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\mathbb{E}_{\varepsilon}\big{[}g^{2}_{\tau}(\varepsilon)\big{]}=\mathbb{V}%
\mbox{ar}\,[\varepsilon] |
\mathbf{\kappa} |
\displaystyle\qquad+\lambda_{n}\sum^{q}_{j=1}\widehat{\omega}_{n,j}\frac{{%
\mathrm{sgn}}(\beta^{0}_{j})u_{j}}{\sqrt{n}}+o_{\mathbb{P}}(1) |
100\frac{1}{M}\sum^{M}_{l=1}\frac{\big{|}\widehat{\cal A}^{c}_{n,l}\cap{\cal A%
}\big{|}}{|{\cal A}|}, |
\widetilde{\tau} |
\mathbb{E}_{\varepsilon}\big{[}h_{\tau}(\varepsilon)\big{]}\mathbf{{u}}^{\top}%
\mathbb{E}_{\mathbf{X}}[\mathbf{X}\mathbf{X}^{\top}]\mathbf{{u}} |
\beta^{0}_{1}=5\log(n) |
l_{0}=0 |
t\in[0,B] |
\widehat{\textrm{$\mathbf{\beta}$}}_{n}-\textrm{$\mathbf{\beta}^{0}$}=O_{%
\mathbb{P}}(n^{-1/2}) |
\displaystyle=\left\{Q_{n}(G_{0},\textrm{$\mathbf{\beta}^{0}$}+n^{-1/2}\mathbf%
{{u}})-Q_{n}(G_{0},\textrm{$\mathbf{\beta}^{0}$})\right\} |
\lambda_{n}n^{(\gamma-1)/2}{\underset{n\rightarrow\infty}{\longrightarrow}}\infty |
{\cal S}^{c} |
\mathbf{{u}}_{\cal A}=-\mathbb{E}^{-1}_{\varepsilon}[h_{\tau}(\varepsilon)]%
\mathbb{E}_{\mathbf{X}}[\mathbf{X}_{{\cal A}}\mathbf{X}^{\top}_{{\cal A}}]^{-1%
}\bigg{(}\textbf{W}_{3}+l_{0}\widehat{\textrm{$\mathbf{\omega}$}}_{n,{\cal A}}%
{\mathrm{sgn}}(\textrm{$\mathbf{\beta}$}^{0}_{\cal A})\bigg{)}. |
\rho_{\tau}(x)=|\tau-1\!\!1_{x<0}|x^{2},\qquad x\in\mathbb{R}, |
\widehat{\omega}_{n,j}=O_{\mathbb{P}}(1) |
\widetilde{\textrm{$\mathbf{\beta}$}}_{n,{\cal A}} |
\delta_{i}\equiv 1\!\!1_{T_{i}\leq{\cal C}_{i}} |
X_{2i}\sim{\cal N}(1,5) |
h_{\tau}(x)\equiv\rho^{\prime\prime}_{\tau}(x-t)|_{t=0}=2\tau 1\!\!1_{x\geq 0}%
+2(1-\tau)1\!\!1_{x<0} |
\widetilde{\textrm{$\mathbf{\beta}$}}_{n}-\textrm{$\mathbf{\beta}^{0}$} |
{\cal T}_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}$})={\cal R}_{n}(\widehat{%
G}_{n},\textrm{$\mathbf{\beta}$})+\lambda_{n}\sum^{p}_{j=1}\widehat{\omega}_{n%
,j}\big{(}|\beta_{j}|-|\beta^{0}_{j}|\big{)}, |
\mathbb{P}[{\cal A}=\widehat{\cal A}_{n}]{\underset{n\rightarrow\infty}{%
\longrightarrow}}1 |
j\in\{6,\cdots,p\} |
\lambda_{n}=n^{1/2-0.01} |
k\in\{2,3\cdots,10\} |
\mathbb{P}_{\cal C} |
