Kyudan/arXiv_latex · Datasets at Hugging Face

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\Lambda_{\cal C}
{\cal Q}_{1n}
\sum^{n}_{i=1}\frac{\delta_{i}}{G_{0}(Y_{i})}g_{\tau}(\varepsilon_{i})\mathbf{% X}^{\top}_{{\cal A}^{c},i}\textrm{$\mathbf{\beta}$}^{(2)}_{{\cal A}^{c}}\frac{% 1}{n}\sum^{n}_{j=1}\int^{B}_{0}1\!\!1_{Y_{i}\geq s}\frac{dM_{j}^{\cal C}(s)}{y% (s)}=O_{\mathbb{P}}(1).
(Y_{i},\delta_{i})_{1\leqslant i\leqslant n}
\textrm{$\mathbf{\omega}$}^{0}_{\cal A}\equiv\lim_{n\rightarrow\infty}\widehat% {\textrm{$\mathbf{\omega}$}}_{n,{\cal A}}=\|\textrm{$\mathbf{\beta}$}^{0}_{\cal A% }\|^{-\gamma}
\mathbf{{u}}=-(\textbf{W}_{1}+\textbf{W}_{2})\mathbb{E}^{-1}_{\varepsilon}\big% {[}h_{\tau}(\varepsilon)\big{]}\mathbb{E}_{\mathbf{X}}[\mathbf{X}\mathbf{X}^{% \top}]^{-1}
\frac{1}{n}\sum^{n}_{i=1}\frac{\delta_{i}}{G_{0}(Y_{i})}g_{\tau}(\varepsilon_{% i})\mathbf{X}^{\top}_{{\cal A}^{c},i}1\!\!1_{Y_{i}\geq s}=O_{\mathbb{P}}(1)% \overset{\mathbb{P}}{\underset{n\rightarrow\infty}{\longrightarrow}}\textrm{$% \mathbf{\kappa}$}_{{\cal A}^{c}}(s).
{\cal W}(0,1)
\displaystyle=\sum^{n}_{i=1}\delta_{i}\bigg{(}\frac{1}{\widehat{G}_{n}(Y_{i})}% -\frac{1}{G_{0}(Y_{i})}\bigg{)}\big{(}\rho_{\tau}(\varepsilon_{i}-n^{1/2}% \mathbf{X}^{\top}_{i}\mathbf{{u}})-\rho_{\tau}(\varepsilon_{i})\big{)}
n^{-1/2}\lambda_{n}=O_{\mathbb{P}}(1)
\textbf{W}_{2}\sim{\cal N}_{p}(\textrm{$\mathbf{0}$}_{p},\textbf{S}_{2})
t=n^{-1/2}\mathbf{X}_{i}^{\top}\mathbf{{u}}
\textbf{S}_{1}
\widetilde{\textrm{$\mathbf{\beta}$}}_{n}
\widehat{\textrm{$\mathbf{\beta}$}}_{n}\equiv\mathop{\mathrm{arg\,min}}_{% \textrm{$\mathbf{\beta}$}\in\mathbb{R}^{p}}\bigg{(}\sum^{n}_{i=1}\frac{\delta_% {i}}{\widehat{G}_{n}(Y_{i})}\rho_{\tau}(\log(Y_{i})-\mathbf{X}_{i}^{\top}% \textrm{$\mathbf{\beta}$})+\lambda_{n}\sum^{p}_{j=1}\widehat{\omega}_{n,j}\|% \beta_{j}\|\bigg{)},
\widehat{\textrm{$\mathbf{\beta}$}}_{n,{\cal A}}
{\cal C}_{i}\sim{\cal U}[0,c_{1}]
\mathbb{P}[\widehat{\textrm{$\mathbf{\beta}$}}_{n}\in{\cal W}_{n}]{\underset{n% \rightarrow\infty}{\longrightarrow}}0
n\geq 2500
{\cal R}_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}$})
\mathbb{P}[\widehat{\cal A}_{n}\subseteq{\cal A}]{\underset{n\rightarrow\infty% }{\longrightarrow}}1
\textrm{$\mathbf{\beta}$}^{(2)}\equiv\big{(}\textrm{$\mathbf{\beta}$}^{(2)}_{% \cal A},\textrm{$\mathbf{\beta}$}^{(2)}_{{\cal A}^{c}}\big{)}\in{\cal W}_{n}
\lambda_{n}=(150k)^{1/2-0.1}
\mathbb{E}_{\varepsilon}[\varepsilon]=0
\\|\mathbf{{u}}\\|=c<\infty
{\cal T}_{n}(G,\textrm{$\mathbf{\beta}$})\equiv Q_{n}(G,\textrm{$\mathbf{\beta% }$})-Q_{n}(G,\textrm{$\mathbf{\beta}^{0}$})+\lambda_{n}\sum^{p}_{j=1}\widehat{% \omega}_{n,j}\big{(}\|\beta_{j}\|-\|\beta^{0}_{j}\|\big{)}.
\textrm{$\mathbf{\beta}$}^{(2)}
{\cal T}_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}$}^{(2)})-{\cal T}_{n}(% \widehat{G}_{n},\textrm{$\mathbf{\beta}$}^{(1)})=Q_{n}(\widehat{G}_{n},\textrm% {$\mathbf{\beta}$}^{(2)})-Q_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}$}^{(1)% })+\lambda_{n}\sum^{p}_{j=1}\widehat{\omega}_{n,j}\big{(}\|\beta^{(2)}_{j}\|-\|% \beta^{(1)}_{j}\|\big{)}.
