Conditional screening for ultrahigh-dimensional survival data in case-cohort studies

Abstract

The case-cohort design has been widely used to reduce the cost of covariate measurements in large cohort studies. In many such studies, the number of covariates is very large, and the goal of the research is to identify active covariates which have great influence on response. Since the introduction of sure independence screening, screening procedures have achieved great success in terms of effectively reducing the dimensionality and identifying active covariates. However, commonly used screening methods are based on marginal correlation or its variants, they may fail to identify hidden active variables which are jointly important but are weakly correlated with the response. Moreover, these screening methods are mainly proposed for data under the simple random sampling and can not be directly applied to case-cohort data. In this paper, we consider the ultrahigh-dimensional survival data under the case-cohort design, and propose a conditional screening method by incorporating some important prior known information of active variables. This method can effectively detect hidden active variables. Furthermore, it possesses the sure screening property under some mild regularity conditions and does not require any complicated numerical optimization. We evaluate the finite sample performance of the proposed method via extensive simulation studies and further illustrate the new approach through a real data set from patients with breast cancer.

Appendix

1.1 Appendix A: regularity conditions

Let $S_T(t|\mathbf{Z}_i)=\exp \{-\varLambda _0(t)\exp (\varvec{\alpha }^{\mathrm{T}}\mathbf{Z}_i)\}$ and $S_C(t|\mathbf{Z}_i)=P(C_i>t|\mathbf{Z}_i)$ denote the survival functions of $T_i$ and $C_i$, $F_T(t|\mathbf{Z}_i)=1-S_T(t|\mathbf{Z}_i)$, $\varLambda _0(t)=\int _0^t\lambda _0(s)\mathrm{d}s$ denotes the cumulative baseline hazard function. For any vector ${\varvec{\nu }}=(\nu _1,\ldots ,\nu _p)\in R^p$, let $\Vert {\varvec{\nu }}\Vert _d=\root d \of {\sum _{j=1}^p|\nu _j|^d}$ be the $L_d$ norm. For any random variables $\zeta : \varOmega \rightarrow R^d$, $\zeta _1: \varOmega \rightarrow R^{d_1}$, $\zeta _2: \varOmega \rightarrow R^{d_2}$ and $\eta : \varOmega \rightarrow R^p$, the conditional linear expectation of $\zeta$ given $\eta$ is defined as $E^{*}(\zeta |\eta )=E(\zeta )+B^{T}\{\eta -E(\eta )\}$, where $B=\mathrm{argmin}_{D\in R^d\times R^p}E[\{\zeta -E(\zeta )-D^{\mathrm{T}}(\eta -E(\eta ))\}^2|\eta ]$. The conditional linear covariance between $\zeta _1$ and $\zeta _2$ given $\eta$ is defined as $Cov^{*}(\zeta _1,\zeta _2|\eta )=E^{*}[\{\zeta _1-E^{*}(\zeta _1|\eta )\} \{\zeta _2-E^{*}(\zeta _2|\eta )\}|\eta ]$. The properties of $E^{*}(\zeta |\eta )$ and $Cov^{*}(\zeta _1,\zeta _2|\eta )$ are presented in “Appendix B”. The regularity conditions listed below are imposed throughout our discussions.

• C1. For each $j\notin {\mathcal {C}}$ and $k\in {\mathcal {C}}\bigcup \{j\}$, there exists a neighborhood ${\mathcal {B}}_j$ of $({\varvec{\beta }}_{{\mathcal {C}},j}^{0},{\beta }_j^{0})^{\mathrm{T}}$ such that

  $$\begin{aligned} \sup _{t\in [0,\tau ],({\varvec{\beta }}_{{\mathcal {C}},j},{\beta _j})^{\mathrm{T}}\in {\mathcal {B}}_j}\Vert S_{j,k}^{(l)}(\varvec{\beta }_{{\mathcal {C}},j},\beta _j,t)- s_{j,k}^{(l)}(\varvec{\beta }_{{\mathcal {C}},j},\beta _j,t)\Vert _2\rightarrow 0 \end{aligned}$$

  in probability as $n\rightarrow \infty$ ($l=0,1$), $s_{j,k}^{(0)}(\varvec{\beta }_{{\mathcal {C}},j},\beta _j,t)$ is bounded away from zero on ${\mathcal {B}}_j\times [0,\tau ]$, $s_{j,k}^{(l)}(\varvec{\beta }_{{\mathcal {C}},j},\beta _j,t)$ are bounded on ${\mathcal {B}}_j\times [0,\tau ]$.

• C2. For all $j=1,\ldots ,p$, $\int _0^{\tau }\lambda _{j,0}(t)\mathrm {d}t< \infty$ and $E\{Y(\tau )\}>0$.

• C3. The covariates $Z_{j}$ ($j=1,\ldots , p$) are independent of time and bounded by a constant $L_0$. Furthermore, $E(Z_j)=0$ for all $j\in \{1,\ldots , p\}$.

