# M-Estimation in Censored Regression Model using Instrumental Variables under
Endogeneity

Swati Shukla
IIT Kanpur
Department of Mathematics and Statistics
Kanpur 208016, India
Email<EMAIL_ADDRESS>
Subhra Sankar Dhar
IIT Kanpur
Department of Mathematics and Statistics
Kanpur 208106, India
Email<EMAIL_ADDRESS>
Shalabh
IIT Kanpur
Department of Mathematics and Statistics
Kanpur 208106, India
Email<EMAIL_ADDRESS>

Abstract

We propose and study M-estimation to estimate the parameters in the censored
regression model in the presence of endogeneity, i.e., the Tobit model. In the
course of this study, we follow two-stage procedures: the first stage consists
of applying control function procedures to address the issue of endogeneity
using instrumental variables, and the second stage applies the M-estimation
technique to estimate the unknown parameters involved in the model. The large
sample properties of the proposed estimators are derived and analyzed. The
finite sample properties of the estimators are studied through Monte Carlo
simulation and a real data application related to women’s labor force
participation.

Key Words: M-estimation, Tobit model, endogeneity, control variables, two-
stage estimation, exogenous variable, instrumental variables.

##  1 Introduction

The regression models in which the dependent variable is censored or limited
to a specific range are referred to as censored regression models. Censoring
in a regression model may arise due to the process that generates the data, or
it may be incorporated into the model by the experimenter. For instance,
expenditure on durable goods may be zero for some households based on their
economic status; on the other side, an experimenter may introduce censoring in
the data by top-coding the income exceeding a certain level. The first type of
censoring in the preceding example is due to the underlying process generating
the data, that is, the individual’s decision not to spend on durable goods,
while the second type of censoring is independent of the individual’s
decision. Tobit models are a particular class of censored regression models
where the dependent variable is censored above or below the threshold zero.
They are also referred to as standard Tobit models or Tobit type-I models.
Tobit models were introduced in the pioneering work by [53] in which Tobin
analyzed the household expenditure on durable goods using a regression model
with a dependent variable (household expenditure), which was censored below
the threshold zero. In this context, the methodology proposed in the Section 2
can be implemented for the non-zero threshold as well.

In this article, we consider Tobit models with endogenous explanatory
variables. An explanatory variable is said to be endogenous if it is
correlated with the error term in the model. Censoring and endogeneity occur
in tandem in many real-life situations. One such real-life example comes from
the standard labor supply model, which assumes that each individual has some
desired hours of work, say $y^{*}$, which is subject to the constraints of how
many hours an individual wants and does not want to work. Thus, the variable
$y^{*}$ is assumed to be the interior solution to a utility maximization
problem subject to constraints on the number of working hours and leisure.
Further, it is also assumed that the firm sets the lower limit of working
hours, say $c$. Therefore, the dependent variable $y$ is the actual number of
hours worked, which is denoted as $y=y^{*}$ if $y^{*}>c$, and $y=c$ if
$y^{*}<c$. In this model, one is interested in examining the effects of other
independent variables, such as non-labor income, education, age, wage, etc.,
on the dependent variable $y$. Therefore, one can fit a regression model to
see the effect of these independent variables on $y$. However, it is usually
understood that wages and the number of hours are correlated across
individuals, which causes the problem of endogeneity in the regression model
due to the reverse causality between wages and the number of hours. Now, since
the dependent variable $y$ is censored below 0 and there is one endogenous
explanatory variable, i.e., wage, in the model, it leads to the problem of
estimating the censored regression model with an endogenous explanatory
variable.

There are various solutions available in the literature to deal with
endogeneity in the Tobit models (see, e.g., [25]). Among them, we consider the
instrumental variable (IV) estimation method using the control function
approach to remove endogeneity from the model. The instrumental variable
method is a standard procedure to remove endogeneity from the model (see,
e.g., [8]). This method introduces instrumental variables or instruments,
which are correlated,in limit, with the endogenous explanatory variables and
uncorrelated,in limit, with the errors in the model. Using the instrumental
variables, we consider a two-stage estimation procedure to have some
appropriate estimators of the unknown parameters involved in the model. In the
first stage, we regress endogenous variables on instrumental variables to find
residuals, which are referred to as control variables. In the second stage, we
include these control variables as additional explanatory variables to remove
the endogeneity from the model under certain assumptions and then estimate the
model with M-estimation. An advantage of using the M-estimation procedures
with the control function approach to address the issue of endogeneity is that
it allows one to include instrumental variables straightforwardly due to its
flexibility in specifying the objective function.

###  1.1 Literature review

Early literature on the estimation of the unknown parameters involved in the
Tobit model uses the maximum likelihood estimation technique, assuming the
normality of the errors in the model. Later on, [2] established the
consistency and asymptotic normality of the maximum likelihood estimator under
the assumption that errors are normally distributed, and [24] extended the
least squares estimation to a two-step estimation procedure for the Tobit
model. The assumption of normality of error is crucial for the estimator to be
consistent, unlike in the standard linear regression model, where the
parameter estimates are consistent for a wide class of non-normal error
distributions. However, if the assumption of normality is violated, then the
parameter estimates are inconsistent for the Tobit models; see [22] and [6],
where the inconsistency of maximum likelihood estimates was demonstrated for
several standard non-normal error distributions. Another important aspect was
highlighted by [31], [39], [5], where it is shown that heteroskedasticity of
the error terms can also lead to inconsistent parameter estimates even when
the functional form of the error density is appropriately specified. The ideas
of [40], [10], [26], and [20] are examples of a class of estimation methods
that combine likelihood-based approaches with the nonparametric estimation of
the distribution function of the error, but this methodology is sensitive to
the assumption of identically distributed error terms.

The semi-parametric and non-parametric approaches to estimation have also been
introduced to overcome the issue of the specification of error distributions.
[45] extends the consistent least absolute deviation (CLAD) estimator to the
Tobit model and establishes consistency and asymptotic normality of the
estimators without assuming any functional form of the error. However,
consistency and asymptotic normality of estimators require stronger
assumptions on the behavior of the regression function than those imposed on a
model with normally distributed error. In a similar spirit, [46] proposed the
symmetrically trimmed least square estimator and established the consistency
and asymptotic normality of the estimator, assuming the errors are coming from
a symmetric distribution. The modified version of the Huber estimator ([29])
was proposed by [38] for the Tobit models, and it was named as winsorized mean
estimators (WME).

It is well known that the use of M-estimation techniques helps in overcoming
the robustness issues with the least squares approach ([30]; [23]) for
uncensored data. [47], [57], [37] constructed M-estimators for the model using
appropriately censored data. The methods described in [47] and [37] are
extensions of the estimators proposed by [10] to general M-estimators. All
these estimators are consistent and asymptotically normal under some
conditions. Although the existence of a solution to the estimating equation
has been demonstrated, this approach’s main drawback is that it neither
provides a unique solution nor guarantees the consistency of all solutions.
[33] developed an M-estimator for the semi-parametric linear model with right-
censored data. A class of asymptotically normal, consistent estimators is
provided by [33]’s method, which was developed for random censoring. After a
few years, [32] developed generalized M-estimators for type-I Tobit models in
high-dimensional settings and established consistency and asymptotic
normality. Besides, there is an extensive amount of study in the literature
addressing the problem of endogeneity in the Tobit models. Early literature
relies on complete models with parametric specifications, as studied in [25],
[3], [52]. These procedures are primarily based on control function approaches
and marginal or conditional maximum likelihood procedures. The semi-parametric
approach using the control function is studied by [15], [8], [13].

The instrumental variable estimation provides a consistent estimator of the
regression coefficients in the linear model when the random error is
correlated with the explanatory variables. The area of instrumental variables
has remained an area of interest for many researchers from various
perspectives, including parametric, semi-parametric, nonparametric, Bayesian,
and non-Bayesian inference. An excellent overview of instrumental variable
models is available in [9]. The robust estimation of instrumental variable
models has been discussed in the literature by several researchers. A robust
instrumental variable estimator is proposed by Cohen et al. (2013). The
robustness-related issues of estimators are studied in [54]. The aspect of the
robustness of bootstrap inference methods for instrumental variable regression
models by considering the test statistics for parameter hypotheses based on
the instrumental variables and the generalized method of trimmed moments
estimators are discussed in [11]. The Bayesian nonparametric instrumental
variables approach to correcting the endogeneity bias in regression models
when the functional form of the covariate effects is unknown is discussed in
[56]. The aspects of the application of two-stage instrumental variable
estimation have been extensively used in real data. For example, [17]
presented a goodness of fit statistics in the instrumental variables model and
applied it to COVID-19 data; see also [4], [12],[1],[35],[21],[16],[43], [51]
etc.

###  1.2 Main Contribution

In this study, we propose a general class of estimators, which are obtained by
applying the M-estimation procedure to a generic loss function $\rho$. One of
the advantages of the M-estimation procedure is that it provides flexibility
in choosing the objective function, which allows one to customize the
estimation procedure according to the complexity of the data and the questions
of the research at hand. This adaptability of objective functions can be
helpful when some of the classical assumptions of the regression models are
violated or there are outliers in the data. One such example can be quantile-
based methodology (see, e.g., [18]), which is a particular case of
M-estimation-based methodology. We establish consistency and asymptotic
normality of the derived estimator under the assumption that $\rho$ is
Lipschitz continuous, having the second-order sub-gradient to be bounded. The
theoretical results in this paper are established under the same assumption on
the behavior of the regression function, as in [45] and [38]. Thus, we bring
many standard semi-parametric estimators, such as CLAD and WME, under the same
umbrella through a generic loss function and also introduce another estimator
by choosing the log cos hyperbolic (log-cosh) loss function. The log-cosh-loss
function possesses an important property, as described in [50] that the log-
cosh-loss function belongs to the class of robust estimators that focus on
solutions near the median rather than the mean. Another point of view is that
robust estimators are more tolerant of outliers in the data set, which is
possibly one of the main reasons to prefer the log-cosh loss function over
others. As demonstrated in the simulation study, the proposed estimator based
on the log-cosh loss function performs equally well as the well-known
estimators CLAD and WME.

The M-estimators in lower dimensions have not yet been established for
censored regression models when data are censored by a fixed constant.
Furthermore, models with fixed censoring are more prone to distributional
misspecification because knowledge of the underlying data-generating mechanism
is seldom available. To get around this, we establish a unified class of
estimators with no distributional assumptions and establish consistency and
asymptotic normality.

###  1.3 Overcoming Mathematical Challenges

One of the main challenges is that the objective function may not be
continuously differentiable; therefore, usual approaches based on Taylor’s
expansion of objective functions are not always applicable. To overcome this
problem, we adopt the same technique discussed in [45] with appropriate
modifications for a generic Lipschitz continuous loss function. [45] modified
the techniques in [28] for the censored outcomes, which approximate the sub-
gradient of the objective function by its continuously differentiable
expectation. Then, it relates it to the first-order condition, which minimizes
the objective function by setting the sub-gradient of the objective function
equal to zero asymptotically. We extend regularity conditions to those
described in [45] for a class of loss functions to establish consistency and
asymptotic normality.

###  1.4 Organisation of the Paper

The model description is presented in Section 2. Section 2.2 describes the
estimation using the control function approach and the proposed M-estimation
procedure. Section 3 presents the main theoretical results and states the main
theorems. Section 4 provides a simulation study to validate the theoretical
results. Section 5 showcases real-life applications of the proposed
methodology on a real data set. Finally, Section 6 provides some concluding
remarks on this work. All the proofs of the theorems are supplied in Section
7.

The R codes of simulated and real data study are available at
https://github.com/swati-1602/CRM.git.

##  2 Model Description and Estimation Procedure

###  2.1 The Model

For given paired data $(y_{i},\boldsymbol{x}^{*}_{i}),\hskip
2.84544pti=1,\ldots,n,$ where $y_{i}$ is the $i^{th}$ observations on the
response variable, defined as

$y_{i}=\begin{cases}y^{*}_{i}&\text{if}\>y^{*}_{i}>\textit{c}\\\
\textit{c}&\text{if}\>y^{*}_{i}\leq\textit{c},\\\ \end{cases}$

where $c$ is a fixed and non-random censoring threshold. Here $y_{i}$ is fully
observed but the latent variable $y_{i}^{*}$ is not fully observed and
modelled as

$y^{*}=\boldsymbol{x}^{T}_{i}\boldsymbol{\alpha}_{0}+w_{i}\gamma_{0}+\epsilon_{i},$
(2.1)

and $\boldsymbol{x}^{*}_{i}=(\boldsymbol{x}_{i},w_{i})$ is the $i^{th}$
observations on a vector of $(p+1)$ dimensional explanatory variables with two
components. One component $\boldsymbol{x}_{i}=(x_{i1},\ldots,x_{ip})^{T}$ is a
vector of $p$ dimensional exogenous explanatory variables that are
uncorrelated with the error term, and $w_{i}$ is the endogenous explanatory
variable that is correlated with the error term in (2.1). The unobservable
random error $\epsilon_{i}$ is assumed to be independent of
$\boldsymbol{x}_{i}$. Here, we have considered only one endogenous variable in
the model for the sake of simplicity in understanding. However, the results
and estimation procedure in this paper are also applicable and can be extended
to a model with more than one endogenous variable. The vector of unknown
parameters associated with the exogenous variable is denoted by
$\boldsymbol{\alpha}_{0}=(\alpha_{10},\ldots,\alpha_{p0})^{T}$ and the
parameter associated with the endogenous explanatory variable is represented
by $\gamma_{0}$.

Now, consider the censored linear regression model with left censoring at a
fixed and non-random threshold $c$, also called the standard Tobit model,
defined as

$y_{i}=\max\leavevmode\nobreak\ (c,y^{*})=\max\leavevmode\nobreak\
\\{c,\boldsymbol{x}^{T}_{i}\boldsymbol{\alpha}_{0}+w_{i}\gamma_{0}+\epsilon_{i}\\},\hskip
11.38092pti=1,\ldots,n.$ (2.2)

This model is often referred to as the semi-parametric censored linear
regression model since the error distribution is not specified. Without loss
of generality, we restrict our study to a censored linear regression model
where the censoring threshold is 0, i.e.,

$y_{i}=\text{ max
}\\{0,\boldsymbol{x}^{T}_{i}\boldsymbol{\alpha}_{0}+w_{i}\gamma_{0}+\epsilon_{i}\\},\hskip
2.84544pti=1,\ldots,n.$ (2.3)

###  2.2 Estimation of Parameters in Presence of Endogeneity

The common approach to estimating the parameters of a regression model in the
presence of one or more endogenous regressors is based on the instrumental
variable (IV) approach. The instrumental variable approach is based on the
principle that, in the first stage, the endogeneity from the model is removed
with the help of instruments or instrumental variables. Then, one estimates
the parameters of the model in the next stage using some suitable estimation
technique. The inherent assumption in the instrumental variable approach is
the validity of the instruments. An instrument is said to be valid if it is
exogenous and is correlated with the endogenous variables. In the present
context, we use instrumental variable estimation based on the control function
approach (see, e.g., [52], [41], and [48]) to remove endogeneity from the
model.

Let $z_{i}$ denote the $i^{th}$ value of the instrumental variable
corresponding to the endogenous variable $w_{i}$ in the model (2.3), for
$i=1,\ldots,n$. Then the control function approach proceeds in the following
steps:

  1. 1.

We regress the observations on endogenous variable $w_{i}$ on the instrumental
variable $z_{i}$, i.e.,

$w_{i}=z_{i}\delta_{1}+\eta_{i},\quad i=1,\ldots,n,$ (2.4)

where $\delta_{1}$ is the unknown parameter associated with $z$ and $\eta_{i}$
is the random error in the $i^{th}$ observation such that
$cov(z_{i},\eta_{i})=0$.

  2. 2.

The source of endogeneity is controlled using $E(\epsilon_{i}|\eta_{i})$ by
decomposing $\epsilon_{i}$ as follows:

$\epsilon_{i}=E(\epsilon_{i}|\eta_{i})+\xi_{i},\quad i=1,\ldots,n,$ (2.5)

where $\xi_{i}$ is a random variable denoting the deviation of $\epsilon_{i}$
from $E(\epsilon_{i}|\eta_{i})$. Note that $E(\epsilon_{i}|\eta_{i})\neq 0$
since $w_{i}$ is endogenous, implying that $cov(\epsilon_{i},\eta_{i})\neq 0$.

  3. 3.

The control function is the functional form of $E(\epsilon_{i}|\eta_{i})$,
which is often chosen to be linear, and we choose
$E(\epsilon_{i}|\eta_{i})=\rho_{10}\eta_{i}$, reducing (2.5) as follows:

$\epsilon_{i}=\rho_{10}\eta_{i}+\xi_{i},\quad i=1,\ldots,n,$ (2.6)

where $\rho_{10}$ is an unknown parameter in (2.6).

  4. 4.

The idea is to incorporate the control function in the original model (see,
(2.3)) to account for endogeneity and use $\xi_{i}$ as the error term instead
of $\epsilon_{i}$. Note that (2.5) is constructed in such a way that
$cov(\xi_{i},\eta_{i})=0$, since all the variation in $\epsilon_{i}$ due to
$\eta_{i}$ is subsumed into $E(\epsilon_{i}|\eta_{i})$. Thus, the following
transformed model will be free from endogeneity and can be estimated using any
suitable technique.

$y_{i}=\text{ max
}\\{0,\boldsymbol{x}^{T}_{i}\boldsymbol{\alpha}_{0}+w_{i}\gamma_{0}+\rho_{10}\eta_{i}+\xi_{i}\\},\quad
i=1,\ldots,n.$ (2.7)

  5. 5.