\mathbb{P}[\widehat{\textrm{$\mathbf{\beta}$}}_{n}\in{\cal A}^{c}\cap\widehat{% \cal A}_{n}]{\underset{n\rightarrow\infty}{\longrightarrow}}0
\widehat{\mathbf{{u}}}_{n,{\cal A}}\equiv\sqrt{n}\big{(}\widehat{\textrm{$% \mathbf{\beta}$}}_{n,{\cal A}}-\textrm{$\mathbf{\beta}$}^{0}_{\cal A}\big{)}% \overset{\cal L}{\underset{n\rightarrow\infty}{\longrightarrow}}{\cal N}_{\|{% \cal A}\|}\bigg{(}-\mathbb{E}^{-1}_{\varepsilon}[h_{\tau}(\varepsilon)]l_{0}{% \textrm{$\mathbf{\omega}$}^{0}_{\cal A}}^{\top}{\mathrm{sgn}}(\textrm{$\mathbf% {\beta}$}^{0}_{\cal A})\mathbb{E}_{\mathbf{X}}[\mathbf{X}_{{\cal A}}\mathbf{X}% ^{\top}_{{\cal A}}]^{-1},\textrm{$\mathbf{\Xi}$}\bigg{)}
\textrm{$\mathbf{\beta}$}=\textrm{$\mathbf{\beta}^{0}$}+n^{-1/2}\mathbf{{u}}
\textrm{$\mathbf{\kappa}$}(s)\equiv\lim_{n\rightarrow\infty}\frac{1}{n}\sum^{n% }_{i=1}\frac{\delta_{i}}{G_{0}(Y_{i})}1\!\!1_{Y_{i}\geq s}\mathbf{X}_{i}g_{% \tau}(\varepsilon_{i}),
\textrm{$\mathbf{\kappa}$}(s)
\mathbb{E}_{\varepsilon}[h_{\tau}(\varepsilon)]
\mathbb{E}_{\mathbf{X}}\bigg{[}\mathbb{E}_{\varepsilon}\bigg{[}\mathbb{E}_{% \cal C}\bigg{[}\frac{\delta_{i}}{G_{0}(Y_{i})}g_{\tau}(\varepsilon_{i})\mathbf% {X}_{i}^{\top}\mathbf{{u}}\|\varepsilon_{i},\mathbf{X}_{i}\bigg{]}\|\mathbf{X}_{% i}\bigg{]}\bigg{]}=\mathbb{E}_{\mathbf{X}}\big{[}\mathbb{E}_{\varepsilon}\big{% [}g_{\tau}(\varepsilon_{i})\mathbf{X}_{i}^{\top}\mathbf{{u}}\|\mathbf{X}_{i}% \big{]}\big{]}=0.
\mathbf{{u}}^{\top}(\textbf{W}_{1}+\textbf{W}_{2})
\displaystyle\qquad+\frac{\mathbf{{u}}^{\top}_{\cal A}}{\sqrt{n}}\sum^{n}_{j=1% }\int^{B}_{0}\frac{\textrm{$\mathbf{\kappa}$}_{\cal A}(s)}{y(s)}dM^{\cal C}_{j% }(s)+\lambda_{n}\sum^{q}_{j=1}\widehat{\omega}_{n,j}\frac{{\mathrm{sgn}}(\beta% ^{0}_{j})u_{j}}{\sqrt{n}}+o_{\mathbb{P}}(1)
\lim_{n\rightarrow\infty}\mathbb{P}\big{[}\widehat{\cal A}_{n}={\cal A}\big{]}=1
\beta_{0}^{0}=0
\displaystyle\quad+\left\{Q_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}^{0}$}+% n^{-1/2}\mathbf{{u}})-Q_{n}(G_{0},\textrm{$\mathbf{\beta}^{0}$}+n^{-1/2}% \mathbf{{u}})\right\}
{\cal V}(\textrm{$\mathbf{\beta}^{0}$})\equiv\big{\{}\textrm{$\mathbf{\beta}$}% ;\\|\textrm{$\mathbf{\beta}$}-\textrm{$\mathbf{\beta}^{0}$}\\|\leq cn^{-1/2}\big% {\}}
n^{-1/2}\sum^{n}_{i=1}\frac{\delta_{i}}{G_{0}(Y_{i})}g_{\tau}(\varepsilon_{i})% \mathbf{X}_{{\cal A}^{c},i}\overset{\cal L}{\underset{n\rightarrow\infty}{% \longrightarrow}}{\cal N}_{\|{\cal A}^{c}\|}(\textrm{$\mathbf{0}$}_{q},\textbf{S% }_{1,{\cal A}^{c}}),
\mathbb{E}_{\mathbf{X}}[\mathbf{X}\mathbf{X}^{\top}]
{1}/{\widehat{G}_{n}(Y_{i})}-{1}/{G_{0}(Y_{i})}
\overset{\cal L}{\underset{n\rightarrow\infty}{\longrightarrow}}
n>2000
\beta^{0}_{0}=0
sd\equiv sd(\widetilde{\textrm{$\mathbf{\beta}$}}_{n}-\textrm{$\mathbf{\beta}^% {0}$})
\beta^{0}_{j}=0
