Conditional screening for ultra-high dimensional covariates with survival outcomes

Abstract

Identifying important biomarkers that are predictive for cancer patients’ prognosis is key in gaining better insights into the biological influences on the disease and has become a critical component of precision medicine. The emergence of large-scale biomedical survival studies, which typically involve excessive number of biomarkers, has brought high demand in designing efficient screening tools for selecting predictive biomarkers. The vast amount of biomarkers defies any existing variable selection methods via regularization. The recently developed variable screening methods, though powerful in many practical setting, fail to incorporate prior information on the importance of each biomarker and are less powerful in detecting marginally weak while jointly important signals. We propose a new conditional screening method for survival outcome data by computing the marginal contribution of each biomarker given priorily known biological information. This is based on the premise that some biomarkers are known to be associated with disease outcomes a priori. Our method possesses sure screening properties and a vanishing false selection rate. The utility of the proposal is further confirmed with extensive simulation studies and analysis of a diffuse large B-cell lymphoma dataset. We are pleased to dedicate this work to Jack Kalbfleisch, who has made instrumental contributions to the development of modern methods of analyzing survival data.

Appendix

The basic properties of the conditional linear expectation are listed in the following proposition.

Proposition 1

${\mathbf {v}}_j(\varvec{\beta }_{{\mathcal {C}},0},0)={\varvec{0}}_{q+1}$ if and only if $v_{j,j}(\varvec{\beta }_{{\mathcal {C}},0},0) = 0$, for all $j\in {\mathcal {C}}$.

The proof is straightforward based on Definitions 2 and 3.

Proposition 2

Let $\varvec{\zeta }$, $\varvec{\zeta }_1$, $\varvec{\zeta }_2$ and $\varvec{\xi }$ be any four random variables in the probability space $(\varOmega ,{\mathcal {F}}, \mathrm {P})$. The following properties hold for the conditional linear expectation $\mathrm {E}^*[\bullet \mid \varvec{\xi }]$ given $\varvec{\xi }$:

• 1. Closed form: $\mathrm {E}^*(\varvec{\zeta }\mid \varvec{\xi }) =\mathrm {E}[\varvec{\zeta }] + \mathrm {Cov}(\varvec{\zeta },\varvec{\xi })\mathrm {Var}[\varvec{\xi }]^{-1} \{\varvec{\xi }-\mathrm {E}(\varvec{\xi })\}$.

• 2. Stability: $\mathrm {E}^*[\varvec{\xi }\mid \varvec{\xi }] = \varvec{\xi }$.

• 3. Linearity: $\mathrm {E}^*[{\mathbf {A}}_1\varvec{\zeta }_1+{\mathbf {A}}_2\varvec{\zeta }_2 \mid \varvec{\xi }] = {\mathbf {A}}_1\mathrm {E}^*[\varvec{\zeta }_1 \mid \varvec{\xi }]+{\mathbf {A}}_2\mathrm {E}^*[\varvec{\zeta }_2 \mid \varvec{\xi }]$, where ${\mathbf {A}}_1$ and ${\mathbf {A}}_2$ are two matrices that are compatible with the equation.

• 4. Law of total expectation: $\mathrm {E}^*[\mathrm {E}^*(\varvec{\zeta }\mid \varvec{\xi })] = \mathrm {E}[\mathrm {E}^*(\varvec{\zeta }\mid \varvec{\xi })] = \mathrm {E}[\varvec{\zeta }]$.

Remark 1

In general, $\mathrm {E}^*(\varvec{\zeta }\mid \varvec{\xi }) \ne \mathrm {E}(\varvec{\zeta }\mid \varvec{\xi })$. Also, $\varvec{\zeta }$ and $\varvec{\xi }$ are independent does not imply $\mathrm {E}^*(\varvec{\zeta }\mid \varvec{\xi }) = 0$, unless $\varvec{\zeta }$ and $\varvec{\xi }$ are jointly normally distributed.

Remark 2

By Proposition 2, we can easily verify the following properties.