The model in (2.7) contains $\eta_{i}$, which is unobserved. Therefore, we
replace $\eta_{i}$ by the residuals of the model in (2.4). Let
$e_{i},i=1,\ldots,n$ denote the residuals of the model in (2.4), then the
model in (2.7) becomes

$y_{i}=\text{ max
}\\{0,\boldsymbol{x}^{T}_{i}\boldsymbol{\alpha}_{0}+w_{i}\gamma_{0}+e_{i}\rho_{10}+\xi_{i}\\}=\text{
max }\\{0,\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}+\xi_{i}\\},$ (2.8)

where $\boldsymbol{X}_{i}=[\boldsymbol{x}_{i},w_{i},e_{i}]^{T}$ and
$\boldsymbol{\beta}_{0}=[\boldsymbol{\alpha}_{0},\gamma_{0},\rho_{10}]^{T}$ is
the $m\times 1$ vector of unknown parameters with $m=p+2$.

####  2.2.1 Implementation of M-estimation Technique

The parameters of the model described in (2.8) can be estimated using any
suitable estimation procedure, such as censored least absolute deviation
(CLAD) ([45]) and winsorized mean estimator (WME) ([38]), to name a few. In
this article, we take a general perspective using a general loss function,
which encapsulates some existing estimation procedures, such as CLAD and WME
as its particular cases. We use the well-known estimation procedure, the
M-estimation procedure, to estimate the parameters in (2.8). The definition of
such an estimator, denoted as $\hat{\boldsymbol{\beta}}_{n}$, is as follows:

$\hat{\boldsymbol{\beta}}_{n}=\underset{\boldsymbol{\beta}\in\mathcal{B}}{\arg\min}\leavevmode\nobreak\
Q_{n}(\boldsymbol{\beta}),$ (2.9)

where

$Q_{n}(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=1}^{n}\rho(y_{i},\boldsymbol{\beta})=\frac{1}{n}\sum_{i=1}^{n}\rho(y_{i}-\text{
max }\\{0,\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\\}).$ (2.10)

Here $\boldsymbol{\beta}=(\boldsymbol{\alpha},\gamma,\rho_{1})^{T},$
$\mathcal{B}$ is the parameter space of $\boldsymbol{\beta},$ and $\rho$ is a
real valued loss function. If $\rho$ in (2.10) is differentiable in
$\boldsymbol{\beta}$ with a continuous derivative
$\frac{\partial\rho(y_{i},\boldsymbol{\beta})}{\partial\boldsymbol{\beta}}$
then, $\hat{\boldsymbol{\beta}}_{n}$ is a root of the equation

$\frac{1}{n}\sum_{i=1}^{n}\frac{\partial(\rho(y_{i},\boldsymbol{\beta}))}{\partial\boldsymbol{\beta}}=0,\hskip
14.22636pt\boldsymbol{\beta}\in\mathcal{B.}$ (2.11)

The class of M-estimators covers the following well-known estimators for the
Tobit model for a given choice of $\rho$:

  1. 1.

The censored least absolute deviation estimator (CLAD) corresponds to

$\rho(y_{i},\boldsymbol{\beta})=|(y_{i}-\text{ max
}\\{0,\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\\}|.$ (2.12)

  2. 2.

The winsorized mean estimator (WME) corresponds to

$\rho(y_{i},\boldsymbol{\beta})=\begin{cases}\frac{1}{2}\left(y_{i}-\text{ max
}\\{0,\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\\}\right)^{2},&\text{if
}|y_{i}-\text{ max }\\{0,\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\\}|\leq w\\\
w\left(|y_{i}-\text{ max
}\\{0,\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\\}|-\frac{w}{2}\right),&\text{otherwise.}\end{cases}$
(2.13)

  3. 3.

The censored log cosh estimator (CLCE) corresponds to

$\rho(y_{i},\boldsymbol{\beta})=log(cosh(y_{i}-\text{max}(0,\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}))).$
(2.14)

##  3 Asymptotic Properties of $\hat{\boldsymbol{\beta}}_{n}$

There is a vast literature on the asymptotic properties, such as consistency
and asymptotic normality, of M-estimators for the usual regression model.
However, to the best of our knowledge, we are unaware of any literature on the
asymptotic properties of M-estimators for regression models when the response
variable is censored at a non-random and fixed threshold in the presence of
endogeneity. In this article, we establish the consistency and asymptotic
normality of $\hat{\boldsymbol{\beta}}_{n}$ described in (2.9).

One of the major issues in applying the M-estimation technique to the model
described in (2.3) is that the objective function need not be convex and
differentiable. The non-convexity of the objective function
$Q_{n}(\boldsymbol{\beta})$ (see, (2.10)) poses the problem of non-
identifiability of the true parameter $\boldsymbol{\beta}_{0}$. For example,
consider the case when all the observations are censored, that is,
$y_{i}=0\;\text{for all}\;i=1,\ldots,n$. Then we have,

$Q_{n}(\boldsymbol{\beta})=\frac{1}{n}\sum\limits_{i=1}^{n}\rho(-max\\{0,\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\\}).$

In such a case, $Q_{n}(\boldsymbol{\beta})=0$ for any
$\boldsymbol{\beta}\in\mathcal{B}$ for which
$\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\leq 0$ for all $i=1,\ldots,n$, under
the assumption that $\rho(0)=0$ (see $\boldsymbol{A.4}$ in Section 3.1).
Therefore, assuming $\rho$ to be a non-negative loss function, any value
$\boldsymbol{\beta}\in\mathcal{B}$ for which
$\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\leq 0$ for all $i=1,\ldots,n$, is a
minimizer of $Q_{n}(\boldsymbol{\beta})$. Thus, $\hat{\boldsymbol{\beta}}_{n}$
is not unique, and the true parameter $\boldsymbol{\beta}_{0}$ is not
identifiable in this case.

To avoid the above-mentioned situation, one can assume that there are “enough”
numbers of uncensored observations for a large sample size of $n$. More
precisely, let $n_{0}$ denote the number of uncensored observations in a
sample of size $n$ such that $n_{0}/n\xrightarrow{}\lambda\in(0,1]$ as
$n\xrightarrow{}\infty$. In this case, for any
$\boldsymbol{\beta}\in\mathcal{B}$ for which
$\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\leq 0$ for all $i=1,\ldots,n$, we
have,

$Q_{n}(\boldsymbol{\beta})=\frac{1}{n}\sum\limits_{i=1}^{n_{0}}\rho(y_{i}).$

Note that for this case as well, one encounters the problem related to the
non-identifiability of $\boldsymbol{\beta}_{0}$ since
$Q_{n}(\boldsymbol{\beta})$ does not depend on $\boldsymbol{\beta}$. As a
consequence, any $\boldsymbol{\beta}\in\mathcal{B}$ for which
$\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\leq 0$ for all $i=1,\ldots,n$, is a
minimizer of $Q_{n}(\boldsymbol{\beta})$. This again leads to the problem of
the non-identifiability of $\boldsymbol{\beta}_{0}$ and the non-uniqueness of
$\hat{\boldsymbol{\beta}}_{n}.$

In light of both the above-described situations, it is necessary to restrict
that $\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0$ for “enough” observations in
a large sample. Therefore, minimization of (2.9) can be viewed as executing
M-estimation only for those sample observations for which
$\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\geq 0$. Thus, if
$\hat{\boldsymbol{\beta}}_{n}$ is to be consistent, the true regression
function $\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}$ must be non-negative
for a significant proportion of the sample. As pointed out in [45], to
overcome the issue of non-identifiability of $\boldsymbol{\beta}_{0}$, it is
necessary to restrict the behavior of the true regression function
$\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}$ such that for large enough
samples $\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0$, and the regressors
are not collinear. This requirement is articulated in the assumption
$\boldsymbol{A.3}$.

###  3.1 Strong Consistency of $\hat{\boldsymbol{\beta}}_{n}$

We establish the consistency of $\hat{\boldsymbol{\beta}}_{n}$ under the
following assumptions on the parameter space, errors, regressors, the true
regression function, and the loss function. Recall the model (2.8); all of the
assumptions discussed below are based on it.

Assumptions:

  1. $\boldsymbol{{A.1}}$

$\mathcal{B}$ defined in (2.9) and (2.10) is a compact space, and
$\boldsymbol{\beta}_{0}\in\mathcal{B}^{o}.$ Here $\boldsymbol{\beta}_{0}$ is
the same as defined in (2.8), and $\mathcal{B}^{o}$ is the interior of
$\mathcal{B}.$

  2. $\boldsymbol{A.2}$

The random variables $\\{\xi_{i}\\}_{1\leq i\leq n}$ defined in (2.5) are
identically and independently distributed with the common probability density
function $f$. Moreover, for all $1\leq i\leq n,$ $\xi_{i}$ is independent of
$\boldsymbol{X}_{i},$ where $\boldsymbol{X}_{i}$ is the same as defined in
(2.8).

  3. $\boldsymbol{A.3}$

For all $1\leq i\leq n,$ the random vectors $\\{\boldsymbol{X}_{i}\\}$ defined
in (2.8) are independent with $E\|\boldsymbol{X}_{i}\|^{4}<K_{0},$ where
$K_{0}$ is some positive constant. Moreover, for some positive
$\epsilon_{0}>0,$

$\underset{n\to\infty}{\lim}E\left[\frac{1}{n}\sum_{i=1}^{n}\mathds{1}(\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}_{0}\geq\epsilon_{0})\boldsymbol{X}_{i}\boldsymbol{X}^{T}_{i}\right]=Q,$
(3.1)

where $Q$ is some positive definite matrix.

  4. $\boldsymbol{A.4}$

$\rho(x)\geq 0$ for all $x\in\mathbb{R},$ $\rho(0)=0,$ $\rho(x)=\rho(-x)$ for
all $x\in\mathbb{R},$ and moreover,

$|\rho(x)-\rho(y)|\leq k.|x-y|,$

where $k$ is some positive constant. Here $\rho$ is the same as defined in
(2.10).

  5. $\boldsymbol{A.5}$

For all ${1\leq i\leq n},$

$E[\rho(y_{i}-max(0,\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}))]<E[\rho(y_{i}-max(0,\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}))].$
(3.2)

######  Remark 3.1.

As required by Lemma 7.2, the assumption $\boldsymbol{A.1}$ guarantees the
existence and measurability of $\hat{\boldsymbol{\beta}}_{n}$ as well as the
uniformity of the almost sure convergence of the minimand over $\mathcal{B}$.
Assumption $\boldsymbol{A.2}$ is self-explanatory. Assumption
$\boldsymbol{A.3}$ ensures that
$\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}_{0}>0$ for ”enough” sample points
and the regressors $\boldsymbol{X}_{i}$’s are not collinear for these
observations. Assumption $\boldsymbol{A.4}$ implies that the loss function is
smooth enough. The condition described in (3.2) is borrowed from [19], which
is necessary to ensure the identifiable uniqueness of the true parameter
$\boldsymbol{\beta}_{0}$. It is indeed true that such a condition holds for a
wide range of $\rho$.

######  Theorem 3.1.

For the model described in (2.8), under the assumptions $\boldsymbol{A.1}$,
$\boldsymbol{A.2}$, $\boldsymbol{A.3}$, $\boldsymbol{A.4}$, and
$\boldsymbol{A.5}$, we have
$||\hat{\boldsymbol{\beta}}_{n}-\boldsymbol{\beta}_{0}||_{\mathbb{R}^{m}}\rightarrow
0$ almost surely as $n\rightarrow\infty$. Here $m=p+2$,
$\hat{\boldsymbol{\beta}}_{n}$ is the same as defined in (2.9), and
$\boldsymbol{\beta}_{0}$ is the same as defined in (2.8).

######  Proof.

See Appendix $B.1.$ ∎

###  3.2 Asymptotic Normality of $\hat{\boldsymbol{\beta}}_{n}$

In this section, we establish the asymptotic distribution of
$\hat{\boldsymbol{\beta}}_{n}$ described in (2.9). Since the sample minimand
$Q_{n}(\boldsymbol{\beta})$ is non-differentiable, the common method for
proving asymptotic normality, which relies on an expansion of the objective
function using Taylor’s expansion, is not immediately applicable to the
situation at hand. The purpose of this paper is to establish general
asymptotic normality results equivalent to those found in [14], [27], [28],
and the following works (see, e.g., [28], [7], [34], [49], [36], [44], and
[45]). As the loss function involved in (2.10) is not differentiable at
$\boldsymbol{\beta}$ when $\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}=0$, we
must rule out sequences of $\boldsymbol{X}_{i}$ values that are orthogonal to
$\boldsymbol{\beta}$ and have a positive frequency to establish the asymptotic
normality of $\hat{\boldsymbol{\beta}}_{n}$. The following assumption is
sufficient for this purpose:

  * $\boldsymbol{A.6}$

For some constants, $K_{1}>0$ and $\zeta_{0}>0$, and for all $1\leq i\leq n,$
the random vector $\boldsymbol{X}_{i}$ defined in (2.8), under the condition
$\|\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\|<\zeta_{0}$, satisfy the
following inequality for all $0\leq z^{*}<\zeta_{0},$ and $r=0,1,2:$

$E\left[\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}|\leq\|\boldsymbol{X}_{i}\|.z^{*}\right)\|\boldsymbol{X}_{i}\|^{r}\right]\leq
K_{1}.z^{*}.$

  * $\boldsymbol{A.7}$

The loss function $\rho$ defined in (2.10) is twice differentiable with
respect to $\boldsymbol{\beta}$, provided
$\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0$ for all $1\leq i\leq n.$
Moreover, for some constants $M_{0}>0$ and $M_{1}>0$, $|\rho^{\prime}(x)|\leq
M_{1}$ and $|\rho^{\operatorname*{\prime\prime}}(x)|\leq M_{0}$ for all
$x\in\mathbb{R},$ and in addition,
$\sum\limits_{i=1}^{\infty}\rho^{\prime}(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0})<\infty$,
almost surely for all $1\leq i\leq n.$ Here $\rho^{{}^{\prime}}$ and
$\rho^{{}^{\prime\prime}}$ denote the first and second derivatives of $\rho$,
respectively.

######  Remark 3.2.

The assumption $\boldsymbol{A.6}$ implies the differentiability of the
minimand defined in (2.10) at $\boldsymbol{\beta}.$ Further, when for all
$1\leq i\leq n$, $\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0,$ with
probability 1, then $\boldsymbol{A.6}$ holds, under some smoothness condition
on the distribution of $\boldsymbol{X}_{i}$. Moreover, the assumption
$\boldsymbol{A.7}$ ensures some degree of smoothness of the loss function
$\rho$ along with some tail property of the function $\rho.$

######  Theorem 3.2.

For the model described in (2.8), under the assumptions $\boldsymbol{A.1}$,
$\boldsymbol{A.6}$ and $\boldsymbol{A.7}$, we have

$D_{n}^{-1/2}H_{n}.\sqrt{n}\left(\hat{\boldsymbol{\beta}}_{n}-\boldsymbol{\beta}_{0}\right)\overset{d}{\to}Z,$

where Z is a $\mathbb{R}^{m}$ valued random vector associated with a standard
m-dimensional multivariate normal distribution,

$H_{n}=\frac{1}{n}\sum_{i=1}^{n}E\left[\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\boldsymbol{X}_{i}\boldsymbol{X}^{T}_{i}\rho^{\operatorname*{\prime\prime}}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\right],$

and $D^{-1/2}_{n}$ is such that

$(D_{n}^{-1/2}D_{n}^{-1/2})^{-1}=\frac{1}{n}\sum_{i=1}^{n}E\left[\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\boldsymbol{X}_{i}\boldsymbol{X}^{T}_{i}\\{\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\\}^{2}\right].$

######  Proof.

See Appendix $B.2.$ ∎

###  3.3 Consistent Estimation of $D_{n}$ and $H_{n}$

Consistent estimators of $D_{n}$ and $H_{n}$ must be derived to use the
asymptotic normality of $\hat{\boldsymbol{\beta}}_{n}$ to construct large
sample hypothesis tests for the parameter vector $\boldsymbol{\beta}_{0}$.
There are “natural” estimators of the matrices $H_{n}$ and $D_{n}$ for the
model (2.8); these are

$\begin{split}\hat{H_{n}}&=\frac{1}{n}\sum_{i=1}^{n}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\hat{\boldsymbol{\beta}}_{n}>0\right)\boldsymbol{X}_{i}\boldsymbol{X}^{T}_{i}\rho^{\operatorname*{\prime\prime}}\left(y_{i}-\boldsymbol{X}^{T}_{i}\hat{\boldsymbol{\beta}}_{n}\right)\\\
&=\frac{1}{n}\sum_{i=1}^{n}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\hat{\boldsymbol{\beta}}_{n}>0\right)\boldsymbol{X}_{i}\boldsymbol{X}^{T}_{i}\rho^{\operatorname*{\prime\prime}}\left(\hat{\xi}_{i}\right)\end{split}$
(3.3)

and

$\hat{D_{n}}=\frac{1}{n}\sum_{i=1}^{n}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\hat{\boldsymbol{\beta}}_{n}>0\right)\boldsymbol{X}_{i}\boldsymbol{X}^{T}_{i}\\{\rho^{\prime}(\hat{\xi_{i}})\\}^{2},$
(3.4)

where,
$\hat{\xi_{i}}=y_{i}-\boldsymbol{X}^{T}_{i}\hat{\boldsymbol{\beta}}_{n}.$

######  Theorem 3.3.

Under the conditions of Theorem 3.2, the estimators $\hat{H}_{n}$ and
$\hat{D}_{n},$ as stated in (3.3) and (3.4), are strongly consistent, i.e.,
$\hat{H}_{n}-H_{n}=o(1)$ and $\hat{D}_{n}-D_{n}=o(1)$ almost surely.

