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\begin{document}
\title{ \bf Supplementary Material on   ``Uniform Bahadur Representation for Nonparametric
Censored Quantile Regression: A Redistribution-of-Mass Approach"}
\author{ Efang Kong \\
%\thanks{%
%\textrm{ School of Mathematics, Statistics   and Actuarial Science, University of Kent at Canterbury, UK. %
%  E-mail: \texttt{E.kong@kent.ac.uk.}}}\\
{\normalsize \textit{University of Electronic Science   and Technology, China}}\\
\normalsize   and  \textit{University of Kent at Canterbury, UK }\\ \\
Yingcun Xia \\
%\thanks{%
%\rm Department of Statistics   and Applied Probability   and RMI, National
%University of Singapore, Singapore.
% E-mail: \texttt{staxyc@nus.edu.sg.}}\\
{\normalsize \textit{National University of Singapore, Singapore}} \\
\normalsize   and  \textit{University of Electronic Science   and Technology, China} }
\date{}
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\maketitle


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This section contains
detailed proofs of all the theoretical results from the main text, together with additional lemmas and propositions. 
Equations which first appear in this section would be labelled as $\mbox{(A??)}$
while those which have already appeared in the main text
 are referenced by the Arabic number already assigned to them.
Also in this section, symbols $\sup\limits_{\x }$, $\sup\limits_{\x,\tau }$
or $\sup\limits_{\x,\tau,\bbeta}$ stand  for $\sup\lim$ taken with respect to
$\x\in \mathcal{D},\tau\in [\tau^*,1-\tau^*],$ and $\bbeta\in R^{n(A)}$.
Similarly, the phrase {\it  uniformly in $\x,\, \tau$} means
{\it  uniformly in $\x\in \mathcal{D},\tau\in [\tau^*,1-\tau^*].$}\\

\noindent{\bf Proof of Lemma 1}.%\ref{Gonza}
  With $h_n\to 0,\ nh_n^p/\log n\to \infty$ as $n\to \infty$, and the sequence of weights $B_{nk}(\x)$ chosen to be the Gasser-Muller's type weights, Theorem 2.2 of  Gonzalez-Manteigaa and  Cadarso-Suarez (1994) states that
$$
\sup\limits_{\x,\tau} |\hat F_{KM}(t|\x)-F_0(t|\x)|
=O\Big(h_n^2+\Big(\frac{\log n}{nh_n^p}\Big)^{1/2}\Big),
$$
$$
\hat F_{KM}(t|\x)-F_0(t|\x)=
\frac{\{1- F_0(t|\x)\}}{nh_n^p}\sum\limits_{k=1}^n \tilde K_{ h_n}(\X_{k{\x}})\varphi(Y_k,d_k,t,\x)
+O\Big(h_n^2+\Big(\frac{\log n}{nh_n^p}\Big)^{3/4} \Big),
$$
%\endn
uniformly in $\x\in\mathcal{D}$ and $t$.
Their line of arguments is still valid with
other forms of weights,  as long as $\sum_{k=1}^n B_{nk}(\x)=1$. Specifically, for
the generalized K-M  estimator $\hat F_{KM}(.|\X_i)$ of
(10) 
%(\ref{favor})
 with local polynomial weights, we have, with probability one
\beginy
\n& \sup\limits_{\x,\tau} |\hat F_{KM}(t|\x)-F_0(t|\x)|
=O\Big(h_n^{s_3}+\Big(\frac{\log n}{nh_n^p}\Big)^{1/2}\Big), & \\
&
\hat F_{KM}(t|\x)-F_0(t|\x)=
\frac{\{1- F_0(t|\x)\}}{nh_n^p}\sum\limits_{k=1}^n \tilde B_{nk}(\x)\varphi(Y_k,d_k,t,\x)
+O\Big(h_n^{s_3}+\Big(\frac{\log n}{nh_n^p}\Big)^{3/4} \Big). &
\label{bean2}
\endy
Note that with $nh_n^{p+4/3s_3}/\log n<\infty$ (Assumption [A7]),
the term  $h_n^{s_3}$ can be absorbed to form part of $O\Big( ({nh_n^p}/{\log n} )^{-3/4} \Big)$.

Plug  (\ref{bean2}) into the definition of $\hat F^S_n(.|.)$, and
we have again with probability one,
\beginy
\n \hat F^S_n(t|\x)&&\hspace{-.5cm}= \int \Big\{F_0(s|\x) + \frac{1}{nh_n^p}
\sum\limits_{k=1}^n\tilde B_{nk}(\x) \{1- F_0(s|\x)\}\varphi(Y_k,d_k,s,\x)\Big\}
 \bar{K}(\frac{s-t}{h_{1n}})\frac{1}{h_{1n}}ds\\
 &&\hspace{-.5cm}=F_0(t|\x)+\frac{\{1- F_0(t|\x)\}}{nh_n^p}\sum\limits_{k=1}^n
 \tilde B_{nk}(\x)\chi(Y_k,d_k,t,\x)
+O\Big(h_{1n}^2+\Big(\frac{\log n}{nh_n^p}\Big)^{3/4} \Big),\ \ \ \label{xihu}
 \endy
 where
\beginn
\chi(Y_k,d_k,t,\x)=\int\{1- F_0(s|\x)\}\varphi(Y_k,d_k,s,\x)
\frac{1}{h_{1n}}\bar{K}\Big(\frac{s-t}{h_{1n}}\Big)ds.
\endn
Since $E[\chi(Y_k,d_k,t,\x)]=0$, $E|\varphi(Y_k,d_k,t+h_{1n}s,\x)-\varphi(Y_k,d_k,t,\x)|=O(h_{1n})$,
and  $h_{1n}=O((nh_n^p/\log n)^{-1/2})$ (Assumption [A8]),
we have, with probability one,
\beginy
&&\n\frac{\{1- F_0(t|\x)\}}{nh_n^p}\sum\limits_{k=1}^n \tilde B_{nk}(\x)\chi(Y_k,d_k,t,\x)-
\frac{\{1- F_0(t|\x)\}}{nh_n^p}\sum\limits_{k=1}^n \tilde B_{nk}(\x)\varphi(Y_k,d_k,t,\x)\\
&&=O(\{\frac{h_{1n}\log n}{nh_n^p}\}^{1/2})
=O(\{\frac{\log n}{nh_n^p}\}^{3/4}). \label{baby}
\endy
With  $h_{1n}^2=O(h_n^{s_3})$ (Assumption [A7]),
Lemma 1 %\ref{Gonza} 
thus follows from (\ref{xihu}) and (\ref{baby}).
  $\hspace{\fill}\square$

To keep the exposition simple, we illustrate by focusing on the case where  $K(.)$  is the uniform density on $[-1,1]^p$.
The arguments could be easily adapted   for the case of a general
symmetric probability density in  $R^p$  with a compact support. Rewrite (9) %(\ref{tiger1}) 
as
\beginy
\sum\limits_{i\in S_n(\x)}\Big[w_{in}(\tau)
\rho_\tau(Y_i-\bbeta^\top\bmu_n(\X_{i\x}) )
+(1-w_{in}(\tau))\rho_\tau(Y^{+\infty}-\bbeta^\top
\bmu_n(\X_{i\x}))\Big],
\label{tiger2}
\endy
where  the index set
$S_n(\x)=\{i:1\le i\le n,\ |\X_{i\x}|\le \delta_n\}$ with cardinality  $N_n(\x)=\sharp(S_n(\x)).$
Denote the minimizer of (\ref{tiger2}) as $\hat{\bbeta}_{n\tau}(\x)$; and to facilitate the discussion on its properties,
 we implicitly assume the following simple facts; see also Chaudhuri (1991) and Kong , Linton and Xia (2013).

[FACT1] For any positive integer $m$,
let $\x$ be a vector in $R^m$ and $p(\x)$ be an arbitrary nonzero polynomial
in $\x$. Then, the Lebesgue measure of the set $\{\x|p(\x)=0\}$
is $0.$

[FACT2] Let $\X^{(1)},\cdots,\X^{(m)}$ be  independent random vectors in $R^m$ with the property that  $Prob(\X^{(i)}\in {\bf H})=0$  for  all $i=1,\cdots,n,$  and any given linear subspace
 ${\bf H}$ of $R^m$ such that $\mbox{dim}({\bf H})\le m-1$. Then the collection
 $\{\X^{(1)},\cdots,\X^{(m)}\}$  is
almost surely linearly independent.

[FACT3] For any $\x\in \mathcal{D}$,  denote by $\omega_{\delta_n}(\t,\x)$, the
conditional density of  $\delta^{-1}_n(\X-{\x})$, given that $|\X-{\x}|\le \delta_n$. Then $\omega_{\delta_n}(\t,\x)$
converges to the uniform density
on $[-1,1]^p$ uniformly
 in $\t$ as well as in $\x\in\mathcal{D}$.