{\cal Q}_{1n}=\sum^{n}_{i=1}\frac{\delta_{i}}{G_{0}(Y_{i})}\bigg{(}g_{\tau}(% \varepsilon_{i})\frac{\mathbf{X}_{i}^{\top}\mathbf{{u}}}{\sqrt{n}}+\frac{1}{2}% h_{\tau}(\varepsilon_{i})\bigg{(}\frac{\mathbf{X}_{i}^{\top}\mathbf{{u}}}{% \sqrt{n}}\bigg{)}^{2}+o_{\mathbb{P}}(n^{-1})\bigg{)}.
\mathbb{E}_{\varepsilon}[h_{\tau}(\varepsilon)]>0
\varepsilon\sim{\cal U}[-1,2]-1/6
X_{ji}\sim{\cal N}(1,1)
{\cal U}[-1,2]-1/6
\mathbb{P}[t\leq T\leq{\cal C}]\geq\zeta_{0}>0
\mathbb{S}_{1}=O_{\mathbb{P}}(1)
{\cal A}\equiv\{j\in\{1,\cdots,p\};\;\beta^{0}_{j}\neq 0\}.
\displaystyle-\bigg{(}g_{\tau}(\varepsilon_{i})\mathbf{X}^{\top}_{{\cal A},i}% \big{(}\textrm{$\mathbf{\beta}$}^{(2)}-\textrm{$\mathbf{\beta}^{0}$}\big{)}_{% \cal A}+\frac{h_{\tau}(\varepsilon_{i})}{2}\big{(}\mathbf{X}^{\top}_{{\cal A},% i}\big{(}\textrm{$\mathbf{\beta}$}^{(2)}-\textrm{$\mathbf{\beta}^{0}$}\big{)}_% {\cal A}\big{)}^{2}+o_{\mathbb{P}}\big{(}\mathbf{X}^{\top}_{{\cal A},i}\big{(}% \textrm{$\mathbf{\beta}$}^{(2)}-\textrm{$\mathbf{\beta}^{0}$}\big{)}_{\cal A}% \big{)}^{2}\bigg{)}\bigg{\}}
Q_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}^{0}$}+n^{-1/2}\mathbf{{u}})-Q_{n% }(\widehat{G}_{n},\textrm{$\mathbf{\beta}^{0}$})=\frac{\mathbb{E}_{\varepsilon% }[h_{\tau}(\varepsilon)]}{2}\mathbf{{u}}^{\top}\mathbb{E}_{\mathbf{X}}[\mathbf% {X}\mathbf{X}^{\top}]\mathbf{{u}}\big{(}1+o_{\mathbb{P}}(1)\big{)}.
j\in\{1,\cdots,q\}
\displaystyle+\sum^{n}_{i=1}\frac{\delta_{i}}{G_{0}(Y_{i})}\bigg{\{}\bigg{(}g_% {\tau}(\varepsilon_{i})\mathbf{X}^{\top}_{i,{\cal A}^{c}}\textrm{$\mathbf{% \beta}$}^{(2)}_{{\cal A}^{c}}+\frac{h_{\tau}(\varepsilon_{i})}{2}\big{(}% \mathbf{X}^{\top}_{i}\big{(}\textrm{$\mathbf{\beta}$}^{(2)}_{{\cal A}}-\textrm% {$\mathbf{\beta}^{0}$}_{\cal A},\textrm{$\mathbf{\beta}$}^{(2)}_{{\cal A}^{c}}% \big{)}\big{)}^{2}+o_{\mathbb{P}}\big{(}\mathbf{X}^{\top}_{i}\big{(}\textrm{$% \mathbf{\beta}$}^{(2)}_{{\cal A}}-\textrm{$\mathbf{\beta}^{0}$}_{\cal A},% \textrm{$\mathbf{\beta}$}^{(2)}_{{\cal A}^{c}}\big{)}\big{)}^{2}\bigg{)}
{\cal Q}_{2n}
\textbf{S}_{3}\equiv\mathbb{E}_{\varepsilon}\big{[}h_{\tau}(\varepsilon)\big{]% }\mathbb{E}_{\mathbf{X}}[\mathbf{X}\mathbf{X}^{\top}]
l_{0}\equiv\lim_{n\rightarrow\infty}n^{-1/2}\lambda_{n}
{\cal W}_{n}\equiv\big{\{}\textrm{$\mathbf{\beta}$}\in{\cal V}(\textrm{$% \mathbf{\beta}^{0}$});\\|\textrm{$\mathbf{\beta}$}_{{\cal A}^{c}}\\|>0\big{\}}
S_{21}
\beta^{0}_{0}=2
j=1,\cdots p
\varepsilon_{i}\sim{\cal G}(0,1)
\{M_{j}^{\cal C}(t)\}
\beta_{1}^{0}=1/\log k-1/k
\mathbf{{u}}\in\mathbb{R}^{p}
\mathbb{E}_{\mathbf{X}}[\mathbf{X}_{{\cal A}}\mathbf{X}^{\top}_{{\cal A}}]
\displaystyle=\sum^{n}_{i=1}\frac{\delta_{i}}{\widehat{G}_{n}(Y_{i})}\bigg{\{}% \bigg{(}g_{\tau}(\varepsilon_{i})\mathbf{X}^{\top}_{i}\big{(}\textrm{$\mathbf{% \beta}$}^{(2)}_{{\cal A}}-\textrm{$\mathbf{\beta}^{0}$}_{\cal A},\textrm{$% \mathbf{\beta}$}^{(2)}_{{\cal A}^{c}}\big{)}+\frac{h_{\tau}(\varepsilon_{i})}{% 2}\big{(}\mathbf{X}^{\top}_{i}\big{(}\textrm{$\mathbf{\beta}$}^{(2)}_{{\cal A}% }-\textrm{$\mathbf{\beta}^{0}$}_{\cal A},\textrm{$\mathbf{\beta}$}^{(2)}_{{% \cal A}^{c}}\big{)}\big{)}^{2}