######  Proof.

See Appendix $B.3.$ ∎

In view of the assertions in Theorems 3.2 and 3.3, we have the following
result:

######  Proposition 3.1.

Under the assumptions in Theorems 3.2 and 3.3, we have

$\hat{D}_{n}^{-1/2}\hat{H}_{n}.\sqrt{n}\left(\hat{\boldsymbol{\beta}}_{n}-\boldsymbol{\beta}_{0}\right)\overset{d}{\to}Z,$

where Z is a $\mathbb{R}^{m}$ valued random vector associated with a standard
m-dimensional multivariate normal distribution, $\hat{D}_{n}$ and
$\hat{H}_{n}$ are the same as defined in (3.4) and (3.3), respectively.

######  Remark 3.3.

Observe that one may conduct testing of hypothesis problems related to
$\boldsymbol{\beta}_{0}$ or construct a confidential interval of
$\boldsymbol{\beta}_{0}$ using the result described in proposition in 3.1 as
$\hat{D}_{n}$, $\hat{H}_{n}$, and $\hat{\boldsymbol{\beta}}_{n}$ are
computable for a given data in principle regardless of the computational
complexity.

##  4 Simulation Study

In this section, Monte Carlo experiments are performed to study the finite
sample performance of the estimators in terms of their empirical bias and mean
square error (MSE). This section compares the effectiveness of the proposed
estimators of the unknown parameters involved in the model (2.8) with an
endogenous regressor. In this study, we compare the performance of the
estimators defined in Section 2.2.1, namely, CLAD (see, (2.12)), WME (see,
(2.13)), and CLCE (see, (2.14)). The following model is considered to generate
the data: For $i=1,\ldots,n$,

$\displaystyle y^{*}_{i}$
$\displaystyle=\beta_{0}+\beta_{1}x_{1i}+\beta_{2}x_{2i}+e_{1i},$ (4.1)
$\displaystyle x_{2i}$ $\displaystyle=\alpha z_{i}+e_{2i},$ (4.2)
$\displaystyle e_{1i}$ $\displaystyle=\rho_{1}e_{2i}+\eta_{i},$ (4.3)
$\displaystyle y_{i}$
$\displaystyle=\text{max}(0,\beta_{0}+\beta_{1}x_{1i}+\beta_{2}x_{2i}+\rho_{1}e_{2i}+\eta_{i}),$
(4.4)

In the model (4.1), the endogenous regressor $x_{2i}$ is regressed on the
instrumental variable $z_{i}$ (see, (4.2)), and the exogenous regressor
$(x_{1i})$ is generated from the standard normal distribution $N(0,1)$. The
random error $e_{2i}$ corresponding to the model (4.2) are generated from
$N(0,1)$, while $z_{i}$ is generated from the uniform distribution over
$(0,1)$, i.e., $U(0,1)$. Consequently, the observations on $x_{2i}$ are
generated for given values of $\alpha=1$. Let
$\beta_{T}=(\beta_{0},\beta_{1},\beta_{2},\rho_{1})=(1,2,3,0.5)$, where
$\beta_{T}$ represents the true parameter values. As mentioned in Section 2.2,
by applying the control function approach, the random error $e_{1i}$ in the
model (4.1) and the random error corresponding to the model (4.2) are
correlated (see (4.3)). Afterwards, models (4.1) and (4.2) lead to the model
(4.4). The homoscedastic random errors $\eta_{i}$ corresponding to the model
(4.4) are generated from the following distributions: standard normal
distribution $N(0,1)$, Laplace distribution $DE(0,1)$, and $t$-distribution
with $n$ degrees of freedom $t_{n}$. The heteroscedastic errors $\xi_{i}$ are
generated from $N(0,\sigma_{X_{i}}^{2})$, where
$\sigma_{X_{i}}^{2}=(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta})^{2},\boldsymbol{X_{i}}=(x_{1i},x_{2i})^{T}$
and $\boldsymbol{\beta}=(\beta_{1},\beta_{2})^{T}$. The impact of non-normally
distributed random errors on the empirical bias and empirical MSE can be
judged by comparing the values from $DE(0,1)$ and $U(0,1)$ with $N(0,1)$. The
performance of the proposed estimators studied by considering various choices
of the loss function $\rho$, while the distributions of $\eta_{i}$ are kept
relatively simple so that the true values of the censored regression
coefficients are tractable. Since $\rho$ is a Lipschitz-Continuous function,
we consider a least absolute error loss function (see (2.12)), a Huber error
loss function with a tuning parameter $w=1.35$ (see (2.13)), and a log-cosh
loss function (see (2.14)) as particular choices of $\rho$. Next, the
estimated values of $\beta_{0},$ $\beta_{1},$ $\beta_{2}$ and $\rho_{1}$ are
computed from the Nelder-Meade simplex algorithm using the “optim” function in
R software for these loss functions. We replicate this process $r=5000$ times
for the sample size $n\in\\{50,100,150,1000\\}$. Then, the empirical bias and
empirical mean squared error (MSE) of the parameter estimates of $\beta_{0}$,
$\beta_{1}$, $\beta_{2}$, and $\beta_{3}=\rho_{1}$ are computed as follows:

$Bias(\hat{\beta}_{j})=\frac{1}{r}\sum_{k=1}^{r}\left(\hat{\beta}_{jk}-\beta_{j}\right),j=0,1,2,3$

and

$MSE(\hat{\beta}_{j})=\frac{1}{r}\sum_{k=1}^{r}\left(\hat{\beta}_{jk}-\beta_{j}\right)^{2},j=0,1,2,3$

where $\hat{\beta}_{ij}$ is the estimate of the parameter $\beta_{j}$ in the
$k^{th},k=1,2,\ldots r$ replicate. For these designs, the overall censoring
probabilities vary between $18\%$ to $48\%$, and the different censoring
percentages do not change any qualitative conclusions.

Tables 1 and 2 report the empirical bias and empirical mean square error of
the proposed estimators along with the probability of censored observations
for varying sample sizes $n\in\\{50,100,500,1000\\}$. Due to keeping the
length of the paper shorter, the empirical bias and empirical MSE are reported
only for $n\in\\{50,100,500,1000\\}$. It is clear from the results that
estimators corresponding to different $\rho$ give different bias and MSE,
which depend on the distribution generating random errors. In all cases, MSEs
are decreasing as the sample size increases. Table 1 demonstrates that the
bias and MSE of the CLAD, WME, and CLCE estimators are considerably higher for
the small sample size in the case of all error distributions. For $n=50$, the
MSE of the WME and CLCE estimators is higher than that of CLAD estimates in
the case of homoscedastic error. In the case of heteroscedastic error,
increasing the variance of the error term does have an adverse impact on CLCE
and WME estimators, while CLAD is a better choice in this scenario. For
$n=100$, it performs well because these estimators are less sensitive to
outliers, and as the sample size increases further, CLCE and CLAD estimators
perform well as compared to WME estimators. For the sample size $n=500,1000$,
the bias and MSE of the proposed estimators are approaching zero in the case
of all error distributions, which is congruous with the fact that the proposed
estimators are asymptotically unbiased and consistent.

Table 1: Empirical bias and MSE of $\hat{\beta_{0}},$ $\hat{\beta_{1}},$
$\hat{\beta_{2}}$ and $\hat{\rho_{1}}$ with $n=50,100.$

Estimators | $\boldsymbol{\beta}$ | Error-distribution
---|---|---
|  | $N(0,1)$ | $DE(0,1)$ | $t_{n}$ |  $N(0,\sigma_{X_{i}}^{2})$, $\sigma_{X_{i}}^{2}=(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta})^{2}$
|  | Bias | MSE | C.P | Bias | MSE | C.P | Bias | MSE | C.P | Bias | MSE | C.P
|  | $n=50$
| $\beta_{0}$ | 0.1036 | 0.5342 |  | 0.1153 | 0.6155 |  | 0.1165 | 0.4199 |  | -0.4188 | 1.9455 |
CLAD | $\beta_{1}$ | 0.0009 | 0.0566 |  22$\%$ | 0.0074 | 0.0615 |  38$\%$ | 0.0104 | 0.0538 |  26$\%$ | 0.2362 | 1.1679 |  48$\%$
| $\beta_{2}$ | -0.2207 | 1.0975 |  | -0.2653 | 1.2673 |  | -0.2791 | 1.6153 |  | 0.0458 | 7.1524 |
| $\rho_{1}$ | 0.2363 | 1.1495 |  | 0.2957 | 1.3139 |  | 0.2975 | 1.6387 |  | -0.2375 | 6.8123 |
| $\beta_{0}$ | 0.0300 | 2.5824 |  | -0.0089 | 5.8653 |  | 0.0874 | 5.3688 |  | –2.3589 | 77.024 |
WME | $\beta_{1}$ | 0.0213 | 0.0568 |  18$\%$ | 0.0295 | 0.0948 |  24$\%$ | 0.0143 | 0.0610 |  34$\%$ | 0.5879 | 6.0178 |  38$\%$
| $\beta_{2}$ | -0.1094 | 13.863 |  | -0.0547 | 20.351 |  | -0.2107 | 20.891 |  | 1.3187 | 184.66 |
| $\rho_{1}$ | 0.1411 | 13.999 |  | 0.0990 | 20.365 |  | 0.2411 | 20.933 |  | -0.0087 | 176.35 |
| $\beta_{0}$ | 0.1500 | 2.5797 |  | 0.1031 | 2.8856 |  | 0.1059 | 2.5419 |  | -0.4741 | 53.334 |
CLCE | $\beta_{1}$ | -0.0161 | 0.0454 |  30$\%$ | -0.0190 | 0.0589 |  38$\%$ | -0.0078 | 0.0434 |  28$\%$ | 0.1414 | 1.7814 |  46$\%$
| $\beta_{2}$ | -0.2498 | 9.1222 |  | -0.1596 | 11.327 |  | -0.1725 | 10.190 |  | 0.0011 | 123.23 |
| $\rho_{1}$ | 0.2209 | 9.0952 |  | 0.1343 | 11.252 |  | 0.1582 | 10.210 |  | 0.5446 | 123.18 |
|  | $n=100$
| $\beta_{0}$ | 0.0725 | 0.3409 |  | 0.0817 | 0.3916 |  | 0.0721 | 0.1807 |  | -0.3198 | 0.8408 |
CLAD | $\beta_{1}$ | -0.0021 | 0.0325 |  25$\%$ | 0.0038 | 0.0302 |  30$\%$ | 0.0082 | 0.0209 |  30$\%$ | 0.1835 | 0.4920 |  28$\%$
| $\beta_{2}$ | -0.1597 | 0.5754 |  | -0.1813 | 0.9068 |  | -0.1728 | 0.4896 |  | 0.1310 | 3.1660 |
| $\rho_{1}$ | 0.1688 | 0.5861 |  | 0.1900 | 0.9347 |  | 0.1948 | 0.4963 |  | -0.4250 | 2.7691 |
| $\beta_{0}$ | 0.0054 | 1.2936 |  | 0.0115 | 0.5094 |  | 0.0325 | 1.4159 |  | -0.9830 | 10.263 |
WME | $\beta_{1}$ | 0.0158 | 0.0116 |  29$\%$ | 0.0147 | 0.0116 |  22$\%$ | 0.0069 | 0.0238 |  28$\%$ | 0.2790 | 1.4183 |  26$\%$
| $\beta_{2}$ | -0.0419 | 4.8663 |  | -0.0824 | 1.2155 |  | -0.0968 | 4.4673 |  | 0.4015 | 19.363 |
| $\rho_{1}$ | 0.0593 | 4.8618 |  | -0.1116 | 1.2207 |  | 0.1077 | 4.4372 |  | 0.2418 | 19.371 |
| $\beta_{0}$ | 0.0882 | 0.9985 |  | 0.1158 | 0.4890 |  | 0.0945 | 0.3628 |  | -0.0594 | 8.4772 |
CLCE | $\beta_{1}$ | -0.0095 | 0.0139 |  28$\%$ | -0.0219 | 0.0136 |  38$\%$ | -0.0164 | 0.0206 |  25$\%$ | 0.0162 | 0.5710 |  39$\%$
| $\beta_{2}$ | -0.0964 | 3.7648 |  | -0.1329 | 1.4031 |  | -0.1186 | 1.0547 |  | -0.1027 | 11.408 |
| $\rho_{1}$ | 0.0682 | 3.7569 |  | 0.0969 | 1.4093 |  | 0.0853 | 1.0793 |  | 0.3050 | 10.933 |

Table 2: Empirical bias and MSE of $\hat{\beta_{0}},$ $\hat{\beta_{1}},$
$\hat{\beta_{2}}$ and $\hat{\rho_{1}}$ with $n=500,1000.$

Estimators | $\boldsymbol{\beta}$ | Error-distribution
---|---|---
|  | $N(0,1)$ | $DE(0,1)$ | $t_{n}$ |  $N(0,\sigma_{X_{i}}^{2})$, $\sigma_{X_{i}}^{2}=(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta})^{2}$
|  | Bias | MSE | C.P | Bias | MSE | C.P | Bias | MSE | C.P | Bias | MSE | C.P
|  | $n=500$
| $\beta_{0}$ | 0.0079 | 0.1496 |  | 0.0079 | 0.1487 |  | 0.0056 | 0.0202 |  | -0.1089 | 0.0749 |
CLAD | $\beta_{1}$ | 0.0018 | 0.0037 |  31$\%$ | -0.0009 | 0.0036 |  26.8$\%$ | 0.0016 | 0.0017 |  27$\%$ | 0.0372 | 0.0567 |  36.8$\%$
| $\beta_{2}$ | -0.0235 | 0.0553 |  | -0.0142 | 0.0545 |  | -0.0136 | 0.0305 |  | -0.0055 | 0.3804 |
| $\rho_{1}$ | 0.0258 | 0.0569 |  | 0.0139 | 0.0538 |  | 0.0133 | 0.0306 |  | -0.4912 | 0.5534 |
| $\beta_{0}$ | 0.0034 | 0.1617 |  | 0.0052 | 0.1604 |  | -0.0044 | 0.0635 |  | -0.1408 | 0.5299 |
WME | $\beta_{1}$ | 0.0010 | 0.0022 |  24.4$\%$ | 0.0009 | 0.0024 |  32$\%$ | 0.0025 | 0.0037 |  27.8$\%$ | 0.0513 | 0.1647 |  39.2$\%$
| $\beta_{2}$ | -0.0159 | 0.0305 |  | -0.0144 | 0.0311 |  | -0.0034 | 0.0479 |  | 0.0786 | 1.1313 |
| $\rho_{1}$ | 0.0193 | 0.0311 |  | 0.0166 | 0.0313 |  | 0.0081 | 0.0474 |  | 0.0312 | 0.9958 |
| $\beta_{0}$ | 0.0681 | 0.1782 |  | 0.0805 | 0.1780 |  | 0.0645 | 0.0722 |  | 0.1281 | 0.1720 |
CLCE | $\beta_{1}$ | -0.0205 | 0.0028 |  27$\%$ | -0.0262 | 0.0026 |  31.6$\%$ | -0.0220 | 0.0041 |  25.8$\%$ | -0.1095 | 0.0833 |  34.8$\%$
| $\beta_{2}$ | -0.0490 | 0.0377 |  | -0.0551 | 0.0380 |  | -0.0443 | 0.0531 |  | -0.0534 | 0.5364 |
| $\rho_{1}$ | 0.0137 | 0.0382 |  | 0.0100 | 0.0385 |  | 0.0073 | 0.0535 |  | -0.0004 | 0.4691 |
|  | $n=1000$
| $\beta_{0}$ | 0.0040 | 0.0130 |  | 0.0011 | 0.1366 |  | 0.0056 | 0.0126 |  | -0.1043 | 0.0393 |
CLAD | $\beta_{1}$ | -0.0015 | 0.0015 |  26$\%$ | 0.0000 | 0.0017 |  29$\%$ | 0.0005 | 0.0007 |  29.4$\%$ | 0.0319 | 0.0283 |  40.5$\%$
| $\beta_{2}$ | -0.0105 | 0.0265 |  | -0.0039 | 0.0252 |  | -0.0098 | 0.0126 |  | 0.0207 | 0.1801 |
| $\rho_{1}$ | 0.0110 | 0.0268 |  | 0.0048 | 0.0251 |  | 0.0113 | 0.0130 |  | -0.5302 | 0.4177 |
| $\beta_{0}$ | 0.0022 | 0.1524 |  | -0.0036 | 0.1539 |  | -0.0006 | 0.0513 |  | -0.0657 | 0.2198 |
WME | $\beta_{1}$ | -0.0001 | 0.0011 |  27.3$\%$ | 0.0025 | 0.0011 |  26.8$\%$ | 0.0013 | 0.0018 |  29.7$\%$ | 0.0281 | 0.0766 |  35.4$\%$
| $\beta_{2}$ | -0.0067 | 0.0139 |  | -0.0014 | 0.0144 |  | -0.0016 | 0.0214 |  | 0.0308 | 0.4919 |
| $\rho_{1}$ | 0.0060 | 0.0142 |  | 0.0057 | 0.0146 |  | 0.0004 | 0.0216 |  | 0.0109 | 0.4331 |
| $\beta_{0}$ | 0.0571 | 0.1659 |  | 0.0765 | 0.1683 |  | 0.0621 | 0.0601 |  | 0.1466 | 0.0887 |
CLCE | $\beta_{1}$ | -0.0205 | 0.0016 |  26.3$\%$ | -0.0249 | 0.0015 |  27.2$\%$ | -0.0221 | 0.0023 |  27.5$\%$ | -0.1192 | 0.0470 |  38.8$\%$
| $\beta_{2}$ | -0.0310 | 0.0183 |  | -0.0504 | 0.0191 |  | -0.0390 | 0.0250 |  | -0.0545 | 0.2459 |
| $\rho_{1}$ | -0.0047 | 0.0187 |  | 0.0057 | 0.0186 |  | 0.0015 | 0.0251 |  | -0.0198 | 0.2183 |

(a) $\hat{\beta}_{0}$

(b) $\hat{\beta}_{1}$

(c) $\hat{\beta}_{2}$

(d) $\hat{\rho}_{1}$

Figure 1: Empirical MSE for CLAD, WME and CLCE estimates of $\hat{\beta}_{0}$,
$\hat{\beta}_{1}$, $\hat{\beta}_{3}$, and $\hat{\rho}_{1}$ when errors follow
$N(0,1)$ for $n=50,100,150,\ldots,1000$

(a) $\hat{\beta}_{0}$

(b) $\hat{\beta}_{1}$

(c) $\hat{\beta}_{2}$

(d) $\hat{\rho}_{1}$

Figure 2: Empirical MSE for CLAD, WME and CLCE estimates of $\hat{\beta}_{0},$
$\hat{\beta}_{1}$,$\hat{\beta}_{3}$ and $\hat{\rho}_{1}$ when errors follow
$DE(0,1)$ for $n=50,100,150,\ldots,1000$.