For any given $\x\in\mathcal{D}$, let $[DX_n(\x)]$ be the $N_n(\x)\times n(A)$ matrix
with rows  $\{\bmu^\top_n(\X_{i\x}),i\in S_n(\x)\}$,
and $VY_n(\x)$ be the $N_n(\x)\times 1$ vector  $\{Y_i,i\in S_n(\x)\}.$
For any subset ${\bf h}\subset S_n(\x)$ such that $\sharp({\bf h})=n(A),$ denote by $[DX_n(\x,{\bf h})]$, the
 $ n(A)\times n(A)$ matrix
with rows $\{\bmu^\top_n(\X_{i\x}),\ i\in {\bf h}\}$, and by $VY_n(\x,{\bf h}),$
the $n(A)\times 1$ vector  $\{Y_i,i\in {\bf h}\}$. Define
\beginn
{\rm H}_n(\x)=\{{\bf h}: {\bf h}\subset S_n(\x),\ \sharp(h)=n(A), [DX_n(\x,{\bf h})]
\mbox{  has full rank}\}.
\endn
The following two propositions describe two critical facts about
the minima of  (\ref{tiger2}).
\begin{Proposition}\label{haiqing}
For any given $\x\in\mathcal{D}$ such that  $[DX_n(\x)]$  has rank $n(A)$, then
there is a subset ${\bf h}\in {\rm H}_n(\x)$,  such that  (\ref{tiger2})
has at least one minimum  of the form
\beginn
\hat{\bbeta}_{n\tau}(\x)=[DX_n(\x,{\bf h})]^{-1}VY_n(\x,{\bf h}).
\endn
\end{Proposition}

\noindent{\bf Proof. } Similar to the proof for   Theorem 3.1 in
Koenker and Bassett (1978),
 this roughly follows from the following linear programming formulation
of the minimization of   (\ref{tiger2}):
\beginy
\min \limits_{u_i^+,\ {u}_i^-}
\sum\limits_{i\in S_n(\x)}
w_{in}(\tau)[\tau u_i^++(1-\tau){u}_i^- ]+\sum\limits_{i\in S_n(\x)} \tau(1-w_{in}(\tau)) (u_i^+-{u}_i^-),
\label{sci}
\endy
subject to
\beginn
&&Y_i-\bbeta^\top\bmu_n(\X_{i\x})=u_i^+-{u}_i^-;\\
&& \bbeta\in R^{n(A)};\ u_i^+,\ {u}_i^-\ge 0,\ i=1,\cdots,N_n(\x).
\endn
Note that we have implicitly used the following fact
\beginn
\rho_\tau\{Y^{+\infty}-\bbeta^\top
\bmu_n(\X_{i\x}) \}=\tau(Y^{+\infty}-Y_i+u_i^+-{u}_i^-).
\endn
An equivalence to problem (\ref{sci}) is
\beginy
\min \limits_{u_i^+,\ {u}_i^-}
\sum\limits_{i\in S_n(\x)}[\tau u_i^++(w_{in}(\tau)-\tau){u}_i^-], \label{P}
\endy
 subject to
  \beginn
&&VY_n(\x)-[DX_n(\x)] \bbeta=\mathbf{ u}^+-\mathbf{ u}^-;\\
&&(\bbeta,\mathbf{ u}^+,\mathbf{ u}^-)\in R^{n(A)}\times  R_+^{2N_n(\x)};\\
&& \mathbf{ u}^+=(u_i^+,i\in S_n(\x));
\quad \mathbf{u}^-=({u}_i^-,i \in S_n(\x)).
 \endn
According to Theorem 7.7.4 in Gill, Murray and Wright (1991),   problem (\ref{P}) has a vertex solution,
i.e. there exists some subset ${\bf h}\in {\rm H}_n(\x)$,  such that
$u_i^+={u}_i^-=0,\ i\in {\bf h}$, which is equivalent to the statement in Proposition \ref{haiqing}.
$\hspace{\fill}\square$


%{\color{blue} More details on the vertex solution. using Proposition 6.1 in Koenker.
%In problem (\ref{P}), the first constraint
%\beginn
%VY_n(\x)-[DX_n(\x)] \bbeta=\mathbf{ u}^+-\mathbf{ u}^-
%\endn
%means that
%\beginn
%\bbeta =  ([DX_n(\x)]^\top[DX_n(\x)])^{-1}
%[DX_n(\x)]^\top VY_n(\x)-\mathbf{ u}^++\mathbf{ u}^-,
%\endn
%assuming  $[DX_n(\x)]$ is of full column rank.
%Substitute this  for $\bbeta$  into the first constraint, we have
%equivalently as the constraints in problem (\ref{P}),
%$$
%\left(
%\begin{array}{cc}
%{\bf I}_n-{\bf P}_X&{\bf P}_X-{\bf I}_n\\
%{\bf P}_X-{\bf I}_n&{\bf I}_n-{\bf P}_X\\
%{\bf I}_n&{\bf 0}\\
%{\bf 0}&{\bf I}_n
%\end{array}\right)
%\left(
%\begin{array}{c}\mathbf{ u}^+\\
%\mathbf{ u}^-
%\end{array}\right)\ge \left(
%\begin{array}{c}
%\{{\bf I}_n-{\bf P}_X\}VY_n(\x)\\
%\{{\bf P}_X-{\bf I}_n\}VY_n(\x)\\
%{\bf 0}\\
%{\bf 0}
%\end{array}\right)
%$$
%where ${\bf P}_X= [DX_n(\x)]([DX_n(\x)]^\top[DX_n(\x)])^{-1}
%[DX_n(\x)]^\top$, and ${\bf I}_n$ and ${\bf 0}$ are respectively $n\times n$ identity matrix and
%$n\times 1$ zero vector.
%Therefore, problem (\ref{P}) is a typical linear optimization problem subject to linear constraint, which
%according to Theorem 7.7.4 in Gill, Murray and Wright (1991), must have a vertex solution.
%A vertex solution is defined as a solution whose submatrix of active constraint contains at least one subset of
%in this case $2n$ linearly independent rows. Due to the fact that
%$rank({\bf I}_n-{\bf P}_X)=n-n(A)$ the special structure of the constraint matrix
%$$
%\left(
%\begin{array}{cc}
%{\bf I}_n-{\bf P}_X&{\bf P}_X-{\bf I}_n\\
%{\bf P}_X-{\bf I}_n&{\bf I}_n-{\bf P}_X\\
%{\bf I}_n&{\bf 0}\\
%{\bf 0}&{\bf I}_n
%\end{array}\right),$$  any submatrix of which must at least contain  $n+n(A)$ rows of ${\bf I}_{2n},$
%were it to contain $2n$ linearly independent rows. This implies
%there must exist ${\bf h}\in {\rm H}_n(\x)$ of cardinality  $n(A)$,  such that
%$u_i^+={u}_i^-=0,\ i\in {\bf h}$.
%
%The dual to (\ref{{sci}}) is
%\beginn
%\max\limits_{ {\bf b}\in R^{2n}} {\bf b}^\top \left(
%\begin{array}{c}
%\{{\bf I}_n-{\bf P}_X\}VY_n(\x)\\
%\{{\bf P}_X-{\bf I}_n\}VY_n(\x)
%\end{array}\right),
%\endn
% subject to
%  \beginn
%\left(
%\begin{array}{cc}
%{\bf I}_n-{\bf P}_X&{\bf P}_X-{\bf I}_n\\
%{\bf P}_X-{\bf I}_n&{\bf I}_n-{\bf P}_X
%\end{array}\right) {\bf b}\le
%\left(
%\begin{array}{c}
%\tau\\
%\vdots\\
%\tau\\
%w_{1n}-\tau\\
%\vdots\\
%w_{nn}-\tau
%\end{array}\right),\quad {\bf b}\ge {\bf 0}
% \endn
% Re-parameterization via $ {\bf a}=[
%{\bf I}_n-{\bf P}_X\ {\bf P}_X-{\bf I}_n]{\bf b}$,
%the dual to (\ref{{sci}}) becomes a bounded variable problem,
%\beginn
%\max\limits_{ {\bf a}\in R^{n}} {\bf a}^\top [VY_n(\x)],
%\endn
% subject to
%  \beginn
%[DX_n(\x)]^\top {\bf a}={\bf 0}\ \mbox{ and }\ {\bf a}\in [\tau-w_{in}(\tau),\tau]^{\otimes i}
% \endn
% where $[\tau-w_{in}(\tau),\tau]^{\otimes i}$ denotes the $n-$ fold Cartesian product of the closed intervals
% $[\tau-w_{in}(\tau),\tau],\ i=1,\cdots,n. $}


Let ${\bf 1}_{n(A)}$ stand for the $n(A)\times 1$ vector of ones,
and for any $\x\in\mathcal{D}$ and ${\bf h}\in {\rm H}_n(\x)$, define
$\underline{\bf w}_n({\bf h}|\tau)=(w_{in}(\tau),\ i\in {\bf h})$, and
\beginn
L_n(\x,{\bf h}|\tau)= \Big[DX_n(\x,{\bf h})\Big]^{-1}
\sum\limits_{i\in \bar {\bf h}}
\Big[{w_{in}(\tau)}I\{ Y_i\le  \bmu_n^\top(\X_{i\x})
 \hat{\bbeta}_{n\tau}(\x) \}-\tau\Big]
\bmu_n(\X_{i\x}),
\endn
where $\bar {\bf h}=S_n(\x)\backslash {\bf h}$
denotes the complement of ${\bf h}$ in $S_n(\x)$.