Proposition 3

The conditional linear covariance defined in Definition 5 has the following properties:

• 1. Linear independence and linear zero correlation:

  $$\begin{aligned} \mathrm {Cov}^*(\varvec{\zeta }_1,\varvec{\zeta }_2 \mid \varvec{\xi }) = 0\qquad \Leftrightarrow \qquad \mathrm {E}^*(\varvec{\zeta }_1 \varvec{\zeta }_2 \mid \varvec{\xi }) = \mathrm {E}^*(\varvec{\zeta }_1 \mid \varvec{\xi }) \mathrm {E}^*(\varvec{\zeta }_2 \mid \varvec{\xi }). \end{aligned}$$

• 2. Expectation of conditional linear covariance:

  $$\begin{aligned} \mathrm {E}[\mathrm {Cov}^*(\varvec{\zeta }_1,\varvec{\zeta }_2\mid \varvec{\xi })] = \mathrm {Cov}(\varvec{\zeta }_1,\varvec{\zeta }_2) - \mathrm {Cov}(\varvec{\zeta }_1,\varvec{\xi }) \mathrm {Var}(\varvec{\xi })^{-1} \mathrm {Cov}(\varvec{\xi },\varvec{\zeta }_2). \end{aligned}$$

• 3. Sign: for any increasing function $h(\cdot ): {\mathbb {R}}\rightarrow {\mathbb {R}}$ and random variable $\eta : \varOmega \rightarrow {\mathbb {R}}$, then

  $$\begin{aligned} \mathrm {Cov}^*(h(\eta ),\eta \mid \varvec{\xi }) \ge 0. \end{aligned}$$

Combining Propositions 1–3 and based on Definition 6, we have the following property.

Proposition 4

$v_{j,j}(\varvec{\beta }_{{\mathcal {C}},0},0) = 0$ if and only if $v_j(\varvec{\beta }_{{\mathcal {C}},0},0) = 0$.

Lemma 1

The solution of ${\mathbf {v}}_j(\varvec{\beta }_{{\mathcal {C}}},\beta ) = {\varvec{0}}_{q+1}$ and the solution of ${\mathbf {v}}_{{\mathcal {C}}}(\varvec{\beta }_{{\mathcal {C}}}) = {\varvec{0}}_q$ are both unique, for any $j \notin {\mathcal {C}}$.

1.1 Proof of Theorem 1

Proof

First we make the connection between $\beta _{j}$ to the expected conditional linear covariance between $Z_j$ and $\mathrm {P}[\delta =1\mid {\mathbf {Z}}]$ given ${\mathbf {Z}}_{\mathcal {C}}$, that is

$$\begin{aligned} \mathrm {E}[\mathrm {Cov}^*(Z_j, \mathrm {P}[\delta =1\mid {\mathbf {Z}}]\mid {\mathbf {Z}}_{{\mathcal {C}}})], \end{aligned}$$

then by Condition 2, we relate it to $\alpha _{j}$. For any $j\notin {\mathcal {C}}$ and $k\in {\mathcal {C}}$, it is straightforward to see that

$$\begin{aligned} s^{(m)}_{k}(t) = \mathrm {E}[Z_{k}^{m}\lambda _0(t) \exp ({\mathbf {Z}}^{\mathrm {T}}\varvec{\alpha }) S_T(t\mid {\mathbf {Z}}) S_C(t)], \end{aligned}$$

(21)

and

$$\begin{aligned} r^{(m)}_{j,k}(t,\varvec{\beta }_{{\mathcal {C}}},\beta ) =\mathrm {E}[Z^m_{k} \exp ({\mathbf {Z}}_{{\mathcal {C}}}^{\mathrm {T}}\varvec{\beta }_{{\mathcal {C}}} + Z_{j}\beta ) S_T(t\mid {\mathbf {Z}}) S_C(t)], \end{aligned}$$

(22)

for $m = 0,1$. Then

$$\begin{aligned}&{v_{j,k}(\varvec{\beta }_{{\mathcal {C}}},\beta ) }\nonumber \\&\quad = \int _0^\tau \mathrm {E}\left[ W_{j,k}(t,\varvec{\beta }_{{\mathcal {C}}},\beta )\exp ({\mathbf {Z}}^{\mathrm {T}}\varvec{\alpha }) S_T(t\mid {\mathbf {Z}}) S_C(t)\lambda _0(t)\right] \mathrm {d}t, \end{aligned}$$

