The variable selection by the Dantzig selector for Cox’s proportional hazards model

Abstract

The proportional hazards model proposed by D. R. Cox in a high-dimensional and sparse setting is discussed. The regression parameter is estimated by the Dantzig selector, which will be proved to have the variable selection consistency. This fact enables us to reduce the dimension of the parameter and to construct asymptotically normal estimators for the regression parameter and the cumulative baseline hazard function.

Appendix

Proof of Lemma 11

Under the condition that $\hat{\beta }_{n \hat{T}_n^c}^{(2)} = 0$, we use Taylor expansion to deduce that

$$\begin{aligned} J_{n \hat{T}_n, \hat{T}_n}(\beta _0) \left( \hat{\beta }_{n \hat{T}_n}^{(2)}-\beta _{0 \hat{T}_n}\right) 1_{\{\hat{T}_n = T_0\}} = U_n(\beta _0)_{\hat{T}_n} 1_{\{\hat{T}_n=T_0\}} +o_p(\Vert \hat{\beta }_{n \hat{T}_n}^{(2)}- \beta _{0 \hat{T}_n}\Vert _2). \end{aligned}$$

Therefore, under Assumption 9, it follows from Lemma 10 that

$$\begin{aligned} \mathcal {I}\left( \hat{\beta }_{n \hat{T}_n}^{(2)}-\beta _{0 \hat{T}_n}\right) 1_{\{\hat{T}_n=T_0\}} = U_n(\beta _0)_{\hat{T}_n}1_{\{\hat{T}_n=T_0\}} +o_p(\Vert \hat{\beta }_{n \hat{T}_n}^{(2)} - \beta _{0 \hat{T}_n}\Vert _2) + o_p(1). \end{aligned}$$

Since $\mathcal {I}$ is assumed to be non-singular and $P(\hat{T}_n = T_0) \rightarrow 1$ as $n \rightarrow \infty$ by Theorem 7, we obtain the conclusion. $\square$

Proof of Theorem 13

It follows from the Taylor expansion that

$$\begin{aligned} \left\{ U_{n \hat{T}_n}(\hat{\beta }_{n \hat{T}_n}^{(2)}) - U_{n T_0}(\beta _{0 T_0})\right\} 1_{\{\hat{T}_n = T_0\}} = -J_{n T_0,T_0}(\beta _{n T_0}^*)(\hat{\beta }_{n \hat{T}_n}^{(2)} - \beta _{0 T_0}) 1_{\{\hat{T}_n = T_0\}}, \end{aligned}$$

where $\beta _n^*$ is the point between $\hat{\beta }_n^{(2)}$ and $\beta _0$. Then, the assertion is obtained by using Slutsky’s theorem and the corresponding result from Andersen and Gill (1982). $\square$

Proof of Theorem 15

We have that

$$\begin{aligned}&\sqrt{n}\{\hat{\Lambda }(t) - \Lambda _0(t)\}1_{\{\hat{T}_n = T_0\}}\\&\quad = \left[ H_{n T_0}(\beta _{n T_0}^*,t)^\top \sqrt{n} (\hat{\beta }_{n \hat{T}_n}^{(2)}-\beta _{0 T_0}) + \sqrt{n} W_n(t)\right] 1_{\{\hat{T}_n = T_0\}} + o_p(1). \end{aligned}$$

We can use the fact (10) to deduce that

$$\begin{aligned}&\sqrt{n}\{\hat{\Lambda }(t)-\Lambda _0(t)\}1_{\{\hat{T}_n = T_0\}}\\&\qquad + \sqrt{n} \int _0^t (\hat{\beta }_{n\hat{T}_n}^{(2)}-\beta _{0T_0})^\top \frac{s^{(1)}}{s^{(0)}}(\beta _{0T_0},s) \lambda _0(s) \mathrm{d}s 1_{\{\hat{T}_n = T_0\}}\\&\quad = \sqrt{n} W_n(t) 1_{\{\hat{T}_n = T_0\}} + o_p(1), \end{aligned}$$

where

$$\begin{aligned} W_n(t) = \sqrt{n} \int _0^t \frac{d\bar{M}(s)}{S_n^{(0)}(\beta _0,s)},\quad t \in [0,1]. \end{aligned}$$

Then, the conclusion is obtained by using Slutsky’s theorem and the corresponding result from Andersen and Gill (1982). $\square$