\begin{Proposition}\label{merkel} If,  for some ${\bf h}\in {\rm H}_n(\x)$,
$ \hat{\bbeta}_{n\tau}(\x)=[DX_n(\x,{\bf h})]^{-1}
VY_n(\x,{\bf h})$
 is a minimum (not necessarily a unique one) of  (\ref{tiger2}), then we must have
\beginy
L_n(\x,{\bf h}|\tau)\in [\tau {\bf 1}_{n(A)}
-\underline{\bf w}_n({\bf h}|\tau),\tau {\bf 1}_{n(A)}],\label{chilli}
\endy
 where
 $
 [\tau {\bf 1}_{n(A)}-\underline{\bf w}_n({\bf h}|\tau),\tau {\bf 1}_{n(A)}]
 $
is the $n(A)-$fold Cartesian product of the closed intervals
 $\{[\tau-w_{in}(\tau),\tau],\ i\in {\bf h}\}$.
Further,  $ \hat{\bbeta}_{n\tau}(\x)$ is a unique minima if and only if
$L_n(\x,{\bf h}|\tau)\in (\tau {\bf 1}_{n(A)}-\underline{\bf w}_n({\bf h}|\tau),\tau {\bf 1}_{n(A)}).$
\end{Proposition}

\noindent{\bf Proof.}
Rewrite optimization problem (\ref{P}) as minimizing with respect to $\bbeta$
the following quantity
\beginn
\psi(\bbeta)=\sum\limits_{i\in S_n(\x)}
[\tau-\frac{w_{in}(\tau)}{2}+\frac{w_{in}(\tau)}{2}
\mbox{sign}(Y_i-\bmu_n^\top(\X_{i\x}) \bbeta)]
[Y_i-\bmu_n^\top(\X_{i\x})  \bbeta].
\endn
Its directional derivative in direction $\underline{a}$, a unit vector in $R^{n(A)}$,
\beginy
\psi'(\bbeta;\underline{a})=
\sum\limits_{i\in S_n(\x)}
\Big\{\frac{w_{in}(\tau)}{2}-\frac{w_{in}(\tau)}{2}
\mbox{sign}^*[Y_i-\bmu_n^\top(\X_{i\x}) \bbeta;
-\bmu_n^\top(\X_{i\x})\underline{a}]-\tau\Big\}
\bmu_n^\top(\X_{i\x})\underline{a},\label{he}
\endy
where
$$\mbox{sign}^*(u;z)=\left\{
\begin{array}{ll}
\mbox{sign} (u)&\mbox{ if }u\ne 0,\\
\mbox{sign} (z)&\mbox{ if }u= 0.
\end{array}\right.$$
This can be derived as follows. For small enough $\delta>0,$
\beginn
\psi(\bbeta+\delta\underline{a})=\sum\limits_{i\in S_n(\x)}
\{\tau-\frac{w_{in}(\tau)}{2}+\frac{w_{in}(\tau)}{2}
\mbox{sign}[Y_i-\bmu_n^\top(\X_{i\x}) (\bbeta+\delta\underline{a})]\}
\{Y_i-\bmu_n^\top(\X_{i\x})  (\bbeta+\delta\underline{a})\}.
\endn
Therefore,
\beginn
&&\psi(\bbeta+\delta\underline{a})-\psi(\bbeta)\\
&=&\sum\limits_{i\in S_n(\x)}
\frac{w_{in}(\tau)}{2}
\Big[\mbox{sign}\{Y_i-\bmu_n^\top(\X_{i\x}) (\bbeta+\delta\underline{a})\}
-\mbox{sign}\{Y_i-\bmu_n^\top(\X_{i\x}) \bbeta\}\Big]
\{Y_i-\bmu_n^\top(\X_{i\x}) \bbeta\}\\
&&-\delta\underline{a}^\top\sum\limits_{i\in S_n(\x)}
[\tau-\frac{w_{in}(\tau)}{2}+\frac{w_{in}(\tau)}{2}
\mbox{sign}\{Y_i-\bmu_n^\top(\X_{i\x}) (\bbeta+\delta\underline{a})\}]
\bmu_n(\X_{i\x}).
\endn
Consider each individual term and as $\delta\to 0,$ the first term vanishes,
 and the second term when divided by $\delta$ converges to $\psi'(\bbeta;\underline{a})$
defined in (\ref{he}).

Now since $\psi(\bbeta)$ is convex in $\bbeta$, it attains a (local) minimum
at $ \hat{\bbeta}_{n\tau}(\x)$
if and only if  $\psi'(\hat{\bbeta}_{n\tau}(\x);\underline{a})>0$
for all $\underline{a}\ne \underline{0}.$ Note that at
$ \hat{\bbeta}_{n\tau}(\x)=[DX_n(\x,{\bf h})]^{-1}
VY_n(\x,{\bf h})$,
\beginn
&\psi'(\hat{\bbeta}_{n\tau}(\x);\underline{a})&=
\sum\limits_{i\in {\bf h}}[\frac{w_{in}(\tau)}{2}+\frac{w_{in}(\tau)}{2}
\mbox{sign}(\bmu_n^\top(\X_{i\x}) \underline{a})-\tau]
\bmu_n^\top(\X_{i\x}) \underline{a}\\
&&+\sum\limits_{i\in \bar {\bf h}}
\Big\{\frac{w_{in}(\tau)}{2}-\frac{w_{in}(\tau)}{2}
\mbox{sign}^*[Y_i-\bmu_n^\top(\X_{i\x}) \hat{\bbeta}_{n\tau}(\x);
-\bmu_n^\top(\X_{i\x})\underline{a}]-\tau\Big\}
\bmu_n^\top(\X_{i\x})\underline{a}.
\endn
Letting $\underline{\nu}=(\nu_i;i\in {\bf h})=[DX_n(\x,{\bf h})]\underline{a}$, we have that
$\psi'(\hat{\bbeta}_{n\tau}(\x);\underline{a})>0$
for all $\underline{a}\ne \underline{0},$ if and only if
\beginn
&0<&\sum\limits_{i\in  {\bf h}}\Big[(\frac{w_{in}(\tau)}{2}-\tau)\nu_i+\frac{w_{in}(\tau)}{2}|\nu_i|\Big]
+\sum\limits_{i\in \bar {\bf h}}
\Big\{\frac{w_{in}(\tau)}{2}-\frac{w_{in}(\tau)}{2}
\mbox{sign}^*\Big[Y_i-\bmu_n^\top(\X_{i\x}) \hat{\bbeta}_{n\tau}(\x);
\\
&&
-\bmu_n(\X_{i\x})[DX_n(\x,{\bf h})]^{-1}\underline{\nu}\Big]-\tau\Big\}
\times\bmu_n^\top(\X_{i\x})^\top[DX_n(\x,{\bf h})]^{-1}\underline{\nu},
\endn
for all $\underline{\nu}\ne 0.$ This is equivalent to
\beginy
\n&\tau {\bf 1}_{n(A)}-\underline{\bf w}_n({\bf h})<&\hspace{-0.3cm}\sum\limits_{i\in \bar {\bf h}}
\Big\{\frac{w_{in}(\tau)}{2}-\frac{w_{in}(\tau)}{2}
\mbox{sign}^*\Big[Y_i-\bmu_n^\top(\X_{i\x})
\hat{\bbeta}_{n\tau}(\x);
-\bmu_n(\X_{i\x})[DX_n(\x,{\bf h})]^{-1}\underline{\nu}\Big]\\
&&-\tau\Big\}
\times[DX_n(\x,{\bf h})]^{-1}\bmu_n(\X_{i\x})<\tau {\bf 1}_{n(A)},
\label{roger}
\endy
for all $\underline{\nu}\ne 0.$ As once given ${\bf h}$ thus $\hat{\bbeta}_{n\tau}(\x)$,
observations $\{(Y_i,\X_i):i\in \bar {\bf h}\}$ behave like independent random vectors, consequently according to [FACT2],
the event that for some $i\in \bar {\bf h}$,
\beginn
Y_i-\bmu_n(\X_{i\x}) ^\top\hat{\bbeta}_{n\tau}(\x)
=Y_i-\bmu_n(\X_{i\x})[DX_n(\x,{\bf h})]^{-1}
VY_n(\x,{\bf h})=\underline{\bf 0}
\endn
has probability zero,  since it means at least $n(A)+1$ independent random vectors    lie on a certain
hyperplane of dimension $n(A)$.
Therefore, (\ref{roger}) reduces to
\beginn
&\tau {\bf 1}_{n(A)}-\underline{\bf w}_n({\bf h})<&\hspace{-0.3cm}\sum\limits_{i\in \bar {\bf h}}
\Big\{\frac{w_{in}(\tau)}{2}-\frac{w_{in}(\tau)}{2}
\mbox{sign}\Big[Y_i-\bmu_n^\top(\X_{i\x})
 \hat{\bbeta}_{n\tau}(\x)\Big]\\
&&-\tau\Big\}
\times[DX_n(\x,{\bf h})]^{-1}\bmu_n(\X_{i\x})<\tau {\bf 1}_{n(A)},
\endn
as required.$\hspace{\fill}\square$


Below are some classical results in kernel smoothing which will be repeatedly referred to throughout  this section:
with probability one,
 \beginy
&&\hspace{-0.8cm}\sup\limits_{\x }|
(n\delta_n^p)^{-1}N_n(\x)- f_{\X}(\x)|=o(1) ,\label{ri}\\
&&\hspace{-0.8cm}\sup\limits_{\x,\tau}| \Sigma_{n\tau} (\x)-[1-G(Q_{\tau}(\x)|\x)]f_{0}(Q_{\tau}(\x)|\x){\Sigma}(A)
 |=O(\delta_n+(nh_n^{p}/\log n)^{-1/2}),  \label{tai}\\
&&\hspace{-0.8cm}\sup\limits_{\x}|\tilde \Sigma_n(\x)-f_{\X}(\x)\Sigma(\tilde A)|
=O\Big((nh_n^p/\log n)^{-1/2}+h_n\Big).
\label{ellen}
\endy
Also for notational simplicity, we shall write $F_0(.|\X_i),\
f_0(.|\X_i), $ $   G(.|\X_i), \  g(.|\X_i)$
as $F_{i0}(.), $ $
f_i(.), $ $   G_i(.)$ and $ g_i(.),$ respectively.
 For any given ${\x}\in R^p,$ $\bbeta\in R^{n(A)}$ and
 $F(.|\x)$ a conditional distribution function defined with any $\x\in \mathcal{D}$,
write
\beginn
&&Z_{ni}({\x},\bbeta,F|\tau)=\Big[w_{i}(\tau|F)
I\{  Y_i\le  \bbeta^\top\bmu_n(\X_{i\x})
\}-\tau\Big] \bmu_n(\X_{i\x}),\ i\in S_n(\x),\\
&&\tilde H_n({\x},\bbeta,F|\tau)=
 E_i[Z_{ni}({\x},\bbeta,F|\tau)],
\endn
where $E_i(.)$ denotes expectation taken with respect to the joint
distribution of $\xi_i=(Y_i,\X_{i},d_i)$.