(23)

where

$$\begin{aligned} W_{j,k}(t,\varvec{\beta }_{{\mathcal {C}}},\beta ) = Z_k- \frac{\mathrm {E}[Z_k \exp ({\mathbf {Z}}_{{\mathcal {C}}}^{\mathrm {T}}\varvec{\beta }_{{\mathcal {C}}} + Z_{j}\beta ) S_T(t\mid {\mathbf {Z}}) S_C(t)]}{\mathrm {E}[\exp ({\mathbf {Z}}_{{\mathcal {C}}}^{\mathrm {T}}\varvec{\beta }_{{\mathcal {C}}} + Z_{j}\beta ) S_T(t\mid {\mathbf {Z}}) S_C(t)]}. \end{aligned}$$

By Proposition 2,

$$\begin{aligned}&\mathrm {E}\left[ W_{j,k}(t,\varvec{\beta }_{{\mathcal {C}}},\beta )\exp ({\mathbf {Z}}^{\mathrm {T}}\varvec{\alpha }) S_T(t\mid {\mathbf {Z}}) S_C(t)\right] \\&\quad =\mathrm {E}\left\{ \mathrm {E}^*\left[ W_{j,k}(t,\varvec{\beta }_{{\mathcal {C}}},\beta )\exp ({\mathbf {Z}}^{\mathrm {T}}\varvec{\alpha }) S_T(t\mid {\mathbf {Z}}) S_C(t) \right] \right\} . \end{aligned}$$

By Definition 6,

$$\begin{aligned} v_j(\varvec{\beta }_{{\mathcal {C}}},\beta )= & {} v_{j,j}(\varvec{\beta }_{\mathcal {C}},\beta ) -\sum _{k\in {\mathcal {C}}} a_k v_{j,k}(\varvec{\beta }_{\mathcal {C}},\beta ) \\= & {} \mathrm {E}\left[ \mathrm {Cov}^{*} (Z_j, \mathrm {P}[\delta =1\mid {\mathbf {Z}}] \mid {\mathbf {Z}}_{{\mathcal {C}}})\right] - g_j(\varvec{\beta }_{{\mathcal {C}}},\beta ), \end{aligned}$$

where

$$\begin{aligned}&\mathrm {E}\left[ \mathrm {Cov}^*(Z_j, \mathrm {P}[\delta =1\mid {\mathbf {Z}}] \mid {\mathbf {Z}}_{{\mathcal {C}}})\right] \\&\quad = \int _0^\tau \mathrm {E}\left[ (Z_j - \mathrm {E}^*[Z_j \mid {\mathbf {Z}}_{{\mathcal {C}}}])\exp ({\mathbf {Z}}^{\mathrm {T}}\varvec{\alpha }) S_T(t\mid {\mathbf {Z}}) S_C(t)\lambda _0(t)\right] \mathrm {d}t, \end{aligned}$$

and

$$\begin{aligned}&g_{j}(\varvec{\beta }_{{\mathcal {C}}},\beta )\\= & {} \int _0^\tau \frac{\mathrm {E}[(Z_j - \mathrm {E}^*[Z_j \mid {\mathbf {Z}}_{{\mathcal {C}}}]) \exp ({\mathbf {Z}}_{{\mathcal {C}}}^{\mathrm {T}}\varvec{\beta }_{{\mathcal {C}}}+Z_j\beta ) S_T(t\mid {\mathbf {Z}})S_C(t)]}{\mathrm {E}[\exp ({\mathbf {Z}}_{{\mathcal {C}}}^{\mathrm {T}}\varvec{\beta }_{{\mathcal {C}}}+Z_j\beta ) S_T(t\mid {\mathbf {Z}})S_C(t)]}\\&\times \mathrm {E}\left[ \exp ({\mathbf {Z}}^{\mathrm {T}}\varvec{\alpha }) S_T(t\mid {\mathbf {Z}})\lambda _0(t) S_C(t)\right] \mathrm {d}t. \end{aligned}$$

By Definition 2, ${\mathbf {v}}_j(\varvec{\beta }_{{\mathcal {C}},j},\beta _j) = {\varvec{0}}_{q+1}$,

$$\begin{aligned} g_j(\varvec{\beta }_{{\mathcal {C}},j}, \beta _j) = \mathrm {E}\left[ \mathrm {Cov}^*(Z_j, \mathrm {P}[\delta =1\mid {\mathbf {Z}}] \mid {\mathbf {Z}}_{{\mathcal {C}}})\right] . \end{aligned}$$

