This paper considers the problem of estimating the distribution of a response variable conditioned on observing some factors. Existing approaches are often deficient in one of the qualities of flexibility, interpretability and tractability. We propose a model that possesses these desirable properties. The proposed model, analogous to classic mixture regression models, models the conditional quantile function as a mixture (weighted sum) of basis quantile functions, with the weight of each basis quantile function being a function of the factors. The model can approximate any bounded conditional quantile model. It has a factor model structure with a closed-form expression. The calibration problem is formulated as convex optimization, which can be viewed as conducting quantile regressions of all confidence levels simultaneously and does not suffer from quantile crossing by design. The calibration is equivalent to minimization of Continuous Probability Ranked Score (CRPS). We prove the asymptotic normality of the estimator. Additionally, based on risk quadrangle framework, we generalize the proposed approach to conditional distributions defined by Conditional Value-at-Risk (CVaR), expectile and other functions of uncertainty measures. Based on CP decomposition of tensors, we propose a dimensionality reduction method by reducing the rank of the parameter tensor and propose an alternating algorithm for estimating the parameter tensor. Our numerical experiments demonstrate the efficiency of the approach.

翻译：本文研究在观测到某些因子后，估计响应变量分布的问题。现有方法在灵活性、可解释性和可计算性三者之间往往存在不足。我们提出了一种兼具这些理想性质的模型。该模型类似于经典混合回归模型，将条件分位数函数建模为基分位数函数的混合（加权和），其中每个基分位数函数的权重是各因子的函数。该模型能够逼近任意有界条件分位数模型，具有因子模型结构且可表示为闭式表达式。校准问题被构造成凸优化形式，可视为同时对所有置信水平进行分位数回归，并从根本上避免分位数交叉问题。优化目标等价于连续概率排名评分（CRPS）的最小化。我们证明了估计量的渐近正态性。此外，基于风险四边形框架，我们将该方法推广到由条件风险价值（CVaR）、期望损失及其他不确定性测度函数定义的分布。基于张量的CP分解，我们提出通过降低参数张量秩的降维方法，并开发了估计参数张量的交替算法。数值实验验证了该方法的高效性。