n\in\{400,600,\cdots,2000\}
\textrm{$\mathbf{\beta}$}^{(1)}_{{\cal A}^{c}}=\textrm{$\mathbf{0}$}_{\|{\cal A% }^{c}\|}
\widehat{\textrm{$\mathbf{\beta}$}}_{n}
\displaystyle=\mathbb{E}_{\mathbf{X}}\bigg{[}\mathbb{E}_{\varepsilon}\bigg{[}% \frac{g^{2}_{\tau}(\varepsilon_{i})}{G_{0}(Y_{i})}\frac{\mathbf{{u}}\mathbf{X}% _{i}\mathbf{X}_{i}^{\top}\mathbf{{u}}}{n}\|\mathbf{X}_{i}\bigg{]}\bigg{]}
\textrm{$\mathbf{\beta}$}_{\cal A}
{\cal Q}_{1n}=\sum^{n}_{i=1}\frac{\delta_{i}}{G_{0}(Y_{i})}\big{(}\rho_{\tau}(% \varepsilon_{i}-n^{-1/2}\mathbf{X}_{i}^{\top}\mathbf{{u}})-\rho_{\tau}(% \varepsilon_{i})\big{)},
n^{(\gamma-1)/2}\lambda_{n}\rightarrow\infty
\textrm{$\mathbf{\beta}$}\in\mathbb{R}^{p}
\begin{split}\sqrt{n}\bigg{(}\frac{1}{\widehat{G}_{n}(Y_{i})}-\frac{1}{G_{0}(Y% _{i})}\bigg{)}&=-\frac{\sqrt{n}\big{(}\widehat{G}_{n}(Y_{i})-G_{0}(Y_{i})\big{% )}}{G^{2}_{0}(Y_{i})}\big{(}1+o_{\mathbb{P}}(1)\big{)}\\ &=\frac{1}{G_{0}(Y_{i})}\frac{1}{\sqrt{n}}\sum^{n}_{j=1}\int^{B}_{0}1\!\!1_{Y_% {i}\geq s}\frac{dM_{j}^{\cal C}(s)}{y(s)}\big{(}1+o_{\mathbb{P}}(1)\big{)}.% \end{split}
\displaystyle-\bigg{(}\frac{h_{\tau}(\varepsilon_{i})}{2}\big{(}\mathbf{X}^{% \top}_{{\cal A},i}\big{(}\textrm{$\mathbf{\beta}$}^{(2)}-\textrm{$\mathbf{% \beta}^{0}$}\big{)}_{\cal A}\big{)}^{2}+o_{\mathbb{P}}\big{(}\mathbf{X}^{\top}% _{{\cal A},i}\big{(}\textrm{$\mathbf{\beta}$}^{(2)}-\textrm{$\mathbf{\beta}^{0% }$}\big{)}_{\cal A}\big{)}\bigg{)}\bigg{\}}\frac{1}{n}\sum^{n}_{j=1}\int^{B}_{% 0}1\!\!1_{Y_{i}\geq s}\frac{dM_{j}^{\cal C}(s)}{y(s)}\big{(}1+o_{\mathbb{P}}(1% )\big{)}
n>2500
{\cal R}_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}$})=Q_{n}(\widehat{G}_{n},% \textrm{$\mathbf{\beta}$})-Q_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}^{0}$})
t=\mathbf{X}^{\top}_{i}(\textrm{$\mathbf{\beta}$}^{(2)}-\textrm{$\mathbf{\beta% }^{0}$})
\mathbb{E}_{\varepsilon}[h_{\tau}(\varepsilon)]\mathbb{E}_{\mathbf{X}}[\mathbf% {X}_{{\cal A}}\mathbf{X}^{\top}_{{\cal A}}]\mathbf{{u}}_{\cal A}+\textbf{W}_{3% }+l_{0}\widehat{\textrm{$\mathbf{\omega}$}}_{n,{\cal A}}{\mathrm{sgn}}(\textrm% {$\mathbf{\beta}$}^{0}_{\cal A})=\textrm{$\mathbf{0}$}_{\|{\cal A}\|},
\\|\widehat{\textrm{$\mathbf{\beta}$}}_{n}-\textrm{$\mathbf{\beta}$}^{0}\\|
(\varepsilon_{i})_{1\leqslant i\leqslant n}
\frac{\partial{\cal T}_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}^{0}$}+n^{-1% /2}\mathbf{{u}})}{\partial\mathbf{{u}}_{\cal A}}=\textrm{$\mathbf{0}$}_{\|{\cal A% }\|},
\textrm{sgn}(x)
\displaystyle{\cal Q}_{3n}-{\cal Q}_{2n}
1\!\!1_{E}
\beta^{0}_{1}=\log(n)
\mathbb{E}_{\cal C}[\delta_{i}]=\mathbb{E}_{\cal C}[1\!\!1_{T_{i}\leq{\cal C}_% {i}}]=\mathbb{P}_{\cal C}[{\cal C}_{i}\geq T_{i}]=G_{0}(Y_{i})
e=\varepsilon_{i}
\widehat{\textrm{$\mathbf{\beta}$}}_{n,{\cal A}}\overset{\mathbb{P}}{\underset% {n\rightarrow\infty}{\longrightarrow}}\textrm{$\mathbf{\beta}$}^{0}_{\cal A}% \neq\textrm{$\mathbf{0}$}_{\|{\cal A}\|}
P(Y\geq t\|X)
\displaystyle x_{max}-x_{min}

