Nonparametric independence feature screening for ultrahigh-dimensional survival data

Abstract

With the explosion of digital information, high-dimensional data is frequently collected in prevalent domains, in which the dimension of covariates can be much larger than the sample size. Many effective methods have been developed to reduce the dimension of such data recently, however, few methods might perform well for survival data with censoring. In this article, we develop a novel nonparametric feature screening procedure based on ultrahigh-dimensional survival data by incorporating the inverse probability weighting scheme to tackle the issue of censoring. The proposed method is model-free and hence can be implemented for extensive survival models. Moreover, it is robust to heterogeneity and invariant to monotone increasing transformations of the response. The sure screening property and ranking consistency property are also established under mild conditions. The competence and robustness of our method is further confirmed through comprehensive simulation studies and an analysis of a real data example.

Appendices

Appendix A

Lemma 1

(Bitouzé et al. 1999, Theorem 1) Let $\{X_i\}_{i=1}^n$ and $\{Y_i\}_{i=1}^n$ be independent sequences of independent identically distributed nonnegative random variables with distribution functions $F(\cdot )$ and $G(\cdot )$. Let $\widehat{F}_{n}$ be the Kaplan–Meier estimator of $F(\cdot )$. There exists a constant $M>0$, for any $\lambda >0$, such that

$$\begin{aligned} Pr\left( \sqrt{n}\Vert (1-G)(\widehat{F}_n-F)\Vert _{\infty }>\lambda \right) \le 2.5\exp \{-2\lambda ^2+M\lambda \}. \end{aligned}$$

Lemma 2

(Serfling 1980, P201, Theorem B) Let $h = h(X_1,X_2,\ldots ,X_m)$ be the kernel of the U-statistic, $\theta = \theta (F)$, with $E\exp \left\{ sh(X_1,X_2,\ldots ,X_m)\right\} <\infty$, $0<s<s_0$. For any $\varepsilon >o$, when $n>m$, there exist $c_1>0$ and $0<\rho <1$ such that

$$\begin{aligned} Pr(U_n-\theta \ge \varepsilon )\le c_1\rho ^n. \end{aligned}$$

Lemma 3

Under Condition (C1), for any $c_2 >0$, when $n \ge M^2c_3^{-1}$, where $\delta >0$ and $c_3 = \frac{1}{9}(\frac{c_2}{1+c_2})^2\delta ^8$,

$$\begin{aligned} Pr\left( \max _{i,j,l}\left| \frac{K(Y_i)K(Y_j)K(Y_l)}{\widehat{K}(Y_i)\widehat{K}(Y_j)\widehat{K}(Y_l)}-1\right| \ge c_2\right) \le 2.5n^3\exp \left\{ -c_3n\right\} . \end{aligned}$$

Proof of Lemma 3

For any x, $y>0$, taking $a_1=\frac{c_2}{1+c_2}$, i.e., $a_1\in (0,1)$, it is easy to show that

$$\begin{aligned} |x^{-1}-y^{-1}|\ge c_2y^{-1} \Rightarrow |x-y|\ge a_1y. \end{aligned}$$

Let $S(t)= 1- F(t)= Pr(T> t)$. Condition (C1) implies that there exist a constant $\delta >0$, such that $\delta \le S(Y_i) \le 1$, $\delta \le K(Y_i) \le 1$ and $0\le \widehat{K}(Y_i)\le 1$ for $i = 1, 2, \ldots , n$, therefore by Lemma 1, it follows that

$$\begin{aligned}&Pr\left( \left| \frac{K(Y_i)K(Y_j)K(Y_l)}{\widehat{K}(Y_i)\widehat{K}(Y_j)\widehat{K}(Y_l)}-1\right| \ge c_2\right) \\&\quad \le Pr\left( |\widehat{K}(Y_i)\widehat{K}(Y_j)\widehat{K}(Y_l)-K(Y_i)K(Y_j)K(Y_l)|\ge a_1K(Y_i)K(Y_j)K(Y_l)\right) \\&\quad \le Pr\left( \widehat{K}(Y_i)\widehat{K}(Y_j)|\widehat{K}(Y_l)-K(Y_l)|+ \widehat{K}(Y_i)K(Y_l)|\widehat{K}(Y_j)-K(Y_j)|\right. \\&\qquad \left. + \, K(Y_j)K(Y_l)|\widehat{K}(Y_i)-K(Y_i)|\ge a_1\delta ^3\right) \\&\quad \le Pr\left( 3\Vert \widehat{K}-K\Vert _{\infty }\ge a_1\delta ^3\right) \le Pr\left( \sqrt{n}\Vert (1-F)(\widehat{K}-K)\Vert _{\infty }\ge \sqrt{n}\frac{a_1\delta ^4}{3}\right) \\&\quad \le 2.5 \exp \left\{ -2n\left( \frac{a_1\delta ^4}{3}\right) ^2+M\sqrt{n}\frac{a_1\delta ^4}{3}\right\} , \end{aligned}$$