\displaystyle+\sum^{n}_{i=1}\delta_{i}\bigg{(}\frac{1}{\widehat{G}_{n}(Y_{i})}%
-\frac{1}{G_{0}(Y_{i})}\bigg{)}\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} |
s,t\in[0,B] |
p\in\{50,150\} |
\beta_{0}^{0}=2 |
Q_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}$}^{(2)})-Q_{n}(\widehat{G}_{n},%
\textrm{$\mathbf{\beta}$}^{(1)})=O_{\mathbb{P}}(1). |
\beta^{0}_{0},\beta^{0}_{1},\cdots,\beta^{0}_{p} |
\mathbb{E}_{\varepsilon}[g_{\tau}(\varepsilon)]=\mathbb{E}_{\varepsilon}[%
\varepsilon]=0 |
(X_{1},\cdots,X_{p}) |
t=\mathbf{X}^{\top}_{i}(\textrm{$\mathbf{\beta}$}^{(1)}-\textrm{$\mathbf{\beta%
}^{0}$}) |
\displaystyle\qquad+\frac{1}{n}\sum^{n}_{i=1}\frac{\delta_{i}}{G_{0}(Y_{i})}g_%
{\tau}(\varepsilon_{i})\frac{\mathbf{X}^{\top}_{{\cal A},i}\mathbf{{u}}_{\cal A%
}}{\sqrt{n}}\sum^{n}_{j=1}\int^{B}_{0}\frac{1\!\!1_{Y_{i}\geq s}}{y(s)}dM^{%
\cal C}_{j}(s)\bigg{\}}\big{(}1+o_{\mathbb{P}}(1)\big{)}+\lambda_{n}\sum^{q}_{%
j=1}\widehat{\omega}_{n,j}\frac{{\mathrm{sgn}}(\beta^{0}_{j})u_{j}}{\sqrt{n}}, |
\displaystyle=\frac{\mathbb{E}_{\varepsilon}[h_{\tau}(\varepsilon)]}{2}\mathbf%
{{u}}^{\top}_{\cal A}\mathbb{E}_{\mathbf{X}}[\mathbf{X}_{{\cal A}}\mathbf{X}^{%
\top}_{{\cal A}}]\mathbf{{u}}_{\cal A}+\frac{\mathbf{{u}}^{\top}_{\cal A}}{%
\sqrt{n}}\sum^{n}_{i=1}\bigg{(}\frac{\delta_{i}}{G_{0}(Y_{i})}g_{\tau}(%
\varepsilon_{i})\mathbf{X}_{{\cal A},i}+\int^{B}_{0}\frac{\textrm{$\mathbf{%
\kappa}$}_{\cal A}(s)}{y(s)}dM^{\cal C}_{i}(s)\bigg{)} |
{\cal N}_{p}(\textbf{m},\textbf{V}) |
\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}_{i,{\cal A%
}}\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{\}} |
n^{1/2}(\widetilde{\textrm{$\mathbf{\beta}$}}_{n}-\textrm{$\mathbf{\beta}^{0}$%
})\overset{\cal L}{\underset{n\rightarrow\infty}{\longrightarrow}}{\cal N}_{p}%
(\textrm{$\mathbf{0}$}_{p},\textbf{S}^{-1}_{3}(\textbf{S}_{1}+\textbf{S}_{2})%
\textbf{S}^{-1}_{3}) |
\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{)}, |
\mathbb{E}_{\varepsilon}[\varepsilon^{4}]<\infty |
n<1500 |
l_{0}>0 |
\widehat{\cal A}_{n,l} |
c=\|\mathbf{{u}}\| |
