delta ,\, \theta _\chi +\delta ]$, allows to write

$$\begin{aligned}{}[\theta _\chi -\delta ,\, \theta _\chi +\delta ]\subset \bigcup _{j=1}^{d_n}\left( y_j-l_n, y_j+l_n\right) \end{aligned}$$

(5.4)

with $\displaystyle l_n=n^{-1/2}$ and $d_n=O\left( n^{1/2}\right)$. We put

$$\begin{aligned} {\mathcal G}_n=\left\{ y_j-l_n,y_j+l_n,1\le j\le d_n\right\} . \end{aligned}$$

(5.5)

We combine (H4) together with the monotony of $\mathbf{E}[\widetilde{\Psi }_N (\chi ,\cdot )]$ and $\widetilde{\Psi }_N (\chi ,\cdot )$ to write

$$\begin{aligned}&\sup _{t\in [\theta _\chi -\delta ,\, \theta _\chi +\delta ]}\left| \widetilde{\Psi }_N (\chi ,t)\right. \nonumber \\&\quad \left. -\,\mathbf{E}\left[ \widetilde{\Psi }_N (\chi ,t)\right] \right| \le \max _{1\le j\le d_n} \max _{z\in \{y_j-l_n, y_j+l_n\}}\left| \widetilde{\Psi }_N (\chi ,z)\right. \nonumber \\&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \left. -\, \mathbf{E}\left[ \widetilde{\Psi }_N (\chi ,z)\right] \right| +2Cl_n. \end{aligned}$$

(5.6)

On the one hand condition (H6) implies that

$$\begin{aligned} l_n=o\left( \sqrt{\frac{\log n}{n\phi _\chi (h)}}\right) . \end{aligned}$$

(5.7)

On the other hand we have, for any $\varepsilon >0$

$$\begin{aligned}&\mathbf{P}\left( \max _{z\in {\mathcal G}_n}\left| \widetilde{\Psi }_N (\chi ,z)-\mathbf{E}\left[ \widetilde{\Psi }_N (\chi ,z)\right] \right|>\varepsilon \right) \nonumber \\&\quad \le \sum _{z\in {\mathcal G}_n} \mathbf{P}\left( \left| \widetilde{\Psi }_N (\chi ,z)-\mathbf{E}\left[ \widetilde{\Psi }_N(\chi ,z)\right] \right| > \varepsilon \right) . \end{aligned}$$

(5.8)

Now, using truncation method to prove our result for general case where $\psi$ is not necessary bounded. Indeed, we consider the following random variable

$$\begin{aligned} \widetilde{\Psi }_N^*(\chi ,t)= \frac{1}{n\mathbf{E}[K(h^{-1}d(\chi ,{\varvec{\chi }}_1))]}\sum _{i=1}^nK(h^{-1}d(\chi ,{\varvec{\chi }}_i))\frac{\tau }{G(Y_{i})}\psi ^*(Y_i,t) \end{aligned}$$

where $\psi ^*(\cdot ,t)=\psi (\cdot ,t)\mathbbm {1}_{(\psi (\cdot ,t)< \gamma _n)}$ with $\gamma _n=n^{a/p}$. So, the claimed result is a consequence of the three intermediates results.

$$\begin{aligned}&\max _{z\in {\mathcal G}_n} \left| \mathbf{E}[\widetilde{\Psi }_N^* (\chi ,z)]- \mathbf{E}[{\widehat{\Psi }} _N (\chi ,z)]\right| = o\left( \sqrt{\frac{\log n}{n\phi _\chi (h)}}\right) , \end{aligned}$$

(5.9)

$$\begin{aligned}&\sum _nd_n\max _{z\in {\mathcal G}_n}{} \mathbf{P}\left( \left| \widetilde{\Psi }_N^* (\chi ,z)- \widetilde{\Psi }_N (\chi ,z)\right|>\epsilon _0\left( \sqrt{\frac{\log n}{n\phi _\chi (h)}}\right) \right) \nonumber \\&\quad <\infty \quad \text{ for } \text{ some } \epsilon _0>0 \end{aligned}$$

(5.10)

and

$$\begin{aligned} \sum _nd_n\max _{z\in {\mathcal G}_n}{} \mathbf{P}\left( \left| \widetilde{\Psi }_N^* (\chi ,z)-\mathbf{E}[\widetilde{\Psi }_N^* (\chi ,z)] \right| >\epsilon _0\left( \sqrt{\frac{\log n}{n\phi _\chi (h)}}\right) \right) <\infty .\quad \end{aligned}$$

(5.11)

$\square$

Proof of (5.9)

