Robust Feature Screening for Ultrahigh-Dimensional Censored Data Subject to Measurement Error

Abstract

Feature screening is commonly used to handle ultrahigh-dimensional data prior to conducting a formal data analysis. While various feature screening methods have been developed in the literature, research gaps still exist. The existing methods usually make an implicit assumption that data are accurately measured. This requirement, however, is frequently violated in applications. In this chapter, we consider error-prone ultrahigh-dimensional survival data and propose a robust feature screening method. We develop an iteration algorithm to improve the performance of retaining all informative covariates. Theoretical results are established for the proposed method. Simulation studies are reported to assess the performance of the proposed method, together with an application of the proposed method to handle a mantle cell lymphoma microarray dataset.

Appendix

1.1 A. Technical Lemmas

In this appendix, we provide some lemmas that are useful to derive the main theorems. The first lemma is the probabilistic bound of the estimated survivor function.

Lemma 1

Let H(t) = P(Y _i > t) denote the cumulative distribution function of Y _i , where $Y_i = \min \{T_i, C_i\}$ . Suppose that there is a finite time point τ, such that H(τ) > η for a positive constant η. Then for ξ > 2⁷ n ⁻¹ η ⁻² , there exist positive constants κ ₁ and κ ₂ such that

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} P\left( \sup \limits_{t \in [0,\tau]} \left| \widehat{F}(t) - F(t) \right| > \xi \right) \leq \kappa_1 \exp\left(-n \xi^2 \eta^4 \kappa_2 \right). \end{array} \end{aligned} $$

(A.1)

This lemma is Theorem 2 of Földes & Rejtö (1981). The second lemma is about the probabilistic bound of the estimator (16).

Lemma 2

Under regularity conditions (C1) and (C2), for any ξ ^∗ > 0, we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} P\left( \sup \limits_{x} \left| \widehat{F}_{adj,j}(x) - F_{X_{(j)}}(x) \right| > \xi^\ast \right) \leq \kappa_3 \exp \left\{- \frac{2n^2 \xi^{\ast2}}{G^2} + o( n^{-\frac{1}{5}}) \right\} \end{array} \end{aligned} $$

(A.2)

for some positive constants G and κ ₃.

Proof

We first write

$$\displaystyle \begin{aligned} \begin{array}{rcl} \widehat{f}_{adj,j}(x) - f_{X_{(j)}}(x) & =&\displaystyle \left\{\widehat{f}_{adj,j}(x) - f_{adj,j}(x) \right\} + \left\{ f_{adj,j}(x) - f_{X_{(j)}}(x) \right\} \\ & =&\displaystyle \widehat{f}_{adj,j}(x) - f_{adj,j}(x),{} \end{array} \end{aligned} $$

(A.3)

because f _adj,j(x) is just a different symbol of the inverse Fourier transformation of $F_{X_{(j)}}(x)$, i.e., $f_{adj,j}(x) - f_{X_{(j)}}(x) = 0$. Therefore, the remaining task is to examine $\widehat {f}_{adj,j}(x) - f_{adj,j}(x)$. By (11) and (15), we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \widehat{f}_{adj,j}(x) - f_{adj,j}(x) = \frac{1}{2\pi} \int_{-\infty}^\infty \frac{\exp\left( -\mathbf{i}u x \right)}{\phi_{\epsilon_{(j)}}(u)} \left\{ \widehat{\phi}_{X_{(j)}^\ast}(u) - \phi_{X_{(j)}^\ast}(u) \right\} du. \end{array} \end{aligned} $$

(A.4)

Note that $\phi _{X_{(j)}^\ast }(u) = E\left \{ \exp \left ( \mathbf {i} u X_{i(j)}^\ast \right ) \right \}$ and $\widehat {\phi }_{X_{(j)}^\ast }(u)$ is given by (14); then

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle \widehat{\phi}_{X_{(j)}^\ast}(u) - \phi_{X_{(j)}^\ast}(u) \\ & &\displaystyle = \left\{\frac{1}{n} \sum \limits_{i=1}^n \exp \left( \mathbf{i} u X_{i(j)}^\ast \right) \right\}\int_{-\infty}^\infty \exp \left( \mathbf{i} u hz \right) K(z) dz - E\left\{ \exp \left( \mathbf{i} u X_{i(j)}^\ast \right) \right\}.\qquad {} \end{array} \end{aligned} $$

(A.5)