(a) $\hat{\beta}_{0}$

(b) $\hat{\beta}_{1}$

(c) $\hat{\beta}_{2}$

(d) $\hat{\rho}_{1}$

Figure 3: Empirical MSE for CLAD, WME and CLCE estimates of $\hat{\beta}_{0},$
$\hat{\beta}_{1}$,$\hat{\beta}_{3}$ and $\hat{\rho}_{1}$ when errors follow
$t_{n}$ for $n=50,100,150,\ldots,1000.$

(a) $\hat{\beta}_{0}$

(b) $\hat{\beta}_{1}$

(c) $\hat{\beta}_{2}$

(d) $\hat{\rho}_{1}$

Figure 4: Empirical MSE for CLAD, WME and CLCE estimates of $\hat{\beta}_{0},$
$\hat{\beta}_{1}$,$\hat{\beta}_{3}$ and $\hat{\rho}_{1}$ when errors follow
$N(0,\sigma_{X_{i}}^{2})$, where
$\sigma_{X_{i}}^{2}=(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta})^{2}$ for
$n=50,100,150,\ldots,1000.$

It is evident from Figures 1, 2 and 3 that in the presence of endogenous
variables, the estimators of the parameters associated with the model (2.8)
are consistent, as it is theoretically established in Theorem 3.1. Next,
Figure 4 represents the comparison of the MSE of these estimators in the case
of heteroskedasticity, and it is observed that the MSEs of WME and CLCE
estimators are higher than that of CLAD for small sample sizes, but as the
sample size increases, eventually the MSEs decrease. So, it is evident that in
the case of heteroskedasticity, the MSEs are approaching zero as the sample
size increases. Hence, these estimators are performing well in the case of all
error distributions, including non-normal distributions, considered here.

Table 3 represents the empirical bias and empirical MSE of symmetrically
censored least squares (SCLS) when the support of the error distribution is
bounded. We perform a simulation study based on truncated normal distribution,
truncated Laplace distribution, truncated $t$-distribution, and
$N(0,\sigma_{X_{i}}^{2})$, where
$\sigma_{X_{i}}^{2}=(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta})^{2}$ as error
distribution. For a small sample size, the values reported in Table 3 indicate
that the MSEs of $\beta_{2}$ and $\rho_{1}$ are higher as compared to that of
$\beta_{1}$ and $\beta_{0}$.

Table 3: Empirical bias and MSE of $\hat{\beta_{0}},$ $\hat{\beta_{1}},$
$\hat{\beta_{2}}$ and $\hat{\rho_{1}}$ with $n=50,100,500,1000.$

Estimators | $\boldsymbol{\beta}$ | Error-distribution
---|---|---
|  | $N(0,1)$ | $DE(0,1)$ | $t_{n}$ |  $N(0,\sigma_{X_{i}}^{2})$, $\sigma_{X_{i}}^{2}=(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta})^{2}$
|  | Bias | MSE | C.P | Bias | MSE | C.P | Bias | MSE | C.P | Bias | MSE | C.P
|  | $n=50$
| $\beta_{0}$ | 0.0436 | 0.9645 |  | 0.0512 | 2.0848 |  | 0.0398 | 3.7091 |  | 0.0220 | 2.5667 |
SCLS | $\beta_{1}$ | 0.0040 | 0.0127 |  26$\%$ | -0.0002 | 0.0102 |  30$\%$ | 0.0051 | 0.0121 |  28$\%$ | 0.0119 | 0.0381 |  38$\%$
| $\beta_{2}$ | -0.1036 | 3.4037 |  | -0.1494 | 9.3995 |  | -0.1429 | 10.237 |  | -0.0435 | 14.448 |
| $\rho_{1}$ | 0.1109 | 3.4103 |  | 0.1499 | 9.3870 |  | 0.1459 | 10.236 |  | 0.0645 | 14.404 |
|  | $n=100$
| $\beta_{0}$ | -0.0024 | 8.7150 |  | 0.1545 | 6.3232 |  | 0.1256 | 2.4670 |  | 0.0654 | 2.2007 |
SCLS | $\beta_{1}$ | 0.0018 | 0.0057 |  25$\%$ | 0.0018 | 0.0051 |  22$\%$ | 0.0004 | 0.0054 |  25$\%$ | 0.0084 | 0.0181 |  33$\%$
| $\beta_{2}$ | -0.0604 | 1.8580 |  | 0.0102 | 4.7568 |  | -0.0527 | 1.8049 |  | -0.0845 | 1.0643 |
| $\rho_{1}$ | 0.0620 | 1.8618 |  | -0.0050 | 4.7580 |  | 0.0514 | 1.8140 |  | 0.0932 | 1.0663 |
|  | $n=500$
| $\beta_{0}$ | 0.0052 | 0.0077 |  | 0.0034 | 0.0067 |  | 0.0055 | 0.0076 |  | 0.0020 | 0.0165 |
SCLS | $\beta_{1}$ | -0.0003 | 0.0011 |  22$\%$ | 0.0018 | 0.0009 |  29$\%$ | 0.0009 | 0.0011 |  24.2$\%$ | 0.0032 | 0.0035 |  28.2$\%$
| $\beta_{2}$ | -0.0113 | 0.0200 |  | -0.0114 | 0.0188 |  | -0.0097 | 0.0201 |  | -0.0096 | 0.0426 |
| $\rho_{1}$ | 0.0119 | 0.0205 |  | 0.0117 | 0.0192 |  | 0.0093 | 0.0203 |  | 0.0115 | 0.0438 |
|  | $n=1000$
| $\beta_{0}$ | 0.0045 | 0.0035 |  | 0.0020 | 0.0033 |  | 0.0020 | 0.0034 |  | 0.0035 | 0.0078 |
SCLS | $\beta_{1}$ | 0.0001 | 0.0005 |  25.7$\%$ | 0.0004 | 0.0005 |  28.4$\%$ | -0.0005 | 0.0005 |  29$\%$ | 0.0007 | 0.0017 |  27.6$\%$
| $\beta_{2}$ | -0.0080 | 0.0090 |  | -0.0053 | 0.0082 |  | -0.0046 | 0.0087 |  | -0.0092 | 0.0194 |
| $\rho_{1}$ | 0.0078 | 0.0092 |  | 0.0060 | 0.0080 |  | 0.0058 | 0.0090 |  | 0.0102 | 0.0199 |

##  5 Real Data Study

To illustrate the applicability of the proposed estimators over other
estimators, we consider a well-known data set, Mroz data. This Mroz (1987)
data set is available in the Wooldridge R package (see
https://rdrr.io/rforge/Ecdat/man/Mroz.html), which pertains to U.S. women’s
labor force participation. The data set contains observations on 753
individuals. The dependent variable is the total number of hours worked per
week. In the data, 325 of the 753 women worked 0 hours; therefore, the
dependent variable is left-censored at zero. Hence, the censoring probability
is approximately 0.43. For this data set, the explanatory variables are years
of education (educ), years of experience (exper) and its square (expersq), age
of the wife (age), number of children under six years old (kidslt6), number of
children equal to or greater than six years old (kidsge6), and non-wife
household income (nwifeinc). We consider nwifeinc to be an endogenous variable
since it may be correlated to unobserved household preferences regarding the
wife’s labor force participation. We add the husband’s years of schooling
(huseduc) as an instrument because it can influence both his income and the
now-wife’s household income, but it should not influence the wife’s decision
to engage in the labor force.

Table 4: Parameter estimates of Tobit model Variables | CLAD | WME | CLCE
---|---|---|---
age | -0.4580 | 0.5077 | -0.2991
educ | 0.3550 | 0.8450 | 0.1719
exper | 1.1123 | 1.5373 | 0.9289
expersq | -0.4014 | -1.9696 | -0.3233
kidslt6 | -0.7913 | 0.0358 | 0.0724
kidsge6 | 0.2221 | 0.0978 | -0.1578
nwifeinc | 0.1249 | -2.5682 | -1.2380
residual | -0.3397 | 2.6877 | 0.8040
Table 5: Bootstrap MSE of CLAD, WME, and CLCE Estimator for $B=500,1000,1500$.

Parameters | BMSE
---|---
| CLAD | WME | CLCE
| $B=500$ | $B=1000$ | $B=1500$ | $B=500$ | $B=1000$ | $B=1500$ | $B=500$ | $B=1000$ | $B=1500$
age | 0.7118 | 0.7011 | 0.6985 | 0.3518 | 0.3604 | 0.3645 | 0.5177 | 0.4941 | 0.4618
educ | 0.5898 | 0.5855 | 0.6129 | 0.4844 | 0.4811 | 0.4676 | 0.6843 | 0.7085 | 0.6672
exper | 0.4927 | 0.5439 | 0.4890 | 2.8544 | 3.1788 | 3.1627 | 1.6108 | 1.6805 | 1.5337
expersq | 1.8319 | 1.7724 | 1.8819 | 3.8145 | 4.3454 | 4.2618 | 1.5171 | 1.6423 | 1.4472
kidslt6 | 2.9191 | 2.8311 | 2.9829 | 0.1231 | 0.1228 | 0.1213 | 0.1535 | 0.1541 | 0.1613
kidsge6 | 0.7251 | 0.7112 | 0.7530 | 0.0904 | 0.1065 | 0.1066 | 0.2150 | 0.1889 | 0.1798
nwifeinc | 0.8463 | 0.8253 | 0.8776 | 4.4537 | 4.5254 | 4.1562 | 2.9621 | 2.9260 | 2.9958
residual | 1.6876 | 1.6326 | 1.7354 | 4.9339 | 5.0020 | 4.6065 | 3.9345 | 3.8406 | 3.8480

To validate the theoretical result on the real data set, estimates and
bootstrap mean square error (MSE) are computed in this part. Following are the
steps to find the bootstrap mean square error (BMSE) of parameters in the
Tobit model using the control function approach:

  * •

First, obtain the residual from the linear regression model by regressing the
endogenous variable (nwifeinc) on the instrumental variable (husedu). To
address endogeneity and create a control function, the residual obtained from
the previous step is incorporated into the Tobit model.

  * •

Estimate the Tobit model parameters, which contain the explanatory variables
and the control function, using the original data set. Here,
$\hat{\theta}_{1},\ldots,\hat{\theta}_{p},$ represents the $p$ true parameter
values.

  * •

Using basic random sampling with replacement, generate $B$ bootstrap samples
$\\{s^{*}_{1},\ldots,s^{*}_{B}\\}$ from the original data set to create a new
data set of the same size as the original data set.

  * •

Estimate the parameters of the Tobit model for each bootstrap sample, which is
denoted as
$\\{\hat{\boldsymbol{\theta}}^{*}_{1},\hat{\boldsymbol{\theta}}^{*}_{2},\ldots,\hat{\boldsymbol{\theta}}^{*}_{B}\\}.$

  * •

Compute the bootstrap mean square error (BMSE) between the bootstrap estimates
and the true parameter value for each bootstrap sample.

  * •

Use the following formula to find the average BMSE for each parameter over all
bootstrap samples:

$BMSE=\frac{1}{B}\sum_{b=1}^{B}\left(\hat{\theta}_{ib}-\hat{\theta}_{i}\right)^{2}$

where $\hat{\theta}_{ib}$ is the $ith$ element of
$\hat{\boldsymbol{\theta}}^{*}_{b}$, $b=1,\ldots,B$ and $i=1,\ldots,p.$

Table 4 contains the results of parameter estimates for the Tobit model. In
the table, the second column gives the CLAD parameter estimates (see, (2.12)),
the third column gives the WME estimates (see, (2.13)), and the last column
gives the CLCE estimates (see, (2.14)). The interpretation of these estimates
shows interesting patterns in the relationship between various factors and the
working hours of married women.

Regarding CLAD and CLCE, the age coefficient is negative, demonstrating that
married women work fewer hours as they get older. Nevertheless, WME has a
positive coefficient, representing that working hours increase with age.
Across all three cases (CLAD, WME, and CLCE), a positive coefficient for
education (educ) and experience (exper) indicates that married women who hold
higher levels of education also work for a longer period of time. Furthermore,
the coefficient for the squared term of experience (expersq) is negative in
all three cases. This implies that, where more working hours are initially
associated with more experience, this effect wanes as experience levels
increase. There is evidence that married women with small children work less
hours; this is indicated by the variable (kidslt6) having a negative
coefficient.

In the overall context, working hours for CLAD and CLCE are positively
impacted by kidsge6, nwifeinc, and the residual term, but negatively impacted
by age, education, experience, and kidslt6. In the case of WME, working hours
are positively influenced by age, education, experience, and kidslt6, but
adversely affected by kidsge6 and nwifeinc. The residual term also has a
positive impact. Table 5 represents the BMSE of parameter estimates. When the
number of bootstrap samples ($B$) increases, CLAD shows consistent improvement
in BMSE across all parameters. Overall, it does quite well. The performance of
WME is mixed. As the bootstrap sample size increases, it shows improvement for
certain parameters (e.g., kidslt6) but increases BMSE for other parameters
(e.g., exper, expersq). In contrast, CLCE consistently performs well in BMSE
as $B$ increases, outperforming WME and CLAD overall.

##  6 Conclusion

In this study, we have proposed and studied the M-estimation for the Tobit
model with an endogenous regressor. By incorporating the control function
technique using instrumental variables, we address endogeneity issues in the
Tobit model. In the presence of left censoring, our proposed method allows the
exploration of M-estimation with a control function technique to accommodate a
general non-convex, non-differentiable, and non-monotone loss function. Our
method is adaptable and suitable for a wide range of applications because the
loss function that is taken into account covers well-known instances such as
Huber, least absolute, and log-cosh.

Through rigorous theoretical analysis, the strong consistency and asymptotic
normality of the M-estimator under some regularity conditions are derived.
Additionally, a consistent estimation procedure for the estimation of the
covariance matrix of the asymptotic distribution of the proposed M-estimators
is developed. The finite sample performance of three different estimators-
CLAD, WME, and CLCE- of the parameters in the Tobit model through an extensive
simulation study with a large sample size. The analyses included both
heteroskedastic and homoskedastic error distributions. The results of the
simulation study indicate that all estimators perform well under various
different conditions. The effectiveness of the suggested methodology, in
particular the CLCE estimator corresponding to the log-cosh loss function, was
demonstrated by the data. It is noteworthy that the proposed estimator CLCE
consistently outperforms over CLAD and WME with respect to MSE.

To further validate the proposed methodology, we conducted a real data study
by representing the bootstrap MSE estimates of parameters. The empirical
results reinforced the improved performance of the estimators, affirming their
robustness in practical applications. This research provides significant
insight into the statistical modeling of endogenous regressors in censored
data. It is possible to extend this concept in the future to identify rank-
based estimators when endogenous regressors are present. In this
investigation, a continuous endogenous regressor has been taken into
consideration. A similar method can also be used to extend this concept to
binary endogenous variables.

Acknowledgement : Subhra Sankar Dhar gratefully acknowledges his core research
grant CRG/2022/001489, Government of India.

##  7 Appendix

$\boldsymbol{B.1.}$ Consistency

In this section, we present the proof of the Theorem 3.1 stated in Section 3
to show strong consistency of the M-estimator for the model (2.8). It is
similar to those used by [45] to demonstrate the strong consistency of the LAD
estimator for the standard Tobit model and [42] to demonstrate the weak
consistency of the LAD estimator for the nonlinear regression model. The
following lemmas of [55] are employed in the following proof:

######  Lemma 7.1 (Lemma 2.3 of [55]).

Let $\\{y_{i}\\}$ be a sequence of independent random variables that takes
value in $(\Omega,\mathcal{F})$,$\mathcal{F}$ is a $\sigma$-field. Let
$\rho\colon(\Omega,\mathcal{B})\to\mathbb{R}$, where
$\mathcal{B}\in\mathbb{R}^{m}$ is compact. Assume that

  1. $\boldsymbol{(a)}$

For each $\boldsymbol{\beta}$ in $\mathcal{B},$
$\rho(\omega,\boldsymbol{\beta})$ is $\mathcal{F}$-measurable.

  2. $\boldsymbol{(b)}$

$\rho(y_{i},\boldsymbol{\beta})$ is continuous on $\mathcal{B}$, uniformly in
i, a.s.