It clear that, for all $z\in {\mathcal G}_n$ we can write

$$\begin{aligned}&\left| \mathbf{E}[\widetilde{\Psi }_N^* (\chi ,z)]- \mathbf{E}[\widetilde{\Psi }_N (\chi ,z)]\right| \\&\quad \le C\frac{1}{\phi _\chi (h)}\mathbf{E}\left[ \left| \psi (Y,z)\right| \mathbbm {1}_{\{\psi (Y,z)\ge \gamma _n\}}K(h^{-1}d(\chi ,X))\right] . \end{aligned}$$

Applying Holder inequality, for $\kappa =\frac{p}{2}$ with $\zeta$ such that $\frac{1}{\kappa }+\frac{1}{\zeta }=1$, to write that, for all $z\in {\mathcal G}_n$

$$\begin{aligned}&\mathbf{E}\left[ \left| \psi (Y,z)\right| \mathbbm {1}_{\{\psi (Y,z)\ge \gamma _n\}}K(h^{-1}d(\chi ,{\varvec{\chi }}_1))\right] \\&\quad \le \mathbf{E}^{1/ \kappa }\left[ \left| \psi ^{\kappa }(Y,z)\right| \mathbbm {1}_{\{\psi (Y,z)\ge \gamma _n\}}\right] \mathbf{E}^{1/ \zeta }\left[ K^{\zeta }(h^{-1}d(\chi ,{\varvec{\chi }}_1))\right] \\&\quad \le \gamma _n^{-1} \mathbf{E}^{1/ \kappa }\left[ \left| \psi ^{2\kappa }(Y,z)\right| \right] \mathbf{E}^{1/ \zeta }\left[ K^{\zeta }(h^{-1}d(\chi ,{\varvec{\chi }}_1))\right] \\&\quad \le C\gamma _n^{-1}\phi _\chi ^{1/ \zeta }(h). \end{aligned}$$

Thus,

$$\begin{aligned} \max _{z\in {\mathcal G}_n} \left| \widetilde{\Psi }_N^* (\chi ,z)-\mathbf{E}[\widetilde{\Psi }_N^* (\chi ,z)] \right| \le n^{-a/p}\phi _\chi ^{(1-\zeta )/\zeta }. \end{aligned}$$

Therefore, (5.9) is consequence of the fact that $a>p>2$.

Proof of (5.10)

To do so, we use the Markov’s inequality $\forall z\in {\mathcal G}_n$, $\forall \epsilon >0$

$$\begin{aligned}&\mathbf{P}\left( \left| \widetilde{\Psi }_N^* (\chi ,z)-\widetilde{\Psi }_N(\chi ,z)\right|> \epsilon \right) \\&\quad \le \sum _{i=1}^n \mathbf{P}\left( \psi (Y_i,z)>n^{a/p}\right) \\&\quad \le n^{1-a}{} \mathbf{E}\left[ \psi ^p(Y,z)\right] .\\ \end{aligned}$$

In particular, for $\epsilon =\epsilon _0\left( \sqrt{\frac{\log n}{n\phi _\chi (h)}}\right)$ and thanks to $a>4$, we have

$$\begin{aligned} d_n\max _{z\in {\mathcal G}_n}{} \mathbf{P}\left( |\widetilde{\Psi }_N (\chi ,z)- \widetilde{\Psi }_N^* (\chi ,z)|>\epsilon _0\left( \sqrt{\frac{\log n}{n\phi _\chi (h)}}\right) \right) \le n^{3/2-a}< Cn^{-1-\nu }. \end{aligned}$$

Proof of (5.11)

We define, for all $z\in {\mathcal G}_n$,

$$\begin{aligned} \Lambda _i(z)=K_i (\chi )\psi ^*(Y_i,z)\frac{\tau }{G(Y_{i})}-\mathbf{E}\left[ K_1(\chi )\frac{\tau }{G(Y_{i})}\psi ^*(Y_i,z)\right] . \end{aligned}$$

Therefore, for all $\epsilon >0$

$$\begin{aligned} \mathbf{P}\left\{ \left| \widetilde{\Psi }_N^* (\chi ,z)-\mathbf{E}\left[ \widetilde{\Psi }_N^* (\chi ,z)\right] \right|>\varepsilon \right\}= & {} \mathbf{P}\left\{ \left| \sum _{i=1}^{n}{\Lambda _{i}(z)}\right| >\varepsilon n\mathbf{E}[K_1(\chi )]\right\} . \end{aligned}$$

