Shrinkage estimation of varying covariate effects based on quantile regression

Abstract

Varying covariate effects often manifest meaningful heterogeneity in covariate-response associations. In this paper, we adopt a quantile regression model that assumes linearity at a continuous range of quantile levels as a tool to explore such data dynamics. The consideration of potential non-constancy of covariate effects necessitates a new perspective for variable selection, which, under the assumed quantile regression model, is to retain variables that have effects on all quantiles of interest as well as those that influence only part of quantiles considered. Current work on l1-penalized quantile regression either does not concern varying covariate effects or may not produce consistent variable selection in the presence of covariates with partial effects, a practical scenario of interest. In this work, we propose a shrinkage approach by adopting a novel uniform adaptive LASSO penalty. The new approach enjoys easy implementation without requiring smoothing. Moreover, it can consistently identify the true model (uniformly across quantiles) and achieve the oracle estimation efficiency. We further extend the proposed shrinkage method to the case where responses are subject to random right censoring. Numerical studies confirm the theoretical results and support the utility of our proposals.

Consistent model identification of varying coefficient quantile regression with BIC tuning parameter selection

“…Let β̃ ( τ ) be the unpenanlized quantile regression estimator at level τ . When ω j ( τ ) = 1/(sup τ ∈Δ | β̃ ( j ) ( τ )|), the proposed estimator reduces to the UAL estimator (Peng et al, 2014). It can be shown that the classical adaptive weight, 1/| β̃ ( j ) ( τ )|, is also a qualified adaptive weight function.…”

“…However, there still remains two important questions: (1) Is there a general type of adaptive weights to achieve oracle properties? (2) The numerical studies in Peng et al (2014) suggest that a BIC-type tuning parameter selector generally yields satisfactory rates of correct model identification. Is there any formal justification for the BIC-type selector?…”

“…Motivated by this consideration, Peng et al (2014) proposed a uniform adaptive-LASSO (UAL) varying coefficient quantile regression over a spectrum of quantiles, which was designed to remove only covariates with no effect on any quantile index of interest. Their key strategy, following the idea of adaptive-LASSO (Zou, 2006), is to penalize the estimation by the maximum departure of the coefficient function from zero.…”

High dimensional censored quantile regression

“…proportional hazards, location shift effects) have been recognized. For example, variable selection based on these models may pose a considerable risk of missing variables that have complex, non-constant survival impact but are scientifically important (Peng et al, 2014). …”

“…With p fixed, several authors also investigated the variable selection problem for CQR. For example, Shows et al (2010) and Peng et al (2014) proposed penalized estimating equations where censoring was handled by inverse probability weighting. Wang et al (2013) developed a robust variable selection method based on Wang and Wang (2009)’s work, adopting a global dimension reduction formulation to facilitate the local weight estimation.…”