# Functional relative error regression under left truncation and right
censoring

A. Boucetta
Faculty of Mathematics
USTHB
Algiers, Algeria
<EMAIL_ADDRESS>
&Z. Guessoum
Faculty of Mathematics
USTHB
Algiers, Algeria
<EMAIL_ADDRESS>
E. Ould-Said
Laboratoire de Math. Pures et Appliquées
Univ. du Littoral Côte d’Opale
Calais, France
<EMAIL_ADDRESS>

(March 8, 2023)

###### Abstract

The nonparametric estimators built by minimizing the mean squared relative
error are gaining in popularity for their robustness in the presence of
outliers in comparison to the Nadaraya Watson estimators. In this paper we
build a relative error regression function estimator in the case of a
functional explanatory variable and a left truncated and right censored scalar
variable. The pointwise and uniform convergence of the estimator is proved and
its performance is assessed by a numerical study in particularly the
robustness which is highlighted using the influence function as a measure of
robustness.

_Keywords_ Censored data $\cdot$ Functional data $\cdot$ Functional regression
$\cdot$ Relative error $\cdot$ Truncated data.

##  1 Introduction

Functional data is a generalization of the multivariate data concept (vectors)
to infinite dimension. This type of data is obtained when an observed
phenomenon varies over a continuum set which is often the case in fields such
as physics, chemistry (spectrometric analysis), health (ECG data for example),
…
To exploit the richness of information brought by the infinite dimension
structure of the data, a set of statistical tools is needed. Functional data
analysis (FDA) is a statistical field that deals with the treatment and
analysis of this particular type of data, it has been popularized mostly by
the monographs of Ramsay and Silverman (2005) and Ferraty and Vieu (2006). The
work of Ramsay and Silverman (2005) is a good introduction to the practical
aspects of FDA and to parametric functional models while the work of Ferraty
and Vieu (2006) focuses on the nonparametric methods applied to functional
data and is considered as a major reference on the topic of nonparametric
functional regression which is the approach on which this work is based. For a
recent account of the theory of FDA we refer the interested reader to Hsing
and Eubank (2015).
In a scalar on functional regression framework the task is to model the link
between a real random variable denoted by $Y$ and a functional explanatory
variable $\boldsymbol{\chi}$ taking its values in $(\mathcal{F},d)$, where
$\mathcal{F}$ is a semi metric space and $d$ its corresponding semi metric,
such that $Y=r(\boldsymbol{\chi})+\epsilon,$ where $r(\cdot)$ is the
regression operator mapping $\mathcal{F}$ to $\mathbb{R}$ and $\epsilon$ an
error random variable. It is worth noting that in some practical cases, the
variable of interest $Y$, can not be fully observed. When we deal with time-
to-event data as in survival analysis, the lifetime may be left truncated
and/or right censored. Right censoring occurs when for some individuals, the
event of interest is observed only if it happens prior to some specified times
and these times vary from an individual to another. We have left truncation
when only the individuals who experience the event of interest after a certain
delayed entry time are included in the study (Klein and Moeschberger, 2003).
Usually $r$ is estimated by minimizing the least squares criterion. The
estimations obtained then are sensitive to outliers since it assumes that all
the variables have an equal weight, which makes them inadequate in some
situations with large outliers. To overcome this limitation we shall use the
following mean quadratic relative error as a loss function:

$\mathbb{E}\left[\left(\frac{Y-r(\boldsymbol{\chi})}{Y}\right)^{2}\mid\boldsymbol{\chi}\right]\text{,
for }Y>0.$ (1)

Park and Stefanski (1998) have shown that the minimizer of (1) is given by:

$r(\chi)=\frac{\mathbb{E}\left[Y^{-1}\mid\boldsymbol{\chi}=\chi\right]}{\mathbb{E}\left[Y^{-2}\mid\boldsymbol{\chi}=\chi\right]}=:\frac{r_{1}(\chi)}{r_{2}(\chi)},$
(2)

where $r_{1}(\chi)$ and $r_{2}(\chi)$ are assumed finite almost surely. The
first nonparametric estimator of equation (2) was proposed by Jones et al.
(2008) in the case when the explanatory variable is a real variable while in
the functional case we refer to Demongeot et al. (2016). Research that treats
nonparametric functional data analysis (or NPFDA) and the relative error
regression estimator with incomplete data includes but not limited to Altendji
et al. (2018), Mechab et al. (2019) and Fetitah et al. (2020).
The aim of this paper is to bring together the areas of NPFDA, robust
estimation and survival analysis by establishing the pointwise and the uniform
convergence with rates of the constructed estimator that takes into account
both right-censoring and left-truncation of the variable of interest and the
functional character of the covariable and highlighting its asymptotic
properties and robustness.
The rest of this paper is organized as follows. In Section 2 we introduce the
LTRC model and its components. Section 3 discusses the construction of the
proposed estimator. In Section 4 and Section 5 the main results and their
assumptions are stated while the proofs are relegated to Section 7. Section 6
is devoted to a deep simulation study of the performance and robustness of the
relative error regression estimator using the influence function.

##  2 LTRC model

Let $(\boldsymbol{\chi}_{i},Y_{i})_{i=1,\cdots,N}$ be an $N$ independent
identically distributed (iid) sample of the pair $(\boldsymbol{\chi},Y)$ with
values in ${\cal F}\times\mathbb{R}_{+}^{*}$ where the sample size $N$ is
deterministic but unknown and $Y$ represents a lifetime variable with
continuous distribution function $F$.
In an LTRC model the variable of interest is subject to a left truncation and
a right censoring. The right censoring is induced by a random variable $S$
supposed strictly positive with continuous distribution function denoted by
$G$, that is, we observe $(\boldsymbol{\chi}_{i},Z_{i},\delta_{i})$ where
$Z_{i}=\min\\{Y_{i},S_{i}\\}$ and $\delta_{i}=\mathds{1}_{\\{Y_{i}\leq
S_{i}\\}}$ the indicator of non censoring, meanwhile, the left truncation is
due to the random variable $T$, that is, we observe $Z_{i}$ only if $Z_{i}\geq
T_{i}$. We denote by $L$ the distribution function of $T$ and by
$\alpha=\mathbb{P}(Z_{i}\geq T_{i})$ the probability of absence of truncation
which is assumed strictly positive (if not then nothing is observed). So
ultimately one observes the quadruple
$(\boldsymbol{\chi}_{i},Z_{i},T_{i},\delta_{i})$ $i=1,...,n$ with $n\leq N$ if
there is no confusion in regard to the index $i$, note that conditionally on
the value of $n$ the observed data are still iid.
We assume that the random variables $Z,T,S$ are mutually independent and it
follows that the distribution function of $Z$ is given by $H=1-(1-F)(1-G)$.
Let $\mathbb{P}$ denote the probability measure associated to the original $N$
sample and $\mathbf{P}$ the probability measure related to the sample observed
conditionally on the event $\\{Z_{i}\geq T_{i}\\}$, also we denote by
$\mathbb{E}$ and $\mathbf{E}$ their respective associated expectation
operators.
Let $C(y)=\mathbf{P}(T\leq y\leq Z)=\mathbf{L}(y)-\mathbf{H}(y)$, where
$\mathbf{L}(y)$, $\mathbf{H}(y)$ and $C(y)$ can be estimated empirically by