By Conditions (C1) and (C2) and the finiteness of $\int _{-\infty }^\infty u^r K(u) du$ for all $r \in \mathbb {N}$, applying the Taylor series expansion of the exponential function gives that $\int _{-\infty }^\infty \exp \left ( \mathbf {i} u h z \right ) K(u) du = 1 + o( n^{-\frac {1}{5}} )$. Combining with (A.5) gives

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \widehat{\phi}_{X_{(j)}^\ast}(u) - \phi_{X_{(j)}^\ast}(u) = \frac{1}{n} \sum \limits_{i=1}^n \exp \left( \mathbf{i} u X_{i(j)}^\ast \right) - E\left\{ \exp \left( \mathbf{i} u X_{i(j)}^\ast \right) \right\} + o( n^{-\frac{6}{5}} ).\qquad \end{array} \end{aligned} $$

(A.6)

Let $Z_i = \exp \left ( \mathbf {i} u X_{i(j)}^\ast \right )$, which is a complex random variable. By Theorem 1.2 of Isaev and McKay (2016), we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \Big| E \left[ \exp \left\{ Z_i - E(Z_i) \right\} \right] - 1 \Big| \leq \exp\left( \frac{G^2}{8} \right) - 1, \end{array} \end{aligned} $$

(A.7)

where G is some constant with $G > \text{diam}Z \triangleq \inf \left \{c \in \mathbb {R}^+ : P\left ( |Z_1 - Z_2| > c \right )\right .$ . Note that $\widehat {\phi }_{X_{(j)}^\ast }(u) = \frac {1}{n} \sum \limits _{i=1}^n Z_i$; then by (A.6), for any ξ ₂ > 0 and ν > 0,

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle P\left( \Big| \widehat{\phi}_{X_{(j)}^\ast}(u) - \phi_{X_{(j)}^\ast}(u) \Big| \geq \xi_2 \right) \\ & \leq&\displaystyle P\left( \frac{1}{n} \sum \limits_{i=1}^n \Big|Z_i - E(Z_i) \Big| \geq \xi_2 + o(n^{-\frac{6}{5}}) \right) \\ & =&\displaystyle P\left[ \exp \left(\nu \sum \limits_{i=1}^n \Big|Z_i - E(Z_i) \Big| \right) \geq \exp \left\{\nu n \xi_2 + o(n^{-\frac{1}{5}}) \right\} \right] \\ & \leq&\displaystyle \exp \left\{-\nu n \xi_2 + o(n^{-\frac{1}{5}}) \right\} E \left\{ \exp \left(\nu \sum \limits_{i=1}^n \Big|Z_i - E(Z_i) \Big| \right) \right\} \\ & =&\displaystyle \left[\exp \left\{-\nu n \xi_2 + o(n^{-\frac{1}{5}}) \right\} \right] \times \left[ \prod \limits_{i=1}^n E \left\{ \exp \left(\nu \Big|Z_i - E(Z_i) \Big| \right) \right\} \right] \\ & =&\displaystyle \exp \left\{ \frac{n \nu^2 G^2}{8} -\nu n \xi_2 + o(n^{-\frac{1}{5}}) \right\},{} \end{array} \end{aligned} $$

(A.8)

where the third step is due to Markov’s inequality, the fourth step is by the independence of the $X_i^\ast$, and the last step comes from (A.7) with Z _i replaced by νZ _i, so that with constant νG satisfying $\nu G > \inf \{\nu c : P\left ( \nu |Z_1 - Z_2| > \nu c \right ) = 0 \}$, we have $E \left \{ \exp \left (\nu | Z_i - E(Z_i) | \right ) \right \} \leq \exp \left ( \frac {\nu ^2 G^2}{8} \right )$.

To get the best upper bound, we take the right-hand side of (A.8) as the function of ν and then minimize it. Specifically, let $\varphi (\nu ) = \frac {n \nu ^2 G^2}{8} -\nu n \xi _2 + o(n^{-\frac {1}{5}})$. Since φ(ν) is a quadratic function, it is easy to check that $\nu ^\ast \triangleq \mathop {\operatorname {argminn}} \limits _\nu \varphi (\nu ) = \frac {4\xi _2}{G^2}$. Then replacing ν by ν ^∗ in (A.8) yields

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} P\left( \left| \widehat{\phi}_{X_{(j)}^\ast}(u) - \phi_{X_{(j)}^\ast}(u) \right| \geq \xi_2 \right) \leq \exp \left\{- \frac{2n \xi_2^2}{G^2} + o(n^{-\frac{1}{5}}) \right\}. \end{array} \end{aligned} $$