\Lambda_{\cal C}

{\cal Q}_{1n}

\sum^{n}_{i=1}\frac{\delta_{i}}{G_{0}(Y_{i})}g_{\tau}(\varepsilon_{i})\mathbf{% X}^{\top}_{{\cal A}^{c},i}\textrm{$\mathbf{\beta}$}^{(2)}_{{\cal A}^{c}}\frac{% 1}{n}\sum^{n}_{j=1}\int^{B}_{0}1\!\!1_{Y_{i}\geq s}\frac{dM_{j}^{\cal C}(s)}{y% (s)}=O_{\mathbb{P}}(1).

(Y_{i},\delta_{i})_{1\leqslant i\leqslant n}

\textrm{$\mathbf{\omega}$}^{0}_{\cal A}\equiv\lim_{n\rightarrow\infty}\widehat% {\textrm{$\mathbf{\omega}$}}_{n,{\cal A}}=|\textrm{$\mathbf{\beta}$}^{0}_{\cal A% }|^{-\gamma}

\mathbf{{u}}=-(\textbf{W}_{1}+\textbf{W}_{2})\mathbb{E}^{-1}_{\varepsilon}\big% {[}h_{\tau}(\varepsilon)\big{]}\mathbb{E}_{\mathbf{X}}[\mathbf{X}\mathbf{X}^{% \top}]^{-1}

\frac{1}{n}\sum^{n}_{i=1}\frac{\delta_{i}}{G_{0}(Y_{i})}g_{\tau}(\varepsilon_{% i})\mathbf{X}^{\top}_{{\cal A}^{c},i}1\!\!1_{Y_{i}\geq s}=O_{\mathbb{P}}(1)% \overset{\mathbb{P}}{\underset{n\rightarrow\infty}{\longrightarrow}}\textrm{$% \mathbf{\kappa}$}_{{\cal A}^{c}}(s).

{\cal W}(0,1)

\displaystyle=\sum^{n}_{i=1}\delta_{i}\bigg{(}\frac{1}{\widehat{G}_{n}(Y_{i})}% -\frac{1}{G_{0}(Y_{i})}\bigg{)}\big{(}\rho_{\tau}(\varepsilon_{i}-n^{1/2}% \mathbf{X}^{\top}_{i}\mathbf{{u}})-\rho_{\tau}(\varepsilon_{i})\big{)}

n^{-1/2}\lambda_{n}=O_{\mathbb{P}}(1)

\textbf{W}_{2}\sim{\cal N}_{p}(\textrm{$\mathbf{0}$}_{p},\textbf{S}_{2})

t=n^{-1/2}\mathbf{X}_{i}^{\top}\mathbf{{u}}

\textbf{S}_{1}

\widetilde{\textrm{$\mathbf{\beta}$}}_{n}

\widehat{\textrm{$\mathbf{\beta}$}}_{n}\equiv\mathop{\mathrm{arg\,min}}_{% \textrm{$\mathbf{\beta}$}\in\mathbb{R}^{p}}\bigg{(}\sum^{n}_{i=1}\frac{\delta_% {i}}{\widehat{G}_{n}(Y_{i})}\rho_{\tau}(\log(Y_{i})-\mathbf{X}_{i}^{\top}% \textrm{$\mathbf{\beta}$})+\lambda_{n}\sum^{p}_{j=1}\widehat{\omega}_{n,j}|% \beta_{j}|\bigg{)},

\widehat{\textrm{$\mathbf{\beta}$}}_{n,{\cal A}}

{\cal C}_{i}\sim{\cal U}[0,c_{1}]

\mathbb{P}[\widehat{\textrm{$\mathbf{\beta}$}}_{n}\in{\cal W}_{n}]{\underset{n% \rightarrow\infty}{\longrightarrow}}0

n\geq 2500

{\cal R}_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}$})

\mathbb{P}[\widehat{\cal A}_{n}\subseteq{\cal A}]{\underset{n\rightarrow\infty% }{\longrightarrow}}1

\textrm{$\mathbf{\beta}$}^{(2)}\equiv\big{(}\textrm{$\mathbf{\beta}$}^{(2)}_{% \cal A},\textrm{$\mathbf{\beta}$}^{(2)}_{{\cal A}^{c}}\big{)}\in{\cal W}_{n}

\lambda_{n}=(150k)^{1/2-0.1}

\mathbb{E}_{\varepsilon}[\varepsilon]=0

\|\mathbf{{u}}\|=c<\infty

{\cal T}_{n}(G,\textrm{$\mathbf{\beta}$})\equiv Q_{n}(G,\textrm{$\mathbf{\beta% }$})-Q_{n}(G,\textrm{$\mathbf{\beta}^{0}$})+\lambda_{n}\sum^{p}_{j=1}\widehat{% \omega}_{n,j}\big{(}|\beta_{j}|-|\beta^{0}_{j}|\big{)}.

\textrm{$\mathbf{\beta}$}^{(2)}

{\cal T}_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}$}^{(2)})-{\cal T}_{n}(% \widehat{G}_{n},\textrm{$\mathbf{\beta}$}^{(1)})=Q_{n}(\widehat{G}_{n},\textrm% {$\mathbf{\beta}$}^{(2)})-Q_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}$}^{(1)% })+\lambda_{n}\sum^{p}_{j=1}\widehat{\omega}_{n,j}\big{(}|\beta^{(2)}_{j}|-|% \beta^{(1)}_{j}|\big{)}.