Regarding the convergence rate of $\hat{\bbeta}_{n\tau}(\x)$
that is uniform in $\x\in\mathcal{D}$ and $ \tau\in [\tau^*,1-\tau^*]$, we have

\begin{Lemma} \label{uni} Under conditions in Theorem 1,
%\ref{yin}, 
we have
 \beginy
 \sup\limits_{\x,\tau}|\hat {\bbeta}_{n\tau}(\x)- {\bbeta}_{n\tau}(\x )|
 =O_p(\tau_n).
 \label{will}
\endy
\end{Lemma}
\noindent {\bf Proof}.
First of all,
according to Proposition \ref{merkel}  there exists some finite constant $\phi_1$ such that
\beginy
\sup\limits_{\x,\tau}|\sum\limits_{i\in S_n(\x)}
Z_{ni}({\x},\hat{\bbeta}_{n\tau}(\x),\hat F_n(.|.)|\tau)|\le \phi_1.\label{mondeo}
\endy
Secondly, according to the first assertion of  Proposition \ref{T}, if $\tilde\tau_n^{1-\alpha }/\{\delta_n^{2p\alpha }\log n\}<\infty$, then
\beginy
\sup\limits_{\x,\tau,\beta}|\sum\limits_{i\in S_n(\x)}[Z_{ni}({\x},\bbeta,\hat F_n(.|.)|\tau)-
Z_{ni}({\x},\bbeta,F_0|\tau)]|
=O_p((n\delta_n^p\log n)^{1/2}) \  \label{ll}
\endy
On the other hand, based on Proposition \ref{diane} below,
 there exists positive constants
$\epsilon_1^*,c_1^*$ and $M_2^*$, such that
\beginy
\Big|\tilde H_n({\x},\bbeta,F_0|\tau)\Big]
\Big|\ge \min(\epsilon_1^*,c_1^*|\bbeta- {\bbeta}_{n\tau}(\x )|) \label{gg}
\endy
uniformly in $\x,\tau$ and $\bbeta$ whenever
\beginn
&& |\bbeta-
{\bbeta}_{n\tau}(\x )|\ge
M_3^*[n\delta_n^p/\log n]^{-1/2},
\endn
where $M_3^*> M_2^*$.


Combining (\ref{mondeo}), (\ref{ll}), (\ref{gg}) and the facts that $\sharp ( {\bf
h})=n(A)$,
 $\min\limits_{\x,\tau}(n\delta_n^p)^{-1}\sharp (\bar {\bf
h})>0$ almost surely, we can conclude that for large enough $K_1(>M_2^*) $,
\beginn
\n&&\hspace{-1cm}\limsup_{n}Prob\{\sup\limits_{\x,\tau}|\hat{\bbeta}_{n\tau}(\x)-
{\bbeta}_{n\tau}(\x )|\ge K_1 [n\delta_n^p/\log n]^{-1/2}\}\\
&&\hspace{-1cm}\le \lim_{n}Prob\{\sup\limits_{\x,\bbeta}
 |\sum\limits_{i} I(|\X_{i\x}|\le \delta_n)[Z_{ni}({\x},\bbeta,F_0|\tau)
-\tilde H_n({\x},\bbeta,F_0|\tau)]|
\ge \frac{K_1}{2} (n\delta_n^p\log n)^{1/2}\}.
\endn
Finally, we invoke Theorem II.37 of Pollard (1984)
to show that  for large enough $K_1$, the probability on the right hand side of the above
 is  of $o(1)$  as $n\to \infty.$ To this aim, first note that the following classes of functions
$(i):\{\bmu_n(\X_{i\x}):\x\in\mathcal{D}\}$,
$(ii):\{I(Y_i\le \bbeta^\top \bmu_n(\X_{i\x})):
\bbeta\in R^{n(A)},\ \x\in\mathcal{D}\}$,
 $(iii):\{I(|\X_{i\x}|\le \delta_n):\x\in\mathcal{D}\}$ and $(iv)\{w_{i}(\tau|F_0):\tau\in [\tau^*,1-\tau^*]\}$  are all Euclidean for a constant envelope (Lemma (2.13) of Pakes and Pollard, 1989; Lemma 18 and Lemma 22 of Nolan and Pollard, 1987).
  The closer properties of Euclidean classes further dictates that
$\{I(|\X_{i\x}|\le \delta_n) Z_{ni}({\x},\bbeta,F_0|\tau):\
\bbeta\in R^{n(A)},\ \x\in\mathcal{D},\tau\in [\tau^*,1-\tau^*]\}$ is also Euclidean,
thus the conditions required by Theorem II.37 of Pollard (1984) are met. The proof is thus complete by noting that
$E[I(|\X_{i\x}|\le \delta_n) Z_{ni}({\x},\bbeta,F_0|\tau)]=O(\delta_n^p)$ uniformly in
 $\bbeta\in R^{n(A)},\ \x\in\mathcal{D}$ and $\tau\in [\tau^*,1-\tau^*]$.
  $\hspace{\fill}\square$


As the random vector $Z_{ni}({\x},\bbeta,F|\tau)$ plays a central role in the proof of Theorem 1,
%\ref{yin},
 let us take a look at its expectation.   Observing that
\beginy
\n&&w_{i}(\tau|F)I\{Y_i\le  \bbeta^\top\bmu_n(\X_{i\x})
\}-\tau\\
\n&=&I\{C_i>  \bbeta^\top\bmu_n(\X_{i\x})
,T_i\le \bbeta^\top\bmu_n(\X_{i\x})
\}+I\{C_i\le  \bbeta^\top\bmu_n(\X_{i\x})
,T_i\le C_i\}\\
&& + I\{C_i\le  \bbeta^\top\bmu_n(\X_{i\x})
,T_i> C_i\}\Big[1-\frac{1-\tau }{1-F_i(C_i) }I\{F_i(C_i)<\tau\}\Big]-\tau,\label{be}
\endy
\vspace{-1cm}
\beginn
&&E\{I(C_i>t,T_i<t)|\X_{i}={\x}\}=[1-G(t|\x)]F_0(t|\x),\\
&&E\{I(C_i>t,T_i<C_i)|\X_{i}={\x}\}=\int_{-\infty}^t F_0(u|\x)g(u|\x)du,
\endn
we have
\beginn
\tilde H_n({\x},\bbeta,F|\tau)=
\int_{[-1,1]^p}\bmu(\t)
\{R_1(\bbeta^\top\bmu(\t)
|{\x}+\delta_n\t)+
R_F(\bbeta^\top\bmu(\t)|
{\x}+\delta_n\t,\tau)-\tau\}\omega_{\delta_n}(\t,\x)d\t,
\endn
where
\beginn
&&R_1(s|\x)=[1-G(s|\x)]F_0(s|\x)
+\int_{-\infty}^s F_0(u|\x)g(u|\x)du,\\
&&R_F(s|\x,,\tau)= \int_{-\infty}^{s}\{1-
F_0(u|\x)\} \Big[1-\frac{1-\tau
}{1-F(u|\x) }I\{F(u|\x)<\tau\}\Big]g(u|\x)du.
\endn
In the case where $F(.|.)=F_0(.|.)$, we have
\beginn
R_1(s|\x)+R_{F_0}(s|\x,\tau)= [1-G(s|\x)]F_0(s|\x)+G(s|\x)
-(1-\tau) \int_{-\infty}^{\min(s,Q_{\tau}(\x))}g(u|\x)du.
\endn


\noindent  {\bf Proof of Theorem 1}. %\ref{yin}
This consists of the following steps.
\begin{description}
\item{\it Step 1:} For any given $\t\in [-1,1]^p$, and $\bbeta$
close enough to $\bbeta_{n\tau}(\x)$,
let $s_1=\bmu(\t)^\top\bbeta_{n\tau}(\x)$ and
$s_2=\bmu(\t)^\top \bbeta$. Bear  in mind the observation that
for $C_i$ lying above $Q_{\tau}(\X_i)$, the quantile fit
will not be affected if the entire weight  is shifted to $Y^{+\infty}$; see also Wang and Wang (2009).
Therefore, assumptions [A2]-[A4] and [FACT3] imply that
\beginy
\n &&R_1(s_2|\x)+
R_{F_0}(s_2|\x,\tau)-R_1(s_1|\x)-
R_{F_0}(s_1|\x,\tau)\\
&=&[1-G(Q_{\tau}(\x)|\x)]f_0(Q_{\tau}(\x)|\x)
\bmu(\t)^\top\{\bbeta-
{\bbeta}_{n\tau}(\x )\}+O(\tau^2_n+|\delta_n|^{2s_2}),\label{monica}
\endy
 and thus for any $\bbeta$
close enough to $\bbeta_{n\tau}(\x)$,
\beginy
\n&&\tilde H_n({\x},\bbeta,F_0|\tau)
-\tilde H_n({\x},{\bbeta}_{n\tau}(\x ),F_0|\tau)\\
&=&\Sigma_n(\x)\{\bbeta-
{\bbeta}_{n\tau}(\x )\}+O(|\bbeta-
{\bbeta}_{n\tau}(\x )|^2+|\delta_n|^{2s_2}),\label{phoebe}
\endy
 uniformly in $\t$, $\x$ and $\tau$.