Now the evaluation of the last quantity is based on the asymptotic behavior of the following quantity

$$\begin{aligned} S_{n}^{'2}=\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{j=1}^n Cov(\Lambda _i(z),\Lambda _j(z))=\displaystyle \sum _{i=1}^{n}\displaystyle \sum _{i\ne j} Cov(\Lambda _{i}(z),\Lambda _{j}(z))+n Var[\Lambda _1(z)]. \end{aligned}$$

For this term we use the same technique developed by Masry (1986) by spliting the sum into two sets defined by

$$\begin{aligned} S'_1=\{ (i,j)\;\ \text{ such } \text{ that }\; \ 1\le i-j\le u_n\} \end{aligned}$$

and

$$\begin{aligned} S'_2=\{ (i,j)\; \ \text{ such } \text{ that }\;\ u_n+1\le i-j\le n-1\}. \end{aligned}$$

We note by $J'_{1,n}$ and $J'_{2,n}$ be the sum of covariance over $S'_1$ and $S'_2$ respectively. On $S'_1$, we have, under (H4)

$$\begin{aligned} J'_{1,n}\le & {} C\sum _{S'_1} \left| \mathbf{E}\left[ K_i(\chi )K_j(\chi )\right] \right| +\left| \mathbf{E}\left[ K_i(\chi )\right] \mathbf{E} \left[ K_j(\chi ) \right] \right| . \end{aligned}$$

Because of (H1), (H5) and (H6) we have

$$\begin{aligned} J'_{1,n} \le C n u_n \left( \frac{\phi _\chi (h)}{n^{2/p}}\right) ^{a/(a-1)}. \end{aligned}$$

Concerning $J'_{2,n}$, we use again Davydov–Rio’s inequality in the $L^\infty$ cases and we obtain

$$\begin{aligned} |Cov(\Lambda _i(z), \Lambda _j(z))|\le C\gamma _n^2\alpha (|i-j|). \end{aligned}$$

Therefore,

$$\begin{aligned} J'_{2,n}=\sum _{S'_2}|Cov(\Lambda _i(z), \Lambda _j(z))|\le \frac{n\gamma _n^2u_n^{-a+1}}{a-1}. \end{aligned}$$

Taking $u_n=\left( \frac{\gamma _n^2}{\left( \frac{\phi _\chi (h)}{n^{2/p}}\right) ^{a/(a-1)}}\right) ^{1/a}$, we prove that

$$\begin{aligned} \sum _{i=1}^{n}\displaystyle \sum _{i\ne j} Cov(\Lambda _{i}(z),\Lambda _{j}(z))=O(n\phi _\chi (h)). \end{aligned}$$

On the other hand, under (H3), we have

$$\begin{aligned} Var(\Lambda _1(z))\le \mathbf{E}\left[ K_i(\chi )\psi ^*(Y_i,z)\right] ^2\le \mathbf{E}\left[ K_i(\chi )\psi (Y_i,z)\right] ^2=O(\phi _\chi (h)). \end{aligned}$$

Under (H6) we have

$$\begin{aligned} S_{n}^{'2}=O(n\phi _\chi (h)). \end{aligned}$$

(5.12)

Now, we apply Fuck–Nagaev’s inequality for $\Lambda _{i}(z)$. We have, for all $\ell >0$ and for all $\varepsilon >0$,

$$\begin{aligned} \mathbf{P}\left\{ \left| \mathbf{E}\left[ {\widehat{\Psi }}_{N}^*(\chi ,z)\right] -{\widehat{\Psi }}_{N}^*(\chi ,z)\right|>\varepsilon \right\}\le & {} \mathbf{P}\left\{ \left| \sum _{i=1}^{n}{\Lambda _{i}}(z)\right| > \varepsilon n\mathbf{E}[K_1(\chi )]\right\} \\\le & {} C (A'_1(\chi )+A'_2(\chi )) \end{aligned}$$

where

$$\begin{aligned} A'_1= \left( 1+\frac{\varepsilon ^{2}n^{2}(\mathbf{E}[K_1(\chi )])^{2}}{S_{n}^{'2}\ell }\right) ^{-\ell /2}\, \text{ and } \; A'_2=n\ell ^{-1}\left( \frac{\ell }{\varepsilon n\mathbf{E}[K_1(\chi )]}\right) ^{a+1}. \end{aligned}$$

In particular, for $\varepsilon =\epsilon _0\frac{\sqrt{n\log n \phi _\chi (h)}}{n\mathbf{E}[K_1(\chi )]}$ and $\ell =C(\log n)^2$, we obtain by (H6)

$$\begin{aligned} d_nA'_2\le C n^{3/2-(a+1)/2}\phi _\chi (h)^{-(a+1)/2}(\log n)^{(3a-1)/2}\le Cn^{-1-\nu '_1} \; \end{aligned}$$