where $\Vert \cdot \Vert _{\infty }$ is the $L_{\infty }$ norm. When $n(\frac{a_1\delta ^4}{3})^2 \ge M\sqrt{n}\frac{a_1\delta ^4}{3}$, i.e. $n\ge M^2(\frac{a_1\delta ^4}{3})^{-2}$, taking $c_3 = (\frac{a_1\delta ^4}{3})^{2}=\frac{1}{9}(\frac{c_2}{c_2+1})^2\delta ^8$, we have

$$\begin{aligned} Pr\left( \max _{i,j,l}\left| \frac{K(Y_i)K(Y_j)K(Y_l)}{\widehat{K}(Y_i)\widehat{K}(Y_j)\widehat{K}(Y_l)}-1\right| \ge c_2\right) \le 2.5n^3\exp \left\{ -c_3n\right\} . \end{aligned}$$

$\square$

Lemma 4

Under Condition (C1), for any $c_4 >0$, when $n \ge M^2c_5^{-1}$, where $c_5 = \frac{1}{9}(\frac{c_4}{1+c_4})^2\delta ^8$,

$$\begin{aligned} Pr\left( \max _{i,j}\left| \frac{K^2(Y_i)K(Y_j)}{\widehat{K}^2(Y_i)\widehat{K}(Y_j)}-1\right| \ge c_4\right) \le 2.5n^2\exp \left\{ -c_5n\right\} . \end{aligned}$$

Proof of Lemma 4

Similar to Lemma 3, taking $a_2=c_4/(1+c_4)$, i.e., $a_2\in (0,1)$, it follows that

$$\begin{aligned}&Pr\left( \left| \frac{K^2(Y_i)K(Y_j)}{\widehat{K}^2(Y_i)\widehat{K}(Y_j)}-1\right| \ge c_4\right) \\&\quad \le Pr\left( \left| \widehat{K}^2(Y_i)\widehat{K}(Y_j) - K^2(Y_i)K(Y_j)\right| \ge a_2K^2(Y_i)K(Y_j)\right) \\&\quad \le Pr\left( \widehat{K}(Y_j)|\widehat{K}^2(Y_i)-K^2(Y_i)|+ K^2(Y_i)|\widehat{K}(Y_j)-K(Y_j)|\ge a_2\delta ^3\right) \\&\quad \le Pr\left( 3\Vert \widehat{K} - K\Vert _{\infty }\ge a_2\delta ^3\right) \le Pr\left( \sqrt{n}\Vert (1-F)(\widehat{K} - K)\Vert _{\infty }\ge \sqrt{n}\frac{a_2\delta ^4}{3} \right) \\&\quad \le 2.5 \exp \left\{ -2n\left( \frac{a_2\delta ^4}{3}\right) ^2+M\sqrt{n}\frac{a_2\delta ^4}{3}\right\} . \end{aligned}$$

When $n(\frac{a_2\delta ^4}{3})^2 \ge M\sqrt{n}\frac{a_2\delta ^4}{3}$, i.e. $n\ge M^2(\frac{a_2\delta ^4}{3})^{-2}$, taking $c_5 = (\frac{a_2\delta ^4}{3})^{2}=\frac{1}{9}(\frac{c_4}{c_4+1})^2\delta ^8$, we have

$$\begin{aligned} Pr\left( \max _{i,j}\left| \frac{K^2(Y_i)K(Y_j)}{\widehat{K}^2(Y_i)\widehat{K}(Y_j)}-1\right| \ge c_4\right) \le 2.5n^2\exp \left\{ -c_5n\right\} . \end{aligned}$$