\item{\it Step 2:} Define the following $n(A)-$dimensional random vector
\beginn
\chi_n({\x},\bbeta, F_0|\tau)&=&\sum\limits_{i}
I(|\X_{i\x}|\le \delta_n)[Z_{ni}({\x},\bbeta, F_0|\tau)
-\tilde H_n({\x},\bbeta,F_0|\tau)]\\
&&-\sum\limits_{i}I(|\X_{i\x}|\le \delta_n)
[Z_{ni}({\x},{\bbeta}_{n\tau}(\x ),F_0|\tau)
-\tilde H_n({\x},\bbeta_{n\tau}(\x),F_0|\tau)].
\endn
As argued in the last paragraph in  the proof of Lemma 3, %\ref{uni}
$\{I(|\X_{i\x}|\le \delta_n) Z_{ni}({\x},\bbeta,F_0|\tau):\
\bbeta\in R^{n(A)},\ \x\in\mathcal{D},\tau\in (0,1)\}$ is  Euclidean,
hence the conditions required by Theorem II.37 of Pollard (1984) are met. As
$E[I(|\X_{i\x}|\le \delta_n) \{Z_{ni}({\x},\bbeta,F_0|\tau)-
Z_{ni}({\x},{\bbeta}_{n\tau}(\x ),F_0|\tau)\}]^2
=O(\delta_n^p\tau_n)$ uniformly in
$ \x,\tau$ and $\bbeta\in R^{n(A)}$,  whenever $|\bbeta-{\bbeta}_{n\tau}(\x )|\le K_1
 \tau_n$, there exists some finite $K_1>0$, such that with probability one,
\beginy
\sup\limits_{\x,\tau,|\bbeta-{\bbeta}_{n\tau}(\x )|\le K_1
 \tau_n}
|\chi_n({\x},\bbeta, F_0|\tau)|=O((n\delta_n^p\tau_n\log n)^{1/2}).\label{kathy}
\endy

\item{\it Step 3:} Combining (\ref{phoebe}) and (\ref{kathy}),
we have
\beginn
\n&&\frac{1}{N_n(\x)}\sum\limits_{i\in S_n(\x)}
\Big[w_{i0}
I\{  Y_i\le  Q_{n\tau}(\X_i,\x)
\}-\tau\Big] \bmu_n(\X_{i\x})\\
&=&\frac{1}{N_n(\x)}\chi_n(\x)+
\tilde H_n({\x},\hat{\bbeta}_{n\tau}(\x),F_0|\tau)
-\tilde H_n({\x},{\bbeta}_{n\tau}(\x ),F_0|\tau)\\
\n&&+\frac{1}{N_n(\x)}\sum\limits_{i\in S_n(\x)}
[w_{i0}(\tau)I\{Y_i\le \hat Q_{n\tau}(\X_i,\x)\}-\tau
]\bmu_n(\X_{i\x})\\
&=&
\Sigma_n (\x)[\hat{\bbeta}_{n\tau}(\x)-{\bbeta}_{n\tau}(\x )]+
\frac{1}{N_n(\x)}\sum\limits_{i\in S_n(\x)}
\bmu_n({\X_{i\x})}[w_{in}(\tau)I\{Y_i\le \hat Q_{n\tau}(\X_i,\x)\}-\tau
]\\
&&+\frac{1}{N_n(\x)}\sum\limits_{i\in S_n(\x)}
\bmu_n(\X_{i\x})(w_{i0}(\tau)-w_{in}(\tau))
\Big[I\{Y_i\le\hat Q_{n\tau}(\X_i,\x)\}
-I\{Y_i\le  Q_{n\tau}(\X_i,\x)\}\Big]\\
&&+ \frac{1}{N_n(\x)}\sum\limits_{i\in S_n(\x)}
\bmu_n(\X_{i\x})(w_{i0}(\tau)-w_{in}(\tau))I\{Y_i\le  Q_{n\tau}(\X_i,\x)\}
+O(\tau_n^{3/4}+\delta_n^{2s_2}).
\endn
\noindent The assertion in Theorem 1
%\ref{yin} 
thus follows from Propositions \ref{T},
 \ref{merkel} and \ref{bmw}.
 $\hspace{\fill}\square$
\end{description}

\noindent{\bf Proof of  Corollary 1}. 
%\ref{non}
We only need to show that the difference between the  covariance matrices
of these two estimators is positive definite. For illustration purposes, we focus on the covariance matrices of
their respective `staple' terms, not in the least because as discussed in Section 4, the `correction' terms will become
relatively negligible if the K-M estimator converges   fast  enough.

To this aim, we start with quantifying the variances of the following two random variables.
$$ (A)\  \frac{d_i} {1-G(Y_i|\X_i)}
 [\tau-I \{Y_i\le Q_{n\tau}({\bf X}_i,\x) \} ]\quad  \quad (B)\
w_{i0}(\tau)
I\{  Y_i\le  Q_{n\tau}(\X_i,\x)
\}-\tau.$$
Given $\X_i$, the second moments of term (A) is in the first order equal to
\beginn
   E \{[\tau-I \{T_i\le Q_{\tau}(\X_i) \}]^2/ (1-G(T_i|\X_i)) \}.
\endn
As for that of term (B), first note that
\beginn
B^2&=&(1-\tau)^2I\{C_i>T_i, T_i<Q_{n\tau}({\bf X}_i,\x)\}+\tau^2 I\{C_i>T_i, T_i>Q_{n\tau}({\bf X}_i,\x)\}\\
&&+\tau^2 I\{C_i<T_i, C_i>Q_{n\tau}({\bf X}_i,\x)\}\\
&&+(1-\tau)^2\frac{F^2_i(C_i|\X_i)}{[1-F_i(C_i|\X_i)]^2}I\{C_i<T_i, C_i<Q_{n\tau}({\bf X}_i,\x)\}\\
&=&B_1+B_2+B_3+B_4
\endn
As $B_2+B_3=\tau^2 I\{C_i>Q_{n\tau}({\bf X}_i,\x)\}I\{T_i>Q_{n\tau}({\bf X}_i,\x)\}$,
we have $E(B_2+B_3)=\tau^2(1-\tau)[1-G(Q_{\tau}({\X_i})|\X_i)]+o(1)$.
For the remaining two terms,
\beginn
B_1=(1-\tau)^2I\{ T_i<Q_{n\tau}({\bf X}_i,\x)\}-(1-\tau)^2 I\{C_i<T_i,T_i<Q_{n\tau}({\bf X}_i,\x)\}
\endn
and consequently,
$E(B_1+B_4)=(1-\tau)^2\tau+(1-\tau)^2E\Big\{I \{C_i\le Q_{\tau}(\X_i)\}\frac{F_0(C_i|\X_i)}{1-F_0(C_i|\X_i)}\Big\}+o(1)$.
Therefore,
\beginn
EB^2=\tau(1-\tau)[1-G(Q_{\tau}({\X_i})|\X_i)]
+(1-\tau)^2E\Big[I \{C_i\le Q_{\tau}(\X_i)\}\frac{F_0(C_i|\X_i)}{1-F_0(C_i|\X_i)}\Big]+o(1).
\endn
Consequently, up to the first order we have
\beginn
\Cov({\bbeta}_{n\tau}(\x )) &=&\frac{\Sigma^{-1}(A)]\tilde\Sigma(A) \Sigma^{-1}(A)}
{\delta_n^pf^2_0(Q_{\tau}(\x)|\x)} \{1-G(Q_{\tau}(\x)|\x)\}^{-2}\\
 &&\times \{\tau(1-\tau)[1-G(Q_{\tau}({\X_i})|\X_i)]
+(1-\tau)^2E\Big[I \{C_i\le Q_{\tau}(\X_i)\}\frac{F_0(C_i|\X_i)}{1-F_0(C_i|\X_i)}\Big]\}\\
&&\le \frac{\Sigma^{-1}(A)]\tilde\Sigma(A) \Sigma^{-1}(A)}
{\delta_n^pf^2_0(Q_{\tau}(\x)|\x)}\frac{\tau(1-\tau)} {\{1-G(Q_{\tau}(\x)|\x)\}^2},\\[1ex]
\Cov(\hat {\bf c}_{n\tau}(\x))&=& \frac{\Sigma^{-1}(A)]\tilde\Sigma(A) \Sigma^{-1}(A)}
{\delta_n^pf^2_0(Q_{\tau}(\x)|\x)}E\Big\{\frac{[\tau-I \{T_i\le Q_{\tau}(\X_i) \}]^2}{1-G(T_i|\X_i)}\Big\}
 \endn
This finishes the proof.
 $\hspace{\fill}\square$




\begin{Proposition}\label{diane} If $\delta_n^{s_2}=O(\tau_n),$ then
there exist   positive constants $\epsilon_1,\  c_1$ and $M_1$, such that for all $\x $ and $\tau $,
\beginy
|\tilde H_n({\x},\bbeta,F_0|\tau)|\ge
 min(\epsilon_1,c_1|\bbeta-\bbeta_{n\tau}(\x)|)\label{chaud}
\endy
 whenever  $|\bbeta-\bbeta_{n\tau}(\x)|\ge
 M_1(n\delta_n^p/\log n)^{-1/2}$.
\end{Proposition}
\noindent{\bf Proof}. This is split into several steps.