(A.9)

Moreover, by (A.4) and (A.9), we observe that with a probability greater than $1-\exp \left \{- \frac {2n \xi _2^2}{G^2} + o(n^{-\frac {1}{5}}) \right \}$,

$$\displaystyle \begin{aligned} \begin{array}{rcl} \sup \limits_x \left| \widehat{f}_{adj,j}(x) - f_{X_{(j)}}(x) \right| & =&\displaystyle \sup \limits_x \left| \widehat{f}_{adj,j}(x) - f_{adj,j}(x) \right| \\ & \leq&\displaystyle \sup \limits_x \left\{\frac{1}{2\pi} \int_{-\infty}^\infty \frac{\exp\left( -\mathbf{i}u x \right)}{\phi_{\epsilon_{(j)}}(u)} \left| \widehat{\phi}_{X_{(j)}^\ast}(u) - \phi_{X_{(j)}^\ast}(u) \right| du \right\} \\ & \leq&\displaystyle \left( \sup \limits_x \frac{1}{2\pi} \int_{-\infty}^\infty \frac{\exp\left( -\mathbf{i}u x \right)}{\phi_{\epsilon_{(j)}}(u)} du \right) \xi_2, \end{array} \end{aligned} $$

where the first equality is due to (A.3), the last step comes from (A.9), and the improper integral $\int _{-\infty }^\infty \frac {\exp \left ( -\mathbf {i}u x \right )}{\phi _{\epsilon _{(j)}}(u)} du$ is shown to converge to a finite value (e.g., Marsden & Hoffman 1999, Proposition 4.3.9).

In other words,

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle P \left\{\sup \limits_x \left| \widehat{f}_{adj,j}(x) - f_{X_{(j)}}(x) \right| \geq \left( \sup \limits_x \frac{1}{2\pi} \int_{-\infty}^\infty \frac{\exp\left( -\mathbf{i}u x \right)}{\phi_{\epsilon_{(j)}}(u)} du \right) \xi_2 \right\}\\ & \leq &\displaystyle \exp \left\{- \frac{2n \xi_2^2}{G^2} + o( n^{-\frac{1}{5}}) \right\}. \end{array} \end{aligned} $$

Specifying $\xi ^\ast = \left ( \sup \limits _x \frac {1}{2\pi } \int _{-\infty }^\infty \frac {\exp \left ( -\mathbf {i}u x \right )}{\phi _{\epsilon _{(j)}}(u)} du \right ) \xi _2$ gives that

$$\displaystyle \begin{aligned} \begin{array}{rcl} P\left( \sup \limits_x \left| \widehat{f}_{adj,j}(x) - f_{X_{(j)}}(x) \right| \geq \xi^\ast \right) & \leq&\displaystyle \kappa_3 \exp \left\{- \frac{2n \xi^{\ast 2}}{G^2} + o( n^{-\frac{1}{5}}) \right\}, \end{array} \end{aligned} $$

where $\kappa _3 \triangleq \exp \left \{ \frac {2n \xi ^{\ast 2}}{G^2} - \frac {2n \xi ^2}{G^2} \right \}$, which is positive. Thus, by the definition of the cumulative distribution function and (16), we conclude the desired result (A.2). □

1.2 B. Proofs of Main Theorems

1.2.1 B.1 Proof of Theorem 1

Part 1

We prove (19).

Since ω _j and $\widehat {\omega }_j$ are formulated in terms of dov(⋅, ⋅) and the associated estimates, to show the desired result, it suffices to examine dov(⋅, ⋅) and its estimates.

Let $\omega _j^\ast \triangleq \widehat {\text{dcov}}(F_{X_{(j)}}(X_{(j)}),F(Y)) = \widetilde {M}_{j,1} + \widetilde {M}_{j,2} - 2 \widetilde {M}_{j,3}$, where $\widetilde {M}_{j,k}$ with k = 1, 2, 3 has the same form as $\widehat {M}_{j,k}$ in (4) with $\widehat {F}_{X_{(j)}}(X_{(j)})$ and $\widehat {F}(Y)$ replaced by $F_{X_{(j)}}(X_{(j)})$ and F(Y ), respectively. Therefore, the difference between $\widehat {\omega }_j$ and ω _j can be expressed as

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \widehat{\omega}_j^\ast - \omega_j = \left( \widehat{\omega}_j^\ast - \omega_j^\ast\right) + \left( \omega_j^\ast - \omega_j\right). \end{array} \end{aligned} $$

(B.1)

Similar to the derivation of Li et al. (2012), we can show that

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} P\left( \max \limits_{j=1,\cdots,p} \left| \omega_j^\ast - \omega_j \right| > \xi \right) = O\left\{ p \exp \left(-\widetilde{c}_1n\xi^2 \right)\right\} \end{array} \end{aligned} $$

(B.2)

for some positive constants $\widetilde {c}_1$ and ξ.