$\mathbf{L}_{n}(y)=\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{T_{i}\leq
y\\}}\;\;\text{,
}\;\mathbf{H}_{n}(y)=\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{Z_{i}\leq
y\\}}\;\text{and }\;C_{n}(y)=\frac{1}{n}\sum_{i=1}^{n}\mathds{1}_{\\{T_{i}\leq
y\leq Z_{i}\\}}.$

In a LTRC model the distribution function $F$ is estimated by the product
limit estimator given in Tsai et al. (1987) called the TJW estimator defined
as $F_{n}(y)=1-\prod_{Z_{i}\leq
y}\left(1-\frac{1}{nC_{n}(Z_{i})}\right)^{\delta_{i}}.$ Note that $C(y)$ can
be rewritten as $C(y)=\mathbb{P}(T\leq y\leq Z\mid Z\geq
T)=\alpha^{-1}L(y)\bar{H}(y)=\alpha^{-1}L(y)\bar{F}(y)\bar{G}(y),$ which
implies

$\alpha=\frac{L(y)\bar{F}(y)\bar{G}(y)}{C(y)}.$ (3)

He and Yang (1998) showed that a consistent estimator of $\alpha$ is given by

$\alpha_{n}=\dfrac{L_{n}(y)\bar{F}_{n}(y)\bar{G}_{n}(y)}{C_{n}(y)}\text{ , for
any $y$ such that }C_{n}(y)\neq 0,$ (4)

where $L_{n}(y)$ is the Lynden-Bell (1971) product limit estimator of the
distribution function L defined as
$L_{n}(y)=\prod_{T_{i}>y}\left(1-\frac{1}{nC_{n}(T_{i})}\right)$ and
$G_{n}(y)$ a TJW-type estimator of the distribution function $G$ defined by
$G_{n}(y)=1-\prod_{Z_{i}\leq
y}\left(1-\dfrac{1}{nC_{n}(Z_{i})}\right)^{1-\delta_{i}}.$

######  Remark 1.

He and Yang (1998) showed that the value of $\alpha_{n}$ does not depend on
the value of $y$.

For any distribution function W denote by $a_{W}=\sup\\{y:W(y)>0\\}$ and
$b_{K}=\inf\\{y:W(y)<1\\}$ its support endpoints. Following Gijbels and Wang
(1993) the distribution function $F$ is identifiable, in a LTRC setting, only
under some conditions on the support of $L$ and $H$ ($a_{L}\leq a_{H}$ and
$b_{L}\leq b_{H}$ ). We assume that $a_{L}<a_{H}$, $b_{L}\leq b_{H}$ and that
for some $b>0$, $b<b_{H}$. Furthermore we suppose that $(T,S)$ and
$(\boldsymbol{\chi},\;Y)$ are independent.

##  3 Construction of the estimator

Under a LTRC setting, one has to account for both left truncation and right
censoring effect and by combining the ideas of Altendji et al. (2018) and
Mechab et al. (2019), we propose the following mean relative error regression
estimator:

$\displaystyle\widehat{r}_{n}(\chi)=\displaystyle\dfrac{\displaystyle\sum\limits_{i=1}^{n}\dfrac{\displaystyle\delta_{i}K(\displaystyle\frac{d(\chi,\boldsymbol{\chi}_{i})}{h})Z_{i}^{-1}}{L_{n}(Z_{i})\bar{G_{n}}(Z_{i})}}{\displaystyle\sum\limits_{i=1}^{n}\dfrac{\displaystyle\delta_{i}K\displaystyle(\frac{d(\chi,\boldsymbol{\chi}_{i})}{h})Z_{i}^{-2}}{L_{n}(Z_{i})\bar{G_{n}}(Z_{i})}},$
(5)

where $K$ is a kernel and $h\coloneqq h_{n}$ is a sequence of positive real
numbers. For technical reasons, we rewrite (5) as

$\displaystyle\widehat{r}_{n}(\chi)=\displaystyle\dfrac{\displaystyle\hat{r}_{1,n}(\chi)}{\hat{r}_{2,n}(\chi)},$

where
$\displaystyle\displaystyle\widehat{r}_{\ell,n}(\chi)\coloneqq\dfrac{1}{n\mathbb{E}\Big{[}K\displaystyle\Big{(}\frac{d(\chi,\boldsymbol{\chi}_{i})}{h}\Big{)}\Big{]}}\displaystyle\sum\limits_{i=1}^{n}\dfrac{\displaystyle\alpha_{n}\delta_{i}K\big{(}\displaystyle\frac{d(\chi,\boldsymbol{\chi}_{i})}{h}\big{)}Z_{i}^{-\ell}}{L_{n}(Z_{i})\bar{G_{n}}(Z_{i})}$
for $\ell=1,\;2.$

##  4 Pointwise almost sure convergence

Let $\chi$ denote an element in $\mathcal{F}$ and $\mathcal{N}_{\chi}$ its
neighborhood, for $\rho>0$ we denote by $B(\chi,\rho)$ the closed ball with
radius $\rho$ and $\chi$ its center that is
$B(\chi,\rho)=\\{\chi^{\prime}:d(\chi,\chi^{\prime})\leq\rho\\}$. For easy
writing we shall denote by $C$, $C^{\prime}$ $c$,… any positive constant whose
value can change from time to time, in absence of confusion.
In what follows, we suppose that the next assumptions hold true:

######  (H1).

$\forall h>0$, $\phi_{\chi}(h)=\mathbb{P}\big{(}\boldsymbol{\chi}\in
B(\chi,h)\big{)}>0$ and $\lim\limits_{h\rightarrow 0}\phi_{\chi}(h)=0$.

######  (H2).

$\forall(\chi_{1},\chi_{2})\in\mathcal{N}_{\chi}\times\mathcal{N}_{\chi}$ and
for some $k_{\ell}>0$ we have $\mid
r_{\ell}(\chi_{1})-r_{\ell}(\chi_{2})\mid\leq
Cd^{k_{\ell}}(\chi_{1},\chi_{2})\text{, }\ \ \ell=1,2.$

######  (H3).

The kernel K is a measurable function with support $\left]0,1\right[$ and
satisfies:

$0<C\mathds{1}_{\left[0,1\right]}(.)\leq K(.)\leq
C^{\prime}\mathds{1}_{\left[0,1\right]}(.)<\infty.$

######  (H4).

The bandwidth $h$ is such that $\lim_{n\longrightarrow+\infty}h=0\text{ and
}\lim_{n\longrightarrow+\infty}\dfrac{\log(n)}{n\phi_{\chi}(h)}=0.$

######  (H5).

For any $m\geq 2$, the conditional moments of $Y^{-m}$ satisfy
$\mathbb{E}\left[Y^{-m}\mid\boldsymbol{\chi}=\chi\right]<C<\infty.$

######  Remark 2.

Assumption (H1) is a classic hypothesis in functional data analysis it means
that the probability to have $\boldsymbol{\chi}$, around the element at which
the regression operator is evaluated, is non null. Assumption (H2) is used to
study the bias term of our estimator while the assumptions (H3) \- (H5) are
technical conditions needed to prove our result.

