Mixture Quantiles Estimated by Constrained Linear Regression
Cheng Peng, Yizhou Li, Stan Uryasev
TL;DR
The paper introduces a mixture-quantiles model that expresses the quantile function as a linear combination of basis quantiles, yielding a convex estimation problem via constrained linear regression. It proves that, with an L_q loss and a full set of probabilities, the estimator is asymptotically equivalent to a minimum q-Wasserstein distance estimator and is asymptotically normal under standard regularity conditions. The approach offers practical regularization and constraints (nonnegativity, cardinality, L_1/L_2 penalties, and P-spline smoothing) to improve finite-sample performance and tail accuracy. Through extensive simulations and real-data case studies (Gaussian mixtures, electricity prices, and financial drawdowns), the method demonstrates superior tail fitting and substantial computational efficiency relative to benchmark approaches while preserving good global fit. Overall, this framework provides a scalable, interpretable, and tail-aware alternative for univariate distribution modeling with broad applicability in risk management and uncertainty quantification.
Abstract
We study the problem of modeling univariate distributions via their quantile functions. We introduce a flexible family of distributions whose quantile function is a linear combination of basis quantiles. Because the model is linear in its parameters, estimation reduces to constrained linear regression, yielding a convex optimization problem that readily accommodates cardinality constraints as well as L1 or smoothness regularization. For Lq-type objectives we show the estimator is asymptotically equivalent to a minimum q-Wasserstein distance estimator and establish asymptotic normality. Experiments on simulated and real-world datasets demonstrate that the proposed method accurately captures both the central body and extreme tails of distributions while requiring substantially less computation than standard benchmark approaches.