$\square$

Proof of Theorem 1

We now proof the first statement. To start with, rewrite

$$\begin{aligned} \widehat{\omega }_k&= \frac{2}{n(n-1)(n-2)}\sum _{j< i< l}^{n}\frac{\varDelta _i\varDelta _j\varDelta _l}{\widehat{K}(Y_i)\widehat{K}(Y_j)\widehat{K}(Y_l)}\left\{ X_{jk}X_{ik}I(Y_j<Y_l)I(Y_i<Y_l) \right. \\&\quad +\,\left. X_{lk}X_{ik}I(Y_l<Y_j)I(Y_i<Y_j)+ X_{jk}X_{lk}I(Y_j<Y_i)I(Y_l<Y_i)\right\} \\&\quad +\,\frac{1}{n(n-1)(n-2)}\sum _{i\ne j}^{n}\frac{\varDelta _i\varDelta _j}{\widehat{K}^2(Y_i)\widehat{K}(Y_j)}X_{ik}^2I(Y_i<Y_j) \\&\triangleq \widehat{\omega }_{k1} + \frac{1}{n-2}\widehat{\omega }_{k2}. \end{aligned}$$

Thus,

$$\begin{aligned} \left| \widehat{\omega }_k - \omega _k \right| \le \left| \widehat{\omega }_{k1} - \omega _k\right| + \left| \frac{1}{n-2}\widehat{\omega }_{k2}\right| \triangleq |I_{k1}| + |I_{k2}|. \end{aligned}$$

(9)

For $I_{k1}$, denote

$$\begin{aligned} \widehat{\omega }_{k1} = U_n\left[ \frac{\varDelta _i\varDelta _j\varDelta _l}{\widehat{K}(Y_i)\widehat{K}(Y_j)\widehat{K}(Y_l)}X_{ik}X_{jk}I(Y_i<Y_l)I(Y_j<Y_l)\right] \triangleq U_nf_1(W_i,W_j,W_l), \end{aligned}$$

where

$$\begin{aligned}&U_nf_1(W_i,W_j,W_l)\\&\quad = \left( C_{n}^{3}\right) ^{-1}\sum _{i<j<l}\frac{1}{3}\left[ f_1(W_i,W_j,W_l)+f_1(W_i,W_l,W_j)+f_1(W_l,W_j,W_i)\right] . \end{aligned}$$

We can prove that

$$\begin{aligned}&E\left[ \frac{\varDelta _i\varDelta _j\varDelta _l}{K(Y_i)K(Y_j)K(Y_l)}X_{ik}X_{jk}I(Y_i<Y_l)I(Y_j<Y_l)\right] \\&\quad = E\left\{ E\left[ \frac{\varDelta _i\varDelta _j\varDelta _l}{K(Y_i)K(Y_j)K(Y_l)}X_{ik}X_{jk}I(Y_i<Y_l)I(Y_j<Y_l)|T_i,T_j,T_l\right] \right\} \\&\quad = E\left[ X_{ik}X_{jk}I(T_i<T_l)I(T_j<T_l)\right] =E\left\{ E\left[ X_{ik}X_{jk}I(T_i<T_l)I(T_j<T_l)|T_l\right] \right\} \\&\quad = E\left\{ E^2\left[ X_{ik}I(T_i<T_l)|T_l\right] \right\} = \omega _k. \end{aligned}$$

Then,

$$\begin{aligned}&\widehat{\omega }_{k1}-\omega _{k} \\&\quad = U_n\left[ \left( \frac{K(Y_i)K(Y_j)K(Y_l)}{\widehat{K}(Y_i)\widehat{K}(Y_j)\widehat{K}(Y_l)}-1\right) \right. \\&\qquad \times \,\left. \frac{\varDelta _i\varDelta _j\varDelta _l}{K(Y_i)K(Y_j)K(Y_l)}X_{ik}X_{jk}I(Y_i<Y_l)I(Y_j<Y_l)\right] \\&\qquad +\,(U_n-E)\left[ \frac{\varDelta _i\varDelta _j\varDelta _l}{K(Y_i)K(Y_j)K(Y_l)}X_{ik}X_{jk}I(Y_i<Y_l)I(Y_j<Y_l)\right] \\&\quad \triangleq J_{k1}+J_{k2}. \end{aligned}$$