The following theorem gives the pointwise almost sure convergence with rate of
the proposed estimator $\hat{r}_{n}(\chi)$ towards $r(\chi)$.

######  Theorem 1.

Under assumptions (H1) \- (H5) , we have:

$\mid\hat{r}_{n}(\chi)-r(\chi)\mid=\mathcal{O}\left(h^{k_{1}}\right)+\mathcal{O}\left(h^{k_{2}}\right)+\mathcal{O}\left(\sqrt{\frac{\log(n)}{n\phi_{\chi}(h)}}\right)\text{
a.s}.$

######  Remark 3.

Our stated rate of convergence is the same as the one obtained by Demongeot et
al. (2016) in the complete data case and by (Mechab et al., 2019) and
(Altendji et al., 2018) respectively for the case where the data is subject to
censoring or truncation.

##  5 Uniform almost sure convergence

In this section we establish the uniform almost sure convergence of our
estimator over some subset $\mathcal{S}_{\mathcal{F}}$ of $\mathcal{F}$. In
odrer to do so, it is necessary to make some additional topological
assumptions since in the functional case, the uniform consistency is not a
straightforward generalization of the pointwise convergence (see Ferraty et
al. (2010)).

######  (H6).

There exists a function $\phi(.)$, such that:
$\forall\chi\in\mathcal{S}_{\mathcal{F}}$, $\forall h>0$,
$0<C\phi(h)<\mathbb{P}({\chi}\in B(\chi,h))<C^{\prime}\phi(h)<\infty$.

######  (H7).

There exists $\beta>0$ such that
$\forall\;\chi_{1},\chi_{2}\in\mathcal{S}_{\mathcal{F}}^{\beta},\quad\mid
r_{\ell}(\chi_{1})-r_{\ell}(\chi_{2})\mid\leq
Cd^{k_{\ell}}(\chi_{1},\chi_{2}),$ where
$\mathcal{S}^{\beta}_{\mathcal{F}}=\left\\{\chi\in\mathcal{F},\exists\chi^{\prime}\in\mathcal{S}_{\mathcal{F}}\text{
such that }d\left(\chi,\chi^{\prime}\right)\leq\beta\right\\}$.

######  (H8).

The kernel K is bounded and Lipschitzian on $\left[0,1\right]$.

######  (H9).

The functions $\phi(.)$ and Kolmogorov’s $\epsilon$-entropy
$\psi_{\mathcal{S}_{\mathcal{F}}}(.)$ of $\mathcal{S}_{\mathcal{F}}$ are such
that:

  * •

There exist $\eta_{0}$ such that $\forall\eta<\eta_{0}$,
$\phi^{\prime}\left(\eta_{0}\right)<C$.

  * •

For $n$ large enough, we have $\frac{\left(\log
n\right)^{2}}{n\phi\left(h\right)}<\psi_{\mathcal{S}_{\mathcal{F}}}\left(\frac{\log
n}{n}\right)<\frac{n\phi\left(h\right)}{\log n}.$

  * •

Kolmogorov’s $\epsilon$-entropy $\psi_{\mathcal{S}_{\mathcal{F}}}(.)$ of
$\mathcal{S}_{\mathcal{F}}$ verifies that
$\sum_{n=1}^{+\infty}\exp\left(\left(1-\gamma\right)\psi_{\mathcal{S}_{\mathcal{F}}}\left(\frac{\log
n}{n}\right)\right)<\infty,$
for at least one $\gamma>1$.

######  (H10).

For any $m\geq 2$,
$\mathbb{E}\left[Y^{-m}\mid\boldsymbol{\chi}=\chi\right]\leq C<\infty$ for all
$\chi\in\mathcal{S}_{\mathcal{F}}$ and
$\inf\limits_{\chi\in\mathcal{S}_{\mathcal{F}}}r_{2}(\chi)\geq C^{\prime}>0$.

######  Remark 4.

Assumptions (H6), (H7), (H10) are the same considered by Demongeot et al.
(2016) and they represent a reformulation of assumptions (H1), (H2), (H5) in
the uniform consistency context, while Assumption (H9) controls the
Kolmogorov’s $\epsilon$-entropy of $\mathcal{S}_{\mathcal{F}}$ and ensures
that $\dfrac{\psi_{\mathcal{S}_{\mathcal{F}}}\left(\frac{\log
n}{n}\right)}{n\phi(h)}$ tends to $0$ when $n\longrightarrow\infty$ (Ferraty
et al., 2010).

######  Theorem 2.

Let Assumptions (H6) \- (H10) hold. Then we have:

$\sup_{\chi\in\mathcal{S}_{\mathcal{F}}}\mid\hat{r}_{n}(\chi)-r(\chi)\mid=\mathcal{O}\left(h^{k_{1}}\right)+\mathcal{O}\left(h^{k_{2}}\right)+\mathcal{O}\left(\sqrt{\frac{\psi_{\mathcal{S}_{\mathcal{F}}}\left(\frac{\log
n}{n}\right)}{n\phi\left(h\right)}}\right)\text{ a.s}.$

######  Remark 5.

The rate of uniform convergence obtained is equivalent to the uniform rate in
the complete data framework from Demongeot et al. (2016) and to the one stated
in the censoring case in the work of Fetitah et al. (2020). To our knowledge,
no research paper has yet dealt with the uniform convergence in a relative
error functional regression context when the variable of interest is left
truncated.

##  6 Simulation study

In order to investigate the performance of the estimator given by (5) a
simulation study must be conducted. First of all we need to generate the
necessary data. We choose the size $n$ and we consider the next curves
generated by:

$\boldsymbol{\chi}_{i}(t)=a\cos(2\pi t)+b\sin(4\pi t)+c(t-0.5)(t-0.25)\text{
with }t\in\left[0,1\right],$ (6)

where $a$, $b$, $c$ follow the uniform law $\mathcal{U}_{\left[0,3\right]}$.
All the curves are discretized on the same grid generated from 100 equidistant
measurements on $\left[0,1\right]$. We also choose the following regression
operator $r$ defined by
$r(\boldsymbol{\chi}_{i})=\int_{0}^{1}\boldsymbol{\chi}_{i}^{2}(t)dt+10.$ To
have $Y_{i}=r(\boldsymbol{\chi}_{i})+\epsilon_{i}$ we generate $\epsilon_{i}$
as a random noise sampled from a $\mathcal{N}(0,1)$. Next, we generate $S_{i}$
from an exponential distribution $\mathcal{E}\left(\mu\right)$ where $\mu$ is
adapted to obtain the desired censoring rate. We set $Z_{i}=\min(Y_{i},S_{i})$
and $\delta_{i}=\mathds{1}_{\left\\{Y_{i}\leq S_{i}\right\\}}$ and ultimately
sample $T_{i}$ from a $\mathcal{N}(\lambda,2)$, where $\lambda$ takes
different values to get the desired truncation rate. Finally the observation
$(\boldsymbol{\chi}_{i},Z_{i},T_{i},\delta_{i})$ is kept only if $Z_{i}\geq
T_{i}$, otherwise it is rejected and the procedure is resumed from the
beginning. This algorithm is repeated until $n$ observations are reached. At
the end we also obtain the value of $N$.