\frac{1}{n}\sum^{n}_{i=1}\frac{\delta_{i}}{G_{0}(Y_{i})}\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}$}^{(2)}_{{\cal A}^{c}}\big{)}\big{)}^{2}=O_{\mathbb{P}}\big{(}\|\textrm%
{$\mathbf{\beta}$}^{(2)}-\textrm{$\mathbf{\beta}^{0}$}\|^{2}\big{)}=O_{\mathbb%
{P}}(n^{-1}). |
\lambda_{n}=n^{1/2} |
{\cal R}_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}$})\equiv\sum^{n}_{i=1}%
\frac{\delta_{i}}{\widehat{G}_{n}(Y_{i})}\bigg{(}\rho_{\tau}\big{(}\log(Y_{i})%
-\mathbf{X}^{\top}_{i}\textrm{$\mathbf{\beta}$}\big{)}-\rho_{\tau}(\varepsilon%
_{i})\bigg{)}=\sum^{n}_{i=1}\frac{\delta_{i}}{\widehat{G}_{n}(Y_{i})}\bigg{(}%
\rho_{\tau}\big{(}\varepsilon_{i}-n^{-1/2}\mathbf{X}^{\top}_{i}\mathbf{{u}}%
\big{)}-\rho_{\tau}(\varepsilon_{i})\bigg{)}. |
\big{(}\widehat{\beta}_{n,1},\cdots,\widehat{\beta}_{n,p}\big{)} |
\textbf{v}_{3}\equiv\frac{\delta}{G_{0}(Y)}g_{\tau}(\varepsilon)\mathbf{X}_{{%
\cal A}}+\int^{B}_{0}\frac{\textrm{$\mathbf{\kappa}$}_{\cal A}(s)}{y(s)}dM^{%
\cal C}(s) |
X_{1i}\sim{\cal N}(1,1) |
l_{0}\neq 0 |
\mathbb{E}_{\mathbf{X}} |
\beta^{0}_{5}=-1 |
\widetilde{\textrm{$\mathbf{\beta}$}}_{n}\equiv\mathop{\mathrm{arg\,min}}_{%
\textrm{$\mathbf{\beta}$}\in\mathbb{R}^{p}}\sum^{n}_{i=1}\frac{\delta_{i}}{%
\widehat{G}_{n}(Y_{i})}\rho_{\tau}(\log(Y_{i})-\mathbf{X}_{i}^{\top}\textrm{$%
\mathbf{\beta}$}), |
L2\equiv\|\widetilde{\textrm{$\mathbf{\beta}$}}_{n}-\textrm{$\mathbf{\beta}^{0%
}$}\| |
{\cal A}^{c}=\{q+1,\cdots,p\} |
1-G_{0}(t) |
{\cal A}=\{1\} |
{\cal V}(\textrm{$\mathbf{\beta}^{0}$}) |
{\cal T}_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}^{0}$}+n^{-1/2}\mathbf{{u}}) |
\beta^{0}_{0}\neq 0 |
\displaystyle=\mathbb{E}_{\varepsilon}\big{[}g^{2}_{\tau}(\varepsilon_{i})\big%
{]}\frac{\mathbf{{u}}^{\top}}{n}\mathbb{E}_{\mathbf{X}}\bigg{[}\frac{\mathbf{X%
}\mathbf{X}^{\top}}{G_{0}(Y)}\bigg{]}\mathbf{{u}}, |
\textbf{S}_{1}\equiv\mathbb{E}_{\varepsilon}\big{[}g^{2}_{\tau}(\varepsilon)%
\big{]}\mathbb{E}_{\mathbf{X}}\big{[}\mathbf{X}\mathbf{X}^{\top}/G_{0}(Y)\big{]} |
n<2000 |
\displaystyle=\sum^{n}_{i=1}\frac{\delta_{i}}{G_{0}(Y_{i})}\bigg{\{}\bigg{(}g_%
{\tau}(\varepsilon_{i})\mathbf{X}^{\top}_{i}\big{(}\textrm{$\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}$}^{(2)}_{{\cal A}^{c}}%