When $\alpha _{j} = 0$, then $\mathrm {E}\left[ \mathrm {Cov}^*(Z_j, \mathrm {P}[\delta =1\mid {\mathbf {Z}}] \mid {\mathbf {Z}}_{{\mathcal {C}}})\right] = 0$ by Condition 2.3. Thus $g_j(\varvec{\beta }_{{\mathcal {C}},j},\beta _j) = 0$. Also, by Propositions 1 and 2, $g_j(\varvec{\beta }_{{\mathcal {C}},0},0) = 0$, then ${\mathbf {v}}_j(\varvec{\beta }_{{\mathcal {C}},0},0) = {\varvec{0}}_{q+1}$. By uniqueness in Lemma 1, $\beta _j = 0$.

When $\alpha _j \ne 0$, by Condition 2, we have

$$\begin{aligned} |g_j(\varvec{\beta }_{{\mathcal {C}},j},\beta _j)| = | \mathrm {E}\left[ \mathrm {Cov}^*( Z_j, \mathrm {P}[\delta =1\mid {\mathbf {Z}}] \mid {\mathbf {Z}}_{{\mathcal {C}}})\right] | > c_1 n^{-\kappa }. \end{aligned}$$

This implies that $g_j(\varvec{\beta }_{{\mathcal {C}},j},\beta _j)$ and $\mathrm {E}\left[ \mathrm {Cov}^*( Z_j, \mathrm {P}[\delta =1\mid {\mathbf {Z}}] \mid {\mathbf {Z}}_{{\mathcal {C}}})\right]$ are both nonzero and have the same signs since they are equal. Next we show for any $\varvec{\beta }_{{\mathcal {C}}}$, $g_j(\varvec{\beta }_{{\mathcal {C}}},0)$ and $\mathrm {E}\left[ \mathrm {Cov}^*( Z_j, \mathrm {P}[\delta =1\mid {\mathbf {Z}}] \mid {\mathbf {Z}}_{{\mathcal {C}}})\right]$ have the opposite signs unless they are equal to zero. This fact implies that $\beta _j\ne 0$. Specifically, note that $\mathrm {P}(\delta = 1\mid {\mathbf {Z}})$ is the probability of occurring the event and $S_T(t\mid {\mathbf {Z}})S_C(t) = \mathrm {P}(X > t \mid {\mathbf {Z}})$ represents the probability at risk at time t. Based on Model (1), for any t,

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

By Proposition 3, $\mathrm {Cov}^*( Z_j, \mathrm {P}[\delta =1\mid {\mathbf {Z}}] \mid {\mathbf {Z}}_{{\mathcal {C}}})$ and $\mathrm {Cov}^*[Z_j, S_T(t\mid {\mathbf {Z}})S_C(t)\mid {\mathbf {Z}}_{\mathcal {C}}]$ have the opposite signs unless they are zero. This further implies that for any $\varvec{\beta }_{{\mathcal {C}}}$,

$$\begin{aligned} g_{j}(\varvec{\beta }_{{\mathcal {C}}},0)= & {} \int _{0}^{\tau }\frac{\mathrm {E}[\exp ({\mathbf {Z}}_{{\mathcal {C}}}^{\mathrm {T}} \varvec{\beta }_{{\mathcal {C}}})\mathrm {Cov}^*[Z_j, S_T(t\mid {\mathbf {Z}})S_C(t)\mid {\mathbf {Z}}_{\mathcal {C}}]]}{\mathrm {E}[\exp ({\mathbf {Z}}_{{\mathcal {C}}}^{\mathrm {T}}\varvec{\beta }_{{\mathcal {C}}}) S_T(t\mid {\mathbf {Z}})S_C(t)]}\\&\times \mathrm {E}\left[ \exp ({\mathbf {Z}}^{\mathrm {T}}\varvec{\alpha }) S_T(t\mid {\mathbf {Z}})\lambda _0(t) S_C(t)\right] \mathrm {d}t, \end{aligned}$$

and $\mathrm {E}\left[ \mathrm {Cov}^*( Z_j, \mathrm {P}[\delta =1\mid {\mathbf {Z}}]\mid {\mathbf {Z}}_{{\mathcal {C}}})\right]$ have opposite signs unless they are equal to zero. Therefore, $\beta _j \ne 0$. $\square$