• C4. All $Z_{j}$, $j\in {\mathcal {A}}_{-{\mathcal {C}}}$ are independent of all $Z_{j}$, $j\notin {\mathcal {A}}_{-{\mathcal {C}}}$ given $\mathbf{Z}_{{\mathcal {C}}}$.

• C5. There exists a constant $L_1$ such that $\Vert \varvec{\alpha }\Vert _1<L_1$ and $\Vert ({\varvec{\beta }}_{{\mathcal {C}},j},{\beta }_j)^{\mathrm{T}}\Vert _1<L_1$.

• C6. There exist constants $c_1>0$ and $0<\kappa <1/2$ such that $\min _{j\in {\mathcal {A}}_{-{\mathcal {C}}}}|E[Cov^{*}(Z_{j},P(\delta =1|\mathbf{Z})|\mathbf{Z}_{{\mathcal {C}}})]| \ge c_1 n^{-\kappa }$.

• C7. There exists a constant $L>0$ such that $n^{-1}\Vert \mathbf{U}_j(\widehat{\varvec{\beta }}_{{\mathcal {C}},j},\widehat{\beta }_j)-\mathbf{U}_j(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})\Vert _2 \ge L\Vert (\widehat{\varvec{\beta }}_{{\mathcal {C}},j},\widehat{\beta }_j)^{\mathrm{T}} -({\varvec{\beta }}_{{\mathcal {C}},j}^{0},{\beta }_j^{0})^{\mathrm{T}}\Vert _2$ for all $j\notin {\mathcal {C}}$.

• C8. Let ${\tilde{n}}=\sum _{i=1}^n\xi _i$ denote the sample size of subcohort, then ${\tilde{n}}/n$ converges to the constant $\pi \in (0,1)$.

Conditions C1 and C2 are common assumptions in survival analysis (Andersen and Gill 1982; Fleming and Harrington 1991). Condition C3 assumes the covariates are bounded, similar condition also used in Hong et al. (2018). Condition C4 is similar to the partial orthogonality assumption of the covariates. Condition C5 controls the total effect size of the covariates, it is reasonable under the sparsity principle. Condition C6 is a typical assumption which has been widely used in the literature of feature screening, such as condition 3 in Fan and Lv (2008), condition 2 in Li et al. (2012b), condition 2 in Song et al. (2014), conditions 2 and 5 in Wu and Yin (2015), etc. Condition C7 is a mild assumption which holds in many situations. Condition C8 is a common assumption on the case-cohort design.

1.2 Appendix B: lemmas and theoretic proofs

Let ${\varvec{\beta }}_{{\mathcal {C}},0}$ be the solution of the equations $\mathbf{u}_{{\mathcal {C}}}(\varvec{\beta }_{{\mathcal {C}}})=[u_{j,k}(\varvec{\beta }_{{\mathcal {C}}},0), k\in {\mathcal {C}}]^{\mathrm{T}}=\mathbf{0}_{q}$. Define $\mathbf{v}_{j}(\varvec{\beta }_{{\mathcal {C}},j},\beta _j)=u_{j,j}(\varvec{\beta }_{{\mathcal {C}},j},\beta _j)- \sum _{k\in {\mathcal {C}}}b_k u_{j,k}(\varvec{\beta }_{{\mathcal {C}},j},\beta _j)$, where vector $\mathbf{b}_{{\mathcal {C}}}=[b_k, k\in {\mathcal {C}}]^{\mathrm{T}}$ such that $E^{*}[Z_j|\mathbf{Z}_{{\mathcal {C}}}]=\sum _{k\in {\mathcal {C}}} b_kZ_k$. As a preparation, we first introduce some lemmas.

Lemma 4

Let $\varvec{\xi }=(\xi _1,\ldots ,\xi _n)$ be a random vector containing ${\tilde{n}}$ ones and $n-{\tilde{n}}$ zeros, with each permutation equally likely. Let $B_i (t)$ $(i = 1, \ldots , n)$ be independent and identically distributed real-valued random processes on $[0,\tau ]$ with $E\{B_i(t)\}=\mu _B(t)$, $var(B_i(\tau ))< \infty$. Let $B(t)=\{B_1(t),\ldots ,B_n(t)\}$ be independent of $\xi$. Suppose that almost all paths of $B_i(t)$ have finite variation. Then, $n^{-1/2}\sum _{i=1}^n\xi _i\{B_i(t)-\mu _B(t)\}$ converges weakly in $l^\infty [0,\tau ]$ to a zero-mean Gaussian process and therefore $n^{-1/2}\sum _{i=1}^n\xi _i\{B_i(t)-\mu _B(t)\}$ converges in probability to zero uniformly in t.

This Lemma is the same as Lemma A1 of Kang and Cai (2009).

Lemma 5

Given that $\xi$ is independent of $\varDelta$ and Y(t), $n^{1/2}\{\widehat{\pi }^{-1}(t)-\pi ^{-1}\}$ converges weakly to a zero-mean Gaussian process.