{\it Steps 1:} We show that there exist constants $M_2>0$ and $\epsilon_1>0$,
such that  for all $\x $ and $\tau$,$|\tilde H_n({\x},\bbeta,F_0|\tau)|\ge \epsilon_1$, whenever
$|\bbeta-\bbeta_{n\tau}(\x)|\ge M_2.$

If this is false, we can construct three sequences of vectors $\{\bbeta_{n^*}\}$
 in $R^{n(A)}$, $\{\x_{n^*}\}$  in $R^p$, $\{\tau_{n^*}\}$  such that with
 $\Delta_{n^*}\equiv\bbeta_{n^*}-\bbeta_{n0}(\x_{n^*})$ and ${n^*}\to \infty,$
we have  $|\Delta_{n^*}|\to\infty$ and
$\tilde H_n({\x_{n^*}},\bbeta_{n^*},F_0|\tau_{n*})\to 0.$
 Without loss of generality, we can assume $ \Delta_{n^*}/\|\Delta_{n^*}\|\to \mbox{ some }\Delta^*\in R^{n(A)}$,
 such that $\|\Delta^*\|=1$, $\x_{n^*}\to \mbox{ some }\x^0\in \mathcal{D}$
 and $\tau_{n*}\to \mbox{ some }\tau^0\in [\tau^*,1-\tau^*]$.

As $\bmu(\t)^\top \bbeta_{n\tau}(\x)$ is uniformly bounded over  $\t\in [-1,1]^p,$
 $\x\in \mathcal{D}$ and $\tau\in [\tau^*,1-\tau^*]$, we have
 for any given $\t\in [-1,1]^p,$ $\bmu(\t)^\top\bbeta_{n^*}$
must tend to either $+\infty$
or $-\infty$, depending on whether $\bmu(\t)^\top \Delta^* $ is positive
or negative, respectively.
Specifically, for those $\t\in [-1,1]^p$ such that $\bmu(\t)^\top \Delta^*<0, $
\beginy
&&R_1(\bmu(\t)^\top
\bbeta_{n^*}|{\x}_{n^*}+\delta_{n^*}\t)+
R_{F_0}(\bmu(\t)^\top
\bbeta_{n^*}|{\x}_{n^*}+\delta_{n^*}\t,\tau_{n*})\to
0,\label{mia1}
\endy
as ${n^*}\to\infty$; while for those $\t\in [-1,1]^p$,
 such that $\bmu(\t)^\top \Delta^*>0, $ we have as ${n^*}\to \infty,$
\beginy
\n&&R_1(\t^\top\bbeta_{n^*}|{\x}_{n^*}+\delta_{n^*}\t)
+R_F(\t^\top\bbeta_{n^*}|{\x}_{n^*}+\delta_{n^*}\t,\tau_{n*})-\tau_{n*}\\
&&\hspace{4cm}\to
(1-\tau_0)\{1-G(Q_{\tau^0}({\x}^0)|{\x}^0)\}.
\label{mia2}
\endy

Since the region $[-1,1]^p\cap \{\t:\bmu(\t)^\top \Delta^*=0\}$
must have Legesque measure zero, if $|\tilde H_n(\bbeta_{n^*},F,\x_{n^*})|\to 0,$
as $n^*\to\infty,$ (\ref{mia1}), (\ref{mia2}) and
a straightforward application of the dominated convergence theorem then yield
\beginn
 \tau^0 \int_{[-1,1]^p\cap \{\t^\top \Delta^*<0\}}\bmu(\t)d\t=
(1-\tau^0)\{1-G(Q_{\tau}({\x}^0)|\x^0)\}
\int_{[-1,1]^p\cap \{\t^\top \Delta^*>0\}}
\bmu(\t)d\t.
\endn
Multiplying either side by $ \Delta^*$, we get
\beginn
\tau^0 \int_{[-1,1]^p\cap \{\bmu(\t)^\top \Delta^*<0\}}
\bmu(\t)^\top \Delta^*d\t=
(1-\tau^0)\{1-G(Q_{\tau^0}({\x^0})|\x^0)\}
\int_{[-1,1]^p\cap \{\bmu(\t)^\top \Delta^*>0\}}
\bmu(\t)^\top \Delta^*d\t.
\endn
As $\tau^0\in [\tau^*,1-\tau^*]$ and whence
$G(Q_{\tau^0}({\x}^0)|\x^0)<1,$
the above equality implies that the two regions
$[-1,1]^p\cap \{\bmu(\t)^\top \Delta^*<0\}$ and
$[-1,1]^p\cap \{\bmu(\t)^\top \Delta^*>0\}$
must both have  Legesque measure zero. This contradicts [FACT1].

{\it Step 2:}
Note that for any $s\in R$  and  ${\x}\in R^p$,
\beginy
\n R_1(s|\x)+R_{F_0}(s|\x,\tau)-\tau&=& \{1-G(s|\x)\}\{F_0(s|\x)-\tau\}\\
&& +(1-\tau) \{G(s|\x)-\int_{-\infty}^{\min(s,Q_{\tau}(\x))}g(u|\x)du\}.
\label{kidman}
\endy
For any given $\bbeta\in R^{n(A)}$, write
$\Delta\equiv\bbeta-\bbeta_{n\tau}(\x)$,  and thus for any given $\t\in [-1,1]^p$,
$\bbeta^\top \bmu(\t)=
\Delta^\top \bmu(\t)+Q_{\tau}({\x}+\t\delta_n)-
r(\t\delta_n,\x),$ and
\beginy
&&F_0(\bbeta^\top \bmu(\t)|{\x}+\delta_n\t)-\tau
=h(\Delta,{\x},\t,\tau)\Delta^\top \bmu(\t)+O(\delta_n^{s_2})\label{tea}
\endy
uniformly in $\t,\x,\tau$ and $\Delta$, where
\beginn
&& h(\Delta,{\x},\t,\tau)\\
&& = \frac{F_0(\Delta^\top
\bmu(\t)+Q_{\tau}({\x}+\t\delta_n)
-r(\t\delta_n,\x)|{\x}+\delta_n\t)
-F_0(Q_{\tau}({\x}+\t\delta_n)-
r(\t\delta_n,\x)|{\x}+\delta_n\t)}
{\Delta^\top \bmu(\t)}
\endn
if  $\Delta^\top
\bmu(\t)\ne 0$, and is defined arbitrarily
otherwise, since for nonzero $\Delta$, the set $[-1,1]^p\cap
\{\t:\bmu(\t)^\top
\Delta =0\}$ has Lebesque measure zero.  In view
of [A3], there exists $b>0$, such that the density
$f_{0}(.|\x)$ is continuous and bounded away from
zero in the interval $[Q_{\tau}(\x)-b,Q_{\tau}(\x)+b]$ uniformly in $\x$ and $\tau\in [\tau^*,1-\tau^*]$.
Suppose $s_2$ in [A2] and the bandwidth $\delta_n$ are such   that
$|r(\t\delta_n,{\x}|=O(\delta_n^{s_2})<b$ for all
$\x$ and $\t\in [-1,1]^p$. Therefore, there exists some $M_3>0,$
such that
\beginy
\inf\limits_{ \begin{array}{c}
|\Delta|\le M_2,\t\in [-1,1]^p\\
\x,\tau\in [\tau^*,1-\tau^*]
\end{array}}h(\Delta,{\x},\t)>M_3. \label{isa}
\endy
Similar to   (\ref{tea}),
it could be shown that
\beginn
&&G(\bbeta^\top \bmu(\t)|{\x}+\t\delta_n)
-Pr\{C_i<\min[Q_{\tau}({\x}+\t\delta_n),
\bbeta^\top \bmu(\t)]|{\x}+\delta_n\t\}\\
 &=& I\{\Delta^\top \bmu(\t)>r(\t\delta_n,\x)\}\tilde h(\Delta,{\x},\t,\tau)\Delta^\top
\bmu(\t)+O(\delta_n^{s_4}),
\endn
again uniformly in $\t,\x,\tau\in [\tau^*,1-\tau^*] $ and $\Delta$, where
\beginn
&& \tilde h(\Delta,{\x},\t)\\
&&=
\frac{G(\Delta^\top \bmu(\t)+
Q_{\tau}({\x}+\t\delta_n)- r(\t\delta_n,\x)|{\x}+\delta_n\t)
- G(Q_{\tau}({\x}+\t\delta_n)- r(\t\delta_n,\x)|{\x}+\delta_n\t)}
{\Delta^\top \bmu(\t)},
\endn
which is always nonnegative.