(5.13)

for some $\nu '_1>0$ and

$$\begin{aligned} d_nA'_1\le C\left( 1+\frac{\epsilon _0^2\log n}{\ell }\right) ^{-\ell /2}\le Cn^{-1-\nu '_2} \quad \text{ for } \text{ some }\quad \nu '_2>0. \end{aligned}$$

(5.14)

From (5.13) and (5.14) we obtain

$$\begin{aligned} \sum _nd_n\max _{z\in {\mathcal G}_n}{} \mathbf{P}\left( \left| \widetilde{\Psi }_N^* (\chi ,z)-\mathbf{E}[\widetilde{\Psi }_N^* (\chi ,z)] \right| >\epsilon _0\left( \sqrt{\frac{\log n}{n\phi _\chi (h)}}\right) \right) <\infty . \end{aligned}$$

Lemma 2

(Derrar et al. 2015) Under Hypotheses (H1), (H2)((i), (iii)) and (H5), we have,

$$\begin{aligned} \sup _{t\in [\theta _\chi -\delta , \theta _\chi +\delta ]}\left| \mathbf{E}\left[ \widetilde{\Psi }_N (\chi ,t)\right] - \tau \Psi (\chi ,t)\right| = o(1). \end{aligned}$$

Furthermore if we add (H2)(ii), we have

$$\begin{aligned} \sup _{t\in [\theta _\chi -\delta , \theta _\chi +\delta ]}\left| \mathbf{E}\left[ \widetilde{\Psi }_N (\chi ,t)\right] - \tau \Psi (\chi ,t)\right| =O\left( h^{b_1}\right) . \end{aligned}$$

Lemma 3

Under Hypotheses (H1) and (H3)–(H5), we have,

$$\begin{aligned} \sup _{t\in [\theta _\chi -\delta , \theta _\chi +\delta ]}\left| {\widehat{\Psi }}_N (\chi ,t)- \widetilde{\Psi }_N (\chi ,t)\right| =\displaystyle O\left( \sqrt{\frac{1}{n}}\right) \quad a.s. \end{aligned}$$

Proof of Lemma  3

By a simple algebra we write for all $t \in [\theta _\chi -\delta , \theta _\chi +\delta ]$

$$\begin{aligned}&\left| {\widehat{\Psi }}_N (\chi ,t)- \widetilde{\Psi }_N (\chi ,t)\right| \\&\quad = \displaystyle \left| \frac{\tau _{n}-\tau }{n\mathbf{E}[K_{1}]}\displaystyle \sum _{i=1}^{n} \frac{1}{G_{n}(Y_{i})}K\left( \frac{d(\chi ,{\varvec{\chi }}_{i})}{h}\right) \psi (Y_{i},t)\right. \\&\qquad + \left. \displaystyle \;\tau \,\left( \frac{G_{n}(Y)-G(Y)}{G_{n}(Y)G(Y)}\right) \frac{1}{n\mathbf{E}[K_{1}]} \displaystyle \sum _{i=1}^{n}K\left( \frac{d(\chi ,{\varvec{\chi }}_{i})}{h}\right) \psi (Y_{i},t)\right| \\&\quad \le \displaystyle \left[ \frac{\big |\tau _{n}-\tau \big |}{G_{n}(a_{F})}+\tau \frac{\sup _{t\in [a,\;b]}\big | G_{n}(t)-G(t)\big |}{G _{n}(a_{F})G(a_{F})}\right] \\&\qquad \times \, \displaystyle \frac{1}{n\mathbf{E}[K_{1}]}\displaystyle \sum _{i=1}^{n}K\left( \frac{d(\chi ,{\varvec{\chi }}_{i})}{h}\right) \psi (Y_{i},t). \end{aligned}$$

Using the result of Ould Saïd and Tatachak (2009) for which we have $\big |\tau _{n}-\tau \big |=O_{a.s.}(n^{-\frac{1}{2}})$ and the remark 6 of Woodroofe 1985) where we obtain $|G_{n}(a_{F})-G(a_{F})|=O_{a.s.}(n^{-\frac{1}{2}})$. Furthermore, for the second part, we recall that this term has been studied by Attouch et al. (2012) where we show that

$$\begin{aligned}&\sup _{t\in [\theta _\chi -\delta , \theta _\chi +\delta ]}\frac{1}{n\mathbf{E}[K_{1}]}\displaystyle \sum _{i=1}^{n}\left[ K\left( \frac{d(\chi ,{\varvec{\chi }}_{i})}{h}\right) \psi (Y_{i},t)\right. \\&\quad \left. -\;\mathbf{E}\left[ K\left( \frac{d(\chi ,{\varvec{\chi }}_{i})}{h}\right) \psi (Y_{i},t)\right] \right] =o\left( 1\right) \end{aligned}$$