This lemma is extracted from lemma A3 of Ni et al. (2016).

Lemma 6

For independent random variables $Y_1,\ldots , Y_n$ with bounded ranges $[-M,M]$ and zero mean,

$$\begin{aligned} P\left( |Y_1+\ldots +Y_n|>y\right) \le 2 \exp \left( -\frac{1}{2}\frac{y^2}{V+My/3}\right) \end{aligned}$$

for $V\ge Var(Y_1+\ldots +Y_n)$.

This lemma is extracted from lemma 2.2.9 of van der Vaart and Wellner (1996).

Lemma 7

Let $\zeta$, $\zeta _1$, $\zeta _2$ and $\eta$ be any four random variables in the probability space $(\varOmega , {\mathcal {F}},P)$, the following properties hold for the conditional linear expectation $E^*(\cdot |\eta )$ given $\eta$:

• 1. $E^*(\zeta |\eta )=E(\zeta )+Cov(\zeta ,\eta )Var(\eta )^{-1}\{\eta -E(\eta )\}$;

• 2. $E^*(\eta |\eta )=\eta$;

• 3. For any matrices $A_1$ and $A_2$, $E^*(A_1\zeta _1+A_2\zeta _2|\eta )=A_1E^*(\zeta _1|\eta )+A_2E^*(\zeta _2|\eta )$;

• 4. $E^*[E^*(\zeta |\eta )]=E[E^*(\zeta |\eta )]=E[\zeta ]$.

This lemma is extracted from proposition 2 of Hong et al. (2018).

Lemma 8

The conditional linear covariance has the following properties:

• 1. $Cov^*(\zeta _1,\zeta _2|\eta )=0\Longleftrightarrow E^*(\zeta _1\zeta _2|\eta )=E^*(\zeta _1|\eta )E^*(\zeta _2|\eta )$;

• 2. $E[Cov^*(\zeta _1,\zeta _2|\eta )]=Cov(\zeta _1,\zeta _2)- Cov(\zeta _1,\eta )Var(\eta )^{-1}Cov(\eta ,\zeta _2)$;

• 3. For any increasing function $h(\cdot ): R \rightarrow R$ and random variable $\xi : \varOmega \rightarrow R$, we have $Cov^*(h(\xi ),\xi |\eta )\ge 0$.

This lemma is extracted from proposition 3 of Hong et al. (2018).

1.2.1 Proof of Lemma 1

Proof

We first relate $\beta _j^{0}$ to $E[Cov^{*}\{Z_{j},P(\delta =1|\mathbf{Z})|\mathbf{Z}_{{\mathcal {C}}}\}]$, then by condition C6, we relate it to $\alpha _j$. For any $j\notin {\mathcal {C}}$ and $k\in {\mathcal {C}}$, straightforward calculations entail that $s_k^{l}(t)=E\{Z_k^l\lambda _0(t)\exp (\varvec{\alpha }^{\mathrm{T}}\mathbf{Z})S_TS_C\}$ and $s_{j,k}^{(l)}(\varvec{\beta }_{{\mathcal {C}},j},\beta _j,t)=E\{Z_k^l\exp (\mathbf{Z}_{{\mathcal {C}}}^{\mathrm{T}}\varvec{\beta }_{{\mathcal {C}},j}+Z_j\beta _j )S_TS_C\}$ $(l=0,1,2)$, then

$$\begin{aligned}&u_{j,k}(\varvec{\beta }_{{\mathcal {C}},j},\beta _j)\\&\quad =\int _0^{\tau }E\left\{ \left[ Z_k-\frac{E\{Z_k\exp (\mathbf{Z}_{{\mathcal {C}}}^{\mathrm{T}}\varvec{\beta }_{{\mathcal {C}},j}+Z_j\beta _j )S_TS_C\}}{E\{\exp (\mathbf{Z}_{{\mathcal {C}}}^{\mathrm{T}}\varvec{\beta }_{{\mathcal {C}},j}+Z_j\beta _j )S_TS_C\}}\right] \lambda _0(t)\exp (\varvec{\alpha }^{\mathrm{T}}\mathbf{Z})S_TS_C\right\} \mathrm{d}t. \end{aligned}$$