{\it Step 3:}  Write $
 \hat Q_{n\tau}({\x}+\t\delta_n,\x)
 =  \hat\beta^\top_{n\tau}(\x)\bmu(\t).$
 Combining the results in {\it Step 2}, we have
\beginy
\n&&\int_{[-1,1]^p}\bmu(\t)
\Big[\{1-G(s|{\x}+\delta_n\t)\}
\{F_0(s|{\x}+\delta_n\t)-\tau\}\\
\n&& \hspace{2cm} + (1-\tau)\{G(s|{\x}+\delta_n\t)-Pr\{C_i<\min(Q_{\tau}({\x}
+\delta_n\t),s)|{\x}\}\}\Big]
d\t\\
\n&\ge& \Big[\int_{[-1,1]^p}\{1-G(s|{\x}+\delta_n\t)\}h(\Delta,{\x},\t,\tau)\bmu(\t)
\bmu(\t)^\top d\t\Big]\Delta+O(\delta_n^{s_4})\\
&\ge& M_3(1-b_2)\Big[\int I\{|\bmu(\t)|\le b_1/2\}\bmu(\t)
\bmu(\t)^\top d\t\Big]\Delta, \label{nicole}
\endy
where the last inequality(\ref{nicole}) follows from [A4] and (\ref{isa}).
Denote  by $\lambda_1(>0)$, the smallest eigenvalue  of the positive definite matrix
\beginn
\int I\{|\bmu(\t)|\le b_1/2\}\bmu(\t)
\bmu(\t)^\top d\t.
\endn
It can be seen that
\beginn
|\tilde H_n({\x},\bbeta,F_0|\tau)|\ge
 \lambda_1M_3(1-b_2)|\bbeta-\bbeta_{n\tau}(\x)|;
\endn
this together with the conclusion in  {\it Step 1}  leads to (\ref{chaud}).
$\hspace{\fill}\square$

\begin{Proposition}\label{bmw} Suppose conditions in Theorem 1 hold.
%\ref{yin} 
We have
\beginn
&&\frac{1}{N_n(\x)}\sum\limits_{i\in S_n(\x)}
\Big[\bmu_n(\X_{i\x})(w_{i0}(\tau)-w_{in}(\tau))I\{Y_i\le  Q_{n\tau}(\X_i,\x)\}\\
&&\hspace{2cm}-E_i\{\bmu_n(\X_{i\x})(w_{i0}(\tau)-w_{in}(\tau))I\{Y_i\le  Q_{n\tau}(\X_i,\x)\}\}\Big]=
O_p\Big\{\tau_n\Big(\frac{\tilde\tau_n^{1-\alpha}}{\delta_n^{p\alpha}\log n}\Big)^{1/2} \Big\}
\endn
uniformly in $\x,\tau$, where $E_i(.)$
 stands for expectation taken with respect
to the joint distribution of $(\X_i, Y_i)$ with the other argument held fixed.
\end{Proposition}

The proof of  Proposition \ref{bmw} is based on the concept of stochastic equicontinuity in empirical process.
To start with, we introduce the definition of $\epsilon-$covering number.
Let ${\bf S}$ be a sample space and $\mathcal{S}$, the sigma field. Let $P$ be a probability measure on
$\mathcal{S}$ and $\mathcal{F}$ be a class of measurable functions from ${\bf S}$ to $R^d$. Denote by
($\mathcal{F}, \|.\|_{\infty})$ and ($\mathcal{F}, \|.\|_{L_2(P)})$,
 the subset of metric spaces equipped with different
norms, the first with the supremum norm and the second, the $L_2(P)$
norm. The $\epsilon-$covering number
$N(\epsilon,\|.\|_{L_2(P)},\mathcal{F})$ is defined to be the
minimum number of balls  of radius $\epsilon$ with respect to the
$L_2(P)$ norm needed to cover the set $\mathcal{F}$.
$N(\epsilon,\|.\|_{\infty},\mathcal{F})$ is similarly defined.

\noindent{\bf Proof}.
A useful identity is  that, for given $\x,\,\tau,$
and a generic conditional distribution function $ F(.|.)$,
\beginy
\n&&(w_{i0}(\tau)-w_{in}(\tau))I\{Y_i\le  Q_{n\tau}(\X_i,\x)\}\bmu_n(\X_{i\x})\\
&&=(1-\tau)[Z_{n1}(\xi_i|\x, F_0,\tau)-Z_{n1}(\xi_i|\x, F,\tau )+Z_{n2}(\xi_i|\x,  F,\tau)],\,i\in S_n(\x),\ \label{amber}
\endy
where $w_{in}(\tau)\equiv w_{i}(\tau|F(.|.))$,
\beginn
Z_{n1}(\xi_i|\x, F,\tau)=\bmu_n(\X_{i\x})
I\{|\X_{i\x}|\le \delta_n\}I\{C_i\le  Q_{n\tau}(\X_i,\x),T_i>C_i\}\frac{I\{F_i(C_i)\le \tau\}}{1-F_{i0}(C_i)},
\endn
\beginn
Z_{n2}(\xi_i|\x, F,\tau)&=&I\{|\X_{i\x}|\le \delta_n\}I\{C_i\le  Q_{n\tau}(\X_i,\x),T_i>C_i,F_{i}(C_i)\le \tau\}\\
&& \times \frac{F_{i0}(C_i)- F_{i}(C_i)}{(1-F_{i0}(C_i))(1-F_{i}(C_i))}\bmu_n(\X_{i\x}).
\endn
We start with proving
\beginy
\n &&\sup\limits_{F(.|.)\in \mathcal{F}_n}\Big|\frac{1}{n\delta_n^p}\sum\limits_{i=1}^n
 \Big[Z_{n1}(\xi_i|\x, F,\tau)-Z_{n1}(\xi_i|\x,F_0,\tau)\\
 &&\hspace{2cm} -E\{Z_{n1}(\xi_i|\x, F,\tau)-Z_{n1}(\xi_i|\x,F_0,\tau)\}\Big]\Big|=
 o_p\{(n\delta_n^p/\epsilon)^{-1/2}\}. \label{autumn}
 \endy
Let $\mathcal{H}=\{Z_{n1}(\xi|\x,F,\tau):\x\in\mathcal{D},\ F(.|.)\in  \mathcal{C}_M^{s_4}( (0,1)\otimes\mathcal{D})\}$.
The proof  follows the same lines as those for  the Equicontinuity Lemma (Pollard, 1984, pp. 150), provided that
 the $\epsilon-$covering number $N(\epsilon,\|.\|_{L_2(P_n)},\mathcal{H})$ satisfies condition (16)
  therein, where $P_n(.)$ denotes the empirical measure that puts mass $ n^{-1}$ at each of
  $Z_{n1}(\xi_i|\x, F,\tau),\ i=1,\cdots,n$.

Next, note the following simple but useful fact:  for any real values
 $|a|<1,\ |b|<1, $ and $ \ x,\, y \in\{0,1\}$,
it holds that $|ax-by|^2\le |x-y|+|a-b|^2.$
An application of this inequality yields
 \beginn
 &&\hspace{-1cm}|Z_{n1}(\xi|\x,F,\tau)-Z_{n1}(\xi|\x',F',\tau)|^2\le
 |\bmu_n(\X_{i\x})I\{|\X_{i\x}|\le \delta_n\}I\{C_i\le  Q_{n\tau}(\X_i,\x)\}-\\
&& \bmu_n(\X_{i\x'})I\{|\X_{i\x'}|\le \delta_n\}I\{C_i\le  Q_{n\tau}(\X_i,\x')\}|
+|I\{F_i(C_i)\le \tau\}-I\{F'_i(C_i)\le \tau\}|.
 \endn
 This in turn implies that the upper bound for $\epsilon-$covering number
 $N(\epsilon,\|.\|_{L_2(P_n)},\mathcal{H})$ is,  the product of
 $N(\epsilon,\|.\|_{L_2(P_n)},\mathcal{F}_1)$,
 the $\frac{\epsilon}{2}-$covering number of
 \beginy
 \mathcal{F}_1=\{\bmu_n(\X_{i\x})I\{|\X_{i\x}|\le \delta_n\}I\{C_i\le  Q_{n\tau}(\X_i,\x)\}:
 \x\in\mathcal{D},\tau\in [\tau^*,1-\tau^*] \}\label{vc}
 \endy
 and ${\epsilon^2}/{4}-$covering number
 $N({\epsilon^2}/{4},\|.\|_{\infty}, \mathcal{C}_M^{s_4}( (0,1)\otimes\mathcal{D}))$.
On one hand,  by Lemmas 2.6.15, 2.6.18 and 2.6.20 of van der Vaart and Wellner (1989)
 and Example 38 of Pollard (1984) (pp.35),
 $\mathcal{F}_1$ of (\ref{vc}) is VC-subgraph class of functions with
 a constant envelope and thus according to  Theorem 2.6.7
 of van der Vaart et al. (1989),
 there exist some universal constants $K_1>0,\ W>0$ such that
\beginy
\sup\limits_{P}N(\epsilon,\mathcal{F}_1,\|.\|_{L_2(P)})\le K_1\epsilon^{-W}
\mbox{ for any }\epsilon>0.\label{chen}
\endy
On the other hand, Theorem 2.7.1
 of van der Vaart  et al. (1989) states that there exists some constant $K_2$ depending only
 on $p,\ s_4$ and the Legesque measure of the set $(0,1)\otimes\mathcal{D}$, such that
  \beginn
 \log N({\epsilon^2}/{4}, \mathcal{C}_M^{s_4}( (0,1)\otimes\mathcal{D}),\|.\|_{\infty})\le
 K_2\epsilon^{-2(p+1)/s_4}\quad \mbox{ for any }\epsilon>0.
 \endn
 This together with (\ref{chen}) implies that there exists some constant $K_3$ such that
 \beginy
\sup\limits_{P} \log N(\epsilon,\|.\|_{L_2(P)},\mathcal{H})\le K_3\epsilon^{-2(p+1)/s_4}.\label{hong}
 \endy
 If $s_4>p+1,$ (\ref{hong}) means that
 the requirement on the covering integral-condition (16) of the
 Equi-continuity Lemma (Pollard, 1984, pp. 150) is  satisfied
 and (\ref{autumn})  consequently holds.

