This paper proposes a new approach to estimating the distribution of a response variable conditioned on factors. We model the conditional quantile function as a mixture (weighted sum) of basis quantile functions, with weights depending on these factors. The estimation problem is formulated as a convex optimization problem. The objective function is equivalent to the continuous ranked probability score (CRPS). This approach can be viewed as conducting quantile regressions for all confidence levels simultaneously while inherently avoiding quantile crossing. We use spline functions of factors as a primary example for the weight function. We prove an approximation property of the model. To address computational challenges, we propose a dimensionality reduction method using tensor decomposition and an alternating algorithm. Our approach offers flexibility, interpretability, tractability, and extendability. Numerical experiments demonstrate its effectiveness.

In this paper, we show that the popular C-SVM, soft-margin support vector classifier is equivalent to minimization of Buffered Probability of Exceedance (bPOE), a recently introduced characterization of uncertainty. To show this, we introduce a new SVM formulation, called the EC-SVM, which is derived from a simple bPOE minimization problem that is easy to interpret with a meaningful free parameter, optimal objective value, and probabilistic derivation. Over the range of its free parameter, the EC-SVM has both a convex and non-convex case which we connect to existing SVM formulations. We first show that the C-SVM, formulated with any regularization norm, is equivalent to the convex EC-SVM. Similarly, we show that the E$\nu$-SVM is equivalent to the EC-SVM over its entire parameter range, which includes both the convex and non-convex case. These equivalences, coupled with the interpretability of the EC-SVM, allow us to gain surprising new insights into the C-SVM and fully connect soft margin support vector classification with superquantile and bPOE concepts. We also show that the EC-SVM can easily be cast as a robust optimization problem, where bPOE is minimized with data lying in a fixed uncertainty set. This reformulation allows us to clearly differentiate between the convex and non-convex case, with convexity associated with pessimistic views of uncertainty and non-convexity associated with optimistic views of uncertainty. Finally, we address some practical considerations. First, we show that these new insights can assist in making parameter selection more efficient. Second, we discuss optimization approaches for solving the EC-SVM. Third, we address the issue of generalization, providing generalization bounds for both bPOE and misclassification rate.

This article has outlined a new approach for the simultaneous calculation of value-at-risk (VaR) and optimization of conditional VaR (CVaR) for a broad class of problems. We have shown that CVaR can be efficiently minimized using LP techniques. Our numerical experiments show that CVaR optimal portfolios are near optimal in VaR terms, i.e., VaR cannot be reduced further more than a few percent. Also, CVaR constraints can be handled efficiently using equivalent linear constraints, which dramatically improves the efficiency of the optimization techniques.