1.2 Proof of Theorem 2

Proof

For any $j\in {\mathcal {M}}_{-{\mathcal {C}}}$, we have $\beta _j \ne 0$ by Theorem 1, by mean value theorem, for some ${{\widetilde{\beta }}}_j \in (0, \beta _j)$,

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

Next we show that $\left| \frac{\partial v_{j}}{\partial \beta }(\varvec{\beta }_{{\mathcal {C}},j},{{\widetilde{\beta }}}_j)\right|$ is bounded. For given any $\varvec{\beta }_{{\mathcal {C}}}$, consider $g_{j}(\varvec{\beta }_{{\mathcal {C}}},\beta )$ as a function of $\beta$, Then

$$\begin{aligned} \frac{\partial g_j}{\partial \beta }(\varvec{\beta }_{\mathcal {C}},\beta ) =\mathrm {E}\left[ \int _0^{\tau }H_j(t,\varvec{\beta }_{{\mathcal {C}}},\beta )S_C(t)\mathrm {d}F_T(t\mid {\mathbf {Z}})\right] , \end{aligned}$$

where

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

By Condition 2.1, $\mathrm {P}(|Z|<K_0) = 1$, then $\sup _{\varvec{\beta }_{\mathcal {C}},\beta }|H_j(t,\varvec{\beta }_{{\mathcal {C}}},\beta )|\le 2 K_0^2$. Thus,

$$\begin{aligned} \left| \frac{\partial v_{j}}{\partial \beta }(\varvec{\beta }_{{\mathcal {C}},j},{{\widetilde{\beta }}}_j)\right| \le \sup _{\varvec{\beta }_{\mathcal {C}},\beta }\left| \frac{\partial g_j}{\partial \beta }(\varvec{\beta }_{\mathcal {C}},\beta )\right| \le 2K_0^2 |\mathrm {E}[\mathrm {E}[S_C(T)\mid {\mathbf {Z}}]] \le 2K_0^2. \end{aligned}$$

By the proof in Theorem 1, $g(\varvec{\beta }_{{\mathcal {C}},j}, 0)$ and $\mathrm {E}\left[ \mathrm {Cov}^*( Z_j, \mathrm {E}\{F_{T}(C \mid {\mathbf {Z}}) \mid {\mathbf {Z}}\} \mid {\mathbf {Z}}_{{\mathcal {C}}})\right]$ have the opposite signs, and by Condition 2,

$$\begin{aligned} |v_{j}(\varvec{\beta }_{{\mathcal {C}},j},0)| = | \mathrm {E}\left[ \mathrm {Cov}^*( Z_j, \mathrm {P}[\delta =1\mid {\mathbf {Z}}]\mid {\mathbf {Z}}_{{\mathcal {C}}})\right] | + |g_{j}(\varvec{\beta }_{{\mathcal {C}},j},0)| > c_1 n^{-\kappa }. \end{aligned}$$

Taking $c_2 = 0.5K_0^{-2} c_1$, $\beta _j> 0.5K_0^{-2} |v_j(\varvec{\beta }_{{\mathcal {C}},j},0)| > c_2 n^{-\kappa }$. This completes the proof. $\square$

1.3 Proof of Theorem 3

Proof

For any $j\notin {\mathcal {C}}$ and $k\in {\mathcal {C}}\cup \{j\}$, by Lin and Wei (1989), we have

$$\begin{aligned} {\overline{{\mathbf {V}}}}_j(\varvec{\beta }_{{\mathcal {C}}},\beta ) = \mathrm {E}_n\{{\mathbf {W}}_{i,j}(\varvec{\beta }_{{\mathcal {C}}},\beta )\} + o_p(1), \end{aligned}$$

where $\mathrm {E}_n[\cdot ]$ denotes the empirical measure, which is defined as $\mathrm {E}_n[\varvec{\xi }_i] = n^{-1} \sum _{i=1}^n \varvec{\xi }_i$ for any random variables $\varvec{\xi }_1,\ldots , \varvec{\xi }_n$, and ${\mathbf {W}}_{i,j}(\varvec{\beta }_{{\mathcal {C}}},\beta )$ are independent over i, and write ${\mathbf {W}}_{i,j}(\varvec{\beta }_{{\mathcal {C}}},\beta ) = [W_{i,j,k}(\varvec{\beta }_{{\mathcal {C}}},\beta ),k\in {\mathcal {C}}\cup \{j\}]^{\mathrm {T}}$ with