\mathbb{P}[\widehat{\textrm{$\mathbf{\beta}$}}_{n}\in{\cal A}^{c}\cap\widehat{% \cal A}_{n}]{\underset{n\rightarrow\infty}{\longrightarrow}}0

\widehat{\mathbf{{u}}}_{n,{\cal A}}\equiv\sqrt{n}\big{(}\widehat{\textrm{$% \mathbf{\beta}$}}_{n,{\cal A}}-\textrm{$\mathbf{\beta}$}^{0}_{\cal A}\big{)}% \overset{\cal L}{\underset{n\rightarrow\infty}{\longrightarrow}}{\cal N}_{|{% \cal A}|}\bigg{(}-\mathbb{E}^{-1}_{\varepsilon}[h_{\tau}(\varepsilon)]l_{0}{% \textrm{$\mathbf{\omega}$}^{0}_{\cal A}}^{\top}{\mathrm{sgn}}(\textrm{$\mathbf% {\beta}$}^{0}_{\cal A})\mathbb{E}_{\mathbf{X}}[\mathbf{X}_{{\cal A}}\mathbf{X}% ^{\top}_{{\cal A}}]^{-1},\textrm{$\mathbf{\Xi}$}\bigg{)}

\textrm{$\mathbf{\beta}$}=\textrm{$\mathbf{\beta}^{0}$}+n^{-1/2}\mathbf{{u}}

\textrm{$\mathbf{\kappa}$}(s)\equiv\lim_{n\rightarrow\infty}\frac{1}{n}\sum^{n% }_{i=1}\frac{\delta_{i}}{G_{0}(Y_{i})}1\!\!1_{Y_{i}\geq s}\mathbf{X}_{i}g_{% \tau}(\varepsilon_{i}),

\textrm{$\mathbf{\kappa}$}(s)

\mathbb{E}_{\varepsilon}[h_{\tau}(\varepsilon)]

\mathbb{E}_{\mathbf{X}}\bigg{[}\mathbb{E}_{\varepsilon}\bigg{[}\mathbb{E}_{% \cal C}\bigg{[}\frac{\delta_{i}}{G_{0}(Y_{i})}g_{\tau}(\varepsilon_{i})\mathbf% {X}_{i}^{\top}\mathbf{{u}}|\varepsilon_{i},\mathbf{X}_{i}\bigg{]}|\mathbf{X}_{% i}\bigg{]}\bigg{]}=\mathbb{E}_{\mathbf{X}}\big{[}\mathbb{E}_{\varepsilon}\big{% [}g_{\tau}(\varepsilon_{i})\mathbf{X}_{i}^{\top}\mathbf{{u}}|\mathbf{X}_{i}% \big{]}\big{]}=0.

\mathbf{{u}}^{\top}(\textbf{W}_{1}+\textbf{W}_{2})

\displaystyle\qquad+\frac{\mathbf{{u}}^{\top}_{\cal A}}{\sqrt{n}}\sum^{n}_{j=1% }\int^{B}_{0}\frac{\textrm{$\mathbf{\kappa}$}_{\cal A}(s)}{y(s)}dM^{\cal C}_{j% }(s)+\lambda_{n}\sum^{q}_{j=1}\widehat{\omega}_{n,j}\frac{{\mathrm{sgn}}(\beta% ^{0}_{j})u_{j}}{\sqrt{n}}+o_{\mathbb{P}}(1)

\lim_{n\rightarrow\infty}\mathbb{P}\big{[}\widehat{\cal A}_{n}={\cal A}\big{]}=1

\beta_{0}^{0}=0

\displaystyle\quad+\left\{Q_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}^{0}$}+% n^{-1/2}\mathbf{{u}})-Q_{n}(G_{0},\textrm{$\mathbf{\beta}^{0}$}+n^{-1/2}% \mathbf{{u}})\right\}

{\cal V}(\textrm{$\mathbf{\beta}^{0}$})\equiv\big{\{}\textrm{$\mathbf{\beta}$}% ;\|\textrm{$\mathbf{\beta}$}-\textrm{$\mathbf{\beta}^{0}$}\|\leq cn^{-1/2}\big% {\}}

n^{-1/2}\sum^{n}_{i=1}\frac{\delta_{i}}{G_{0}(Y_{i})}g_{\tau}(\varepsilon_{i})% \mathbf{X}_{{\cal A}^{c},i}\overset{\cal L}{\underset{n\rightarrow\infty}{% \longrightarrow}}{\cal N}_{|{\cal A}^{c}|}(\textrm{$\mathbf{0}$}_{q},\textbf{S% }_{1,{\cal A}^{c}}),

\mathbb{E}_{\mathbf{X}}[\mathbf{X}\mathbf{X}^{\top}]

{1}/{\widehat{G}_{n}(Y_{i})}-{1}/{G_{0}(Y_{i})}

\overset{\cal L}{\underset{n\rightarrow\infty}{\longrightarrow}}

n>2000

\beta^{0}_{0}=0

sd\equiv sd(\widetilde{\textrm{$\mathbf{\beta}$}}_{n}-\textrm{$\mathbf{\beta}^% {0}$})

\beta^{0}_{j}=0

{\cal Q}_{1n}=\sum^{n}_{i=1}\frac{\delta_{i}}{G_{0}(Y_{i})}\bigg{(}g_{\tau}(% \varepsilon_{i})\frac{\mathbf{X}_{i}^{\top}\mathbf{{u}}}{\sqrt{n}}+\frac{1}{2}% h_{\tau}(\varepsilon_{i})\bigg{(}\frac{\mathbf{X}_{i}^{\top}\mathbf{{u}}}{% \sqrt{n}}\bigg{)}^{2}+o_{\mathbb{P}}(n^{-1})\bigg{)}.