  3. $\boldsymbol{(c)}$

There exists a measurable $m_{i}\colon\Omega\to\mathbb{R}$ for which
$|\rho(\omega,\boldsymbol{\beta})|<m_{i}(\omega),\hskip
2.84544pt\forall\boldsymbol{\beta}\in\mathcal{B},$ and for all i,
$E\left[m_{i}(y_{i})\right]^{1+\delta^{*}}\leq M<\infty,$ and some
$\delta^{*}>0.$ Then

    1. $\bullet$

$E\left[\rho(y_{i},\boldsymbol{\beta})\right]$ is continuous on $\mathcal{B}$,
uniformly in i.

    2. $\bullet$

$\underset{\boldsymbol{\beta}\;\in\;\mathcal{B}}{\text{sup}}\;\left|\frac{1}{n}\sum\limits_{i=1}^{n}\rho(y_{i},\boldsymbol{\beta})-\overline{S}_{n}(\boldsymbol{\beta})\right|\overset{a.s}{\to}0,$
where
$\overline{S}_{n}(\boldsymbol{\beta})=\frac{1}{n}\sum\limits_{i=1}^{n}E\left[\rho(y_{i},\boldsymbol{\beta})\right].$

######  Lemma 7.2 (Lemma 2.2 of [55]).

Let $S_{n}(\omega,\boldsymbol{\beta})$ be a measurable function on a
measurable space $(\Omega,\mathcal{F})$ and a continuous function of
$\boldsymbol{\beta}$ on a compact set $\mathcal{B}$ for each $\omega$ in
$\Omega.$ Then there exists a measurable function $\hat{\beta}_{n}(\omega)$
such that

$S_{n}\left[\omega,\hat{\boldsymbol{\beta}}_{n}(\omega)\right]=\underset{\boldsymbol{\beta}\;\in\;\mathcal{B}}{\text{inf}}S_{n}\left(\omega,\boldsymbol{\beta}\right)$

for all $\omega$ in $\Omega.$ if
$\left|S_{n}(\omega,\boldsymbol{\beta})-\overline{S}_{n}(\boldsymbol{\beta})\right|\overset{a.s}{\to}0.$
uniformly for all $\boldsymbol{\beta}$ in $\mathcal{B}$ and if
$\overline{S}_{n}(\boldsymbol{\beta})$ has a minimum at
$\boldsymbol{\beta}_{0}$ which is identifiably unique(i.e., for any
$\epsilon^{*}>0$,

$\underset{n\geq
n_{0}}{\text{inf}}\underset{\boldsymbol{\beta}}{\text{min}}\left\\{\overline{S}_{n}(\boldsymbol{\beta})-\overline{S}_{n}(\boldsymbol{\beta}_{0})\colon\boldsymbol{\beta}\in\mathcal{B},\|\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\|>\epsilon^{*}\right\\}>0$

for some positive integer $n_{0}$), then $\hat{\boldsymbol{\beta}}_{n}$
converges to $\boldsymbol{\beta}_{0}$ almost surely $(\omega).$

###### Proof of Theorem 3.1..

Suppose

$\xi_{i}^{*}=y_{i}-max\\{0,\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}_{0}\\},$
(7.1)

and define

$h_{i}=h_{i}(\boldsymbol{\beta},\boldsymbol{\beta}_{0})=max\\{0,\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}\\}-max\\{0,\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}_{0}\\},$
(7.2)

where $\boldsymbol{\delta}^{*}=\boldsymbol{\beta}-\boldsymbol{\beta}_{0}$,
First, note that minimizing $Q_{n}(\boldsymbol{\beta})$ is equivalent to
minimizing the normalized function

$S_{n}(\boldsymbol{\beta})=Q_{n}(\boldsymbol{\beta})-Q_{n}(\boldsymbol{\beta}_{0}),$
(7.3)

which can be written as

$S_{n}(\boldsymbol{\beta})=\frac{1}{n}\sum\limits_{i=1}^{n}\rho\left(y_{i}-max\\{0,\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}\\}\right)-\frac{1}{n}\sum\limits_{i=1}^{n}\rho\left(y_{i}-max\\{0,\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}_{0}\\}\right)$
(7.4)

Since $Q_{n}(\boldsymbol{\beta}_{0})$ is independent of $\boldsymbol{\beta}$,
normalization has no impact on the sequence of minimizing values
$\\{\hat{\beta}_{n}\\}.$ Now we will show that $S_{n}(\boldsymbol{\beta})$
will satisfy the conditions of the lemma (7.1). Then, using (7.4), we obtain

$\begin{split}S_{n}(\boldsymbol{\beta})&=\frac{1}{n}\sum\limits_{i=1}^{n}\rho\left(\xi_{i}^{*}-h_{i}\right)-\frac{1}{n}\sum\limits_{i=1}^{n}\rho\left(\xi_{i}^{*}\right)\\\
&=\frac{1}{n}\sum\limits_{i=1}^{n}\left[\rho(\xi_{i}^{*}-h_{i})-\rho(\xi_{i}^{*})\right]\\\
&\leq\frac{1}{n}\sum\limits_{i=1}^{n}\left|\rho(\xi_{i}^{*}-h_{i})-\rho(\xi_{i}^{*})\right|\\\
&\leq\frac{k}{n}\sum\limits_{i=1}^{n}\left|\xi_{i}^{*}-h_{i}-\xi_{i}^{*}\right|=\frac{k}{n}\sum\limits_{i=1}^{n}\left|h_{i}\right|\end{split}$
(7.5)
$\begin{split}\text{and}\;|h_{i}|=|h_{i}(\boldsymbol{\beta},\boldsymbol{\beta}_{0})|&=\left|max\\{0,\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}\\}-max\\{0,\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}_{0}\\}\right|\\\
&=\left|\frac{1}{2}\left[\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}+|\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}|\right]-\frac{1}{2}\left[\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}_{0}+|\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}_{0}|\right]\right|\\\
&=\left|\frac{1}{2}\left[\boldsymbol{X}_{i}^{T}(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})\right]+\frac{1}{2}\left[|\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}|-|\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}_{0}|\right]\right|\\\
&\leq\frac{1}{2}\left|\boldsymbol{X}_{i}^{T}\left(\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\right)\right|+\frac{1}{2}\left|\boldsymbol{X}_{i}^{T}\left(\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\right)\right|=\left|\boldsymbol{X}_{i}^{T}\left(\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\right)\right|\\\
&\leq\|\boldsymbol{X}_{i}\|\;\|\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\|\\\
&\leq\|\boldsymbol{X}_{i}\|\left(\|\boldsymbol{\beta}\|+\|\boldsymbol{\beta}_{0}\|\right)=\mathcal{O}(\|\boldsymbol{X}_{i}\|).\\\
\end{split}$ (7.6)

Because summand ($|h_{i}|$) in ((7.5)) is bounded (in magnitude) by
$\|\boldsymbol{X}_{i}\|\left(\|\boldsymbol{\beta}\|+\|\boldsymbol{\beta}_{0}\|\right)$,
which is $\mathcal{O}(\|\boldsymbol{X}_{i}\|)$ due to the compactness of
$\mathcal{B}$. Assumptions $\boldsymbol{A.2}$ and $\boldsymbol{A.3}$, as well
as Lemma 7.1, lead to

$\underset{n\to\infty}{lim}\left|S_{n}\left(\boldsymbol{\beta}\right)-E\left[S_{n}\left(\boldsymbol{\beta}\right)\right]\right|=0\hskip
5.69046pta.s.,$ (7.7)

uniformly in $\boldsymbol{\beta}$. Since
$E\left[S_{n}(\boldsymbol{\beta}_{0})\right]=0$, according to Lemma 7.2,
$\hat{\boldsymbol{\beta}}_{n}$ is strongly consistent if for every
$\epsilon^{*}>0,$ $E\left[S_{n}\left(\boldsymbol{\beta}\right)\right]$ is
bounded away from 0, uniformly in $\boldsymbol{\beta}$ for
$\|\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\|\geq\epsilon^{*}$ and for all
sufficiently large $n$. Since
$E\left[Q_{n}\left(\boldsymbol{\beta}\right)\right]$ is not convex in
$\boldsymbol{\beta}$. Therefore, to demonstrate that
$E\left[Q_{n}\left(\boldsymbol{\beta}\right)-Q_{n}\left(\boldsymbol{\beta}_{0}\right)\right]$
has unique minima whenever $\boldsymbol{\beta}$ near $\boldsymbol{\beta}_{0},$
the conditional expectation of $S_{n}(\boldsymbol{\beta})$ can be expressed as

$E\left[S_{n}\left(\boldsymbol{\beta}\right)|\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{n}\right]=E\left[Q_{n}\left(\boldsymbol{\beta}\right)|\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{n}\right]-E\left[Q_{n}\left(\boldsymbol{\beta}_{0}\right)|\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{n}\right].$
(7.8)

We determine $E\left[Q_{n}\left(\boldsymbol{\beta}\right)\right]$ and
$E\left[Q_{n}\left(\boldsymbol{\beta}_{0}\right)\right]$ individually, then we
subtract it to demonstrate that the individual term is positive.

$\begin{split}E\left[Q_{n}\left(\boldsymbol{\beta}\right)|\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{n}\right]&=\frac{1}{n}\sum\limits_{i=1}^{n}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)E_{\xi|\boldsymbol{X}}\left[\rho\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\right)\right]+\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}<0\right)E_{\xi|\boldsymbol{X}}\left[\rho\left(y_{i}\right)\right]\\\
&=\frac{1}{n}\sum\limits_{i=1}^{n}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)\left\\{\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)E_{\xi|\boldsymbol{X}}\left[\rho\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\right)\right]\right\\}\\\
&+\frac{1}{n}\sum\limits_{i=1}^{n}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)\left\\{\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}<0\right)E_{\xi|\boldsymbol{X}}\left[\rho\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\right)\right]\right\\}\\\
&+\frac{1}{n}\sum\limits_{i=1}^{n}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}<0\right)\bigr{\\{}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)E_{\xi|\boldsymbol{X}}\left[\rho\left(y_{i}\right)\right]\\\
&+\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}<0\right)E_{\xi|\boldsymbol{X}}\left[\rho\left(y_{i}\right)\right]\bigr{\\}}.\end{split}$
(7.9)

Similarly,

$\begin{split}E\left[Q_{n}(\boldsymbol{\beta}_{0})|\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{n}\right]&=\frac{1}{n}\sum\limits_{i=1}^{n}\bigr{\\{}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)E_{\xi|\boldsymbol{X}}\left[\rho\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\right]\\\
&+\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}<0\right)E_{\xi|\boldsymbol{X}}\left[\rho\left(y_{i}\right)\right]\bigr{\\}}\\\
&=\frac{1}{n}\sum\limits_{i=1}^{n}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\left\\{\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)E_{\xi|\boldsymbol{X}}\left[\rho\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\right]\right\\}\\\
&+\frac{1}{n}\sum\limits_{i=1}^{n}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\left\\{\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}<0\right)E_{\xi|\boldsymbol{X}}\left[\rho\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\right]\right\\}\\\
&+\frac{1}{n}\sum\limits_{i=1}^{n}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}<0\right)\bigr{\\{}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)E_{\xi|\boldsymbol{X}}\left[\rho\left(y_{i}\right)\right]\\\
&+\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}<0\right)E_{\xi|\boldsymbol{X}}\left[\rho\left(y_{i}\right)\right]\bigr{\\}}.\end{split}$
(7.10)

Subtract the (7.10) from the (7.9), and we have

$E\left[Q_{n}\left(\boldsymbol{\beta}\right)-Q_{n}\left(\boldsymbol{\beta}_{0}\right)|\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{n}\right]=E\left[S_{n}\left(\boldsymbol{\beta}\right)|\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{n}\right]$
$\begin{split}&=\frac{1}{n}\sum\limits_{i=1}^{n}\left\\{\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\right\\}E_{\xi|\boldsymbol{X}}\left[\rho\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\right)-\rho\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\right]\\\
&+\frac{1}{n}\sum\limits_{i=1}^{n}\left\\{\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}<0\right)\right\\}E_{\xi|\boldsymbol{X}}\left[\rho\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\right)-\rho\left(y_{i}\right)\right]\\\
&+\frac{1}{n}\sum\limits_{i=1}^{n}\left\\{\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}<0\right)\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\right\\}E_{\xi|\boldsymbol{X}}\left[\rho\left(y_{i}\right)-\rho\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\right].\\\
\end{split}$ (7.11)

Since all terms in this expression are positive, using the assumption
$\boldsymbol{A.5}$, there exists $k_{1}>0$ such that we have the following:

$\begin{split}E\left[S_{n}\left(\boldsymbol{\beta}\right)|\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{n}\right]&\geq\frac{k_{1}}{n}\sum\limits_{i=1}^{n}\\{\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\\\
&+\frac{k_{1}}{n}\sum\limits_{i=1}^{n}\left\\{\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}<0\right)\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\right\\}\\\
&\geq\frac{k_{1}}{n}\sum\limits_{i=1}^{n}\\{\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right).\end{split}$
(7.12)

Now, taking expectation with respect to random vector $\boldsymbol{X}$ defined
in (2.8), and for any $\epsilon_{0}>0$, we have

$\displaystyle E\left[S_{n}\left(\boldsymbol{\beta}\right)\right]$
$\displaystyle\geq\frac{k_{1}\epsilon_{0}}{n}\sum\limits_{i=1}^{n}E_{\boldsymbol{X}}\left\\{\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>\epsilon_{0}\right)\right\\}$
$\displaystyle\geq\frac{k_{1}\epsilon_{0}}{n}\sum\limits_{i=1}^{n}E_{\boldsymbol{X}}\left\\{\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>\epsilon_{0}\right)\right\\}^{3}\left\\{E_{\boldsymbol{X}}\|\boldsymbol{X}_{i}\|^{3}/K_{0}^{3/4}\right\\}^{2}$
(From Assumption $\boldsymbol{A.3}$)
$\displaystyle\geq\frac{k_{1}\epsilon_{0}K_{0}^{-3/2}}{n}\sum\limits_{i=1}^{n}\left[E_{\boldsymbol{X}}\left\\{\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>\epsilon_{0}\right)\|\boldsymbol{X}_{i}\|^{2}\right\\}\right]^{3}$
(Using Hölder’s inequality and Jensen’s inequality)
$\displaystyle\geq\left(k_{1}\epsilon_{0}K_{0}^{-3/2}\right)\left[\frac{1}{n}\sum\limits_{i=1}^{n}E_{\boldsymbol{X}}\left\\{\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>\epsilon_{0}\right)\left(\boldsymbol{X}^{T}_{i}\delta^{*}\right)^{2}\|\delta^{*}\|^{-2}\right\\}\right]^{3}$
(Using Cauchy-Schwarz inequality)
$\displaystyle\geq\left(k_{1}\epsilon_{0}K_{0}^{-3/2}\right)\left[(\delta^{*})^{T}Q\delta^{*}\|\delta^{*}\|^{-2}\right]^{3}>0.$
(Using Assumption $\boldsymbol{A.3}$)

Since $Q$ is a positive definite matrix according to the assumption,
$\boldsymbol{A.3}$. It means the above expression’s right-hand side is
strictly positive. This demonstrates that
$E\left[S_{n}(\boldsymbol{\beta})\right]$ is uniformly bounded away from zero
for large n and
$\|\delta^{*}\|=\|\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\|>\epsilon^{*}$ as
needed. This completes the proof. ∎

$\boldsymbol{B.2.}$ Asymptotic Normality

The proof of asymptotic normality of $\hat{\boldsymbol{\beta}}_{n}$ is based
on the [45] and [28] techniques, which have been appropriately adapted to
apply to the non-identically distributed case. This technique gives sufficient
conditions under which a sequence of consistent estimator
$\\{\hat{\boldsymbol{\beta}}_{n}\\}$ satisfy some first-order condition,

$\frac{1}{n}\sum_{i=1}^{n}\psi\left(y_{i},\hat{\boldsymbol{\beta}}_{n}\right)=o_{p}(1),$
(7.13)

is asymptotically normal. where $\\{y_{i}\\}$ are mutually independent but not
identically distributed and $\psi:\Omega\times\mathcal{B}\to\mathbb{R}^{m}$ is
some given function for $\mathcal{B},$ an open subset of Euclidean space
$\mathbb{R}^{m}$. Define

$\lambda_{n}(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=1}^{n}E\left[\psi_{i}\left(y_{i},\boldsymbol{\beta}\right)\right],$
(7.14)

and let

$\mu_{i}(y_{i},\boldsymbol{\beta},d^{*}_{0})=\underset{\boldsymbol{\tau}}{sup}\left\\{\|\psi_{i}\left(y_{i},\boldsymbol{\tau}\right)-\psi_{i}\left(y_{i},\boldsymbol{\beta}\right)\|:\|\boldsymbol{\tau}-\boldsymbol{\beta}\|\leq
d^{*}_{0}\right\\}.$ (7.15)

The conditions imposed on $\psi_{i}$, $\lambda_{n}$, and $\mu_{i}$ are as
follows:

  * $\boldsymbol{L.1.}$

For each fixed $\boldsymbol{\beta}\in\mathcal{B}$ and each i,
$\psi_{i}\left(y_{i},\boldsymbol{\beta}\right)$ is $\mathcal{F}$-measurable
and $\psi\left(y_{i},\boldsymbol{\beta}\right)$ is separable in the sense of
Doob for all i: i.e., there is a countable subset
$\mathcal{B^{*}}\subset\mathcal{B}$ such that for every open set
$U\subset\mathcal{B}$ and every closed interval $\mathcal{A}$, the sets
$\left\\{y_{i}:\|\psi\left(y_{i},\boldsymbol{\beta}\right)\|\in\mathcal{A},\text{for
all}\;\boldsymbol{\beta}\subset U\right\\}$ and
$\left\\{y_{i}:\|\psi\left(y_{i},\boldsymbol{\beta}\right)\|\in\mathcal{A},\text{for
all}\;\boldsymbol{\beta}\subset U\cap\mathcal{B^{*}}\right\\}$ differ at most
by a set of (Probability) measure zero.

