quantile regression modeling

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.

Quantile crossing—the violation of monotonicity in conditional quantile functions—is a persistent issue in quantile regression. Method: This paper proposes a state-space-based Bayesian joint quantile modeling framework. It introduces a quantile-varying-parameter (QVP) model that embeds non-crossing constraints directly into the prior structure; employs a fused shrinkage prior to adaptively regularize variability across multiple quantile functions; and derives the posterior from a generalized Bayesian decision-theoretic perspective, yielding a natural state-space interpretation. The approach incorporates time-varying parameter extensions and a customized MCMC algorithm. Contribution/Results: Simulation studies and multivariate quantile forecasting experiments in macroeconomics demonstrate substantial improvements in parameter recovery accuracy and out-of-sample predictive performance, consistently surpassing state-of-the-art Bayesian and frequentist quantile regression methods.

This study addresses the challenge of jointly modeling the probability of zero outcomes and the heterogeneous, nonlinear effects in the positive component of semi-continuous data—characterized by a substantial mass at zero and a continuous positive part—along with their complex dependence structure. To this end, we propose the first copula-based semiparametric two-part quantile regression framework. The approach separately models the occurrence of zeros and the magnitude of positive values via quantile regression and flexibly captures their nonlinear dependence across quantiles using a copula function. Theoretical analysis establishes large-sample asymptotic properties, and simulations demonstrate superior performance over existing methods under high zero-inflation and nonlinear scenarios. An empirical application to healthcare data reveals heterogeneous and nonlinear effects of social deprivation on uncompensated and charitable care burdens.

This study addresses the bias in quantile regression estimation arising from the coexistence of endogeneity and additive measurement error in the dependent variable. It is the first to achieve nonparametric identification of conditional quantile coefficient functions and other distributional parameters within a triangular system by integrating the control function approach with Copula modeling. To this end, the paper proposes a two-step sieve maximum likelihood estimator: first, the control function is estimated nonparametrically; then, it is treated as a generated regressor and incorporated into the sieve likelihood via Copula-based weights for maximization, with inference conducted using the bootstrap. Monte Carlo simulations demonstrate that the proposed method substantially reduces estimation bias and exhibits superior performance in scenarios where existing approaches fail.

This study addresses the challenge of identifying and inferring integral functionals of conditional distributions with discontinuous outcomes by proposing a novel ReLU regression framework. The approach constructs closed-form estimators through covariate projection of ReLU-transformed outcome variables and recovers integrated conditional quantile functions via their convex conjugates obtained through the Legendre–Fenchel transform. By introducing ReLU activation into regression, this method enables direct identification of conditional distribution features under only weak distributional assumptions, accommodates discontinuous outcomes, and identifies average quantile treatment effects over arbitrary probability intervals. Leveraging Hadamard directional differentiability and the Delta method, the authors establish a unified theory for the consistent asymptotic distribution of the proposed estimators, substantially expanding the set of distributional parameters identifiable in empirical research.

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.

Existing open-source tools struggle to efficiently support dynamic quantile regression modeling and uncertainty quantification, particularly in time series contexts. This work proposes the R package exdqlm, which establishes a unified framework for dynamic quantile linear models based on the extended asymmetric Laplace distribution. The framework accommodates both static and dynamic specifications and incorporates regularized priors, transition functions, and cross-quantile posterior predictive synthesis. By integrating MCMC with a fast Laplace–delta variational Bayesian inference scheme, the method substantially enhances computational efficiency for long time series while preserving the accuracy of uncertainty quantification. The resulting tool enables efficient modeling, precise forecasting, and comprehensive diagnostic capabilities, thereby addressing a critical gap in practical implementations of complex time series quantile regression analysis.