For $J_{k1}$,

$$\begin{aligned} |J_{k1}|\le & {} \max _{i,j,l}\left| \frac{K(Y_i)K(Y_j)K(Y_l)}{\widehat{K}(Y_i)\widehat{K}(Y_j)\widehat{K}(Y_l)}-1\right| \\&\times \, U_n\left[ \frac{\varDelta _i\varDelta _j\varDelta _l}{K(Y_i)K(Y_j)K(Y_l)}|X_{ik}||X_{jk}|I(Y_i<Y_l)I(Y_j<Y_l)\right] \\\le & {} \max _{i,j,l}\left| \frac{K(Y_i)K(Y_j)K(Y_l)}{\widehat{K}(Y_i)\widehat{K}(Y_j)\widehat{K}(Y_l)}-1\right| \\&\times \, \left| (U_n-E)\left[ \frac{\varDelta _i\varDelta _j\varDelta _l}{K(Y_i)K(Y_j)K(Y_l)}|X_{ik}X_{jk}|I(Y_i<Y_l)I(Y_j<Y_l)\right] \right| \\&+\,\max _{i,j,l}\left| \frac{K(Y_i)K(Y_j)K(Y_l)}{\widehat{K}(Y_i)\widehat{K}(Y_j)\widehat{K}(Y_l)}-1\right| \\&\times \, \left| E\left[ \frac{\varDelta _i\varDelta _j\varDelta _l}{K(Y_i)K(Y_j)K(Y_l)}|X_{ik}X_{jk}|I(Y_i<Y_l)I(Y_j<Y_l)\right] \right| \\&\triangleq J_{k11}+J_{k12}. \end{aligned}$$

Let

$$\begin{aligned} f_2(W_i,W_j,W_l)=\frac{\varDelta _i\varDelta _j\varDelta _l}{K(Y_i)K(Y_j)K(Y_l)}|X_{ik}||X_{jk}|I(Y_i<Y_l)I(Y_j<Y_l), \end{aligned}$$

with Conditions (C1) and (C2), it follows that $E\exp \{sf_2(W_i,W_j,W_l)\} < \infty$. Thus by Lemma 2, when $n > 3$, we have

$$\begin{aligned} Pr\left( |U_n-E|f_2(W_i,W_j,W_l)\ge \frac{1}{4}cn^{-\kappa }\right) \le 2c_1\rho ^n. \end{aligned}$$

(10)

By Lemma 3 , let $c_2 = 1$, then $c_3 = \delta ^8/{36}$, when $n\ge 36M^2\delta ^{-8}$, we have

$$\begin{aligned} Pr\left( \max _{i,j,l}\left| \frac{K(Y_i)K(Y_j)K(Y_l)}{\widehat{K}(Y_i)\widehat{K}(Y_j)\widehat{K}(Y_l)}-1\right| \ge 1\right) \le 2.5n^3\exp \left\{ -c_3n\right\} . \end{aligned}$$

(11)

Therefore, by Eqs. (10) and (11),

$$\begin{aligned}&Pr\left( \left| J_{k11}\right| \ge \frac{1}{4}cn^{-\kappa }\right) \nonumber \\&\quad \le Pr\left( \max _{i,j,l}\left| \frac{K(Y_i)K(Y_j)K(Y_l)}{\widehat{K}(Y_i)\widehat{K}(Y_j)\widehat{K}(Y_l)}-1\right| \ge 1\right) \nonumber \\&\qquad +\,Pr\left( |U_n-E|f_2(W_i,W_j,W_l)\ge \frac{1}{4}cn^{-\kappa }\right) \nonumber \\&\quad \le 2.5n^{3}\exp \left\{ -c_3n\right\} +2c_1\rho ^n. \end{aligned}$$

(12)

Since $Ef_2(W_i,W_j,W_l)\le \frac{1}{\delta ^3}\sup _{k}E^2|X_{k}|$, by Lemma 3 , when $\frac{\delta ^3cn^{-\kappa }}{4\sup _kE^2|X_k|} \le 1$ and $n\ge 36M^2\delta ^{-8}$,

$$\begin{aligned} Pr\left( |J_{k12}|\ge \frac{1}{4}cn^{-\kappa }\right)\le & {} Pr\left( \frac{\sup _kE^2|X_k|}{\delta ^3}\max _{i,j,l}\left| \frac{K(Y_i)K(Y_j)K(Y_l)}{\widehat{K}(Y_i)\widehat{K}(Y_j)\widehat{K}(Y_l)}-1\right| \ge \frac{1}{4}cn^{-\kappa }\right) \nonumber \\\le & {} 2.5n^3\exp \left\{ -c_3n\right\} . \end{aligned}$$

(13)

For $J_{k2}$, denote

$$\begin{aligned} f_3(W_i,W_j,W_l)=\frac{\varDelta _i\varDelta _j\varDelta _l}{K(Y_i)K(Y_j)K(Y_l)}X_{ik}X_{jk}I(Y_i<Y_l)I(Y_j<Y_l), \end{aligned}$$

it is verified that $E\exp \{sf_3(W_i,W_j,W_l)\} < \infty$, then by Lemma 2,

$$\begin{aligned} Pr\left( |J_{k2}|\ge \frac{1}{4}cn^{-\kappa }\right) =Pr\left( |U_n-E|f_3(W_i,W_j,W_l)\ge \frac{1}{4}cn^{-\kappa }\right) \le 2c_1\rho ^n. \end{aligned}$$