###  6.1 Finite sample performance

We have observed the behavior of our estimator
$\widehat{r}_{n}(\chi)=:\hat{r}_{RER}$ under different configurations : sample
size, truncation and censoring rates. We have at the same time compared it to
the classical (Nadaraya-Watson) regression estimator noted
$\hat{r}_{NW}(\chi)$. The semi-metric used in this study is the usual $L_{2}$
distance between two curves as a measure of proximity. For the kernel choice
we picked the asymmetrical quadratic kernel defined by
$K(u)=\frac{3}{2}(1-u^{2})\mathds{1}_{\left]0,1\right[}$, and the optimal
bandwidth $h$ is computed using the leave-one-out cross-validation method. We
computed the Global Mean Squared Error (GMSE) for B replicates of the
estimators and given by:

$GMSE=\frac{1}{B\times
m}\sum\limits_{u=1}^{B}\sum\limits_{k=1}^{m}\left(\hat{r}_{n}^{\left(u\right)}\left(\chi_{k}\right)-r\left(\chi_{k}\right)\right)^{2}.$

Here $B=200$, $m=20$ is the number of curves at which the estimator is
evaluated and $\widehat{r}_{n}^{\left(u\right)}\left({\chi}_{k}\right)$ is the
estimate obtained using sample number $u$. The obtained results are reported
in Table 1.

Table 1: GMSE of the relative error regression estimator and the classic regression Censoring Rate $\approx$ | Truncation Rate $\approx$ | Sample size | RER | Classic regression
---|---|---|---|---
20% | 20% | 100 | 0.2936 | 3.4374
20% | 20% | 300 | 0.1561 | 2.9494
20% | 20% | 500 | 0.0993 | 1.1001
10% | 20% | 100 | 0.2898 | 2.4836
20% | 20% | 100 | 0.2936 | 3.4374
40% | 20% | 100 | 0.3577 | 4.0566
20% | 10% | 100 | 0.2930 | 3.1910
20% | 20% | 100 | 0.2936 | 3.4374
20% | 40% | 100 | 0.3230 | 3.8820

We can see from Table 1 that in terms of the GMSE the relative error
regression estimator has a better quality of fit than the classical regression
one and a better stability since it is less affected by high percentages of
censoring and truncation which comforts the good behavior of our estimator.

###  6.2 Influence function

In this part we shall investigate the robustness of the relative error
estimator in the presence of an atypical observation in the data using the
influence function (IF). The influence function measures the effect induced by
outliers on an estimate. Note that an estimator can be written as a functional
of the distribution function $F$ of the data for some functional $\mathcal{T}$
(see Maronna et al. (2019) and the examples within) that is
$\mathcal{T}\left(F\right)$. For an extra data point $z_{0}\in\mathbb{R}$,
define $F_{\epsilon}=\left(1-\epsilon\right)F+\epsilon\Delta_{z_{0}}$, which
represents the distribution $F$ contaminated by a small fraction $\epsilon$ of
outliers with $\Delta_{z_{0}}$ being the Dirac mass at the added point
$z_{0}$. Hence, the influence function defined by:

$IF\left(z_{0},\mathcal{T}\left(F\right)\right)=\lim\limits_{\epsilon\longrightarrow
0^{+}}\dfrac{\mathcal{T}\left(F_{\epsilon}\right)-\mathcal{T}\left(F\right)}{\epsilon},$

represents the sensitivity of $\mathcal{T}\left(F\right)$ to $z_{0}$. From now
on, the functional $\mathcal{T}$ is taken to be $\widehat{r}_{RER}$ or
$\widehat{r}_{NW}\left(\chi\right)$. The influence function can be seen as an
asymptotic version of the standardized sensitivity curve (SSC) defined by

$SSC_{n}\left(z_{0},\hat{r}_{n}\left(\chi_{0}\right)\right)=\left(n+1\right)\left(\widehat{r}_{n+1}\left(\chi_{0};z_{1},\cdots,z_{n},z_{0}\right)-\widehat{r}_{n}\left(\chi_{0};z_{1},\cdots,z_{n}\right)\right)$

where $\chi_{0}$ is a fixed curve in $\mathcal{F}$ and
$\widehat{r}_{n+1}\left(\chi_{0};z_{1},\cdots,z_{n},z_{0}\right)$(resp.
$\widehat{r}_{n}\left(\chi_{0};z_{1},\cdots,z_{n}\right)$)is the estimator
built using the (n+1)-sample $z_{1},\cdots,z_{n},z_{0}$ (resp. the (n)-sample
$z_{1},\cdots,z_{n},z_{0}$) evaluated at $\chi_{0}$ . Indeed, if we add the
extra data point $z_{0}$ to the n-sample $z_{1},\cdots,z_{n}$ therefore the
fraction of contamination $\epsilon$ is equal to $\frac{1}{n+1}$. Then define
the empirical influence function as
$EIF_{\widehat{r}_{n}}(\chi)=SSC_{n}(z_{0},\widehat{r}_{n}\left(\chi\right))$.
Note that for a robust estimator (meaning that an outlier has a very limited
effect on the estimator) the empirical influence function must be close to 0.

####  6.2.1 Effect of an outlier

In order to study the impact of an outlier $z_{0}$ on the relative error
estimator and the classic regression one, we shall compare the
$EIF_{\widehat{r}_{n}}(\chi)$ obtained for both estimators after generating 20
curves as in (6) and a curve $\chi_{0}$. Then we take (arbitrarily)
$z_{0}=300$ with $\delta_{0}=1$ (the non-censoring indicator), and we plot the
obrained $EIF$ against the distance between each curve and the curve
$\chi_{0}$ for $n=300$, the censoring rate $CR=20\%,40\%$ and the truncation
rate $TR=20\%,40\%$.

(a)

(b)

(c)

Figure 1: Empirical influence function in the neighborhood of the curve
$\chi_{0}$ for a single outlier $z_{0}=300$ for n=300, and (a) $CR\approx
20\%TR\approx 20\%$, (b) $CR\approx 20\%TR\approx 40\%$, (c) $CR\approx
40\%TR\approx 20\%$

##  7 Proofs

### Proof of Theorem 1

The proof of Theorem 1 is based on the following decomposition and the results
from Lemma 1, Lemma 2 and Lemma 3 below.:

$\displaystyle\mid\widehat{r}_{n}(\chi)-r(\chi)\mid$
$\displaystyle\leq\dfrac{1}{\widehat{r}_{2,n}(\chi)}\Big{(}\mid\widehat{r}_{1,n}(\chi)-\tilde{r}_{1,n}(\chi)\mid+\mid\tilde{r}_{1,n}(\chi)-\mathbf{E}(\tilde{r}_{1,n}(\chi))\mid+\mid\mathbf{E}(\tilde{r}_{1,n}(\chi))-r_{1}(\chi)\mid\Big{)}$
$\displaystyle+\dfrac{r(\chi)}{\widehat{r}_{2,n}(\chi)}\Big{(}\mid
r_{2}(\chi)-\mathbf{E}(\tilde{r}_{2,n}(\chi))\mid+\mid\mathbf{E}(\tilde{r}_{2,n}(\chi))-\tilde{r}_{2,n}(\chi)\mid+\mid\tilde{r}_{2,n}(\chi)-\widehat{r}_{2,n}(\chi)\mid\Big{)}.$
(7)