On the other hand, we examine $\widehat {\omega }_j - \omega _j^\ast$ by writing

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \widehat{\omega}_j - \omega_j^\ast = \left( \widehat{M}_{j,1} - \widetilde{M}_{j,1} \right) + \left( \widehat{M}_{j,2} - \widetilde{M}_{j,2} \right) - 2\left( \widehat{M}_{j,3} - \widetilde{M}_{j,3} \right). \end{array} \end{aligned} $$

(B.3)

Since the derivations of $\widehat {M}_{j,2} - \widetilde {M}_{j,2}$ and $\widehat {M}_{j,3} - \widetilde {M}_{j,3}$ are similar to those of $\widehat {M}_{j,1} - \widetilde {M}_{j,1}$, we only present the argument for the latter case.

By adding and subtracting $\frac {1}{n^2} \sum \limits _{i=1}^n \sum \limits _{k=1}^n \Big \{\left | \widehat {F}_{adj,j}(X_{i(j)}) - \widehat {F}_{adj,j}(X_{k(j)}) \right | \left |F(Y_i) - \right .$ $\left . F(Y_k) \right | \Big \}$, we obtain that

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} & &\displaystyle \widehat{M}_{j,1} - \widetilde{M}_{j,1} \\ & =&\displaystyle \Bigg[ \widehat{M}_{j,1} - \frac{1}{n^2} \sum \limits_{i=1}^n \sum \limits_{k=1}^n \Big\{\left| \widehat{F}_{adj,j}(X_{i(j)}) - \widehat{F}_{adj,j}(X_{k(j)}) \right| \left|F(Y_i) - F(Y_k) \right| \Big\} \Bigg] \\ & &\displaystyle + \Bigg[ \frac{1}{n^2} \sum \limits_{i=1}^n \sum \limits_{k=1}^n \Big\{\left| \widehat{F}_{adj,j}(X_{i(j)}) - \widehat{F}_{adj,j}(X_{k(j)}) \right| \left|F(Y_i) - F(Y_k) \right| \Big\} - \widetilde{M}_{j,1} \Bigg] \\ & =&\displaystyle \frac{1}{n^2} \sum \limits_{i=1}^n \sum \limits_{k=1}^n \left\{ \left| \widehat{F}_{adj,j}(X_{i(j)}) - \widehat{F}_{adj,j}(X_{k(j)}) \right| \left( \left| \widehat{F}(Y_i) - \widehat{F}(Y_k) \right| - \left| F(Y_i) - F(Y_k) \right| \right) \right\} \\ & &\displaystyle + \frac{1}{n^2} \sum \limits_{i=1}^n \sum \limits_{k=1}^n \left\{ \left|F(Y_i) - F(Y_k) \right| \left( \left| \widehat{F}_{adj,j}(X_{i(j)}) - \widehat{F}_{X_{(j)}}(X_{k(j)}) \right| \right. \right. \\ & &\displaystyle \left. \left. - \left| F_{adj,j}(X_{i(j)}) - F_{X_{(j)}}(X_{k(j)}) \right| \right) \right\} \\ & \triangleq&\displaystyle S_1 + S_2. \end{array} \end{aligned} $$

(B.4)

First, we examine S ₁. Since $\widehat {F}_{adj,j}(x)$ is the estimated cumulative distribution function with $0 \leq \widehat {F}_{adj,j}(x) \leq 1$, then for any i and k, we have that

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \left| \widehat{F}_{adj,j}(X_{i(j)}) - \widehat{F}_{adj,j}(X_{k(j)}) \right| & \leq &\displaystyle 1. \end{array} \end{aligned} $$

(B.5)

By the triangle inequality, we have that

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \left| \widehat{F}(Y_i) - \widehat{F}(Y_k) \right| - \left| F(Y_i) - F(Y_k) \right| \leq \left| \widehat{F}(Y_i) - F(Y_i) \right| + \left| \widehat{F}(Y_k) - F(Y_k) \right|.\\ \end{array} \end{aligned} $$

