quantile function estimation

In multiple quantile regression, conventional multiple testing procedures fail to control the family-wise error rate (FWER) rigorously. To address this, we propose a multivariate joint test based on rank scores, embedded within a closed testing procedure to guarantee strong FWER control. This work constitutes the first extension of rank-score-based inference to simultaneous quantile regression, overcoming the low statistical power inherent in Bonferroni-type corrections. We establish the asymptotic validity of the proposed test under mild regularity conditions. Extensive Monte Carlo simulations demonstrate that the method maintains the nominal FWER level precisely across diverse simulation designs while delivering substantially higher statistical power than existing approaches.

This study addresses the inefficiency of traditional quantile estimation under light-tailed, heavy-tailed, or asymmetric distributions and its difficulty in smoothly bridging central and tail regions. The authors propose a unified interpolation-based quantile estimation framework that incorporates quadratic, Huber, or Tukey bisquare regularization into the check loss function, enabling continuous control of the effective quantile level via an interpolation parameter. They derive, for the first time, a closed-form parametrization of the effective quantile level under quadratic interpolation and establish a complete asymptotic theory, revealing the dependence of estimation efficiency on distributional shape. Theoretical and simulation results demonstrate that the proposed method reduces asymptotic variance by up to 36% under light-tailed distributions and by up to 57% under heavy-tailed or asymmetric distributions. Empirical analysis of daily log-returns confirms its superior performance in tail risk estimation.

This study addresses the challenge of obtaining consistent estimators in quantile regression when covariates are subject to normal measurement error, a setting complicated by the discontinuity and nonlinearity of the check loss function. The authors propose a novel estimation approach applicable to both linear and nonlinear models, which employs kernel smoothing to handle discontinuities and leverages complex-domain extensions together with moment-generating functions to manage nonlinearity—without requiring joint modeling across multiple quantiles. Within a general quantile regression framework, this work establishes, for the first time, an estimator that achieves root-n consistency and asymptotic normality. Theoretical analysis confirms the standard convergence rate, while numerical simulations and an empirical application to the 2024 Japanese cherry blossom bloom dates demonstrate the method’s practical effectiveness.

This paper addresses the challenge of quantile regression modeling between functional responses and functional predictors. We propose two novel functional partial quantile regression (FPQR) algorithms. Methodologically, we pioneer the integration of partial quantile regression into the function-on-function regression framework, combining functional principal component analysis (FPCA) for dimension reduction with basis function expansions to transform the infinite-dimensional quantile coefficient function estimation into a finite-dimensional multivariate quantile regression problem. Theoretical analysis and Monte Carlo simulations demonstrate that the proposed methods substantially improve estimation accuracy and robustness, particularly in small-sample settings. Empirical studies further confirm their practical effectiveness. The algorithms are implemented in the R package *ffpqr*, ensuring reproducibility and facilitating broad applicability in functional data analysis.

This work addresses the limitation of traditional quantile regression in modeling pairwise similarity responses—such as image similarity scores in face recognition—by introducing, for the first time, a pairwise quantile regression framework. The proposed method predicts conditional quantiles of similarity scores for input pairs by minimizing a pairwise pinball loss. On the theoretical front, sharp concentration inequalities based on U-processes are employed to establish generalization error bounds, yielding fast learning rates under mild assumptions. Empirically, the approach is validated through both simulated data and real-world face recognition similarity scoring tasks, demonstrating its effectiveness and practical utility.

This study addresses the computational and inferential challenges of applying predictive augmented inference to quantile regression in data-limited settings characterized by few high-quality labels and abundant proxy labels. The authors introduce convolution smoothing into this framework for the first time, proposing two computable estimators—along with an ensemble approach—by smoothing the check loss function. This strategy effectively mitigates optimization difficulties arising from the non-differentiability of the original objective and substantially reduces over-coverage in confidence intervals. Theoretical analysis establishes the asymptotic distribution under model misspecification, while numerical experiments and an application to housing data demonstrate that the proposed method is computationally efficient, yields accurate inference, and offers both practical utility and superior performance.

This study addresses the estimation of parameters of the form θ₀ = E[F_Y⁻¹∘F_Z(X)] in the “changes-in-changes” model, for which existing methods lack theoretical guarantees when variables are unbounded. The authors construct a plug-in estimator based on empirical quantiles and establish its √n-consistency and asymptotic normality under assumptions weaker than those in the current literature. They further propose a novel consistent estimator for the asymptotic variance. The theoretical analysis leverages empirical process theory and plug-in methods for quantile functions. Monte Carlo simulations demonstrate that the proposed variance estimator substantially outperforms existing alternatives, leading to markedly improved inference accuracy.

This study addresses the high computational cost of quantile regression for large-scale longitudinal data by proposing an efficient estimation method based on optimal Poisson subsampling. For the first time, optimal Poisson subsampling is integrated into the longitudinal quantile regression framework, combined with a weighted smoothed quantile generalized estimating equation and regularization techniques to achieve sparse parameter estimation. The authors establish the corresponding asymptotic theory to support the proposed approach. Numerical experiments and real data analysis demonstrate that the method significantly outperforms uniform Poisson subsampling in both estimation accuracy and computational efficiency, while the regularized estimator exhibits strong variable selection performance.