\big{)}\big{)}^{2} |
g_{\tau}(x)\equiv\rho^{\prime}_{\tau}(x-t)|_{t=0}=-2\tau x1\!\!1_{x\geq 0}-2(1%
-\tau)x1\!\!1_{x<0} |
\beta^{0}_{0} |
\textrm{$\mathbf{\Xi}$}=\textbf{S}^{-1}_{3,{\cal A}}(\textbf{S}_{1,{\cal A}}+%
\textbf{S}_{2,{\cal A}})\textbf{S}^{-1}_{3,{\cal A}}) |
\|\textrm{$\mathbf{\beta}^{0}$}\| |
\textbf{S}_{2}\equiv\mathbb{E}\big{[}\int^{B}_{0}{\textrm{$\mathbf{\kappa}$}(s%
)\textrm{$\mathbf{\kappa}$}^{\top}(s)}/{y(s)}d\Lambda_{C}(s)\big{]} |
(\widehat{\textrm{$\mathbf{\beta}$}}_{n}-\textrm{$\mathbf{\beta}^{0}$})_{\cal A} |
\mathbb{P}_{\mathbf{X}}[\|\mathbf{X}\|<c]=1 |
\displaystyle Q_{n}(\widehat{G}_{n},\textrm{$\mathbf{\beta}$}^{(2)})-Q_{n}(%
\widehat{G}_{n},\textrm{$\mathbf{\beta}$}^{(1)})=\sum^{n}_{i=1}\frac{\delta_{i%
}}{\widehat{G}_{n}(Y_{i})}\bigg{(}\rho_{\tau}\big{(}\log(Y_{i})-\mathbf{X}_{i}%
^{\top}\textrm{$\mathbf{\beta}$}^{(2)}\big{)}-\rho_{\tau}\big{(}\log(Y_{i})-%
\mathbf{X}_{i}^{\top}\textrm{$\mathbf{\beta}$}^{(1)}\big{)}\bigg{)} |
\begin{split}\lambda_{n}\sum^{p}_{j=1}\widehat{\omega}_{n,j}\big{(}|\beta^{(2)%
}_{j}|-|\beta^{(1)}_{j}|\big{)}&=\lambda_{n}\sum^{p}_{j=q+1}\widehat{\omega}_{%
n,j}|\beta^{(2)}_{j}|\\
&\geq O_{\mathbb{P}}\big{(}\lambda_{n}n^{\gamma/2}n^{-1/2}\big{)}=O_{\mathbb{P%
}}(\lambda_{n}n^{(\gamma-1)/2}){\underset{n\rightarrow\infty}{\longrightarrow}%
}\infty.\end{split} |
\rho_{\tau}(e-t)=\rho_{\tau}(e)+g_{\tau}(e)t+h_{\tau}(e)t^{2}/2+o(t^{2}). |
n=150k |
\textbf{W}_{1}+\mathbb{E}_{\varepsilon}\big{[}h_{\tau}(\varepsilon)\big{]}%
\mathbb{E}_{\mathbf{X}}[\mathbf{X}\mathbf{X}^{\top}]\mathbf{{u}}+\textbf{W}_{2%
}=\textrm{$\mathbf{0}$}_{p}. |
(Y_{i},\mathbf{X}_{i},\delta_{i})_{1\leqslant i\leqslant n} |
\displaystyle=\frac{\mathbb{E}_{\varepsilon}[h_{\tau}(\varepsilon)]}{2}\mathbf%
{{u}}^{\top}_{\cal A}\mathbb{E}_{\mathbf{X}}[\mathbf{X}_{{\cal A}}\mathbf{X}^{%
\top}_{{\cal A}}]\mathbf{{u}}_{\cal A}+\mathbf{{u}}^{\top}_{\cal A}\textbf{W}_%
{3}+l_{0}\widehat{\textrm{$\mathbf{\omega}$}}_{n,{\cal A}}{\mathrm{sgn}}(%
\textrm{$\mathbf{\beta}$}^{0}_{\cal A})^{\top}\mathbf{{u}}_{\cal A}, |
S_{21}=O_{\mathbb{P}}(1) |