Where

$\displaystyle\tilde{r}_{\ell,n}(\chi)=\dfrac{1}{n\mathbb{E}\left(K(\frac{d(\chi,\boldsymbol{\chi}_{1})}{h})\right)}\sum_{i=1}^{n}\frac{\alpha\delta_{i}K(\frac{d(\chi,\boldsymbol{\chi}_{i})}{h})Z_{i}^{-\ell}}{L(Z_{i})\bar{G}(Z_{i})}\text{
for }\ell=1,2.$

######  Lemma 1.

Under assumptions (H1) \- (H4) we have for $\ell=1,2$ :

$\mid\mathbf{E}(\tilde{r}_{\ell,n}(\chi))-r_{\ell}(\chi)\mid=\mathcal{O}\left(h^{k_{\ell}}\right).$

###### Proof.

We have that
$\mid\mathbf{E}(\tilde{r}_{\ell,n}(\chi))-r_{\ell}(\chi)\mid=\left|\dfrac{1}{\mathbb{E}(K(\frac{d(\chi,\boldsymbol{\chi}_{1})}{h}))}\mathbf{E}\left(\dfrac{\alpha\delta_{1}K(\frac{d(\chi,\boldsymbol{\chi}_{1})}{h})Z_{1}^{-\ell}}{L(Z_{1})\bar{G}(Z_{1})}\right)-r_{\ell}(\chi)\right|,$
and we can show that:

$\displaystyle\mathbf{E}\left(\dfrac{\alpha\delta_{1}K(\frac{d(\chi,\boldsymbol{\chi}_{1})}{h})Z_{1}^{-\ell}}{L(Z_{1})\bar{G}(Z_{1})}\right)=\mathbb{E}\left(K(\frac{d(\chi,\boldsymbol{\chi}_{1})}{h})r_{\ell}(\boldsymbol{\chi}_{1})\right).$
(8)

Hence and under Assumption (H2) we obtain:

$\displaystyle\mid\mathbf{E}(\tilde{r}_{\ell,n}(\chi))-r_{\ell}(\chi)\mid\leq\dfrac{1}{\mathbb{E}(K(\frac{d(\chi,\boldsymbol{\chi}_{1})}{h}))}\mathbb{E}\left[K(\frac{d(\chi,\boldsymbol{\chi}_{1})}{h})Cd^{k_{\ell}}(\chi,\boldsymbol{\chi}_{1})\right]\leq
Ch^{k_{\ell}}.$

∎

######  Lemma 2.

Under assumptions (H1), (H3) \- (H5) we have for $\ell=1,2$:

$\mid\tilde{r}_{\ell,n}(\chi)-\mathbf{E}(\tilde{r}_{\ell,n}(\chi))\mid=\mathcal{O}\left(\sqrt{\frac{\log(n)}{n\phi_{\chi}(h)}}\right)\text{
a.s}.$

###### Proof.

Set
$\tilde{r}_{\ell,n}(\chi)-\mathbf{E}(\tilde{r}_{\ell,n}(\chi))=\frac{1}{n}\sum\limits_{i=1}^{n}\frac{1}{\mathbb{E}\left(K(\frac{d(\chi,\boldsymbol{\chi}_{1})}{h})\right)}\left[\frac{\alpha\delta_{i}K(\frac{d(\chi,\boldsymbol{\chi}_{i})}{h})Z_{i}^{-\ell}}{L(Z_{i})\bar{G}(Z_{i})}-\mathbf{E}\left(\frac{\alpha\delta_{i}K(\frac{d(\chi,\boldsymbol{\chi}_{i})}{h})Z_{i}^{-\ell}}{L(Z_{i})\bar{G}(Z_{i})}\right)\right]=:\frac{1}{n}\sum\limits_{i=1}^{n}W_{i,\ell}$.
We use an exponential type inequality given in Corrolary A8.ii in Ferraty and
Vieu (2006), and for that we need to evaluate $\mathbf{E}(\mid
W_{i,\ell}^{m}\mid)$ for $m\geq 2$. Assumptions (H3), (H5) and the binomial
expansion formula allow to show that:

$\displaystyle\mathbf{E}(\mid W_{i,\ell}^{m}\mid)$
$\displaystyle\leq\max\limits_{k\in\\{0,1,2,...,m\\}}\left\\{\binom{m}{k}C^{m-k}\right\\}\sum\limits_{k=0}^{m}\mathbf{E}\left|\frac{\left(\frac{\alpha\delta_{i}K(\frac{d(\chi,\boldsymbol{\chi}_{i})}{h})Z_{i}^{-\ell}}{L(Z_{i})\bar{G}(Z_{i})}\right)}{\mathbb{E}\left(K(\frac{d(\chi,\boldsymbol{\chi}_{)}}{h})\right)}\right|^{k}$
$\displaystyle\leq
C\max\limits_{k\in\\{0,1,2,...,m\\}}\phi_{\chi}^{-k+1}(h)\leq
C\phi_{\chi}^{-m+1}(h).$

We apply the mentioned inequality with $a=\phi_{\chi}^{-\frac{1}{2}}(h)$ and
$u_{n}=\frac{a^{2}\log(n)}{n}$, and we obtain by setting
$\epsilon=\epsilon_{0}\sqrt{u_{n}}$:

$\displaystyle\mathbf{P}\left(\mid\tilde{r}_{\ell,n}(\chi)-\mathbf{E}(\tilde{r}_{\ell,n}(\chi))\mid>\epsilon_{0}\sqrt{u_{n}}\right)=\mathbf{P}\left(\frac{1}{n}\left|\sum_{i=1}^{n}W_{i,\ell}\right|>\epsilon_{0}\sqrt{u_{n}}\right)\leq
2\exp\left(-\frac{\epsilon_{0}^{2}\log(n)}{2(1+\epsilon_{0}\sqrt{u_{n}})}\right)\leq
2n^{-C\epsilon_{0}}.$

An appropriate choice of $\epsilon_{0}$ and Borel-Cantelli’s lemma allow us to
conclude the proof. ∎

######  Lemma 3.

Under assumptions (H1), (H3) \- (H5) we have for $\ell=1,2$:

$\mid\widehat{r}_{\ell,n}(\chi)-\tilde{r}_{\ell,n}(\chi)\mid=\mathcal{O}\left((n^{-1}\log\log(n))^{1/2})\right)\text{
a.s}.$