By the definition, we have

$$\begin{aligned} \mathbf{v}_{j}(\varvec{\beta }_{{\mathcal {C}},j},\beta _j)=u_{j,j}(\varvec{\beta }_{{\mathcal {C}},j},\beta _j)- \sum _{k\in {\mathcal {C}}}b_k u_{j,k}(\varvec{\beta }_{{\mathcal {C}},j},\beta _j) \equiv F_{1j}(\varvec{\beta }_{{\mathcal {C}},j},\beta _j)-F_{2j}(\varvec{\beta }_{{\mathcal {C}},j},\beta _j), \end{aligned}$$

where

$$\begin{aligned} F_{1j}(\varvec{\beta }_{{\mathcal {C}},j},\beta _j)= & {} \int _0^{\tau }E\{(Z_j-\sum _{k\in {\mathcal {C}}}b_kZ_k)\lambda _0(t)\exp (\varvec{\alpha }^{\mathrm{T}}\mathbf{Z})S_TS_C\}\mathrm{d}t\\= & {} \int _0^{\tau }E[\{Z_j-E^*(Z_j|\mathbf{Z}_{\mathcal {C}})\}\lambda _0(t)\exp (\varvec{\alpha }^{\mathrm{T}}\mathbf{Z})S_TS_C]\mathrm{d}t\\= & {} E[Cov^{*}\{Z_{j},P(\delta =1|\mathbf{Z})|\mathbf{Z}_{{\mathcal {C}}}\}], \end{aligned}$$

and

$$\begin{aligned}&F_{2j}(\varvec{\beta }_{{\mathcal {C}},j},\beta _j)\\&=\int _0^{\tau }\left[ \frac{E\{Z_j\exp (\mathbf{Z}_{{\mathcal {C}}}^{\mathrm{T}}\varvec{\beta }_{{\mathcal {C}},j}+Z_j\beta _j )S_TS_C\}}{E\{\exp (\mathbf{Z}_{{\mathcal {C}}}^{\mathrm{T}}\varvec{\beta }_{{\mathcal {C}},j}+Z_j\beta _j )S_TS_C\}}\right. \\&\qquad \left. - \sum _{k\in {\mathcal {C}}}b_k \frac{E\{Z_k\exp (\mathbf{Z}_{{\mathcal {C}}}^{\mathrm{T}}\varvec{\beta }_{{\mathcal {C}},j}+Z_j\beta _j )S_TS_C\}}{E\{\exp (\mathbf{Z}_{{\mathcal {C}}}^{\mathrm{T}}\varvec{\beta }_{{\mathcal {C}},j}+Z_j\beta _j )S_TS_C\}}\right] \\&\qquad \times E\left\{ \lambda _0(t)\exp (\varvec{\alpha }^{\mathrm{T}}\mathbf{Z})S_TS_C\right\} \mathrm{d}t\\&\quad =\int _0^{\tau }\frac{E[\{Z_j-E^*(Z_j|\mathbf{Z}_{\mathcal {C}})\}\exp (\mathbf{Z}_{{\mathcal {C}}}^{\mathrm{T}}\varvec{\beta }_{{\mathcal {C}},j}+Z_j\beta _j )S_TS_C]}{E\{\exp (\mathbf{Z}_{{\mathcal {C}}}^{\mathrm{T}}\varvec{\beta }_{{\mathcal {C}},j}+Z_j\beta _j )S_TS_C\}}\\&\qquad \times E\left\{ \lambda _0(t)\exp (\varvec{\alpha }^{\mathrm{T}}\mathbf{Z})S_TS_C\right\} \mathrm{d}t. \end{aligned}$$

By the definition of $(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})$, we have $\mathbf{u}_j(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})=\mathbf{0}_{q+1}$, then $u_{j,k}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})=0$ for any $k\in {\mathcal {C}}\cup \{j\}$, $\mathbf{v}_{j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})=u_{j,j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})- \sum _{k\in {\mathcal {C}}}b_k u_{j,k}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})=0$, $F_{2j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})=F_{1j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0}) =E[Cov^{*}\{Z_{j},P(\delta =1|\mathbf{Z})|\mathbf{Z}_{{\mathcal {C}}}\}]$. When $\alpha _j=0$, $E[Cov^{*}\{Z_{j},P(\delta =1|\mathbf{Z})|\mathbf{Z}_{{\mathcal {C}}}\}]=0$, thus $F_{2j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})=0$. Because of $F_{2j}(\varvec{\beta }_{{\mathcal {C}},0},0)=0$, $\mathbf{v}_{j}(\varvec{\beta }_{{\mathcal {C}},0},0)=E[Cov^{*}\{Z_{j},P(\delta =1|\mathbf{Z})|\mathbf{Z}_{{\mathcal {C}}}\}] -F_{2j}(\varvec{\beta }_{{\mathcal {C}},0},0)=\mathbf{0}_{q+1}$. By the uniqueness of the solution of $\mathbf{v}_{j}(\varvec{\beta }_{{\mathcal {C}}},\beta )$, we have $\beta _j^{0}=0$.