It follows that

$$\begin{aligned} \sup _{t\in [\theta _\chi -\delta , \theta _\chi +\delta ]}\frac{1}{n\mathbf{E}[K_{1}]}\displaystyle \sum _{i=1}^{n}K\left( \frac{d(\chi ,{\varvec{\chi }}_{i})}{h}\right) \psi (Y_{i},t)=O_{a.s.}(1). \end{aligned}$$

Thus, we have

$$\begin{aligned} \left| {\widehat{\Psi }}_N (\chi ,t)- \widetilde{\Psi }_N (\chi ,t)\right| =\displaystyle O_{a.s.}\Big (n^{-\frac{1}{2}}\Big ) \end{aligned}$$

which completes the proof of Lemma 3.

Lemma 4

Under Hypotheses (H1), (H4)–(H6), we have,

$$\begin{aligned} \widetilde{\Psi }_{D}(\chi )-\mathbf{E}\left[ \widetilde{\Psi }_{D}(\chi )\right] = O\left( \sqrt{\frac{\log n}{n\phi _\chi (h)}}\right) \quad a.co. \end{aligned}$$

Proof of Lemma 4

The proof is similar to Lemma 1 by replacing the function $\psi$ by 1. $\square$

Lemma 5

(Derrar et al. 2015) Under Hypotheses (H1), (H4) and (H5), we have,

$$\begin{aligned} \mathbf{E}\left[ \widetilde{\Psi }_{D}(\chi )\right] =1. \end{aligned}$$

Lemma 6

Under Hypotheses (H1), (H4) and (H5), we have,

$$\begin{aligned} {\widehat{\Psi }}_{D}(\chi )-\widetilde{\Psi }_{D}(\chi )= O_{a.s.}\left( \sqrt{\frac{1}{n}}\right) .\quad \end{aligned}$$

Proof of Lemma 6

Again the proof is similar to Lemma 3 by replacing $\psi$ by 1. $\square$

Corollary 1

Under Assumptions (H1), (H2) ((i)–(ii)), (H3)–(H5) and if $\Psi ^{\prime }(\chi ,\theta _\chi )\not =0$, then $\widehat{\theta }_\chi$ exists a.s. for all sufficiently large n such that

$$\begin{aligned} \exists C>0 \quad \Psi ^\prime (\chi ,\xi _n)> C \end{aligned}$$

where $\xi _n$ is between $\theta _\chi$ and $\widehat{\theta }_\chi$.

Proof of Corollary 1

It is clear that, if $\psi _\chi (Y,.)$ is increasing, we have for all $\epsilon >0$

$$\begin{aligned} \Psi (\chi , \theta _\chi -\epsilon )\le \Psi (\chi ,\theta _\chi )\le \Psi (\chi , \theta _\chi +\epsilon ). \end{aligned}$$

$\square$

The results of Lemmas 1–6 show that

$$\begin{aligned} {\widehat{\Psi }}(\chi ,t)\longrightarrow \Psi (\chi ,t) \; \; \hbox {a. s.} \end{aligned}$$

for all real fixed $t\in [\theta _\chi -\delta ,\theta _\chi +\delta ]$. So, for sufficiently large n and for all $\epsilon \le \delta$

$$\begin{aligned} {\widehat{\Psi }}(\chi ,\theta _\chi -\epsilon )\le 0\le {\widehat{\Psi }}(\chi ,\theta _\chi +\epsilon ) \quad \text{ a. } \text{ s. }. \end{aligned}$$

Since ${\widehat{\Psi }}(\chi ,t)$ is a continuous function of t, there exists a $\widehat{\theta _\chi }\in [\theta _\chi -\epsilon ,\theta _\chi +\epsilon ]$ such that ${\widehat{\Psi }}(\chi ,\widehat{\theta _\chi })=0.$

Concerning, the uniqueness of $\widehat{\theta _\chi }$, we point out that the latter is a direct consequence of the strict monotonicity of $\psi _\chi$, wrt the second component and the positivity of K.

Finally, the second part of this corollary is a direct consequence of the regularity assumption (H2) (i) on $\Psi (\chi ,\cdot )$ and the convergence

$$\begin{aligned} \widehat{\theta _\chi } \rightarrow \theta _\chi \;\; \hbox {a.s.} \;\; \hbox {as}\;\;\; n \longrightarrow \infty . \end{aligned}$$