###### Proof.

When replacing $\alpha,\alpha_{n}$ by their expressions (3) and (4), we have

$\displaystyle\mid\widehat{r}_{\ell,n}(\chi)-\tilde{r}_{\ell,n}(\chi)\mid$
$\displaystyle\leq\sup_{a_{H}\leq t\leq
b}\Bigg{|}\frac{\bar{F}_{n}(t)}{C_{n}(t)}-\frac{\bar{F}(t)}{C(t)}\Bigg{|}\frac{1}{n\mathbb{E}\left(K(\frac{d(\chi,\boldsymbol{\chi}_{)}}{h})\right)}\sum_{i=1}^{n}K(\frac{d(\chi,\boldsymbol{\chi}_{i})}{h})\delta_{i}Z_{i}^{-\ell}$
$\displaystyle\leq\underbrace{\Bigg{|}\frac{\sup\limits_{a_{H}\leq t\leq
b}\mid F_{n}(t)-F(t)\mid}{\inf\limits_{a_{H}\leq t\leq b}\mid
C(t)\mid-\sup\limits_{a_{H}\leq t\leq b}\mid
C_{n}(t)-C(t)\mid}+\frac{\sup\limits_{a_{H}\leq t\leq b}\mid
C_{n}(t)-C(t)\mid}{\inf\limits_{a_{H}\leq t\leq b}\mid
C(t)\mid(\inf\limits_{a_{H}\leq t\leq b}\mid C(t)\mid-\sup\limits_{a_{H}\leq
t\leq b}\mid C_{n}(t)-C(t)\mid)}\Bigg{|}}_{\mathcal{I}_{1}}$
$\displaystyle\times\underbrace{\frac{1}{n\mathbb{E}\left(K(\frac{d(\chi,\boldsymbol{\chi}_{)}}{h})\right)}\sum_{i=1}^{n}K(\frac{d(\chi,\boldsymbol{\chi}_{i})}{h})\delta_{i}Z_{i}^{-\ell}}_{\mathcal{I}_{2}}=\mathcal{I}_{1}\times\mathcal{I}_{2}.$

On the one hand by Corollary 1 in Gijbels and Wang (1993) we have that
$\sup\limits_{a_{H}\leq t\leq b}\mid
F_{n}(t)-F(t)\mid=\mathcal{O}\left((n^{-1}\log\log(n))^{1/2})\right)$. Also
from Lemma 3.3 in Zhou and Yip (1999) we have$\sup\limits_{a_{H}\leq t\leq
b}\mid C_{n}(t)-C(t)\mid=\mathcal{O}\left((n^{-1}\log\log(n))^{1/2})\right)$
a.s, and using the fact that $C(t)>0$ so $\exists$ $\beta>0$ such that
$C(t)\geq\beta>0$ we obtain:

$\mathcal{I}_{1}=\mathcal{O}\left((n^{-1}\log\log(n))^{1/2})\right)\text{
a.s}.$ (9)

On the other hand by stationarity we have:

$\displaystyle\mathcal{I}_{2}$
$\displaystyle=\underbrace{\frac{1}{n\mathbb{E}\left(K(\frac{d(\chi,\boldsymbol{\chi}_{)}}{h})\right)}\sum_{i=1}^{n}\left[K(\frac{d(\chi,\boldsymbol{\chi}_{i})}{h})\delta_{i}Z_{i}^{-\ell}-\mathbf{E}\left(K(\frac{d(\chi,\boldsymbol{\chi}_{i})}{h})\delta_{i}Z_{i}^{-\ell}\right)\right]}_{\mathcal{I}_{2.1}}+\underbrace{\frac{1}{\mathbb{E}\left(K(\frac{d(\chi,\boldsymbol{\chi}_{)}}{h})\right)}\mathbf{E}\left(K(\frac{d(\chi,\boldsymbol{\chi}_{i})}{h})\delta_{i}Z_{i}^{-\ell}\right)}_{\mathcal{I}_{2.2}}.$

Proceeding as in the proof of Lemma 2 one can show that
$\mathcal{I}_{2.1}=\mathcal{O}\left(\sqrt{\frac{\log(n)}{n\phi_{\chi}(h)}}\right)$
a.s. Next,

$\displaystyle\mathcal{I}_{2.2}\leq\frac{\alpha^{-1}}{\mathbb{E}\left(K(\frac{d(\chi,\boldsymbol{\chi}_{)}}{h})\right)}\sup\limits_{a_{H}\leq
t\leq b}L(t)\bar{G}(t)\iint
K(\frac{d(\chi,u)}{h})y^{-\ell}f_{Y\mid\boldsymbol{\chi}}(y\mid
u)dyd\mathbb{P}^{\boldsymbol{\chi}}(u)\leq\alpha^{-1}\sup\limits_{a_{H}\leq
t\leq b}L(t)\bar{G}(t)C.$

Then $\mathcal{I}_{2.2}=\mathcal{O}\left(1\right)$, so
$\mathcal{I}_{2}=\mathcal{O}\left(1\right)\text{ a.s}$. We conclude
that$\mid\widehat{r}_{\ell,n}(\chi)-\tilde{r}_{\ell,n}(\chi)\mid=\mathcal{O}\left((n^{-1}\log\log(n))^{1/2})\right)\text{
a.s}.$ ∎

Theorem 1 is proved. $\square$

### Proof of Theorem 2

To prove Theorem 2 we need some additional lemmas.

######  Lemma 4.

Under assumptions (H6)\- (H8) we have for $\ell=1,2$ :

$\sup_{\chi\in\mathcal{S}_{\mathcal{F}}}\mid\mathbf{E}(\tilde{r}_{\ell,n}(\chi))-r_{\ell}(\chi)\mid=\mathcal{O}\left(h^{k_{\ell}}\right).$

###### Proof.

Obtained immediately from Lemma 1. ∎

######  Lemma 5.

Under assumptions (H6)\- (H8) we have for $\ell=1,2$:

$\sup_{\chi\in\mathcal{S}_{\mathcal{F}}}\mid\widehat{r}_{\ell,n}(\chi)-\tilde{r}_{\ell,n}(\chi)\mid=\mathcal{O}\left((n^{-1}\log\log(n))^{1/2})\right)\text{
a.s}.$

###### Proof.

By following the same steps as in the proof of Lemma 3 and then taking the
supremum over all $\chi$ in $\mathcal{S}_{\mathcal{F}}$ we get the desired
result. ∎

######  Lemma 6.

Under assumptions (H6), (H8) \- (H10) we have for $\ell=1,2$:

$\sup_{\chi\in\mathcal{S}_{\mathcal{F}}}\mid\tilde{r}_{\ell,n}(\chi)-\mathbf{E}(\tilde{r}_{\ell,n}(\chi))\mid=\mathcal{O}\left(\sqrt{\frac{\psi_{\mathcal{S}_{\mathcal{F}}}\left(\frac{\log
n}{n}\right)}{n\phi(h)}}\right)\text{ a.s}.$

###### Proof.

Let $\chi_{1}$, $\chi_{2}$, $\cdots$,
$\chi_{N_{\epsilon}(\mathcal{S}_{\mathcal{F}})}$ a finite set of elements in
$\mathcal{F}$ where $N_{\epsilon}(\mathcal{S}_{\mathcal{F}})$ represents the
minimal number of open balls in $\mathcal{F}$ of radius $\epsilon$ needed to
cover $\mathcal{S}_{\mathcal{F}}$, that is
$\mathcal{S}_{\mathcal{F}}\subset\bigcup_{k=1}^{N_{\epsilon}(\mathcal{S}_{\mathcal{F}})}B(\chi_{k},\epsilon)\text{
and we take }\epsilon=\frac{\log n}{n}.$ Set
$k(\chi)=arg\min\limits_{k\in\left\\{1,2,\cdots,N_{\epsilon}(\mathcal{S}_{\mathcal{F}})\right\\}}d(\chi,\boldsymbol{\chi}_{k})$
for all $\chi\in\mathcal{F}$ and
$K_{i}\left(\chi\right):=K\left(\frac{d(\chi,\boldsymbol{\chi}_{i})}{h}\right)$.
We have