(14)

Using triangle inequality and Eqs. (12)–(14), when $n\ge m_0$, where $m_0 = \max \{3, 36M^2\delta ^{-8}, [\frac{c\delta ^3}{4\sup _kE^2|X_k|}]^{\frac{1}{\kappa }}\}$, it follows that

$$\begin{aligned}&Pr\left( |I_{k1}|\ge \frac{3}{4}cn^{-\kappa }\right) \nonumber \\&\quad \le Pr\left( |J_{k11}|\ge \frac{1}{4}cn^{-\kappa }\right) + Pr\left( |J_{k12}|\ge \frac{1}{4}cn^{-\kappa }\right) + Pr\left( |J_{k2}|\ge \frac{1}{4}cn^{-\kappa }\right) \nonumber \\&\quad \le 5n^3\exp \left\{ -c_3n\right\} +4c_1\rho ^n. \end{aligned}$$

(15)

For $I_{k2}$, under Condition (C1), it follows that

$$\begin{aligned} |\widehat{\omega }_{k2}|= & {} \left| \frac{1}{n(n-1)}\sum _{i\ne j}^n\frac{\varDelta _i\varDelta _j}{\widehat{K}^2(Y_i)\widehat{K}(Y_j)}X_{ik}^2I(Y_i<Y_j)\right| \\\le & {} \frac{1}{n(n-1)}\sum _{i\ne j}^n\left| \frac{K^2(Y_i)K(Y_j)}{\widehat{K}^2(Y_i)\widehat{K}(Y_j)}-1\right| \frac{\varDelta _i\varDelta _j}{K^2(Y_i)K(Y_j)}X_{ik}^2I(Y_i<Y_j) \\&+\, \frac{1}{n(n-1)}\sum _{i\ne j}^n\frac{\varDelta _i\varDelta _j}{K^2(Y_i)K(Y_j)}X_{ik}^2I(Y_i<Y_j)\\\le & {} \max _{i,j}\left| \frac{K^2(Y_i)K(Y_j)}{\widehat{K}^2(Y_i)\widehat{K}(Y_j)}-1\right| \frac{1}{n\delta ^3}\sum _{i=1}^nX_{ik}^2 +\frac{1}{n\delta ^3}\sum _{i=1}^nX_{ik}^2. \end{aligned}$$

Then taking $c_4 = 1$ and $c_5 = \delta ^8/36$, when $n \ge 36M^2\delta ^{-8}$, with Condition (C2) and Lemma 4, we have

$$\begin{aligned}&Pr\left( \left| I_{k2}\right| \ge \frac{c}{4}n^{-\kappa }\right) \nonumber \\&\quad \le Pr\left( \max _{i,j}\left| \frac{K^2(Y_i)K(Y_j)}{\widehat{K}^2(Y_i)\widehat{K}(Y_j)}-1\right| \ge 1\right) +2Pr\left( \frac{1}{n\delta ^3}\sum _{i=1}^{n}X_{ik}^2\ge \frac{c}{8}(n-2)n^{-\kappa }\right) \nonumber \\&\quad \le 2.5n^2\exp \left\{ -c_5n\right\} +2\exp \left\{ -\frac{c}{8}\delta ^3n^{-\kappa }(n-2)\right\} E\exp \left\{ \frac{1}{n}\sum _{i=1}^{n} X_{ik}^2\right\} \nonumber \\&\quad \le 2.5n^2\exp \left\{ -c_5n\right\} +2M_0\exp \left\{ -\frac{c\delta ^3}{8}n^{-\kappa }(n-2)\right\} . \end{aligned}$$

(16)

Therefore, by Eqs. (9), (15) and (16), when $n > m_0$, it follows that

$$\begin{aligned}&Pr\left( \max _{1\le k\le p}|\widehat{\omega }_k-\omega _k|\ge cn^{-\kappa } \right) \le p\left\{ Pr\left( \left| I_{k1}\right| + |I_{k2}|\ge cn^{-\kappa }\right) \right\} \\&\quad \le p\left\{ 4c_1\rho ^n+5n^3\exp \left( -c_3n\right) +2.5n^2\exp \left( -c_5n\right) + 2M_0\exp \left[ -c_6n^{-\kappa }(n-2)\right] \right\} , \end{aligned}$$

where $c_6 = c\delta ^3/8$.

