Factor Model of Mixtures

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.