A tuning-free efficient test for marginal linear effects in high-dimensional quantile regression

Abstract

This work is concerned with testing the marginal linear effects of high-dimensional predictors in quantile regression. We introduce a novel test that is constructed using maxima of pairwise quantile correlations, which permit consistent assessment of the marginal linear effects. The proposed testing procedure is computationally efficient with the aid of a simple multiplier bootstrap method and does not involve any need to select tuning parameters, apart from the number of bootstrap replications. Other distinguishing features of the new procedure are that it imposes no structural assumptions on the unknown dependence structures of the predictor vector and allows the dimension of the predictor vector to be exponentially larger than sample size. To broaden the applicability, we further extend the preceding analysis to the censored response case. The effectiveness of our proposed approach in the finite samples is illustrated through simulation studies.

Appendix: Necessary Lemmas

We start with providing technical lemmas used repeatedly in the online Supplementary Material.

Lemma 3

(Shao and Zhang 2014, Proposition 2) Suppose the distribution function of Y satisfies Assumption (C3), then there exist $\epsilon _0>0$ and $c>0$ such that for any $\epsilon \in (0,\epsilon _0)$,

$$\begin{aligned} \mathbb {P}\left[ n^{-1}\sum \limits _{i=1}^{n}\left| \psi _{\tau }\{Y_{i}- \widehat{Q}_{\tau }(Y)\}-\psi _{\tau }\{Y_{i}- Q_{\tau }(Y)\}\right| > \epsilon \right] \le 3\exp \left( -2cn\epsilon ^2\right) . \end{aligned}$$

Lemma 4

(Chernozhukov et al. 2015, Corollary 5.1) Let $Z, Z_1,\ldots , Z_n$ be i.i.d. random variables taking values in a measurable space $(S, \mathcal {S}),$ Q denote a probability measure on the measurable space, $\mathcal {F}\subset L^{2}(Q)$ be a pointwise measurable class of real-valued functions on S with measurable envelope F, and $\mathcal {N}(\mathcal {F}, \Vert \cdot \Vert _{Q,2}, \delta )$ denote the $\delta$-covering number for $\mathcal {F}$ with respect to the $L^{2}(Q)$-seminorm $\Vert \cdot \Vert _{Q,2}.$ Suppose that $\mathcal {F}$ is VC type, i.e., there exist constants $A\ge e$ and $V\ge 1$ such that $\sup _{Q}\mathcal {N}(\mathcal {F}, \Vert \cdot \Vert _{Q,2}, \epsilon \Vert F\Vert _{Q,2})\le (A/\epsilon )^{V}$ where $\sup _{Q}$ is taken over all finitely discrete distributions on S. Furthermore, suppose that $0<E\{F^{2}(Z)\}<\infty$, and let $\sigma ^{2}>0$ be any positive constant such that $\sup _{f\in \mathcal {F}}E\{f^{2}(Z)\}\le \sigma ^{2}\le E\{F^{2}(Z)\}.$ Define $B=E^{1/2}\{\max _{1\le i\le n}F^{2}(Z_i)\}$. Then

$$\begin{aligned}{} & {} E\left( \Vert n^{-1/2}\sum \limits _{i=1}^{n}\left[ f(Z_i)-E\{f(Z)\}\right] \Vert _{\mathcal {F}}\right) \\{} & {} \lesssim \left( V\sigma ^{2}\log \left[ AE^{1/2}\{F^{2}(Z)\}/\sigma \right] \right) ^{1/2}+n^{-1/2}VB\log \left[ AE^{1/2}\{F^{2}(Z)\}/\sigma \right] , \end{aligned}$$

up to a universal constant.

Lemma 5

(Einmahl and Li 2008, Theorem 3.1) Let $\textbf{z}_1,\ldots , \textbf{z}_n$ be independent centered random vectors in $\mathbb {R}^p$ where $p\ge 2$. Write $\textbf{z}_i=(Z_{i1},\ldots ,Z_{ip})^{{\tiny {\textrm{T}}}}$ for $i=1,\ldots ,n$. If $E(\max _{1\le k\le p}\mid Z_{ik}\mid ^{r})<\infty$ for $i=1,\cdots ,n,$ and some $r>2$, then

$$\begin{aligned}{} & {} \mathbb {P} \left \{\max _{1\le k\le p}\mid \sum \limits _{i=1}^{n}Z_{ik}\mid \ge 2E(\max _{1\le k\le p}\mid \sum \limits _{i=1}^{n}Z_{ik}\mid )+t \right \}\\{} & {} \lesssim \exp \{-t^{2}/(3n\max \limits _{1\le i\le n}\max \limits _{1\le k\le p}E\mid Z_{ik}\mid ^{2})\}+t^{-r}\sum \limits _{i=1}^{n}E(\max \limits _{1\le k\le p}| Z_{ik}|^{r}). \end{aligned}$$