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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L_1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> or smoothness regularization. For <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L_q\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>-type objectives we show the estimator is asymptotically equivalent to a minimum <i>q</i>-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.</p>

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Mixture quantiles estimated by constrained linear regression

  • 期刊论文

  • 发表日期：

    2026年4月20日

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  • Cheng Peng,
  • Yizhou Li,
  • Stan Uryasev

摘要

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 \(L_1\) L 1 or smoothness regularization. For \(L_q\) L q -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.