When $\alpha _j\ne 0$, by condition C6, we have $F_{2j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0}) =E[Cov^{*}\{Z_{j},P(\delta =1|\mathbf{Z})|\mathbf{Z}_{{\mathcal {C}}}\}]\ge c_1n^{-\kappa }$. This implies that $F_{2j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})$ and $E[Cov^{*}\{Z_{j},P(\delta =1|\mathbf{Z})|\mathbf{Z}_{{\mathcal {C}}}\}]$ are both nonzero and have the same signs since they are equal. Specifically, $P(\delta =1|\mathbf{Z})$ is the probability of occurrence of the event and $S_TS_C=P(X>t|\mathbf{Z})$ represents the probability at risk at time t. For any t, we have

$$\begin{aligned} \frac{\partial P(\delta =1|\mathbf{Z})}{\partial Z_j}\times \frac{\partial P(X>t|\mathbf{Z})}{\partial Z_j} \le 0. \end{aligned}$$

By lemma 8, $Cov^*\{Z_j,P(\delta =1|\mathbf{Z})|\mathbf{Z}_{{\mathcal {C}}}\}$ and $Cov^*(Z_j,S_TS_C|\mathbf{Z}_{{\mathcal {C}}})$ have the opposite signs unless they are zero. This further implies that

$$\begin{aligned} F_{2j}(\varvec{\beta }_{{\mathcal {C}},0},0)&= \int _0^{\tau }\frac{E\{\exp (\mathbf{Z}_{{\mathcal {C}}}^{\mathrm{T}}\varvec{\beta }_{{\mathcal {C}},0})Cov^{*}(Z_j,S _TS_C|\mathbf{Z}_{{\mathcal {C}}})\}}{E\{\exp (\mathbf{Z}_{{\mathcal {C}}}^{\mathrm{T}}\varvec{\beta }_{{\mathcal {C}},0})S_TS_C\}}\\&\times E\left\{ \lambda _0(t)\exp (\varvec{\alpha }^{\mathrm{T}}\mathbf{Z})S_TS_C\right\} \mathrm{d}t, \end{aligned}$$

and $E[Cov^{*}\{Z_{j},P(\delta =1|\mathbf{Z})|\mathbf{Z}_{{\mathcal {C}}}\}]$ have opposite signs unless they are equal to zero. So $F_{2j}(\varvec{\beta }_{{\mathcal {C}},0},0)\ne F_{2j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})$, therefore, $\beta _j^{0}\ne 0$. $\square$

1.2.2 Proof of Lemma 2

Proof

By lemma 1, for any $j\in {\mathcal {A}}_{-{\mathcal {C}}}$, we have $\beta _j^{0}\ne 0$. By Taylor expansion, there exists $\widetilde{\beta }_j\in (0, \beta _j^{0})$ such that

$$\begin{aligned} |\mathbf{v}_{j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},0)|= |\mathbf{v}_{j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})-\mathbf{v}_{j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},0)| =|\frac{\partial \mathbf{v}_{j} }{\partial \beta _j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\widetilde{\beta }_j)||\beta _j^{0}|. \end{aligned}$$

By the proof of lemma 1, $\mathbf{v}_{j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})=E[Cov^{*}\{Z_{j},P(\delta =1|\mathbf{Z})|\mathbf{Z}_{{\mathcal {C}}}\}] -F_{2j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0}).$ Given $\varvec{\beta }_{{\mathcal {C}},j}^{0}$, consider $F_{2j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j)$ as a function of $\beta _j$, then

$$\begin{aligned} \frac{\partial {F_{2j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j)}}{\partial \beta _j}= & {} \int _0^{\tau }H_{j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j,t)E\left\{ \lambda _0(t)\exp (\varvec{\alpha }^{\mathrm{T}}\mathbf{Z})S_TS_C\right\} \mathrm{d}t\\= & {} E\left\{ \int _0^{\tau }H_{j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j,t)S_C \mathrm{d}F_T(t|\mathbf{Z})\right\} , \end{aligned}$$

where

$$\begin{aligned}&H_{j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j,t)\\&\quad =\frac{E[Z_j\{Z_j-E^*(Z_j|\mathbf{Z}_{\mathcal {C}})\}\exp (\mathbf{Z}_{{\mathcal {C}}}^{\mathrm{T}}\varvec{\beta }_{{\mathcal {C}},j}^{0}+Z_j\beta _j )S_TS_C]}{E\{\exp (\mathbf{Z}_{{\mathcal {C}}}^{\mathrm{T}}\varvec{\beta }_{{\mathcal {C}},j}^{0}+Z_j\beta _j )S_TS_C\}}\\&\qquad -\frac{E[\{Z_j-E^*(Z_j|\mathbf{Z}_{\mathcal {C}})\}\exp (\mathbf{Z}_{{\mathcal {C}}}^{\mathrm{T}}\varvec{\beta }_{{\mathcal {C}},j}^{0}+Z_j\beta _j )S_TS_C]E\{Z_j\exp (\mathbf{Z}_{{\mathcal {C}}}^{\mathrm{T}}\varvec{\beta }_{{\mathcal {C}},j}^{0}+Z_j\beta _j )S_TS_C\}}{[E\{\exp (\mathbf{Z}_{{\mathcal {C}}}^{\mathrm{T}}\varvec{\beta }_{{\mathcal {C}},j}^{0}+Z_j\beta _j )S_TS_C\}]^2}. \end{aligned}$$