  * $\boldsymbol{L.2.}$

There is some $\boldsymbol{\beta}_{0}\in\mathcal{B}$ such that
$\lambda_{n}\left(\boldsymbol{\beta}_{0}\right)=0$ for n.

  * $\boldsymbol{L.3.}$

There are (strictly positive) numbers, $a^{*}$, $b^{*}$, $c^{*}$, $d^{*}$, and
$n_{0}$ such that, for all $1\leq i\leq n$,

  * $\boldsymbol{(i)}$

For $\|\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\|\leq d^{*}$ and $n\geq
n_{0},$ the inequality $\|\lambda_{n}\left(\boldsymbol{\beta}\right)\|\geq
a^{*}.\|\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\|$ holds.

  * $\boldsymbol{(ii)}$

For $\|\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\|+d_{0}^{*}\leq d^{*}$ and
$d^{*}_{0}\geq 0$, the inequality
$E\left[\mu_{i}\left(y_{i},\boldsymbol{\beta},d^{*}_{0}\right)\right]\leq
b^{*}.d^{*}_{0}$ holds.

  * $\boldsymbol{(iii)}$

For $\|\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\|+d_{0}^{*}\leq d^{*}$ and
$d_{0}^{*}\geq 0,$ the inequality
$E\left[\mu_{i}\left(y_{i},\boldsymbol{\beta},d^{*}_{0}\right)\right]^{2}\leq
c^{*}.d^{*}_{0}$ holds.

  * $\boldsymbol{L.4.}$

For all $1\leq i\leq n$ and some $K>0$, the inequality
$E\|\psi_{i}\left(y_{i},\boldsymbol{\beta}_{0}\right)\|^{2}\leq K$ holds.

######  Lemma 7.3.

Assume that the assumptions $\boldsymbol{L.1}$ to $\boldsymbol{L.4}$ are true
and

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\psi_{i}\left(y_{i},\hat{\boldsymbol{\beta}}_{n}\right)=o_{p}(1)\hskip
11.38092pt\text{for}\hskip
11.38092pt\hat{\boldsymbol{\beta}}_{n}=\hat{\boldsymbol{\beta}}_{n}\left(y_{1},\ldots,y_{n}\right).$

if $\hat{\boldsymbol{\beta}}_{n}$ is consistent for $\boldsymbol{\beta}_{0}$
$\left(i.e.,Pr\left\\{\|\hat{\boldsymbol{\beta}}_{n}-\boldsymbol{\beta}_{0}\|>\eta_{0}\right\\}=o(1)\;\text{for
any}\;\eta_{0}>0\right),$ then

$\left[\sqrt{n}.\lambda_{n}(\boldsymbol{\beta})\right]_{\boldsymbol{\beta}=\hat{\boldsymbol{\beta}}_{n}}=-\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\psi\left(y_{i},\boldsymbol{\beta}_{0}\right)+o_{p}(1).$

######  Proof.

The proof technique closely follows the approach in Section 4 of [28]. At each
step of establishing Lemma 3 in [28], expressions of the form

$n.\lambda\left(\boldsymbol{\beta}\right),\hskip
8.5359ptn.E\left[{\mu}\left(y,\boldsymbol{\beta},d^{*}_{0}\right)\right],\hskip
8.5359ptetc.,$

can be substituted with their corresponding ”average” counterparts

$n.\lambda_{n}\left(\boldsymbol{\beta}\right),\hskip
8.5359pt\sum_{i=1}^{n}E\left[{\mu}_{i}\left(y_{i},\boldsymbol{\beta},d^{*}_{0}\right)\right],\hskip
8.5359ptetc.,$

without compromising the argument’s validity, given that $n$ is large enough.
∎

######  Proof of Theorem 3.2..

A one-sided gradient for non-smooth minimand has been used in the literature
to verify this condition (see [45], [46],[38], and [49]). The sub-gradient
exists only and is specified by $\rho^{\prime}$ since $\rho$ is differentiable
whenever $\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0$ for all $i$. Define the
following function:

$\Psi\left(\boldsymbol{\beta}\right)=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\psi\left(y_{i},\boldsymbol{X}_{i},\boldsymbol{\beta}\right),$

where

$\begin{split}\psi\left(y_{i},\boldsymbol{X}_{i},\boldsymbol{\beta}\right)&=\left(-\boldsymbol{X}_{i}\right)\mathds{1}\left(\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}>0\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\right)\\\
&=\left(-\boldsymbol{X}_{i}\right)\mathds{1}\left(\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}>0\right)\rho^{\prime}\left(max\left\\{\xi_{i}-\boldsymbol{X}_{i}^{T}\boldsymbol{\delta}^{*},-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\right\\}\right).\\\
\end{split}$ (7.16)

Firstly, we will prove that $\hat{\boldsymbol{\beta}}_{n}$ satisfies the
following condition:

$\displaystyle\Psi\left(\hat{\boldsymbol{\beta}}_{n}\right)$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\psi\left(y_{i},\boldsymbol{X}_{i},\hat{\boldsymbol{\beta}}_{n}\right)=o_{p}(1).$
(B.1)

Consider the $jth$ component of $\Psi\left(\boldsymbol{\beta}\right)$ and
denote $\\{e_{j}\\}_{j=1}^{m}$ be the standard basis of $\mathbb{R}^{m}$.
Then, by using the monotonicity of the one-sided sub-gradient, we have

$\displaystyle|\Psi_{n,j}\left(\hat{\boldsymbol{\beta}}_{n}\right)|$
$\displaystyle\leq\underset{\tau^{*}\to
0}{lim}\left[\Psi_{n,j}\left(\hat{\boldsymbol{\beta}}_{n}+\tau^{*}e_{j}\right)-\Psi_{n,j}\left(\hat{\boldsymbol{\beta}}_{n}-\tau^{*}e_{j}\right)\right]$
$\displaystyle\leq\underset{\tau^{*}\to
0}{lim}\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left(-X_{i,j}\right)\Biggr{[}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\hat{\boldsymbol{\beta}}_{n}+\tau^{*}X_{i,j}>0\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\hat{\boldsymbol{\beta}}_{n}-\tau^{*}X_{i,j}\right)$
$\displaystyle-\mathds{1}\left(\boldsymbol{X}^{T}_{i}\hat{\boldsymbol{\beta}}_{n}-\tau^{*}X_{i,j}>0\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\hat{\boldsymbol{\beta}}_{n}+\tau^{*}X_{i,j}\right)\Biggr{]}$
(Using continuity of $\rho^{\prime}$)
$\displaystyle\leq\left[\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left|X_{i,j}\right|\mathds{1}\left(\boldsymbol{X}^{T}_{i}\hat{\boldsymbol{\beta}}_{n}=0\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\hat{\boldsymbol{\beta}}_{n}\right)\right]$
$\displaystyle\leq\frac{1}{\sqrt{n}}\left(\underset{1\leq i\leq
n}{max}\|\boldsymbol{X}_{i}\|\right)\sum_{i=1}^{n}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\hat{\boldsymbol{\beta}}_{n}=0\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\hat{\boldsymbol{\beta}}_{n}\right).$
(7.17)

Due to the assumption $\boldsymbol{A.3}$, $\underset{1\leq i\leq
n}{max}\|\boldsymbol{X}_{i}\|$ is finite. Under the assumptions
$\boldsymbol{A.6}$, $\boldsymbol{A.7}$, and the strong consistency of
$\hat{\boldsymbol{\beta}}_{n}$, the sum on the right side of (7.17) is finite
with probability one for any sufficiently large $n$. So, condition (B.1) is
true.

The next step is to establish the conditions of Lemma 7.3, namely
$\boldsymbol{L.1}$ through $\boldsymbol{L.4}$. The other condition can be
easily proved in this case, so only $\boldsymbol{L.3}$ and $\boldsymbol{L.4}$
will be validated here. To verify $\boldsymbol{L.3(i)}$ holds, we need to show
the following condition:

$\lambda_{n}\left(\boldsymbol{\beta}\right)=-H_{n}\left(\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\right)+O\|\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\|^{2},$

where $H_{n}$ is the matrix specified in the Theorem 3.2. we have,

$\begin{split}\lambda_{n}\left(\boldsymbol{\beta}\right)+H_{n}\left(\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\right)&=\frac{1}{n}\sum_{i=1}^{n}E\left[\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)\left(-\boldsymbol{X}_{i}\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\right)\right]\\\
&+\frac{1}{n}\sum_{i=1}^{n}E\bigr{[}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\boldsymbol{X}_{i}\boldsymbol{X}^{T}_{i}\rho^{\operatorname*{\prime\prime}}\bigr{(}y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\bigr{)}\bigr{]}\boldsymbol{\delta}^{*}\\\
&=\frac{1}{n}\sum_{i=1}^{n}E\bigr{\\{}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)\left(-\boldsymbol{X}_{i}\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\right)\\\
&-\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\left(-\boldsymbol{X}_{i}\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\right)\\\
&+\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\left(-\boldsymbol{X}_{i}\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\right)\\\
&+\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\left(\boldsymbol{X}_{i}\right)\left(\boldsymbol{X}^{T}_{i}\right)\rho^{\operatorname*{\prime\prime}}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\boldsymbol{\delta}^{*}\bigr{\\}}\\\
&=\frac{1}{n}\sum_{i=1}^{n}E\bigr{\\{}\bigr{[}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)-\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\bigr{]}\left(-\boldsymbol{X}_{i}\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\right)\\\
&+\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\left(-\boldsymbol{X}_{i}\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\right)\\\
&+\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\left(\boldsymbol{X}_{i}\right)\left(\boldsymbol{X}^{T}_{i}\right)\rho^{\operatorname*{\prime\prime}}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\boldsymbol{\delta}^{*}\bigr{\\}}\\\
&\leq\frac{1}{n}\sum_{i=1}^{n}E\bigr{\\{}\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}|\leq|\boldsymbol{X}^{T}_{i}(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})|\right)\left(-\boldsymbol{X}_{i}\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\right)\\\
&+\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\left(-\boldsymbol{X}_{i}\right)\bigr{[}\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)-\boldsymbol{X}^{T}_{i}\left(\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\right)\rho^{\operatorname*{\prime\prime}}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\\\
&+O\|\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\|^{2}\bigr{]}+\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\left(\boldsymbol{X}_{i}\right)\left(\boldsymbol{X}^{T}_{i}\right)\rho^{\operatorname*{\prime\prime}}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\boldsymbol{\delta}^{*}\bigr{\\}}.\end{split}$

According to the first-order condition of minimization (see,
$\boldsymbol{L.2}$), the second term on the right-hand side becomes zero.
Additionally, as $\boldsymbol{\beta}$ nears $\boldsymbol{\beta}_{0}$, the
third term converges to zero for all values of $\boldsymbol{\beta}$.
Consequently, we obtain the following expression:

$\begin{split}\lambda_{n}\left(\boldsymbol{\beta}\right)+H_{n}\left(\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\right)&\leq\frac{1}{n}\sum_{i=1}^{n}E\left[\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}|\leq\|\boldsymbol{X}_{i}\|\|(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})\|\right)\left(-\boldsymbol{X}_{i}\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\right)\right]\\\
&+\frac{2}{n}\sum_{i=1}^{n}E\left[\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\left(\boldsymbol{X}_{i}\right)\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\delta}^{*}\right)\rho^{\operatorname*{\prime\prime}}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\right]\\\
&\leq\frac{M_{1}}{n}\sum_{i=1}^{n}E\left[\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}|\leq\|\boldsymbol{X}_{i}\|\|(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})\|\right)\left(-\boldsymbol{X}_{i}\right)\right]\\\
&+\frac{2.M_{0}}{n}\sum_{i=1}^{n}E\left[\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\left(\boldsymbol{X}_{i}\right)\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\delta}^{*}\right)\right].\\\
\end{split}$

Applying the norm on both sides and using the assumption $\boldsymbol{A.6}$,
and$\boldsymbol{A.7}$, then we get

$\begin{split}\|\lambda_{n}\left(\boldsymbol{\beta}\right)+H_{n}\left(\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\right)\|&\leq\frac{2.M_{0}}{n}\sum_{i=1}^{n}E\left[\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\|\boldsymbol{X}_{i}\||\boldsymbol{X}^{T}_{i}\boldsymbol{\delta}^{*}|\right]\\\
&+\frac{M_{1}}{n}\sum_{i=1}^{n}E\left[\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}|\leq\|\boldsymbol{X}_{i}\|\|\boldsymbol{\delta}^{*}\|\right).\|\boldsymbol{X}_{i}\|\right]\\\
&\leq\frac{2.M_{0}}{n}\sum_{i=1}^{n}E\left[\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\|\boldsymbol{X}_{i}\|^{2}\|\boldsymbol{\delta}^{*}\|\right]+M_{1}.K_{1}.\|\boldsymbol{\delta}^{*}\|\\\
&\leq\frac{2.M_{0}.\|\boldsymbol{\delta}^{*}\|}{n}\sum_{i=1}^{n}E\left[\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}|\leq\|\boldsymbol{X}_{i}\|\|\boldsymbol{\delta}^{*}\|\right).\|\boldsymbol{X}_{i}\|^{2}\right]+M_{1}.K_{1}.\|\boldsymbol{\delta}^{*}\|\\\
&\leq
2.M_{0}.K_{1}.\|\boldsymbol{\delta}^{*}\|^{2}+M_{1}.K_{1}.\|\boldsymbol{\delta}^{*}\|\\\
&\leq
2.M_{0}.K_{1}.\|\boldsymbol{\delta}^{*}\|^{2}+M_{1}.K_{1}.\|\boldsymbol{\delta}^{*}\|^{2}\\\
&\leq\left(2.M_{0}.K_{1}+M_{1}.K_{1}\right).\|\boldsymbol{\delta}^{*}\|^{2}.\end{split}$

Thus,

$\lambda_{n}\left(\boldsymbol{\beta}\right)=-H_{n}\left(\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\right)+O\|\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\|^{2},$

Given that $H_{n}$ is positive definite for large $n$ by Assumption
$\boldsymbol{A.3}$, and for all $\boldsymbol{\beta}$ in a sufficiently small
neighborhood of $\boldsymbol{\beta}_{0}$, we can assert the following:

$\begin{split}\lambda_{n}\left(\boldsymbol{\beta}\right)&=-H\boldsymbol{\delta}^{*}\\\
\Rightarrow\boldsymbol{\delta}^{T}\lambda_{n}\left(\boldsymbol{\beta}\right)&=-\boldsymbol{\delta}^{T}H\boldsymbol{\delta}^{*}\\\
\Rightarrow\frac{\boldsymbol{\delta}^{T}\lambda_{n}\left(\boldsymbol{\beta}\right)}{\|\boldsymbol{\delta}^{*}\|^{2}}&\geq-\frac{\boldsymbol{\delta}^{T}H\boldsymbol{\delta}^{*}}{\|\boldsymbol{\delta}^{*}\|^{2}}\geq\underset{\boldsymbol{\delta}^{*}}{-min}\left(\frac{(\boldsymbol{\delta}^{*})^{T}H\boldsymbol{\delta^{*}}}{\|\boldsymbol{\delta}^{*}\|^{2}}\right)=-\nu_{1}\\\
\Rightarrow\frac{|\boldsymbol{\delta}^{T}\lambda_{n}\left(\boldsymbol{\beta}\right)|}{\|\boldsymbol{\delta}^{*}\|^{2}}&\geq|-\nu_{1}|=\nu_{1}.\\\
\end{split}$

Using Cauchy-Schwarz inequality, then

$\displaystyle|\boldsymbol{\delta}^{T}\lambda_{n}\left(\boldsymbol{\beta}\right)|$
$\displaystyle\leq\|\boldsymbol{\delta}^{*}\|\|\lambda_{n}\left(\boldsymbol{\beta}\right)\|$
$\displaystyle\Rightarrow\|\lambda_{n}\left(\boldsymbol{\beta}\right)\|$
$\displaystyle\geq\frac{|\boldsymbol{\delta}^{T}\lambda_{n}\left(\boldsymbol{\beta}\right)|}{\|\boldsymbol{\delta}^{*}\|}\geq\frac{|\boldsymbol{\delta}^{T}\lambda_{n}\left(\boldsymbol{\beta}\right)|}{\|\boldsymbol{\delta}^{*}\|^{2}}>\nu_{1}>0$
$\displaystyle\Rightarrow\|\lambda_{n}\left(\boldsymbol{\beta}\right)\|\|\boldsymbol{\beta}-\boldsymbol{\beta}_{0}\|^{-1}$
$\displaystyle>\nu_{1}>0.$

Since $\nu_{1}$ is the smallest characteristic root of $H,$ as stated in
Theorem 3.2, and is bounded above by zero, inequality $\boldsymbol{L.3(i)}$
satisfy for all $\boldsymbol{\beta}$ near $\boldsymbol{\beta}_{0}.$ To verify
the condition $\boldsymbol{L.3(ii)}$, we have