$\displaystyle\sup_{\chi\in\mathcal{S}_{\mathcal{F}}}\mid\tilde{r}_{\ell,n}(\chi)-\mathbf{E}(\tilde{r}_{\ell,n}(\chi))\mid$
$\displaystyle\leq\underbrace{\sup_{\chi\in\mathcal{S}_{\mathcal{F}}}\mid\tilde{r}_{\ell,n}(\chi)-\tilde{r}_{\ell,n}(\chi_{k(\chi)})}_{\mathcal{L}_{1}}\mid+\underbrace{\sup_{\chi\in\mathcal{S}_{\mathcal{F}}}\mid\tilde{r}_{\ell,n}(\chi_{k(\chi)})-\mathbf{E}\left(\tilde{r}_{\ell,n}(\chi_{k(\chi)})\right)}_{\mathcal{L}_{2}}\mid$
$\displaystyle+\underbrace{\sup_{\chi\in\mathcal{S}_{\mathcal{F}}}\mid\mathbf{E}\left(\tilde{r}_{\ell,n}(\chi_{k(\chi)})\right)-\mathbf{E}(\tilde{r}_{\ell,n}(\chi))}_{\mathcal{L}_{3}}\mid.$

In order to achieve the proof we must evaluate the convergence rate of each
term $\mathcal{L}_{j}$ for $j=1,2,3$. So

$\mathcal{L}_{1}\leq\sup_{\chi\in\mathcal{S}_{\mathcal{F}}}\frac{1}{n}\sum_{i=1}^{n}\left|\dfrac{\alpha\delta_{i}K_{i}(\chi)Z_{i}^{-\ell}}{\mathbb{E}\left(K_{i}(\chi)\right)L(Z_{i})\bar{G}(Z_{i})}-\dfrac{\alpha\delta_{i}K_{i}(\chi_{k(\chi)})Z_{i}^{-\ell}}{\mathbb{E}\left(K_{i}(\chi_{k(\chi)})\right)L(Z_{i})\bar{G}(Z_{i})}\right|$

By combining assumptions (H6) and (H8) we get
$\forall\chi\in\mathcal{S}_{\mathcal{F}}\text{,
}0<C\phi\left(h\right)<\mathbb{E}\left(K_{i}(\chi)\right)<C^{\prime}\phi\left(h\right),$
then:

$\displaystyle\mathcal{L}_{1}$
$\displaystyle\leq\sup_{\chi\in\mathcal{S}_{\mathcal{F}}}\frac{c}{n\phi(h)}\sum_{i=1}^{n}\left|K_{i}(\chi)-K_{i}(\chi_{k(\chi)})\right|\times\frac{\alpha\delta_{i}Z_{i}^{-\ell}}{L(Z_{i})\bar{G}(Z_{i})}\leq\sup_{\chi\in\mathcal{S}_{\mathcal{F}}}\frac{c}{nh\phi(h)}\sum_{i=1}^{n}\left|d\left(\chi,\boldsymbol{\chi}_{i}\right)-d\left(\chi_{i},\chi_{k(\chi)}\right)\right|\times\frac{\alpha\delta_{i}Z_{i}^{-\ell}}{L(Z_{i})\bar{G}(Z_{i})}$
$\displaystyle\leq\sup_{\chi\in\mathcal{S}_{\mathcal{F}}}\frac{c}{n}\sum_{i=1}^{n}\dfrac{\epsilon\alpha\delta_{i}Z_{i}^{-\ell}}{h\phi(h)L(Z_{i})\bar{G}(Z_{i})}=:\sup_{\chi\in\mathcal{S}_{\mathcal{F}}}\frac{c}{n}\sum_{i=1}^{n}F_{i}.$

Next, we evaluate $\mathbf{E}\left(F_{i}^{m}\right)$, $m\geq 2$, for applying
an exponential type inequality for unbounded random variables (see Corollary
A8 in Ferraty and Vieu (2006)). We have under assumption (H10)

$\displaystyle\mathbf{E}\left(F_{i}^{m}\right)$
$\displaystyle=\frac{\epsilon^{m}}{h^{m}\phi(h)^{m}}\mathbf{E}\left(\frac{\alpha^{m}\delta_{i}Z_{i}^{-m\ell}}{L(Z_{i})^{m}\bar{G}(Z_{i})^{m}}\right)=\frac{\epsilon^{m}}{h^{m}\phi(h)^{m}}\mathbb{E}\left(\frac{\alpha^{m-1}Y^{-m\ell}}{L(Y)^{m-1}\bar{G}(Y)^{m-1}}\right)\leq\frac{C\epsilon^{m}}{h^{m}\phi(h)^{m}}.$

We apply the fore-mentioned exponential inequality with
$a^{2}=\dfrac{\epsilon}{h\phi(h)}=\frac{\log n}{nh\phi(h)}$ and Assumption
(H9) to get
$\mathcal{L}_{1}=\mathcal{O}\left((\frac{\psi_{\mathcal{S}_{\mathcal{F}}}\left(\frac{\log
n}{n}\right)}{n\phi(h)})^{\frac{1}{2}}\right)$ .
For the term $\mathcal{L}_{2}$, and for all $\eta>0$ we have:

$\displaystyle\mathbf{P}\left(\mathcal{L}_{2}>\eta(\frac{\psi_{\mathcal{S}_{\mathcal{F}}}(\epsilon)}{n\phi(h)})^{\frac{1}{2}}\right)\leq
N_{\epsilon}(\mathcal{S}_{\mathcal{F}})\max\limits_{k(\chi)\in\left\\{1,2,\cdots,N_{\epsilon}\right\\}}\mathbf{P}\left(\mid\tilde{r}_{\ell,n}(\chi_{k(\chi)})-\mathbf{E}(\tilde{r}_{\ell,n}(\chi_{k(\chi)}))\mid>\eta\sqrt{\frac{\psi_{\mathcal{S}_{\mathcal{F}}}\left(\epsilon\right)}{n\phi(h)}}\right).$

By using the same arguments as in Lemma 2 (taking $a^{2}=\phi(h)^{-m+1}$) we
can show that:

$\mathbf{P}\left(\left|\tilde{r}_{\ell,n}(\chi_{k(\chi)})-\mathbf{E}\left(\tilde{r}_{\ell,n}(\chi_{k(\chi)})\right)\right|>\eta\sqrt{\frac{\psi_{\mathcal{S}_{\mathcal{F}}}\left(\epsilon\right)}{n\phi(h)}}\right)\leq
2\exp\left(-C\eta^{2}\psi_{\mathcal{S}_{\mathcal{F}}}\left(\epsilon\right)\right).$