By condition C3, $|Z_j|\le L_0$, then $\sup _{\beta _j}|H_{j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j,t)| \le 2L_0^2$. So

$$\begin{aligned} \left| \frac{\partial \mathbf{v}_{j} }{\partial \beta _j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\widetilde{\beta }_j)\right| \le \sup _{\beta _j}\left| \frac{\partial {F_{2j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j)}}{\partial \beta _j}\right| \le 2L_0^2\left| E[E\{S_C(T)|\mathbf{Z}\}]\right| \le 2L_0^2. \end{aligned}$$

By the proof in lemma 1, $F_{2j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},0)$ and $E[Cov^{*}\{Z_{j},P(\delta =1|\mathbf{Z})|\mathbf{Z}_{{\mathcal {C}}}\}]$ have opposite signs, combining it with condition C6,

$$\begin{aligned} |\mathbf{v}_{j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},0)|=|E[Cov^{*}\{Z_{j},P(\delta =1|\mathbf{Z})|\mathbf{Z}_{{\mathcal {C}}}\}]| +|F_{2j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},0)|\ge c_1n^{-\kappa }. \end{aligned}$$

So

$$\begin{aligned} \left| \beta _j^{0}\right| =\left| \frac{\partial \mathbf{v}_{j} }{\partial \beta _j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\widetilde{\beta }_j)\right| ^{-1}\left| \mathbf{v}_{j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},0)\right| \ge (2L_0^2)^{-1}c_1n^{-\kappa }. \end{aligned}$$

Taking $c_2=0.5L_0^{-2}c_1$, we have

$$\begin{aligned} \min _{j\in {\mathcal {A}}_{-{\mathcal {C}}}}|\beta _j^{0}|\ge c_2n^{-\kappa }, \end{aligned}$$

which completes the proof. $\square$

1.2.3 Proof of Lemma 3

Proof

Denote $\bar{\mathbf{U}}_j(\varvec{\beta }_{{\mathcal {C}},j},\beta _j)=n^{-1}\mathbf{U}_j(\varvec{\beta }_{{\mathcal {C}},j},\beta _j)$. By the definition of $(\widehat{\varvec{\beta }}_{{\mathcal {C}},j},\widehat{\beta }_j)^{\mathrm{T}}$, we have

$$\begin{aligned} \left\| \bar{\mathbf{U}}_j(\widehat{\varvec{\beta }}_{{\mathcal {C}},j},\widehat{\beta }_j)- \bar{\mathbf{U}}_j({\varvec{\beta }}_{{\mathcal {C}},j}^{0},{\beta }_j^{0})\right\| = \left\| \bar{\mathbf{U}}_j({\varvec{\beta }}_{{\mathcal {C}},j}^{0},{\beta }_j^{0})\right\| . \end{aligned}$$

For any $j\notin {\mathcal {C}}$ and $k\in {\mathcal {C}}\cup \{j\}$, using the similar method of Lin and Wei (1989), by lemmas 4 and 5, we can obtain that

$$\begin{aligned} \bar{\mathbf{U}}_j(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})=n^{-1}\sum _{i=1}^n\mathbf{W}_{i,j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})+o_p(1), \end{aligned}$$

where $\mathbf{W}_{i,j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})$ $(i=1,\ldots , n)$ are independent, $E\{\mathbf{W}_{i,j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})\}=\mathbf{0}$ and $\mathbf{W}_{i,j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})= [{W}_{i,j,k}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0}), k\in {\mathcal {C}}\cup \{j\}]^{\mathrm{T}}$ with

$$\begin{aligned}&{W}_{i,j,k}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})\\&\quad =\int _{0}^{\tau }\Big [ Z_{ik}-\frac{E\{Z_{ik} \exp (\varvec{\beta }_{{\mathcal {C}},j}^{0}\mathbf{Z}_{i,{\mathcal {C}}}+ \beta _j^{0} Z_{ij})S_TS_C\}}{E\{ \exp (\varvec{\beta }_{{\mathcal {C}},j}^{0}\mathbf{Z}_{i,{\mathcal {C}}}+ \beta _j^{0} Z_{ij})S_TS_C\}}\Big ]\mathrm{d}N_i(t)\\&\qquad -\int _{0}^{\tau }\frac{Y_i(t)\exp (\varvec{\beta }_{{\mathcal {C}},j}^{0}\mathbf{Z}_{i,{\mathcal {C}}}+ \beta _j^{0} Z_{ij})}{E\{ \exp (\varvec{\beta }_{{\mathcal {C}},j}^{0}\mathbf{Z}_{i,{\mathcal {C}}}+ \beta _j^{0} Z_{ij})S_TS_C\}} \Big [ Z_{ik}\\&\qquad -\frac{E\{Z_{ik} \exp (\varvec{\beta }_{{\mathcal {C}},j}^{0}\mathbf{Z}_{i,{\mathcal {C}}}+ \beta _j^{0} Z_{ij})S_TS_C\}}{E\{ \exp (\varvec{\beta }_{{\mathcal {C}},j}^{0}\mathbf{Z}_{i,{\mathcal {C}}}+ \beta _j^{0} Z_{ij})S_TS_C\}}\Big ] E\{\mathrm{d}N_i(t)\}. \end{aligned}$$