$\begin{split}\mu_{i}\left(y_{i},\boldsymbol{\beta},d^{*}_{0}\right)&=\underset{\|\boldsymbol{\tau}-\boldsymbol{\beta}\|\leq
d^{*}_{0}}{sup}\left(\|\psi\left(y_{i},\boldsymbol{\beta}\right)-\psi\left(y_{i},\boldsymbol{\tau}\right)\|\right)\\\
&=\underset{\|\boldsymbol{\tau}-\boldsymbol{\beta}\|\leq
d^{*}_{0}}{sup}\|\left(-\boldsymbol{X}_{i}\right)\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\right)\\\
&-\left(-\boldsymbol{X}_{i}\right)\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}>0\right)\rho^{\prime}\bigr{(}y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}\bigr{)}\|\\\
&\leq\underset{\|\boldsymbol{\tau}-\boldsymbol{\beta}\|\leq
d^{*}_{0}}{sup}\|\boldsymbol{X}_{i}\|\left|\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\right)-\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}>0\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}\right)\right|\\\
&\leq\underset{\|\boldsymbol{\tau}-\boldsymbol{\beta}\|\leq
d^{*}_{0}}{sup}\|\boldsymbol{X}_{i}\|\bigr{|}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\right)+\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}\right)\\\
&-\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}\right)-\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}>0\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}\right)\bigr{|}\\\
&\leq\underset{\|\boldsymbol{\tau}-\boldsymbol{\beta}\|\leq
d^{*}_{0}}{sup}\|\boldsymbol{X}_{i}\|\bigr{|}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)\left\\{\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}\right)-\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}\right)\right\\}\\\
&+\left\\{\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)-\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}>0\right)\right\\}\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}\right)\bigr{|}.\\\
\end{split}$

Consider any two real numbers, denoted as a and b. We can observe that the
inequality $\mathds{1}(a>0)-\mathds{1}(b>0)\leq\mathds{1}(|b|\leq|a-b|),$
holds. Additionally, using the mean value theorem, we can express
$t^{*}_{i}\in\left(l_{1}(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta})+(1-l_{1})(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\tau})\right)$,
where $l_{1}\in(0,1)$. Consequently, we have

$\begin{split}\mu_{i}\left(y_{i},\boldsymbol{\beta},d^{*}_{0}\right)&\leq\underset{\|\boldsymbol{\tau}-\boldsymbol{\beta}\|\leq
d^{*}_{0}}{sup}\|\boldsymbol{X}_{i}\|\bigr{\\{}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)|-\boldsymbol{X}^{T}_{i}\left(\boldsymbol{\beta}-\boldsymbol{\tau}\right)\rho^{\operatorname*{\prime\prime}}\left(t^{*}_{i}\right)|\\\
&+\left|\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}\right)\right|\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}|\leq\|\boldsymbol{X}_{i}\|\|\boldsymbol{\tau}-\boldsymbol{\beta}\|\right)\bigr{\\}}\\\
&\leq\underset{\|\boldsymbol{\tau}-\boldsymbol{\beta}\|\leq
d^{*}_{0}}{sup}\|\boldsymbol{X}_{i}\|^{2}\|\boldsymbol{\beta}-\boldsymbol{\tau}\|.|\rho^{\operatorname*{\prime\prime}}\left(t^{*}_{i}\right)|\\\
&+\left|\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}\right)\right|\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}|\leq\|\boldsymbol{X}_{i}\|\|\boldsymbol{\tau}-\boldsymbol{\beta}\|\right)\|\boldsymbol{X}_{i}\|\\\
&\leq\|\boldsymbol{X}_{i}\|^{2}|\rho^{\operatorname*{\prime\prime}}\left(t^{*}_{i}\right)|.d^{*}_{0}+\left|\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}\right)\right|\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}|\leq\|\boldsymbol{X}_{i}\|.d^{*}_{0}\right)\|\boldsymbol{X}_{i}\|\\\
&\leq\|\boldsymbol{X}_{i}\|^{2}.M_{0}.d^{*}_{0}+M_{1}.\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}|\leq\|\boldsymbol{X}_{i}\|.d^{*}_{0}\right).\|\boldsymbol{X}_{i}\|.\end{split}$

From the assumption $\boldsymbol{A.7}$ and taking the expectations on both
sides, we get

$\begin{split}E\left[\mu_{i}\left(y_{i},\boldsymbol{\beta},d^{*}_{0}\right)\right]&\leq\left\\{E\|\boldsymbol{X}_{i}\|^{4}\right\\}^{1/2}M_{0}.d^{*}_{0}+M_{1}.E\left[\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}|\leq\|\boldsymbol{X}_{i}\|.d^{*}_{0}\right).\|\boldsymbol{X}_{i}\|\right]\\\
&\leq K_{0}^{1/2}.M_{0}.d^{*}_{0}+M_{1}.K_{1}.d^{*}_{0}.\\\
&\leq\left(K_{0}^{1/2}.M_{0}+M_{1}.K_{1}\right).d^{*}_{0}.\end{split}$

For $\boldsymbol{L.3(iii)},$

$\begin{split}\left[\mu_{i}\left(y_{i},\boldsymbol{\beta},d^{*}_{0}\right)\right]^{2}&\leq\|\boldsymbol{X}_{i}\|^{4}.M_{0}^{2}.(d^{*}_{0})^{2}+M_{1}^{2}.\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}|\leq\|\boldsymbol{X}_{i}\|.d^{*}_{0}\right).\|\boldsymbol{X}_{i}\|^{2}\\\
&+2.\\{\|\boldsymbol{X}_{i}\|^{2}.M_{0}.d^{*}_{0}.M_{1}.\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}|\leq\|\boldsymbol{X}_{i}\|.d^{*}_{0}\right).\|\boldsymbol{X}_{i}\|\\}\\\
&\leq
2.\left\\{\|\boldsymbol{X}_{i}\|^{4}.M_{0}^{2}.(d^{*}_{0})^{2}+M_{1}^{2}.\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}|\leq\|\boldsymbol{X}_{i}\|.d^{*}_{0}\right).\|\boldsymbol{X}_{i}\|^{2}\right\\}.\\\
\end{split}$

Applying the AM-GM inequality and then Taking expectations on both sides, we
get

$\begin{split}E\left[\left[\mu_{i}\left(y_{i},\boldsymbol{\beta},d^{*}_{0}\right)\right]^{2}\right]&\leq
2.M_{0}^{2}.(d^{*}_{0})^{2}.E\left[\|\boldsymbol{X}_{i}\|^{4}\right]+2.M^{2}_{1}.E\left[\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\tau}|\leq\|\boldsymbol{X}_{i}\|.d^{*}_{0}\right).\|\boldsymbol{X}_{i}\|^{2}\right]\\\
&\leq 2.M_{0}^{2}.(d^{*}_{0})^{2}.K_{0}+2.M_{1}^{2}.K_{1}.d^{*}_{0}\\\ &\leq
2.\left(M^{2}_{0}.d^{*}_{0}.K_{0}+M^{2}_{1}.K_{1}\right).d^{*}_{0}.\end{split}$

So both are $O(d^{*}_{0})$ for $d^{*}_{0}$ near zero. For $\boldsymbol{L.4}$,
we have

$\begin{split}\|\psi\left(y_{i},\boldsymbol{X}_{i},\boldsymbol{\beta}_{0}\right)\|^{2}&=\|\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)(-\boldsymbol{X}_{i})\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\|^{2}\\\
&=\|\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)(-\boldsymbol{X}_{i})\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\\\
&-\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)(-\boldsymbol{X}_{i})\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\\\
&+\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)(-\boldsymbol{X}_{i})\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\|^{2}\\\
&=\|\left\\{\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)-\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)\right\\}\left(-\boldsymbol{X}_{i}\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\\\
&+\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)\left(-\boldsymbol{X}_{i}\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\|^{2}\\\
&\leq\|\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}|\leq\|\boldsymbol{X}_{i}\|\|\boldsymbol{\delta}^{*}\|\right)\left(-\boldsymbol{X}_{i}\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\\\
&+\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)\left(-\boldsymbol{X}_{i}\right)\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\|^{2}\\\
&\leq\bigr{\\{}\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}|\leq\|\boldsymbol{X}_{i}\|\|\boldsymbol{\delta}^{*}\|\right).\|\left(\boldsymbol{X}_{i}\right)\|.|\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)|\\\
&+\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right)\|\left(\boldsymbol{X}_{i}\right)\||\rho^{\prime}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)|\bigr{\\}}^{2}.\\\
\end{split}$

From the the assumption $\boldsymbol{A.7},$ we have

$\begin{split}\|\psi\left(y_{i},\boldsymbol{X}_{i},\boldsymbol{\beta}_{0}\right)\|^{2}&\leq\bigr{\\{}M_{1}.\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}|\leq\|\boldsymbol{X}_{i}\|\|\boldsymbol{\delta}^{*}\|\right).\|\left(\boldsymbol{X}_{i}\right)\|+M_{1}.\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right).\|\left(\boldsymbol{X}_{i}\right)\|\bigr{\\}}^{2}\\\
&\leq
2\bigr{\\{}M_{1}^{2}.\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}|\leq\|\boldsymbol{X}_{i}\|\|\boldsymbol{\delta}^{*}\|\right).\|\left(\boldsymbol{X}_{i}\right)\|^{2}+M_{1}^{2}.\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}>0\right).\|\left(\boldsymbol{X}_{i}\right)\|^{2}\bigr{\\}}\\\
&\leq
2\bigr{\\{}M_{1}^{2}.\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}|\leq\|\boldsymbol{X}_{i}\|\|\boldsymbol{\delta}^{*}\|\right).\|\left(\boldsymbol{X}_{i}\right)\|^{2}\\\
&.+M_{1}^{2}.\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}|\leq\|\boldsymbol{X}_{i}\|\|\boldsymbol{\delta}^{*}\|\right)\|\left(\boldsymbol{X}_{i}\right)\|^{2}\bigr{\\}}.\end{split}$

Taking expectations on both sides and then we get

$\begin{split}E\left[\|\psi\left(y_{i},\boldsymbol{X}_{i},\boldsymbol{\beta}_{0}\right)\|^{2}\right]&\leq
4.M_{1}^{2}.K_{1}.\|\boldsymbol{\delta}^{*}\|=K.\end{split}$

Hence, $\boldsymbol{L.3}$, $\boldsymbol{L.4}$ and other conditions of Lemma
7.3 hold, so

$\displaystyle-\sqrt{n}.\lambda_{n}\left(\hat{\boldsymbol{\beta}}_{n}\right)=H_{n}.\sqrt{n}.\left(\hat{\boldsymbol{\beta}}_{n}-\boldsymbol{\beta}\right)=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\psi\left(y_{i},\boldsymbol{X}_{i},\boldsymbol{\beta}_{0}\right)+o_{p}(1).$
(7.18)

Now apply Lyapuanv’s CLT on
$\psi\left(y_{i},\boldsymbol{X}_{i},\boldsymbol{\beta}_{0}\right)$ with mean
$E\left[\psi\left(y_{i},\boldsymbol{X}_{i},\boldsymbol{\beta}_{0}\right)\right]=0,$(using
first-order condition given in $\boldsymbol{L.2}$) and

$s_{n}^{2}=\sum\limits_{i=1}^{n}E\left[\mathds{1}(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0)\boldsymbol{X}_{i}\boldsymbol{X}^{T}_{i}\\{\rho(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0})\\}^{2}\right],$

where $s_{n}^{2}$ is a positive definite matrix for each $n$. To verify
Lyapunav’s condition for some $\eta^{*}>0$,

$\underset{n\to\infty}{\lim}(s_{n})^{-(2+\eta^{*})}\sum\limits_{i=1}^{n}E\|\psi(y_{i},\boldsymbol{X}_{i},\boldsymbol{\beta}_{0})\|^{2+\eta^{*}}=0.$

Now the following

  1. $(1)$

Using assumptions $\boldsymbol{A.7}$ and $\boldsymbol{A.3}$, this leads to
$E\|\psi(y_{i},\boldsymbol{X}_{i},\boldsymbol{\beta}_{0})\|^{2+\eta^{*}}<\infty.$

  2. $(2)$

Since $\frac{s_{n}}{n}\to A$, which is positive definite by the assumption
$\boldsymbol{A.3}.$

Using (1) and (2), Lyapunav’s Condition is

$\underset{n\to\infty}{\lim}\left(\frac{s_{n}}{n}\right)^{-(2+\eta^{*})}\left(\frac{1}{n^{2+\eta^{*}}}\right)\sum\limits_{i=1}^{n}E\|\psi(y_{i},\boldsymbol{X}_{i},\boldsymbol{\beta}_{0})\|^{2+\eta^{*}}=0$

it implies that

$(s_{n}^{2})^{-1/2}\sum\limits_{i=1}^{n}\psi\left(y_{i},\boldsymbol{X}_{i},\boldsymbol{\beta}_{0}\right)\xrightarrow{D}N\left(0,I\right).$

From (7.18), we have

$\displaystyle\left(\frac{s_{n}^{2}}{n}\right)^{-1/2}H_{n}.\sqrt{n}\left(\hat{\boldsymbol{\beta}}_{n}-\boldsymbol{\beta}\right)$
$\displaystyle=\left(s_{n}^{2}\right)^{-1/2}\sum\limits_{i=1}^{n}\psi\left(y_{i},\boldsymbol{X}_{i},\boldsymbol{\beta}_{0}\right)+o_{p}(1).$

By Slutsky’s theorem,

$\Rightarrow
D_{n}^{-1/2}.H_{n}.\sqrt{n}\left(\hat{\boldsymbol{\beta}}_{n}-\boldsymbol{\beta}\right)\xrightarrow{D}N\left(0,I\right).$

∎

$\textbf{B.3}.$ Strong Consistency of $\hat{D}_{n}$ and $\hat{H}_{n}$

###### Proof of Theorem 3.3..

In this demonstration, only the strong consistency of $\hat{H_{n}}$ defined in
(3.3) will be presented, and the consistency of $\hat{D_{n}}$ defined in (3.4)
can be demonstrated similarly. Consider the $(p,q)th$ element of
$\hat{H}_{n}-H_{n}$, which is defined as,

$\begin{split}a_{p,q}&=\frac{1}{n}\sum\limits_{i=1}^{n}\mathds{1}\left(\boldsymbol{X}_{i}^{T}\hat{\boldsymbol{\beta}}_{n}>0\right)X_{pi}X_{qi}\left\\{\rho^{\operatorname*{\prime\prime}}\left(\xi_{i}-\boldsymbol{X}^{T}_{i}\hat{\boldsymbol{\delta}_{n}}\right)\right\\}\\\
&-\frac{1}{n}\sum\limits_{i=1}^{n}E\Bigr{[}\mathds{1}\left(\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}_{0}>0\right)X_{pi}X_{qi}\left\\{\rho^{\operatorname*{\prime\prime}}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\right\\}\Bigr{]}\end{split}$

where
$\hat{\boldsymbol{\delta}}_{n}=\hat{\boldsymbol{\beta}}_{n}-\boldsymbol{\beta}_{0}$.
Now define,

$\Delta_{n}=\frac{1}{n}\sum\limits_{i=1}^{n}\mathds{1}\left(\boldsymbol{X_{i}}^{T}\boldsymbol{\beta}_{0}>0\right)\boldsymbol{X}_{i}\boldsymbol{X}_{i}^{T}\left\\{\rho^{\operatorname*{\prime\prime}}\left(y_{i}-\boldsymbol{X}_{i}^{T}\boldsymbol{\beta}_{0}\right)\right\\},$

Then, by adding and subtracting the $(p,q)th$ element of $\Delta_{n}$ in (7),
we get

$\begin{split}a_{p,q}&=\frac{1}{n}\sum\limits_{i=1}^{n}\bigr{[}\mathds{1}\bigr{(}\boldsymbol{X}^{T}_{i}\hat{\boldsymbol{\beta}}_{n}>0\bigr{)}-\mathds{1}\bigr{(}\boldsymbol{X}^{T}_{i}\hat{\boldsymbol{\beta}}_{0}>0\bigr{)}\bigr{]}X_{pi}X_{qi}\bigr{\\{}\rho^{\operatorname*{\prime\prime}}\bigr{(}\xi_{i}-\boldsymbol{X}_{i}^{T}\hat{\boldsymbol{\delta}_{n}}\bigr{)}-\rho^{\operatorname*{\prime\prime}}\left(\xi_{i}\right)\bigr{\\}}\\\
&-\frac{1}{n}\sum\limits_{i=1}^{n}\bigr{[}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)X_{pi}X_{qi}\bigr{\\{}\rho^{\operatorname*{\prime\prime}}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\bigr{\\}}\bigr{]}\\\
&-E\bigr{[}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)X_{pi}X_{qi}\bigr{\\{}\rho^{\operatorname*{\prime\prime}}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\bigr{\\}}\bigr{]}\\\
\Rightarrow|a_{p,q}|&=\frac{1}{n}\sum\limits_{i=1}^{n}\bigr{\\{}\mathds{1}\bigr{(}|\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}|\leq\|\boldsymbol{X}_{i}\|\|\hat{\boldsymbol{\delta}}_{n}\|\bigr{)}.\|\boldsymbol{X}_{i}\|^{2}\bigr{\\}}\bigr{|}\rho^{\operatorname*{\prime\prime}}\bigr{(}\xi_{i}-\boldsymbol{X}^{T}_{i}\hat{\boldsymbol{\delta}}_{n}\bigr{)}-\rho^{\operatorname*{\prime\prime}}\left(\xi_{i}\right)\bigr{|}\\\
&+\Bigr{|}\frac{1}{n}\sum\limits_{i=1}^{n}\left[\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\|\boldsymbol{X}_{i}\|^{2}\left\\{\rho^{\operatorname*{\prime\prime}}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\right\\}\right]\\\
&-E\bigr{[}\mathds{1}\bigr{(}\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\bigr{)}\|\boldsymbol{X}_{i}\|^{2}\bigr{\\{}\rho^{\operatorname*{\prime\prime}}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\bigr{\\}}\bigr{]}\Bigr{|}\\\
&\leq\frac{2M_{0}}{n}\sum\limits_{i=1}^{n}\bigr{\\{}\mathds{1}\bigr{(}|\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}|\leq\|\boldsymbol{X}_{i}\|\|\hat{\boldsymbol{\delta}}_{n}\|\bigr{)}.\|\boldsymbol{X}_{i}\|^{2}\bigr{\\}}\\\
&+\Bigr{|}\frac{1}{n}\sum\limits_{i=1}^{n}\bigr{[}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\|\boldsymbol{X}_{i}\|^{2}\bigr{\\{}\rho^{\operatorname*{\prime\prime}}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\bigr{\\}}\bigr{]}\\\
&-E\left[\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\|\boldsymbol{X}_{i}\|^{2}\left\\{\rho^{\operatorname*{\prime\prime}}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\right\\}\right]\Bigr{|}.\\\
\end{split}$ (7.19)