V_{n}=O_{\mathbb{P}}(U_{n}) |
c>0\,\text{so that}\,\underset{n\rightarrow\infty}{\text{lim}}\mathbb{P}(|{U_{%
n}}/{V_{n}}|>c)<e |
\textrm{$\mathbf{\beta}^{0}$}=(\beta^{0}_{1},\cdots,\beta^{0}_{p}) |
\displaystyle=\sum^{n}_{i=1}\frac{\delta_{i}}{G_{0}(Y_{i})}\bigg{\{}\bigg{(}g_%
{\tau}(\varepsilon_{i})\mathbf{X}^{\top}_{{\cal A}^{c},i}\textrm{$\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}$}^{(2)}_{{\cal A}^{c}}\big{)}\big{)}\bigg{)} |
\textrm{$\mathbf{\beta}^{0}$}=\big{(}\textrm{$\mathbf{\beta}$}^{0}_{\cal A},%
\textrm{$\mathbf{0}$}_{|{\cal A}^{c}|}\big{)} |
S_{22} |
\frac{1}{n}\sum^{n}_{i=1}\frac{\delta_{i}}{G_{0}(Y_{i})}h_{\tau}(\varepsilon_{%
i})\mathbf{{u}}^{\top}\mathbf{X}_{i}\mathbf{X}^{\top}_{i}\mathbf{{u}}\overset{%
\mathbb{P}}{\underset{n\rightarrow\infty}{\longrightarrow}}\mathbb{E}\big{[}h_%
{\tau}(\varepsilon)\mathbf{{u}}^{\top}\mathbf{X}\mathbf{X}^{\top}\mathbf{{u}}%
\big{]}=\mathbb{E}_{\varepsilon}\big{[}h_{\tau}(\varepsilon)\big{]}\mathbf{{u}%
}^{\top}\mathbb{E}_{\mathbf{X}}\big{[}\mathbf{X}\mathbf{X}^{\top}\big{]}%
\mathbf{{u}}. |
\displaystyle\qquad\qquad+o_{\mathbb{P}}\big{(}\mathbf{X}^{\top}_{i}\big{(}%
\textrm{$\mathbf{\beta}$}^{(2)}_{{\cal A}}-\textrm{$\mathbf{\beta}^{0}$}_{\cal
A%
},\textrm{$\mathbf{\beta}$}^{(2)}_{{\cal A}^{c}}\big{)}\big{)}^{2}\bigg{)} |
SD\equiv sd((\widehat{\textrm{$\mathbf{\beta}$}}_{n}-\textrm{$\mathbf{\beta}^{%
0}$})_{\cal A}) |
T^{*}_{i}=\beta^{0}_{0}+\sum^{p}_{j=1}\beta^{0}_{j}X_{ji}+\varepsilon_{i},%
\qquad i=1,\cdots,n, |
\lambda_{n}=n^{1/2-0.1} |
\frac{1}{\sqrt{n}}\sum^{n}_{j=1}\int^{B}_{0}\frac{\textrm{$\mathbf{\kappa}$}_{%
{\cal A}^{c}}(s)}{y(s)}dM^{\cal C}_{j}(s)\overset{\cal L}{\underset{n%
\rightarrow\infty}{\longrightarrow}}\textbf{W}_{2,{\cal A}^{c}}\sim{\cal N}_{|%
{\cal A}^{c}|}\big{(}\textrm{$\mathbf{0}$}_{|{\cal A}^{c}|},...\big{)}, |
T_{i}=\exp(T_{i}^{*}) |
\widehat{\omega}_{n,j}\equiv|\widetilde{\beta}_{n,j}|^{-\gamma} |
\textbf{S}_{1,{\cal A}} |
\textbf{W}_{1}+\textbf{W}_{2}\sim{\cal N}_{p}(\textrm{$\mathbf{0}$}_{p},%
\textbf{S}_{1}+\textbf{S}_{2}) |
\mathbb{E}_{\varepsilon}[h_{0.5}(\varepsilon)]=1 |