Let $E_n$ denote the empirical measure, we can write

$$\begin{aligned} \bar{\mathbf{U}}_j(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})=E_n[\mathbf{W}_{i,j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})]+o_p(1). \end{aligned}$$

For any given i, j, k, by conditions C1, C3, C5, there exists a constant $L_2$ such that $|{W}_{i,j,k}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})|\le L_2$, by the fact that $E[{W}_{i,j,k}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})]=0$, we have $Var[{W}_{i,j,k}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})]= E[|{W}_{i,j,k}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})|^2]\le L_2^2$. By lemma 6, for any $t>0$, $j\notin {\mathcal {C}}$ and $k\in {\mathcal {C}}\cup \{j\}$, we have

$$\begin{aligned} P\left( \left| E_n({W}_{i,j,k}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0}))\right| >\frac{t}{n}\right) \le 2\exp \left( -\frac{1}{2}\frac{t^2}{nL_2^2+L_2t/3}\right) . \end{aligned}$$

By Bonferroni inequality, we have

$$\begin{aligned} P\left( \Vert E_n(\mathbf{W}_{i,j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0}))\Vert _2>\frac{t(q+1)}{n}\right) \le 2(q+1)\exp \left( -\frac{1}{2}\frac{t^2}{nL_2^2+L_2t/3}\right) . \end{aligned}$$

As $\Vert \bar{\mathbf{U}}_j(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})-E_n[\mathbf{W}_{i,j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})]\Vert _2=o_p(1)$, for any $\epsilon _1>0$ and $\epsilon _2>0$, there exists $N_1$ such that for any $n>N_1$, we have

$$\begin{aligned} P\left( \left\| \bar{\mathbf{U}}_j(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})-E_n[\mathbf{W}_{i,j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})]\right\| _2>Lc_2\epsilon _1/2\right) <\epsilon _2. \end{aligned}$$

Taking $t=\frac{c_2Ln^{1-\kappa }}{2(q+1)}>0$, then $\frac{t(q+1)}{n}=\frac{c_2Ln^{-\kappa }}{2}$. By Triangle inequality and Bonferroni inequality, we have

$$\begin{aligned}&P\left( \left\| \bar{\mathbf{U}}_j(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})\right\| _2>Lc_2(n^{-\kappa }+\epsilon _1)/2\right) \\&\le P\left( \left\| E_n\{\mathbf{W}_{i,j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})\}\right\| _2>Lc_2n^{-\kappa }/2\right) \\&+ P\left( \left\| \bar{\mathbf{U}}_j(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})-E_n\{\mathbf{W}_{i,j}(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})\}\right\| _2>Lc_2\epsilon _1/2\right) \\&\le 2(q+1)\exp \left( -\frac{1}{2}\frac{c_2^2L^2n^{2-2\kappa }/4(q+1)^2}{nL_2^2+L_2c_2Ln^{1-\kappa }/6(q+1)}\right) +\epsilon _2. \end{aligned}$$

Taking $N=\max \{(L_2/3)^{1/\kappa },N_1\}$, then for any $n>N$, $n^{-\kappa }<3/L_2$, so we have

$$\begin{aligned} P\left( \Vert \bar{\mathbf{U}}_j(\varvec{\beta }_{{\mathcal {C}},j}^{0},\beta _j^{0})\Vert _2>Lc_2(n^{-\kappa }+\epsilon _1)/2\right) \le 2(q+1)\exp \left( -c_3n^{1-2\kappa }\right) +\epsilon _2, \end{aligned}$$

where $c_3=\frac{c_2^2L^2}{8L_2^2(q+1)^2+4c_2L(q+1)}$. By condition C7, we have

$$\begin{aligned}&P\left( \left| \widehat{\beta }_j-\beta _j^{0}\right|>c_2(n^{-\kappa }+\epsilon _1)/2\right) \\&\le P\left( \left\| (\widehat{\varvec{\beta }}_{{\mathcal {C}},j},\widehat{\beta }_j)^{\mathrm{T}}-({\varvec{\beta }}_{{\mathcal {C}},j}^{0},{\beta }_j^{0})^{\mathrm{T}}\right\| _2>c_2(n^{-\kappa }+\epsilon _1)/2\right) \\&\le P\left( \left\| \bar{\mathbf{U}}_j(\widehat{\varvec{\beta }}_{{\mathcal {C}},j},\widehat{\beta }_j)^{\mathrm{T}}-\bar{\mathbf{U}}_j({\varvec{\beta }}_{{\mathcal {C}},j}^{0},{\beta }_j^{0})\right\| _2>Lc_2(n^{-\kappa }+\epsilon _1)/2\right) \\&= P\left( \left\| \bar{\mathbf{U}}_j({\varvec{\beta }}_{{\mathcal {C}},j}^{0},{\beta }_j^{0})\right\| _2>Lc_2(n^{-\kappa }+\epsilon _1)/2\right) \\&\le 2(q+1)\exp \left( -c_3n^{1-2\kappa }\right) +\epsilon _2. \end{aligned}$$