(B.6)

Then by (B.6), we have that

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle P\bigg\{ \bigg| \left| \widehat{F}(Y_i) - \widehat{F}(Y_k) \right| - \left| F(Y_i) - F(Y_k) \right| \bigg| > \xi \bigg\} \\ & \leq&\displaystyle P\left\{ \left( \left| \widehat{F}(Y_i) - F(Y_i) \right| + \left| \widehat{F}(Y_k) - F(Y_k) \right| \right) > \xi \right\} \\ & \leq&\displaystyle P\left\{ \sup \limits_{t \in [0,\tau]} \left| \widehat{F}(t) - F(t) \right| + \sup \limits_{t \in [0,\tau]} \left| \widehat{F}(t) - F(t) \right| > \xi \right\} \\ & =&\displaystyle P\left\{ 2\sup \limits_{t \in [0,\tau]} \left| \widehat{F}(t) - F(t) \right| > \xi \right\} \\ & =&\displaystyle P\left\{ \sup \limits_{t \in [0,\tau]} \left| \widehat{F}(t) - F(t) \right| > \frac{\xi}{2} \right\} \\ & \leq&\displaystyle \kappa_1 \exp\left(- \frac{1}{4} n \xi^2 \eta^4 \kappa_2 \right),{} \end{array} \end{aligned} $$

(B.7)

where the last step is due to Lemma 1 with ξ in the right-hand side of (A.1) replaced by $\frac {\xi }{2}$. Therefore, combining (B.5) and (B.7) gives

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} P\left( \big| S_1 \big| > \xi \right) \leq \kappa_1 \exp\left(- \frac{1}{4} n \xi^2 \eta^4 \kappa_2 \right). \end{array} \end{aligned} $$

(B.8)

Next, we examine S ₂ in a similar manner. Similar to (B.5), we have that

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \left|F(Y_i) - F(Y_k) \right| \leq 1. \end{array} \end{aligned} $$

(B.9)

Similar to the arguments for (B.7), we obtain that

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} & &\displaystyle P\bigg\{ \bigg| \left| \widehat{F}_{adj,j}(X_{i(j)}) - \widehat{F}_{X_{(j)}}(X_{k(j)}) \right| - \left| F_{adj,j}(X_{i(j)}) - F_{X_{(j)}}(X_{k(j)}) \right| \bigg| > \xi \bigg\} \\ & \leq&\displaystyle \kappa_3 \exp \left\{- \frac{n \xi^2}{2G^2} + o( n^{-\frac{1}{5}}) \right\}. \end{array} \end{aligned} $$

(B.10)

Therefore, combining (B.9) and (B.10) yields

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} P\left( \big|S_2\big| > \xi \right) \leq \kappa_3 \exp \left\{- \frac{n \xi^2}{2G^2} + o( n^{-\frac{1}{5}}) \right\}. \end{array} \end{aligned} $$

(B.11)

Finally, combining (B.4), (B.8), and (B.11), the probabilistic bound of $\widehat {M}_{j,1} - \widetilde {M}_{j,1}$ is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} P\left( \left|\widehat{M}_{j,1} - \widetilde{M}_{j,1} \right| > \xi \right) & =&\displaystyle P\left( \big|S_1 + S_2 \big|> \xi \right) {}\\ & \leq&\displaystyle P\left( \big|S_1 \big| + \big| S_2 \big|> \xi \right) \\ & \leq&\displaystyle P\left( \big|S_1\big| > \frac{\xi}{2} \right) + P\left( \big|S_2\big| > \frac{\xi}{2} \right) \\ & \leq&\displaystyle \kappa_1 \exp\left(- \frac{1}{16} n \xi^2 \eta^4 \kappa_2 \right) + \kappa_3 \exp \left\{- \frac{n \xi^2}{8G^2} + o( n^{-\frac{1}{5}}) \right\}. \end{array} \end{aligned} $$

(B.12)

Furthermore, similar derivations show that

$$\displaystyle \begin{aligned} \begin{array}{rcl} P\left( \left|\widehat{M}_{j,2} - \widetilde{M}_{j,2} \right| > \xi \right) & \leq&\displaystyle \kappa_1 \exp\left(- \frac{1}{16} n \xi^2 \eta^4 \kappa_2 \right) \\ & &\displaystyle + \kappa_3 \exp \left\{- \frac{n \xi^2}{8G^2} + o( n^{-\frac{1}{5}}) \right\}{} \end{array} \end{aligned} $$