$$\begin{aligned} W_{i,j,k}(\varvec{\beta }_{{\mathcal {C}}},\beta )= & {} \int _0^\tau \left\{ Z_{i,k} - \frac{r^{(1)}_{j,k}(\varvec{\beta }_{{\mathcal {C}}},\beta , t)}{r^{(0)}_{j,k}(\varvec{\beta }_{{\mathcal {C}}},\beta ,t)}\right\} \mathrm {d}N_i(t) \\&- \int _0^\tau \frac{Y_i(t)\exp ({\mathbf {Z}}_{i,{\mathcal {C}}}\varvec{\beta }^{\mathrm {T}}_{{\mathcal {C}}}+ Z_{i,j}\beta )}{r^{(0)}_{j,k}(\varvec{\beta }_{{\mathcal {C}}},\beta ,t)}\left\{ Z_{i,k} - \frac{r^{(1)}_{j,k}(\varvec{\beta }_{{\mathcal {C}}},\beta , t)}{r^{(0)}_{j,k}(\varvec{\beta }_{{\mathcal {C}}},\beta ,t)}\right\} \mathrm {d}\mathrm {E}[N_i(t)]. \end{aligned}$$

Note that given any i, j, k, with probability one $|W_{i,j,k}(\varvec{\beta }_{{\mathcal {C}}},\beta )|$ are uniformly bounded. Specifically, by Conditions 1.2, 2.1 and 3, with probability one, for all $t\in [0,\tau ]$, $(\varvec{\beta }_{{\mathcal {C}}}^{\mathrm {T}},\beta )^{\mathrm {T}}\in {\mathcal {B}}_j$,

$$\begin{aligned} \left| Z_{i,k} - \frac{r^{(1)}_{j,k}(\varvec{\beta }_{{\mathcal {C}}},\beta , t)}{r^{(0)}_{j,k}(\varvec{\beta }_{{\mathcal {C}}},\beta ,t)}\right|\le & {} |Z_{i,k}| + K_0,\\ \left| \frac{Y_i(t)\exp ({\mathbf {Z}}_{i,{\mathcal {C}}}\varvec{\beta }^{\mathrm {T}}_{{\mathcal {C}}}+ Z_{i,j}\beta )}{r^{(0)}_{j,k}(\varvec{\beta }_{{\mathcal {C}}},\beta ,t)}\right|\le & {} \exp \{K_0(K_1+\delta )-\log (L)\}, \end{aligned}$$

and

$$\begin{aligned} \left| \int _0^{\tau } \mathrm {d}\mathrm {E}[N_i(t)]\right| \le \Lambda _0(\tau )\exp (K_0K_1). \end{aligned}$$

Thus, with probability one,

$$\begin{aligned} |W_{i,j,k}(\varvec{\beta }_{{\mathcal {C}}},\beta )| \le K_2, \end{aligned}$$

where $K_2 = 2K_0(1+\Lambda _0(\tau )\exp (2K_0K_1+K_0\delta -\log L))$. By the fact that $\mathrm {E}[W_{i,j,k}(\varvec{\beta }_{{\mathcal {C}}},\beta )] = 0$,

$$\begin{aligned} \mathrm {Var}[W_{i,j,k}(\varvec{\beta }_{{\mathcal {C}}},\beta )] = \mathrm {E}[|W_{i,j,k}(\varvec{\beta }_{{\mathcal {C}}},\beta )|^2] < K_2^2. \end{aligned}$$

By Lemma 2.2.9 (Bernsterin’s inequality) of Vaart and Wellner (1996), for any $t>0$, for all j, k, $\varvec{\beta }_{{{\mathcal {C}}}}$ and $\beta$, we have

$$\begin{aligned} \mathrm {P}\left( |\mathrm {E}_n(W_{i,j,k}(\varvec{\beta }_{{\mathcal {C}}},\beta ))|>\frac{t}{n}\right) \le 2 \exp \left( -\frac{1}{2}\frac{t^2}{n K^2_2+K_2 t/3}\right) . \end{aligned}$$

