Mixture Quantiles Estimated by Constrained Linear Regression

TL;DR

This paper introduces a flexible, linear-parameter-based quantile modeling approach using constrained linear regression, which efficiently captures distribution features including tails, and is supported by theoretical and empirical validation.

Contribution

It proposes a novel family of distribution models with quantile functions as linear combinations of basis quantiles, solvable via convex constrained linear regression.

Findings

Accurately models distribution tails and centers.

Requires less computation than standard methods.

Establishes asymptotic properties of the estimator.

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.