Now note that since $\hat{\boldsymbol{\delta}}_{n}\overset{a.s}{\to}0$ as
$n\to\infty.$ For any given $\boldsymbol{\tau_{0}}>0$, $\exists\;n_{0}\in N$
$s.t$ $\|\hat{\boldsymbol{\delta}}_{n}\|<\boldsymbol{\tau_{0}},\;a.s\;\forall
n>n_{0}.$ Thus, for $n\geq n_{0},$ we have

$\begin{split}|a_{pq}|&\leq\frac{2M_{0}}{n}\sum\limits_{i=1}^{n}\mathds{1}\left(|\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}|\leq\|\boldsymbol{X}_{i}\|.\boldsymbol{\tau_{0}}\right).\|\boldsymbol{X}_{i}\|^{2}\\\
&+\Bigr{|}\frac{1}{n}\sum\limits_{i=1}^{n}\bigr{[}\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\|\boldsymbol{X}_{i}\|^{2}\left\\{\rho^{\operatorname*{\prime\prime}}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\right\\}\bigr{]}\\\
&-E\left[\mathds{1}\left(\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}>0\right)\|\boldsymbol{X}_{i}\|^{2}\left\\{\rho^{\operatorname*{\prime\prime}}\left(y_{i}-\boldsymbol{X}^{T}_{i}\boldsymbol{\beta}_{0}\right)\right\\}\right]\Bigr{|}.\end{split}$

By the assumption $\boldsymbol{A.6}$ and using the Strong Law of Large
numbers,

$|a_{pq}|\leq 2.M_{0}K_{1}.\tau_{0}.$

Since $\tau_{0}$ is arbitrary, one can make $|a_{pq}|$ arbitrary small by
choosing $\tau_{0}$ arbitrarily small.Thus, as $\tau_{0}\to 0$, we have
$a_{pq}\overset{a.s}{\to}0$ as $n\to\infty.$ Hence
$\hat{H_{n}}-H_{n}\overset{a.s}{\to}0,$ as $n\to\infty.$

∎

## References

  * Acemoglu et al. [2012] Daron Acemoglu, Simon Johnson, and James A. Robinson.  The colonial origins of comparative development: An empirical investigation: Reply.  _The American Economic Review_ , 102(6):3077–3110, 2012.  ISSN 00028282.  URL http://www.jstor.org/stable/41724682.
  * Amemiya [1973] Takeshi Amemiya.  Regression analysis when the dependent variable is truncated normal.  _Econometrica_ , 41(6):997–1016, 1973.  ISSN 00129682, 14680262.  URL http://www.jstor.org/stable/1914031.
  * Amemiya [1979] Takeshi Amemiya.  The estimation of a simultaneous-equation tobit model.  _International economic review_ , pages 169–181, 1979.
  * Angrist [1990] Joshua D. Angrist.  Lifetime earnings and the vietnam era draft lottery: Evidence from social security administrative records.  _The American Economic Review_ , 80(3):313–336, 1990.  ISSN 00028282.  URL http://www.jstor.org/stable/2006669.
  * Arabmazar and Schmidt [1981] Abbas Arabmazar and Peter Schmidt.  Further evidence on the robustness of the tobit estimator to heteroskedasticity.  _Journal of Econometrics_ , 17(2):253–258, 1981\.
  * Arabmazar and Schmidt [1982] Abbas Arabmazar and Peter Schmidt.  An investigation of the robustness of the tobit estimator to non-normality.  _Econometrica_ , 50(4):1055–1063, 1982.  ISSN 00129682, 14680262.  URL http://www.jstor.org/stable/1912776.
  * Bickel [1975] P. J. Bickel.  One-step huber estimates in the linear model.  _Journal of the American Statistical Association_ , 70(350):428–434, 1975.  ISSN 01621459.  URL http://www.jstor.org/stable/2285834.
  * Blundell and Powell [2007] Richard Blundell and James L Powell.  Censored regression quantiles with endogenous regressors.  _Journal of Econometrics_ , 141(1):65–83, 2007\.
  * Bowden Roger J. [Roger John] D. A. (Darrell A.) Bowden Roger J. (Roger John) Turkington.  _Instrumental variables / Roger J. Bowden and Darrell A. Turkington_.  Cambridge University Press, 1984.  ISBN 0521385822.
  * Buckley and James [1979] Jonathan Buckley and Ian James.  Linear regression with censored data.  _Biometrika_ , 66(3):429–436, 1979.  ISSN 00063444.  URL http://www.jstor.org/stable/2335161.
  * Camponovo [2015] Taisuke Camponovo, Lorenzo; Otsu.  Robustness of bootstrap in instrumental variable regression.  _Econometric Reviews_ , 32:352–393., 03 2015.
  * Card [1995] David Card.  Using geographic variation in college proximity to estimate the return to schooling.  NBER Working Papers 4483, National Bureau of Economic Research, Inc, 1995\.  URL https://EconPapers.repec.org/RePEc:nbr:nberwo:4483.
  * Chernozhukov et al. [2015] Victor Chernozhukov, Iván Fernández-Val, and Amanda E Kowalski.  Quantile regression with censoring and endogeneity.  _Journal of Econometrics_ , 186(1):201–221, 2015\.
  * Cramer [(1946] H.(1946) Cramer.  _L aTeX: Mathematical Methods of Statistics_.  Princeton University Press, Princeton, (1946).
  * Das [2002] Mitali Das.  _Estimators and inference in a censored regression model with endogenous covariates_.  Department of Economics, Columbia University, 2002.
  * Davies et al. [2013] Neil M. Davies, George Davey Smith, Frank Windmeijer, and Richard M. Martin.  Cox-2 selective nonsteroidal anti-inflammatory drugs and risk of gastrointestinal tract complications and myocardial infarction: An instrumental variable analysis.  _Epidemiology_ , 24(3):352–362, 2013.  ISSN 10443983.  URL http://www.jstor.org/stable/23486748.
  * Dhar and Shalabh [2022] Subhra Sankar Dhar and Shalabh.  GIVE statistic for goodness of fit in instrumental variables models with application to COVID data.  _Nature Scientific Reports_ , page 9472, 2022.  doi: 10.3150/22-BEJ1509.  URL https://doi.org/10.3150/22-BEJ1509.
  * Dhar and Wu [2023] Subhra Sankar Dhar and Weichi Wu.  Comparing time varying regression quantiles under shift invariance.  _Bernoulli_ , 29(2):1527 – 1554, 2023.  doi: 10.3150/22-BEJ1509.  URL https://doi.org/10.3150/22-BEJ1509.
  * Dimitriadis et al. [2023] Timo Dimitriadis, Tobias Fissler, and Johanna Ziegel.  Characterizing M-estimators.  _Biometrika_ , page asad026, 05 2023.  ISSN 1464-3510.  doi: 10.1093/biomet/asad026.  URL https://doi.org/10.1093/biomet/asad026.
  * Ducan [1986] GM Ducan.  A robust censored regression estimator.  _Journal of Econometrics_ , 32:5–34, 1986.
  * Fish et al. [2010] Fish, Jason, Ettner Susan, Ang, Alfonso, and Brown Arleen.  Association of perceived neighborhood safety on body mass index.  _American Journal of public health_ , 100:2296–303, 11 2010\.  doi: 10.2105/AJPH.2009.183293.
  * Goldberger [1983] Arthur S Goldberger.  Abnormal selection bias.  In _Studies in econometrics, time series, and multivariate statistics_ , pages 67–84. Elsevier, 1983.
  * Hampel et al. [1986] Frank Hampel, Elvezio Ronchetti, Peter Rousseeuw, and Werner Stahel.  _Robust Statistics: The Approach Based on Influence Functions_.  John Wiley and Sons, 03 1986.  ISBN 9780471735779.  doi: 10.1002/9781118186435.
  * Heckman [1976] James J. Heckman.  The common structure of statistical models of truncation, sample selection and limited dependent variables and a simple estimator for such models.  _Annals of Economic and Social Measurement_ , 5(4):475 – 492, 1976.  URL http://www.nber.org/books/aesm76-4.
  * Heckman [1978] James J. Heckman.  Dummy endogenous variables in a simultaneous equation system.  _Econometrica_ , 46(4):931–959, 1978.  ISSN 00129682, 14680262.  URL http://www.jstor.org/stable/1909757.
  * Horowitz [1986] Joel L. Horowitz.  A distribution-free least squares estimator for censored linear regression models.  _Journal of Econometrics_ , 32(1):59–84, 1986\.  ISSN 0304-4076.  doi: https://doi.org/10.1016/0304-4076(86)90012-6.  URL https://www.sciencedirect.com/science/article/pii/0304407686900126.
  * Huber [1964] Peter J. Huber.  Robust estimation of a location parameter.  _The Annals of Mathematical Statistics_ , 35(1):73–101, 1964.  ISSN 00034851.  URL http://www.jstor.org/stable/2238020.
  * Huber [1967] Peter J Huber.  Under nonstandard conditions.  In _Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability: Weather Modification; University of California Press: Berkeley, CA, USA_ , page 221, 1967.
  * Huber [1973] Peter J. Huber.  Robust regression: Asymptotics, conjectures and monte carlo.  _The Annals of Statistics_ , 1(5):799–821, 1973\.  ISSN 00905364.  URL http://www.jstor.org/stable/2958283.
  * Huber et al. [1981] P.J. Huber, J. Wiley, and W. InterScience.  _Robust statistics_.  Wiley New York, 1981.
  * Hurd [1979] Michael Hurd.  Estimation in truncated samples when there is heteroscedasticity.  _Journal of Econometrics_ , 11(2-3):247–258, 1979\.
  * Jelena Bradic [2019] Jiaqi Guo Jelena Bradic.  Generalized m-estimators for high-dimensional tobit i models.  _Electronic Journal of Statistics_ , 13(1):582–645, 2019.  ISSN 1935-7524.  doi: https://doi.org/10.1214/18-EJS1463.
  * Jin [2007] Zhezhen Jin.  M-estimation in regression models for censored data.  _Journal of Statistical Planning and Inference_ , 137(12):3894–3903, 2007.  ISSN 0378-3758.  doi: https://doi.org/10.1016/j.jspi.2007.04.008.  URL https://www.sciencedirect.com/science/article/pii/S0378375807001589.  5th St. Petersburg Workshop on Simulation, Part II.
  * Jureckova [1977] Jana Jureckova.  Asymptotic relations of m-estimates and r-estimates in linear regression model.  _The Annals of Statistics_ , 5(3):464–472, 1977\.  ISSN 00905364.  URL http://www.jstor.org/stable/2958897.
  * Kim et al. [2011] Daniel Kim, Christopher Baum, Michael Ganz, S Subramanian, and Ichiro Kawachi.  The contextual effects of social capital on health: A cross-national instrumental variable analysis.  _Social science and medicine (1982)_ , 73:1689–97, 10 2011\.  doi: 10.1016/j.socscimed.2011.09.019.
  * Koenker and Bassett [1982] Roger W. Koenker and Gilbert W. Jr. Bassett.  Robust tests for heteroscedasticity based on regression quantiles.  _Econometrica_ , 50:43–61, 1982.
  * Lai and Ying [1994] Tze Leung Lai and Zhiliang Ying.  A missing information principle and m-estimators in regression analysis with censored and truncated data.  _The Annals of Statistics_ , 22(3):1222–1255, 1994.  ISSN 00905364.  URL http://www.jstor.org/stable/2242224.
  * Lee [1992] Myoung-Jae Lee.  Winsorized mean estimator for censored regression.  _Econometric Theory_ , 8(3):368–382, 1992.  ISSN 02664666, 14694360.  URL http://www.jstor.org/stable/3532354.
  * Maddala and Nelson [1975] Gangadharrao S Maddala and Forrest D Nelson.  _Specification errors in limited dependent variable models_.  National Bureau of Economic Research, 1975.
  * Miller [1976] Rupert G. Miller.  Least squares regression with censored data.  _Biometrika_ , 63(3):449–464, 1976.  ISSN 00063444.  URL http://www.jstor.org/stable/2335722.
  * Newey [1987] Whitney K. Newey.  Efficient estimation of limited dependent variable models with endogenous explanatory variables.  _Journal of Econometrics_ , 36(3):231–250, 1987\.  ISSN 0304-4076.  doi: https://doi.org/10.1016/0304-4076(87)90001-7.  URL https://www.sciencedirect.com/science/article/pii/0304407687900017.
  * Oberhofer [1982] Walter Oberhofer.  The Consistency of Nonlinear Regression Minimizing the $L_{1}$-Norm.  _The Annals of Statistics_ , 10(1):316 – 319, 1982.  doi: 10.1214/aos/1176345716.  URL https://doi.org/10.1214/aos/1176345716.
  * Pokropek [2016] Artur Pokropek.  Introduction to instrumental variables and their application to large-scale assessment data.  _Large-scale Assessments in Education_ , 4, 12 2016.  doi: 10.1186/s40536-016-0018-2.
  * Powell [1983] James L. Powell.  The asymptotic normality of two-stage least absolute deviations estimators.  _Econometrica_ , 51(5):1569–1575, 1983.  ISSN 00129682, 14680262.  URL http://www.jstor.org/stable/1912290.
  * Powell [1984] James L Powell.  Least absolute deviations estimation for the censored regression model.  _Journal of Econometrics_ , 25(3):303–325, 1984\.  ISSN 0304-4076.  doi: https://doi.org/10.1016/0304-4076(84)90004-6.  URL https://www.sciencedirect.com/science/article/pii/0304407684900046.
  * Powell [1986] James L. Powell.  Symmetrically trimmed least squares estimation for tobit models.  _Econometrica_ , 54(6):1435–1460, 1986.  ISSN 00129682, 14680262.  URL http://www.jstor.org/stable/1914308.
  * Ritov [1990] Ya’acov Ritov.  Estimation in a linear regression model with censored data.  _The Annals of Statistics_ , 18, 03 1990.  doi: 10.1214/aos/1176347502.
  * Rivers and Vuong [1988] Douglas Rivers and Quang H. Vuong.  Limited information estimators and exogeneity tests for simultaneous probit models.  _Journal of Econometrics_ , 39(3):347–366, 1988\.  ISSN 0304-4076.  doi: https://doi.org/10.1016/0304-4076(88)90063-2.  URL https://www.sciencedirect.com/science/article/pii/0304407688900632.
  * Ruppert and Carroll [1980] David Ruppert and Raymond J. Carroll.  Trimmed least squares estimation in the linear model.  _Journal of the American Statistical Association_ , 75(372):828–838, 1980.  ISSN 01621459.  URL http://www.jstor.org/stable/2287169.
  * Saleh and Saleh [2022] Resve A. Saleh and A. K. Md. Ehsanes Saleh.  Statistical properties of the log-cosh loss function used in machine learning, 2022.
  * Shalabh and Sankar [2023] Shalabh and Dhar Subhra Sankar.  Testing the goodness of fit in instrumental variables models.  In _G Families of Probability Distributions_ , pages 330–343. Taylor & Francis, 2023.
  * Smith and Blundell [1986] Richard J. Smith and Richard W. Blundell.  An exogeneity test for a simultaneous equation tobit model with an application to labor supply.  _Econometrica_ , 54(3):679–685, 1986.  ISSN 00129682, 14680262.  URL http://www.jstor.org/stable/1911314.
  * Tobin [1958] J. Tobin.  Liquidity preference as behavior towards risk.  _The Review of Economic Studies_ , 25(2):65–86, 1958.  ISSN 00346527, 1467937X.  URL http://www.jstor.org/stable/2296205.
  * Vansteelandt and Didelez [2018] Stijn Vansteelandt and Vanessa Didelez.  Improving the robustness and efficiency of covariate‐adjusted linear instrumental variable estimators.  _Scandinavian Journal of Statistics_ , 45:941 – 961, 2018\.  URL https://api.semanticscholar.org/CorpusID:88515561.
  * White [1980] Halbert White.  Nonlinear regression on cross-section data.  _Econometrica_ , 48(3):721–746, 1980.  ISSN 00129682, 14680262.  URL http://www.jstor.org/stable/1913132.
  * Wiesenfarth et al. [2014] Manuel Wiesenfarth, Matias Hisgen, Thomas Kneib, and Carmen Cadarso-Suárez.  Bayesian nonparametric instrumental variables regression based on penalized splines and dirichlet process mixtures.  _Journal of Business and Economic Statistics_ , 32:468–482, 07 2014.  doi: 10.1080/07350015.2014.907092.
  * Zhou [1992] Mai Zhou.  M-estimation in censored linear models.  _Biometrika_ , 79(4):837–841, 1992.  ISSN 00063444.  URL http://www.jstor.org/stable/2337240.