Note that the above inequality holds for every $j\notin {\mathcal {C}}$ and $k\in {\mathcal {C}}\cup \{j\}$. By Bonferroni inequality,

$$\begin{aligned} \mathrm {P}\left( \Vert \mathrm {E}_n({\mathbf {W}}_{i,j}(\varvec{\beta }_{{\mathcal {C}}},\beta ))\Vert _2>\frac{t}{(q+1)n}\right) \le 2 (q+1)\exp \left( -\frac{1}{2}\frac{t^2}{n K^2_2+K_2 t/3}\right) . \end{aligned}$$

Since,

$$\begin{aligned} \Vert {\overline{{\mathbf {V}}}}_{j}(\varvec{\beta }_{{\mathcal {C}}},\beta )-\mathrm {E}_n({\mathbf {W}}_{i,j} (\varvec{\beta }_{{\mathcal {C}}},\beta ))\Vert _2 = o_p(1). \end{aligned}$$

Then for any $\epsilon _1>0$ and $\epsilon _2>0$, there exits $N_1$, such that for any $n>N_1$,

$$\begin{aligned} \mathrm {P}(\Vert {\overline{{\mathbf {V}}}}_{j}(\varvec{\beta }_{{\mathcal {C}}},\beta ) -\mathrm {E}_n({\mathbf {W}}_{i,j}(\varvec{\beta }_{{\mathcal {C}}},\beta ))\Vert _2 > M\epsilon _1/2 ) < \epsilon _2, \end{aligned}$$

where M is the same value in Condition 4. By Triangle inequality and Bonferroni inequality, we have

$$\begin{aligned}&\mathrm {P}\left( \Vert {\overline{{\mathbf {V}}}}_{j}(\varvec{\beta }_{{\mathcal {C}}},\beta )\Vert _2> \frac{t}{(q+1)n} \right) \\\le & {} \mathrm {P}\left( \Vert \mathrm {E}_n({\mathbf {W}}_{i,j}(\varvec{\beta }_{{\mathcal {C}}},\beta ))\Vert _2> \frac{t}{(q+1)n} -M\epsilon _1/2\right) \\&+\, \mathrm {P}(\Vert {\overline{{\mathbf {V}}}}_{j}(\varvec{\beta }_{{\mathcal {C}}},\beta ) -\mathrm {E}_n({\mathbf {W}}_{i,j}(\varvec{\beta }_{{\mathcal {C}}},\beta ))\Vert _2 > M\epsilon _2/2 ). \end{aligned}$$

When $n\rightarrow \infty$, take $t = c_2M(q+1)n^{1-\kappa }/2>0$ on both side of the inequality, where $c_2$ is the same value in Theorem 2, we have

$$\begin{aligned}&\mathrm {P}\left( \Vert {\overline{{\mathbf {V}}}}_{j}(\varvec{\beta }_{{\mathcal {C}}},\beta )\Vert _2 > \frac{Mc_2}{2}(n^{-\kappa } -\epsilon _1)\right) \\&\quad \le 2 (q+1)\exp \left( -\frac{M^2c_2^2}{8(q+1)^2}\frac{n^{1-2\kappa }}{K^2_2+K_2 n^{-\kappa }/3}\right) +\epsilon _2. \end{aligned}$$

Take $N =\max \{ \lceil (K_2/3)^{1/\kappa }\rceil ,N_1\}$, then for any $n>N$, $n^{-\kappa }<3/K_2$, and

$$\begin{aligned} \mathrm {P}\left( \Vert {\overline{{\mathbf {V}}}}_{j}(\varvec{\beta }_{{\mathcal {C}}},\beta )\Vert _2 > \frac{Mc_2}{2}(n^{-\kappa } -\epsilon _1)\right) \le 2 (q+1)\exp \left( -\frac{M^2c_2^2}{8(q+1)^2} \frac{n^{1-2\kappa }}{K^2_2+1}\right) +\epsilon _2. \end{aligned}$$