For the second statement, take $\gamma _n = c_0n^{-\kappa }$, where $c_0\le c$, then $\gamma _n\le cn^{-\kappa }$, therefore,

$$\begin{aligned} \widehat{\mathscr {A}}=\{k:\widehat{\omega }_k\ge \gamma _n\}\supset \left\{ k:\widehat{\omega }_k\ge cn^{-\kappa }\right\} , \end{aligned}$$

let $\mathscr {A}_n = \left\{ \max _{k\in \mathscr {A}}|\widehat{\omega }_k-\omega _k|\le cn^{-\kappa }\right\}$, if $\mathscr {A}\nsubseteq \widehat{\mathscr {A}}$, there exist some $k\in \mathscr {A}$, such that $\widehat{\omega }_k<cn^{-\kappa }$, by Assumption (C3),

$$\begin{aligned} |\widehat{\omega }_k-\omega _k|> cn^{-\kappa }\Rightarrow & {} \{\mathscr {A}\nsubseteq \widehat{\mathscr {A}}\}\subset \{|\widehat{\omega }_k-\omega _k|> cn^{-\kappa }, \exists k\in \mathscr {A}\} \\\Rightarrow & {} \mathscr {A}_n\subseteq \{\mathscr {A}\subseteq \widehat{\mathscr {A}}\}. \end{aligned}$$

Therefore,

$$\begin{aligned}&Pr(\mathscr {A}\subseteq \widehat{\mathscr {A}})\\&\quad \ge 1-Pr\left( \max _{k\in \mathscr {A} }|\widehat{\omega }_k-\omega _k|> cn^{-\kappa }\right) \ge 1-sPr\left( |\widehat{\omega }_k-\omega _k|> cn^{-\kappa }\right) \\&\quad \ge 1-s\left\{ 4c_1\rho ^n+5n^3\exp \left( -c_3n\right) +2.5n^2\exp \left( -c_5n\right) + 2M_0\exp \left[ -c_6n^{-\kappa }(n-2)\right] \right\} , \end{aligned}$$

where s is the cardinality of $\mathscr {A}$, here we complete the proof of Theorem 1. $\square$

Proof of Theorem 2

Recall $w_k(t)$, by Condition (C4), for $k\in \mathscr {I}$ and $t \in \varPsi _T$, we can prove that,

$$\begin{aligned} w_k(t)= & {} E\left\{ E\left[ \frac{\varDelta }{K(Y)}X_kI(Y<t)|\mathbf {X}\right] \right\} = E\{X_kE[I(T<t)|\mathbf {X}]\}\\= & {} E\{X_kE[I(T<t)|\mathbf {X}_\mathscr {A}]\} = 0, \end{aligned}$$

and thus $\omega _k = 0$. It follows from Condition (C3) that $\min _{k\in \mathscr {A}} \omega _k - \max _{k\in \mathscr {I}} \omega _k > 2cn^{-\kappa }$. Thus,

$$\begin{aligned}&Pr\left( \min _{k\in \mathscr {A}} \widehat{\omega }_{k} \le \max _{k\in \mathscr {I}}\widehat{\omega }_{k} \right) \\&\quad = Pr\left( \min _{k\in \mathscr {A}}\widehat{\omega }_{k} - \min _{k\in \mathscr {A}}\omega _{k} + 2cn^{-\kappa } \le \max _{k\in \mathscr {I}} \widehat{\omega }_{k} - \max _{k\in \mathscr {I}}\omega _{k}\right) \\&\quad \le Pr\left( \sup _{k\in \mathscr {A}}|\widehat{\omega }_k - \omega _k| \ge cn^{-\kappa }\right) + Pr\left( \sup _{k\in \mathscr {I}}|\widehat{\omega }_k - \omega _k| \ge cn^{-\kappa }\right) \\&\quad \le 2p\left\{ 4c_1\rho ^n+5n^3\exp \left( -c_3n\right) +2.5n^2\exp \left( -c_5n\right) + 2M_0\exp \left[ -c_6n^{-\kappa }(n-2)\right] \right\} , \end{aligned}$$

here we complete the proof of Theorem 2.$\square$

Appendix B

See Table 12.

Table 12 Simulation results for Example 3: $P_{all}$ and $P_k$ corresponding to $X_{1}-X_{p_1}$