We have $\psi_{\mathcal{S}_{\mathcal{F}}}\left(\epsilon\right)=\log
N_{\epsilon}(\mathcal{S}_{\mathcal{F}})$ as defined in Ferraty et al. (2010)
then:

$\displaystyle\mathbf{P}\left(\left|\tilde{r}_{\ell,n}(\chi_{k(\chi)})-\mathbf{E}\left(\tilde{r}_{\ell,n}(\chi_{k(\chi)})\right)\right|>\eta\sqrt{\frac{\psi_{\mathcal{S}_{\mathcal{F}}}\left(\epsilon\right)}{n\phi(h)}}\right)$
$\displaystyle\leq 2\exp\left(-C\eta^{2}\log
N_{\epsilon}(\mathcal{S}_{\mathcal{F}})\right)\leq
2N_{\epsilon}(\mathcal{S}_{\mathcal{F}})^{-C\eta^{2}},$

it follows that:

$\displaystyle
N_{\epsilon}(\mathcal{S}_{\mathcal{F}})\max\limits_{k(\chi)\in\\{1,2,\cdots,N_{\epsilon}\\}}$
$\displaystyle\mathbf{P}\left(\mid\tilde{r}_{\ell,n}(\chi_{k(\chi)})-\mathbf{E}\left(\tilde{r}_{\ell,n}(\chi_{k(\chi)})\right)\mid>\eta\sqrt{\frac{\psi_{\mathcal{S}_{\mathcal{F}}}\left(\epsilon\right)}{n\phi(h)}}\right)\leq
2N_{\epsilon}(\mathcal{S}_{\mathcal{F}})^{1-C\eta^{2}}.$

It is sufficient to take $\eta=\sqrt{\frac{\gamma}{C}}$ which under assumption
(H9) and by Borel-Cantelli’s lemma gives:

$\displaystyle\sum_{n=1}^{\infty}\mathbf{P}\left(\mathcal{L}_{2}>\eta\sqrt{\frac{\psi_{\mathcal{S}_{\mathcal{F}}}\left(\epsilon\right)}{n\phi(h)}}\right)$
$\displaystyle\leq
2\sum_{n=1}^{\infty}N_{\epsilon}(\mathcal{S}_{\mathcal{F}})^{1-\gamma}<\infty\Rightarrow\mathcal{L}_{2}=\mathcal{O}\left(\sqrt{\frac{\psi_{\mathcal{S}_{\mathcal{F}}}\left(\epsilon\right)}{n\phi(h)}}\right)$

Regarding the term $\mathcal{L}_{3}$ it is obvious by Jensen’s inequality
that:

$\mathcal{L}_{3}\leq\mathbf{E}\left(\sup_{\chi\in\mathcal{S}_{\mathcal{F}}}\mid\tilde{r}_{\ell,n}(\chi)-\tilde{r}_{\ell,n}(\chi_{k(\chi)})\mid\right),$

then the proof for the term $\mathcal{L}_{1}$ remains valid for showing that
$\mathcal{L}_{3}=\mathcal{O}\left(\sqrt{\dfrac{\psi_{\mathcal{S}_{\mathcal{F}}}\left(\epsilon\right)}{n\phi(h)}}\right)$
since $\epsilon=\frac{\log n}{n}$, this concludes the proof of Lemma 6. ∎

The proof of Theorem 2 is now complete. $\square$

## References

  * Ramsay and Silverman [2005] J. Ramsay and B.W. Silverman.  _Functional Data Analysis_.  Springer Series in Statistics. Springer, 2005.  ISBN 9780387400808.
  * Ferraty and Vieu [2006] Frédéric Ferraty and Philippe Vieu.  _Nonparametric functional data analysis: theory and practice_ , volume 76.  Springer, 2006.
  * Hsing and Eubank [2015] Tailen Hsing and Randall Eubank.  _Theoretical foundations of functional data analysis, with an introduction to linear operators_.  Wiley, 2015.
  * Klein and Moeschberger [2003] John P Klein and Melvin L Moeschberger.  _Survival analysis: techniques for censored and truncated data_ , volume 1230.  Springer, 2003.
  * Park and Stefanski [1998] Heungsun Park and LA Stefanski.  Relative-error prediction.  _Statistics & probability letters_, 40(3):227–236, 1998.
  * Jones et al. [2008] MC Jones, Heungsun Park, Key-Il Shin, SK Vines, and Seok-Oh Jeong.  Relative error prediction via kernel regression smoothers.  _Journal of Statistical Planning and Inference_ , 138(10):2887–2898, 2008.
  * Demongeot et al. [2016] Jacques Demongeot, Ali Hamie, Ali Laksaci, and Mustapha Rachdi.  Relative-error prediction in nonparametric functional statistics: Theory and practice.  _Journal of Multivariate Analysis_ , 146:261–268, 2016\.
  * Altendji et al. [2018] Belkais Altendji, Jacques Demongeot, Ali Laksaci, and Mustapha Rachdi.  Functional data analysis: estimation of the relative error in functional regression under random left-truncation model.  _Journal of Nonparametric Statistics_ , 30(2):472–490, 2018.
  * Mechab et al. [2019] Boubaker Mechab, Nesrine Hamidi, and Samir Benaissa.  Nonparametric estimation of the relative error in functional regression and censored data.  _Chilean Journal of Statistics (ChJS)_ , 10(2), 2019.
  * Fetitah et al. [2020] Omar Fetitah, Ibrahim M Almanjahie, Mohammed Kadi Attouch, and Ali Righi.  Strong convergence of the functional nonparametric relative error regression estimator under right censoring.  _Mathematica Slovaca_ , 70(6):1469–1490, 2020\.
  * Tsai et al. [1987] Wei-Yann Tsai, Nicholas P Jewell, and Mei-Cheng Wang.  A note on the product-limit estimator under right censoring and left truncation.  _Biometrika_ , 74(4):883–886, 1987.
  * He and Yang [1998] Shuyuan He and Grace L Yang.  Estimation of the truncation probability in the random truncation model.  _Annals of Statistics_ , pages 1011–1027, 1998.
  * Lynden-Bell [1971] D Lynden-Bell.  A method of allowing for known observational selection in small samples applied to 3cr quasars.  _Monthly Notices of the Royal Astronomical Society_ , 155(1):95–118, 1971.
  * Gijbels and Wang [1993] Irène Gijbels and JANE-LING Wang.  Strong representations of the survival function estimator for truncated and censored data with applications.  _Journal of Multivariate analysis_ , 47(2):210–229, 1993.
  * Ferraty et al. [2010] Frédéric Ferraty, Ali Laksaci, Amel Tadj, and Philippe Vieu.  Rate of uniform consistency for nonparametric estimates with functional variables.  _Journal of Statistical planning and inference_ , 140(2):335–352, 2010.
  * Maronna et al. [2019] Ricardo A. Maronna, R. Douglas Martin, Victor J. Yohai, and Matías Salibián-Barrera.  _Robust statistics: theory and methods (with R)_.  John Wiley & Sons, 2019.
  * Zhou and Yip [1999] Yong Zhou and Paul SF Yip.  Nonparametric estimation of quantile density function for truncated and censored data.  _Journal of Nonparametric Statistics_ , 12(1):17–39, 1999.