\mathbb{E}_{\varepsilon}[h_{\tau}(\varepsilon)]>0

\varepsilon\sim{\cal U}[-1,2]-1/6

X_{ji}\sim{\cal N}(1,1)

{\cal U}[-1,2]-1/6

\mathbb{P}[t\leq T\leq{\cal C}]\geq\zeta_{0}>0

\mathbb{S}_{1}=O_{\mathbb{P}}(1)

{\cal A}\equiv\{j\in\{1,\cdots,p\};\;\beta^{0}_{j}\neq 0\}.

\displaystyle-\bigg{(}g_{\tau}(\varepsilon_{i})\mathbf{X}^{\top}_{{\cal A},i}% \big{(}\textrm{$\mathbf{\beta}$}^{(2)}-\textrm{$\mathbf{\beta}^{0}$}\big{)}_{% \cal A}+\frac{h_{\tau}(\varepsilon_{i})}{2}\big{(}\mathbf{X}^{\top}_{{\cal A},% i}\big{(}\textrm{$\mathbf{\beta}$}^{(2)}-\textrm{$\mathbf{\beta}^{0}$}\big{)}_% {\cal A}\big{)}^{2}+o_{\mathbb{P}}\big{(}\mathbf{X}^{\top}_{{\cal A},i}\big{(}% \textrm{$\mathbf{\beta}$}^{(2)}-\textrm{$\mathbf{\beta}^{0}$}\big{)}_{\cal A}% \big{)}^{2}\bigg{)}\bigg{\}}

Q_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}^{0}$}+n^{-1/2}\mathbf{{u}})-Q_{n% }(\widehat{G}_{n},\textrm{$\mathbf{\beta}^{0}$})=\frac{\mathbb{E}_{\varepsilon% }[h_{\tau}(\varepsilon)]}{2}\mathbf{{u}}^{\top}\mathbb{E}_{\mathbf{X}}[\mathbf% {X}\mathbf{X}^{\top}]\mathbf{{u}}\big{(}1+o_{\mathbb{P}}(1)\big{)}.

j\in\{1,\cdots,q\}

\displaystyle+\sum^{n}_{i=1}\frac{\delta_{i}}{G_{0}(Y_{i})}\bigg{\{}\bigg{(}g_% {\tau}(\varepsilon_{i})\mathbf{X}^{\top}_{i,{\cal A}^{c}}\textrm{$\mathbf{% \beta}$}^{(2)}_{{\cal A}^{c}}+\frac{h_{\tau}(\varepsilon_{i})}{2}\big{(}% \mathbf{X}^{\top}_{i}\big{(}\textrm{$\mathbf{\beta}$}^{(2)}_{{\cal A}}-\textrm% {$\mathbf{\beta}^{0}$}_{\cal A},\textrm{$\mathbf{\beta}$}^{(2)}_{{\cal A}^{c}}% \big{)}\big{)}^{2}+o_{\mathbb{P}}\big{(}\mathbf{X}^{\top}_{i}\big{(}\textrm{$% \mathbf{\beta}$}^{(2)}_{{\cal A}}-\textrm{$\mathbf{\beta}^{0}$}_{\cal A},% \textrm{$\mathbf{\beta}$}^{(2)}_{{\cal A}^{c}}\big{)}\big{)}^{2}\bigg{)}

{\cal Q}_{2n}

\textbf{S}_{3}\equiv\mathbb{E}_{\varepsilon}\big{[}h_{\tau}(\varepsilon)\big{]% }\mathbb{E}_{\mathbf{X}}[\mathbf{X}\mathbf{X}^{\top}]

l_{0}\equiv\lim_{n\rightarrow\infty}n^{-1/2}\lambda_{n}

{\cal W}_{n}\equiv\big{\{}\textrm{$\mathbf{\beta}$}\in{\cal V}(\textrm{$% \mathbf{\beta}^{0}$});\|\textrm{$\mathbf{\beta}$}_{{\cal A}^{c}}\|>0\big{\}}

S_{21}

\beta^{0}_{0}=2

j=1,\cdots p

\varepsilon_{i}\sim{\cal G}(0,1)

\{M_{j}^{\cal C}(t)\}

\beta_{1}^{0}=1/\log k-1/k

\mathbf{{u}}\in\mathbb{R}^{p}

\mathbb{E}_{\mathbf{X}}[\mathbf{X}_{{\cal A}}\mathbf{X}^{\top}_{{\cal A}}]

\displaystyle=\sum^{n}_{i=1}\frac{\delta_{i}}{\widehat{G}_{n}(Y_{i})}\bigg{\{}% \bigg{(}g_{\tau}(\varepsilon_{i})\mathbf{X}^{\top}_{i}\big{(}\textrm{$\mathbf{% \beta}$}^{(2)}_{{\cal A}}-\textrm{$\mathbf{\beta}^{0}$}_{\cal A},\textrm{$% \mathbf{\beta}$}^{(2)}_{{\cal A}^{c}}\big{)}+\frac{h_{\tau}(\varepsilon_{i})}{% 2}\big{(}\mathbf{X}^{\top}_{i}\big{(}\textrm{$\mathbf{\beta}$}^{(2)}_{{\cal A}% }-\textrm{$\mathbf{\beta}^{0}$}_{\cal A},\textrm{$\mathbf{\beta}$}^{(2)}_{{% \cal A}^{c}}\big{)}\big{)}^{2}

n\in\{400,600,\cdots,2000\}

\textrm{$\mathbf{\beta}$}^{(1)}_{{\cal A}^{c}}=\textrm{$\mathbf{0}$}_{|{\cal A% }^{c}|}