Note that the above inequality holds for all $(\varvec{\beta }_{{\mathcal {C}}}^{\mathrm {T}},\beta )^{\mathrm {T}} \in {\mathcal {B}}_j$, particularly for $(\varvec{\beta }_{{\mathcal {C}},j}^{\mathrm {T}},\beta _j)^{\mathrm {T}}$, $j \notin {\mathcal {C}}$. Also, we have ${\overline{{\mathbf {V}}}}_{j}({{\widehat{\varvec{\beta }}}}_{{\mathcal {C}},j},{{\widehat{\beta }}}_j) = {\varvec{0}}_{q+1}$. By Condition 4, we have

$$\begin{aligned}&\mathrm {P}\left( |{{\widehat{\beta }}}_j-\beta _j|>\frac{c_2}{2}(n^{-\kappa } -\epsilon _1)\right) \\&\quad \le \mathrm {P}\left( \Vert ({{\widehat{\varvec{\beta }}}}_{{\mathcal {C}},j}^{\mathrm {T}},{{\widehat{\beta }}}_j)^{\mathrm {T}} -(\varvec{\beta }_{{\mathcal {C}},j}^{\mathrm {T}},\beta _j)^{\mathrm {T}} \Vert _2 >\frac{c_2}{2}(n^{-\kappa } -\epsilon _1)\right) \\&\quad \le 2 (q+1)\exp \left( -\frac{M^2c_2^2}{8(q+1)^2} \frac{n^{1-2\kappa }}{K^2_2+1}\right) +\epsilon _2. \end{aligned}$$

Taking $c_3 = \frac{M^2c_2^2}{8(q+1)^2(K^2_2+1)}$ and by Bonferroni completes the proof for part 1.

For part 2, by Theorem 2,

$$\begin{aligned} \min _{j\in {\mathcal {M}}_{-{\mathcal {C}}}}|\beta _j| > c_2 n^{-\kappa }. \end{aligned}$$

Note that, for any $j\in {\mathcal {M}}_{-{\mathcal {C}}}$, event

$$\begin{aligned}&\left\{ |{{\widehat{\beta }}}_j - \beta _j| \le c_2 n^{-\kappa }/2 - \epsilon _1\right\} \\&\quad \subseteq \left\{ |{{\widehat{\beta }}}_j| \ge |\beta _j| - c_2 n^{-\kappa }/2+\epsilon _1\right\} \\&\quad \subseteq \left\{ |{{\widehat{\beta }}}_j| \ge c_2 n^{-\kappa }/2+\epsilon _1\right\} . \end{aligned}$$

Take $\gamma _n = c_4 n^{-\kappa }$ with $c_4 = c_2/4$,

$$\begin{aligned}&\left\{ \max _{j\in {\mathcal {M}}_{-{\mathcal {C}}}}|{{\widehat{\beta }}}_j - \beta _j| \le c_2 n^{-\kappa }/2 - \epsilon _1\right\} \\&\quad \subseteq \left\{ \min _{j\in {\mathcal {M}}_{-{\mathcal {C}}}}|{{\widehat{\beta }}}_j| \ge c_2 n^{-\kappa }/2+\epsilon _1\right\} \\&\quad \subseteq \left\{ \min _{j\in {\mathcal {M}}_{-{\mathcal {C}}}}|{{\widehat{\beta }}}_j| \ge \gamma _n +\epsilon _1\right\} . \end{aligned}$$

Thus,

$$\begin{aligned}&\mathrm {P}\left[ {\mathcal {M}}_{-{\mathcal {C}}} \subseteq {{\widehat{{\mathcal {M}}}}}_{-{\mathcal {C}}}\right] \\&\quad = \mathrm {P}\left[ \min _{j\in {\mathcal {M}}_{-{\mathcal {C}}}}|{{\widehat{\beta }}}_j|> \gamma _n \right] \\&\quad \ge \mathrm {P}\left[ \min _{j\in {\mathcal {M}}_{-{\mathcal {C}}}}|{{\widehat{\beta }}}_j| > \gamma _n +\epsilon _1\right] \\&\quad \ge 1 - \mathrm {P}\left[ \max _{j\in {\mathcal {M}}_{-{\mathcal {C}}}}|{{\widehat{\beta }}}_j - \beta _j| \le c_2 n^{-\kappa }/2 - \epsilon _1\right] \\&\quad \ge 1 -2w(q+1)\exp (-c_3 n^{1-2\kappa }) - \epsilon _2. \end{aligned}$$

Let $n\rightarrow \infty$, we have for any $\epsilon _2>0$,

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathrm {P}\left[ {\mathcal {M}}_{-{\mathcal {C}}} \subseteq {{\widehat{{\mathcal {M}}}}}_{-{\mathcal {C}}}\right] \ge 1 - \epsilon _2. \end{aligned}$$

Note that the left side of the above equation does not depends on n any more. Taking $\epsilon _2\rightarrow 0$ completes proof. $\square$