The handling of  $ \ Z_{n2}(\xi_i|\x, F,\tau) \ $ is much simpler and leads to  results similar to (\ref{autumn}).
$\hspace{\fill}\square$

\begin{Proposition}\label{buy} Suppose conditions in Theorem 1 hold.
% \ref{yin}
 We have
\beginy
&&\n E_i[(w_{i0}(\tau)-w_{in}(\tau))I\{Y_i\le  Q_{n\tau}(\X_i,\x)\}\bmu_n(\X_{i\x})]\\
&&=\frac{1}{n}\sum\limits_{j }E_{i}\Big[\tilde B_{h_n}(\X_j;\X_i)I\{|\X_{i\x}|\le \delta_n\}
\bmu_n(\X_{i\x})\Phi(\X_i,Y_j,d_j|\tau) \Big]\label{Jene}\\
&&\n\quad +O\Big(\delta_n^p\|F(.|.)-F_0(.|.)\|^2+\delta_n^p(nh_n^p/\log n)^{-3/4}+\delta_n^ph_n^{s_4}\Big).
\endy
where the term $O(.)$ is uniform in $\x,\,\tau$ and $F(.|.)\in \mathcal{F}_n.$
\end{Proposition}
\noindent{\bf Proof}. This implies dealing with the  terms introduced in (\ref{amber}).
First note that through arguments used to derive (\ref{phoebe}),
for $F(.|.)$ such that $\|F(.|.)-F_0(.|.)\|$ can be arbitrarily small, we have
\beginy
\n &&E_i[Z_{n1}(\xi_i|\x, F,\tau)-Z_{n1}(\xi_i|\x,F_0,\tau)]\label{april}\\
\n&=&E_i\Big[I\{i\in S_n(\x)\}\bmu_n(\X_{i\x})
\Big\{G(F^{-1}(\tau|\X_i)|\X_i)-G(Q_{\tau}(\X_i)|\X_i)\Big\}\Big] \\
\n&=&E_i\Big[I\{i\in S_n(\x)\}\bmu_n(\X_{i\x})\{\frac{g}{f_0}\}(Q_{\tau}(\X_i)|\X_i)
\{\tau-F(Q_{\tau}(\X_i)|\X_i)\}\Big]\\
\n&&+O(\delta_n^p\|F(.|.)-F_0(.|.)\|^2 ),
\endy
where  the second equality follows from the following observations
$$
G(F^{-1}(\tau|\x)|\x)-G(Q_{\tau}(\x)|\x)= g(Q_{\tau}(\x)|\x)\{F^{-1}(\tau|\x)-F_0^{-1}(\tau|\x)\}\{1+o(1)\},
$$
$$
F^{-1}(\tau|\x)-F_0^{-1}(\tau|\x)=\frac{\tau-F(Q_{\tau}(\x)|\x)}{f_0(Q_{\tau}(\x)|\x)}+O(\|F(.|.)-F_0(.|.)\|^2),
$$
where again the  terms $O(.)$ and $o(.)$ are both unform in $\x,\,\tau$ and $F(.|.)\in \mathcal{F}_n.$
Note that second equality above was proved by van der Vaart (1998) and was also used in Wang et al. (2009) for linear CQR.
 With $F(.|.)=\hat F_n^S$,
 invoke Lemma 1
 %\ref{Gonza} 
 with $t=Q_{\tau}(\X_i),\ \x=\X_i$, plug it into (\ref{april}) and we have
\beginn
&&E_i[Z_{n1}(\xi_i|\x, F,\tau)-Z_{n1}(\xi_i|\x,F_0,\tau)]\\
&&=\frac{1}{nh^p_n}\sum\limits_{j=1}^n E_i\Big[I\{i\in
S_n(\x)\}\bmu_n(\X_{i\x})\tilde
B_{nj}(\X_{j};\X_i)\{\frac{g}{f_0}\}(Q_{\tau}(\X_i)|\X_i)
\varphi(Y_j,d_j,Q_{\tau}(\X_i),\X_i)\Big]\\
&&\hspace{1cm} +O\{ \delta_n^p\|F(.|.)-F_0(.|.)\|^2
+\delta_n^p\Big(\frac{\log n}{nh_n^p}\Big)^{3/4}+\delta_n^ph_n^{s_4}\},
\endn
once again unform in $\x,\,\tau$ and $F(.|.)\in \mathcal{F}_n.$
Similar  results can be proved for  the expectation of $Z_{n2}(\xi_i|\x, F,\tau)$. Note that
\beginn
&&\hspace{-0.5cm}\frac{1}{n}\sum\limits_{j }
E_i\Big[\tilde B_{h_n}(\X_j;\X_i)I\{|\X_{i\x}|\le \delta_n\}\bmu_n(\X_{i\x})
I\{C_i\le  Q_{n\tau}(\X_i,\x),F_{i}(C_i)\le \tau\}
\frac{F_{i0}(C_i)- F_{i}(C_i)}{(1-F_{i}(C_i))}\Big]\\
&&\hspace{-0.5cm}=\frac{1}{n}\sum\limits_{j }E_i\Big[
\tilde B_{h_n}(\X_j;\X_i)I\{|\X_{i\x}|\le \delta_n\}\bmu_n(\X_{i\x})
I\{C_i\le Q_\tau(\X_i)\}\frac{F_{i0}(C_i)- F_{i}(C_i)}{(1-F_{i0}(C_i))}\Big]\\
&&+O(\delta_n^{s_2}\|F(.|.)-F_0(.|.)\|+\|F(.|.)-F_0(.|.)\|^2),
\endn
and consequently (\ref{Jene}) holds.
Depending on the ratio $\delta_n/h_n$,
the leading term in (\ref{Jene}) above admits the following asymptotic expressions:
\begin{itemize}
\item  $\delta_n=o(h_n)$,
\beginy
\bgamma(A)f_{\X}(\x)\frac{\delta_n^p}{n}\sum\limits_{j } \tilde B_{h_n}(\X_j;\x)\Phi(\x,Y_j,d_j)
+O(\delta_n^{p+1}\tilde\tau_n);\label{reith1}
\endy
\item if $\delta_n/h_n\to \infty$,
\beginy
\frac{1}{n}\sum\limits_{j\in S_n(\x) }f_{\X}(\X_j)
\bmu_n(\X_{j\x})\Phi(\X_j,Y_j,d_j)
+O(\delta_n^ph_n\tau_n/\delta_n);\label{reith2}
\endy
\item $\delta_n\asymp h_n$,
\beginy
f_{\X}(\x)\frac{\delta_n^p}{n}\sum\limits_{j } B_{h_n}(\X_j;\x)
\Phi(\x,Y_j,d_j)
+O(\delta_n^{p+1}\tilde\tau_n),\label{reith3}
\endy
where $ B_{h_n}(\X_j;\x)=\int_{[-1,1]^p}  \tilde B_{h_n}(\X_j;\x+\delta_n \t)\mu(\t)d\t$. $\hspace{\fill}\square$
\end{itemize}
\begin{Proposition}\label{T} Suppose conditions in Theorem 1 hold.
%\ref{yin} 
 We have
\beginn
&&\sup\limits_{\tau,\x,\bbeta}\frac{1}{N_n(\x)}
\Big|\sum\limits_{i\in S_n(\x)}[Z_{ni}({\x},\bbeta,\hat F_n|\tau)-
Z_{ni}({\x},\bbeta,F_0|\tau)]\Big|
=O_p\Big\{\tau_n
+\Big(\frac{\tilde \tau_n^{(1-\alpha)}} {\delta_n^{p\alpha}\log n}\Big)^{1/2}\tau_n\Big\};\\
&&\sup\limits_{\tau,\x,\bbeta}\frac{1}{N_n(\x)}
\Big|\sum\limits_{i\in S_n(\x)}
\bmu_n(\X_{i\x})(w_{i0}(\tau)-w_{in}(\tau))
\Big[I\{Y_i\le\bbeta^\top \bmu_n(\X_{i\x})\}
-I\{Y_i\le  Q_{n\tau}(\X_i,\x)\}\Big]\Big|\\
&&\hspace{8.5cm}=O_p\{\tau_n\tilde \tau_n+\Big(\frac{\tilde \tau_n^{(1-\alpha)}}
 {\delta_n^{p\alpha}\log n}\Big)^{1/2}\tau_n^{(3-\alpha)/2}\}.
\endn
\end{Proposition}
{\bf Proof}. The arguments are exactly the same as those used to prove Propositions \ref{bmw}
 and   \ref{buy}.
 $\hspace{\fill}\square$
 
 \
 
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 \end{document}