(B.13)

and

$$\displaystyle \begin{aligned} \begin{array}{rcl} P\left( \left|\widehat{M}_{j,3} - \widetilde{M}_{j,3} \right| > \xi \right) & \leq&\displaystyle \kappa_1 \exp\left(- \frac{1}{16} n \xi^2 \eta^4 \kappa_2 \right) \\ & &\displaystyle + \kappa_3 \exp \left\{- \frac{n \xi^2}{8G^2} + o( n^{-\frac{1}{5}}) \right\}.{} \end{array} \end{aligned} $$

(B.14)

Noting that the upper bounds in (B.12)–(B.14) are dominated by $\exp \left (- c^\ast n \xi ^2 \right )$ for certain constant c ^∗, we apply (B.12)–(B.14) to (B.3) and obtain that

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} P\left( \left| \widehat{\omega}_j - \omega_j^\ast \right| > \xi \right) = O \left\{ \exp\left(- \widetilde{c}_2 n \xi^2 \right) \right\} \end{array} \end{aligned} $$

(B.15)

for some $\widetilde {c}_2 >0$. Thus, combining (B.2) and (B.15) with (B.1) and specifying ξ = cn ^−ζ for the constants c and ζ described in Condition (C4) yield the desired result.

Part 2

We prove (20).

Let $J = \min \limits _{j \in \mathcal {I}} \big |\omega _j\big | - \max \limits _{j \in \mathcal {I}^c} \big |\omega _j \big |$. The left-hand side of (20) can be expressed as

$$\displaystyle \begin{aligned} \begin{array}{rcl} P\left( \max \limits_{j \in \mathcal{I}^c} \big| \widehat{\omega}_j \big| \geq \min \limits_{j \in \mathcal{I}} \big|\widehat{\omega}_j\big| \right) & =&\displaystyle P\left( \max \limits_{j \in \mathcal{I}^c} \big| \widehat{\omega}_j \big| - \max \limits_{j \in \mathcal{I}^c} \big|\omega_j \big| \geq \min \limits_{j \in \mathcal{I}} \big|\widehat{\omega}_j \big| - \max \limits_{j \in \mathcal{I}^c} \big|\omega_j \big| \right) \\ & =&\displaystyle P\left( \max \limits_{j \in \mathcal{I}^c} \big| \widehat{\omega}_j \big| - \max \limits_{j \in \mathcal{I}^c} \big|\omega_j \big| \geq \min \limits_{j \in \mathcal{I}} \big|\widehat{\omega}_j\big| - \min \limits_{j \in \mathcal{I}} \big|\omega_j\big| + J \right) \\ & \leq&\displaystyle P\left( \max \limits_{j \in \mathcal{I}^c} \left| \widehat{\omega}_j - \omega_j \right| + \max \limits_{j \in \mathcal{I}} \left| \widehat{\omega}_j - \omega_j \right| \geq J \right) \\ & \leq&\displaystyle P\left( 2 \max \limits_{j = 1,\cdots,p} \left| \widehat{\omega}_j - \omega_j \right| \geq J \right) \\ & =&\displaystyle O \left\{ \exp\left(- \frac{1}{4} D n v_0^2 \right) \right\}, \end{array} \end{aligned} $$

where the last step comes from the result in Part 1 and Condition (C5). $\hfill \square$

1.2.2 B.2 Proof of Theorem 2

Similar to the derivations of Li et al. (2012), one can obtain that

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left\{ \max \limits_{j \in \mathcal{I}} \left| \widehat{\omega}_j - \omega_j \right| \leq cn^{-\zeta} \right\} \subseteq \left\{ \mathcal{I} \subseteq \widehat{\mathcal{I}} \right\}. \end{array} \end{aligned} $$

It gives

$$\displaystyle \begin{aligned} \begin{array}{rcl} P\left( \mathcal{I} \subseteq \widehat{\mathcal{I}} \right) & \geq &\displaystyle 1 - P\left( \max \limits_{j \in \mathcal{I}} \left| \widehat{\omega}_j - \omega_j \right| \leq cn^{-\zeta} \right) \\ & \geq &\displaystyle 1 - q P\left( \left| \widehat{\omega}_j - \omega_j \right| \leq cn^{-\zeta} \right) \\ & \geq &\displaystyle 1 - O\left\{ q \exp\left(-Dn^{1-2\zeta} \right) \right\}, \end{array} \end{aligned} $$

where the last step comes from Theorem 1. □