Factor Model of Mixtures

TL;DR

This paper introduces a flexible, interpretable mixture model for estimating conditional distributions, employing convex optimization and tensor decomposition, with extensions to risk measures like CVaR and expectiles.

Contribution

It presents a novel mixture-based approach for conditional quantile estimation that avoids crossing and extends to various risk measures using tensor methods.

Findings

Effective in modeling conditional distributions

Avoids quantile crossing issues

Extensible to CVaR and expectile frameworks

Abstract

This paper proposes a new approach to estimating the distribution of a response variable conditioned on observing some factors. The proposed approach possesses desirable properties of flexibility, interpretability, tractability and extendability. The conditional quantile function is modeled by a mixture (weighted sum) of basis quantile functions, with the weights depending on factors. The calibration problem is formulated as a convex optimization problem. It can be viewed as conducting quantile regressions for all confidence levels simultaneously while avoiding quantile crossing by definition. The calibration problem is equivalent to minimizing the continuous ranked probability score (CRPS). Based on the canonical polyadic (CP) decomposition of tensors, we propose a dimensionality reduction method that reduces the rank of the parameter tensor and propose an alternating algorithm for estimation. Additionally, based on Risk Quadrangle framework, we generalize the approach to conditional distributions defined by Conditional Value-at-Risk (CVaR), expectile and other functions of uncertainty measures. Although this paper focuses on using splines as the weight functions, it can be extended to neural networks. Numerical experiments demonstrate the effectiveness of our approach.