\widehat{\textrm{$\mathbf{\beta}$}}_{n}

\displaystyle=\mathbb{E}_{\mathbf{X}}\bigg{[}\mathbb{E}_{\varepsilon}\bigg{[}% \frac{g^{2}_{\tau}(\varepsilon_{i})}{G_{0}(Y_{i})}\frac{\mathbf{{u}}\mathbf{X}% _{i}\mathbf{X}_{i}^{\top}\mathbf{{u}}}{n}|\mathbf{X}_{i}\bigg{]}\bigg{]}

\textrm{$\mathbf{\beta}$}_{\cal A}

{\cal Q}_{1n}=\sum^{n}_{i=1}\frac{\delta_{i}}{G_{0}(Y_{i})}\big{(}\rho_{\tau}(% \varepsilon_{i}-n^{-1/2}\mathbf{X}_{i}^{\top}\mathbf{{u}})-\rho_{\tau}(% \varepsilon_{i})\big{)},

n^{(\gamma-1)/2}\lambda_{n}\rightarrow\infty

\textrm{$\mathbf{\beta}$}\in\mathbb{R}^{p}

\begin{split}\sqrt{n}\bigg{(}\frac{1}{\widehat{G}_{n}(Y_{i})}-\frac{1}{G_{0}(Y% _{i})}\bigg{)}&=-\frac{\sqrt{n}\big{(}\widehat{G}_{n}(Y_{i})-G_{0}(Y_{i})\big{% )}}{G^{2}_{0}(Y_{i})}\big{(}1+o_{\mathbb{P}}(1)\big{)}\\ &=\frac{1}{G_{0}(Y_{i})}\frac{1}{\sqrt{n}}\sum^{n}_{j=1}\int^{B}_{0}1\!\!1_{Y_% {i}\geq s}\frac{dM_{j}^{\cal C}(s)}{y(s)}\big{(}1+o_{\mathbb{P}}(1)\big{)}.% \end{split}

\displaystyle-\bigg{(}\frac{h_{\tau}(\varepsilon_{i})}{2}\big{(}\mathbf{X}^{% \top}_{{\cal A},i}\big{(}\textrm{$\mathbf{\beta}$}^{(2)}-\textrm{$\mathbf{% \beta}^{0}$}\big{)}_{\cal A}\big{)}^{2}+o_{\mathbb{P}}\big{(}\mathbf{X}^{\top}% _{{\cal A},i}\big{(}\textrm{$\mathbf{\beta}$}^{(2)}-\textrm{$\mathbf{\beta}^{0% }$}\big{)}_{\cal A}\big{)}\bigg{)}\bigg{\}}\frac{1}{n}\sum^{n}_{j=1}\int^{B}_{% 0}1\!\!1_{Y_{i}\geq s}\frac{dM_{j}^{\cal C}(s)}{y(s)}\big{(}1+o_{\mathbb{P}}(1% )\big{)}

n>2500

{\cal R}_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}$})=Q_{n}(\widehat{G}_{n},% \textrm{$\mathbf{\beta}$})-Q_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}^{0}$})

t=\mathbf{X}^{\top}_{i}(\textrm{$\mathbf{\beta}$}^{(2)}-\textrm{$\mathbf{\beta% }^{0}$})

\mathbb{E}_{\varepsilon}[h_{\tau}(\varepsilon)]\mathbb{E}_{\mathbf{X}}[\mathbf% {X}_{{\cal A}}\mathbf{X}^{\top}_{{\cal A}}]\mathbf{{u}}_{\cal A}+\textbf{W}_{3% }+l_{0}\widehat{\textrm{$\mathbf{\omega}$}}_{n,{\cal A}}{\mathrm{sgn}}(\textrm% {$\mathbf{\beta}$}^{0}_{\cal A})=\textrm{$\mathbf{0}$}_{|{\cal A}|},

\|\widehat{\textrm{$\mathbf{\beta}$}}_{n}-\textrm{$\mathbf{\beta}$}^{0}\|

(\varepsilon_{i})_{1\leqslant i\leqslant n}

\frac{\partial{\cal T}_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}^{0}$}+n^{-1% /2}\mathbf{{u}})}{\partial\mathbf{{u}}_{\cal A}}=\textrm{$\mathbf{0}$}_{|{\cal A% }|},

\textrm{sgn}(x)

\displaystyle{\cal Q}_{3n}-{\cal Q}_{2n}

1\!\!1_{E}

\beta^{0}_{1}=\log(n)

\mathbb{E}_{\cal C}[\delta_{i}]=\mathbb{E}_{\cal C}[1\!\!1_{T_{i}\leq{\cal C}_% {i}}]=\mathbb{P}_{\cal C}[{\cal C}_{i}\geq T_{i}]=G_{0}(Y_{i})

e=\varepsilon_{i}

\widehat{\textrm{$\mathbf{\beta}$}}_{n,{\cal A}}\overset{\mathbb{P}}{\underset% {n\rightarrow\infty}{\longrightarrow}}\textrm{$\mathbf{\beta}$}^{0}_{\cal A}% \neq\textrm{$\mathbf{0}$}_{|{\cal A}|}

P(Y\geq t|X)

\displaystyle x_{max}-x_{min}