[Paper Review] From Halfspace M-depth to Multiple-output Expectile Regression

Abdelaati Daouia, Davy Paindaveine|arXiv (Cornell University)|May 29, 2019

Advanced Statistical Methods and Models48 references4 citations

TL;DR

This paper introduces hyperplane-valued multivariate M-quantiles and a new statistical depth, halfspace M-depth, which generalizes Tukey's halfspace depth by replacing standard quantile outlyingness with M-quantile outlyingness. The method enables multiple-output expectile regression, with expectile depth showing smoother behavior and better monotonicity than Tukey depth, while multivariate expectiles satisfy coherency axioms for risk measures and support efficient computation and regression modeling.

ABSTRACT

Despite the renewed interest in the Newey and Powell (1987) concept of expectiles in fields such as econometrics, risk management, and extreme value theory, expectile regression---or, more generally, M-quantile regression---unfortunately remains limited to single-output problems. To improve on this, we introduce hyperplane-valued multivariate M-quantiles that show strong advantages, for instance in terms of equivariance, over the various point-valued multivariate M-quantiles available in the literature. Like their competitors, our multivariate M-quantiles are directional in nature and provide centrality regions when all directions are considered. These regions define a new statistical depth, the halfspace M-depth, whose deepest point, in the expectile case, is the mean vector. Remarkably, the halfspace M-depth can alternatively be obtained by substituting, in the celebrated Tukey (1975) halfspace depth, M-quantile outlyingness for standard quantile outlyingness, which supports a posteriori the claim that our multivariate M-quantile concept is the natural one. We investigate thoroughly the properties of the proposed multivariate M-quantiles, of halfspace M-depth, and of the corresponding regions. Since our original motivation was to define multiple-output expectile regression methods, we further focus on the expectile case. We show in particular that expectile depth is smoother than the Tukey depth and enjoys interesting monotonicity properties that are extremely promising for computational purposes. Unlike their quantile analogs, the proposed multivariate expectiles also satisfy the coherency axioms of multivariate risk measures. Finally, we show that our multivariate expectiles indeed allow performing multiple-output expectile regression, which is illustrated on simulated and real data.

Motivation & Objective

• Address the limitation of expectile regression to single-output problems in econometrics and risk management.
• Develop a multivariate extension of M-quantile regression that preserves desirable properties like equivariance and coherency.
• Propose a new statistical depth, halfspace M-depth, which generalizes Tukey depth using M-quantile outlyingness.
• Enable multiple-output expectile regression by defining hyperplane-valued multivariate expectiles.
• Demonstrate the computational and theoretical advantages of expectile depth over Tukey depth, including smoother behavior and monotonicity.

Proposed method

• Define hyperplane-valued multivariate M-quantiles as directional extensions of univariate M-quantiles, using asymmetric quadratic loss (L2) for expectiles.
• Construct halfspace M-depth by substituting M-quantile outlyingness for standard quantile outlyingness in Tukey’s halfspace depth framework.
• Establish that the deepest point of halfspace M-depth corresponds to the mean vector in the expectile case.
• Prove that expectile depth is smoother and exhibits monotonicity properties beneficial for optimization and computation.
• Derive and verify coherency axioms for multivariate expectiles, ensuring compatibility with risk measure theory.
• Implement multiple-output expectile regression using the proposed multivariate expectiles, validated on simulated and real data.

Experimental results

Research questions

• RQ1 Can M-quantile regression be extended to multivariate responses while preserving equivariance and coherence?
• RQ2 How does halfspace M-depth compare to Tukey depth in terms of smoothness and computational tractability?
• RQ3 Do multivariate expectiles satisfy the coherency axioms required for coherent risk measures?
• RQ4 Can the proposed multivariate expectiles support efficient and stable multiple-output regression?
• RQ5 What are the monotonicity and continuity properties of expectile depth that facilitate numerical optimization?

Key findings

• Halfspace M-depth generalizes Tukey depth by replacing standard quantile outlyingness with M-quantile outlyingness, yielding a natural multivariate extension.
• The deepest point of halfspace M-depth is the mean vector in the expectile case, aligning with intuitive centrality.
• Expectile depth is smoother than Tukey depth and exhibits monotonicity, which supports efficient numerical computation.
• Multivariate expectiles satisfy the coherency axioms of risk measures, making them suitable for risk management applications.
• Multiple-output expectile regression is successfully implemented using the proposed multivariate expectiles, with demonstrated performance on real and simulated data.
• The method avoids nonparametric density estimation, enabling inference without smoothing, unlike quantile regression.