Then we have

$$\begin{aligned} P\left( \max _{j\in {\mathcal {A}}_{-{\mathcal {C}}}}\left| \widehat{\beta }_j-\beta _j^{0}\right| >c_2(n^{-\kappa }+\epsilon _1)/2\right) \le 2a(q+1)\exp (-c_3n^{1-2\kappa })+a\epsilon _2, \end{aligned}$$

where $a=|{\mathcal {A}}_{-{\mathcal {C}}}|=\sum _{j\notin {\mathcal {C}}}I(\alpha _j\ne 0)$ is the size of $|{\mathcal {A}}_{-{\mathcal {C}}}|$. $\square$

1.2.4 Proof of Theorem 1

Proof

By the definition of $\widehat{{\mathcal {A}}}_{-{\mathcal {C}}}$ and condition C7, there exists a positive constant $c_4$ such that

$$\begin{aligned} P\left( {\mathcal {A}}_{-{\mathcal {C}}}\subseteq \widehat{{\mathcal {A}}}_{-{\mathcal {C}}}\right) =P\left( \min _{j\in {\mathcal {A}}_{-{\mathcal {C}}}}|\widehat{\beta }_j|/\widehat{\sigma }_j\ge \gamma \right) \ge 1-P\left( \min _{j\in {\mathcal {A}}_{-{\mathcal {C}}}}|\widehat{\beta }_j|<n^{-1/2}c_4\gamma \right) . \end{aligned}$$

Following lemma 2, for any $j\in {\mathcal {A}}_{-{\mathcal {C}}}$, we have $|\beta _j^{0}-\widehat{\beta }_j|\ge |\beta _j^{0}|-|\widehat{\beta }_j|\ge c_2n^{-\kappa }-|\widehat{\beta }_j|$. Suppose $\min _{j\in {\mathcal {A}}_{-{\mathcal {C}}}}|\widehat{\beta }_j|<n^{-1/2}c_4\gamma$, then $\max _{j\in {\mathcal {A}}_{-{\mathcal {C}}}}|\beta _j^{0}-\widehat{\beta }_j|\ge c_2n^{-\kappa }-n^{-1/2}c_4\gamma$. If we have $\gamma <c_2(n^{-\kappa }-\epsilon _1)n^{1/2}/(2c_4)$, we can obtain

$$\begin{aligned} P\left( \min _{j\in {\mathcal {A}}_{-{\mathcal {C}}}}\left| \widehat{\beta }_j\right|<n^{-1/2}c_4\gamma \right) < P\left( \max _{j\in {\mathcal {A}}_{-{\mathcal {C}}}}\left| \beta _j^{0}-\widehat{\beta }_j\right| \ge c_2(n^{-\kappa }+\epsilon _1)/2\right) . \end{aligned}$$

Then $P({\mathcal {A}}_{-{\mathcal {C}}}\subseteq \widehat{{\mathcal {A}}}_{-{\mathcal {C}}}) \ge 1-2a(q+1)\exp (-c_3n^{1-2\kappa })-a\epsilon _2$. Let $n\rightarrow \infty$, for any $\epsilon _2>0$, we have $\lim _{n\rightarrow \infty }P({\mathcal {A}}_{-{\mathcal {C}}}\subseteq \widehat{{\mathcal {A}}}_{-{\mathcal {C}}}) \ge 1-a\epsilon _2$, the right side of the above equation does not depend on n any more. Taking $\epsilon _2\rightarrow 0$, we have $\lim _{n\rightarrow \infty }P({\mathcal {A}}_{-{\mathcal {C}}}\subseteq \widehat{{\mathcal {A}}}_{-{\mathcal {C}}})=1$. $\square$

1.2.5 Proof of Theorem 2

Proof

For any $j\in {\mathcal {A}}_{-{\mathcal {C}}}$, we have $\alpha _j\ne 0$. From lemma 1, we know that $|\beta _j^{0}|> 0$. Similarily, we have $|\beta _j^{0}|=0$ if $j\notin {\mathcal {A}}_{-{\mathcal {C}}}$. As $\widehat{\beta }_j$ is a consistent estimator of $\beta _j^{0}$ and $M_{{\mathcal {C}},j}=|\widehat{\beta }_j|/\widehat{\sigma }_j$, we can easily conclude that $P(\max _{j\notin {\mathcal {A}}_{-{\mathcal {C}}}}M_{{\mathcal {C}},j}<\min _{j\in {\mathcal {A}}_{-{\mathcal {C}}}}M_{{\mathcal {C}},j})\rightarrow 1$ when $n\rightarrow \infty$, which completes the proof of theorem 2. $\square$