Biased Mean Quadrangle and Applications

Paper: arXiv 2603.26901

Authors: Anton Malandii, Stan Uryasev

Abstract

This paper introduces \emph{biased mean regression}, estimating the \emph{biased mean}, i.e., $\mathbb{E}[Y] + x$, where $x \in \mathbb{R}$. The approach addresses a fundamental statistical problem that covers numerous applications. For instance, it can be used to estimate factors driving portfolio loss exceeding the expected loss by a specified amount (e.g., $ x=$10 billion$) or to estimate factors impacting a specific excess release of radiation in the environment, where nuclear safety regulations specify different severity levels. The estimation is performed by minimizing the so-called \emph{superexpectation error}. We establish two equivalence results that connect the method to popular paradigms: (i) biased mean regression is equivalent to quantile regression for an appropriate parameterization and is equivalent to ordinary least squares when $x=0$; (ii) in portfolio optimization, minimizing \emph{superexpectation risk}, associated with the superexpectation error, is equivalent to CVaR optimization. The approach is computationally attractive, as minimizing the superexpectation error reduces to linear programming (LP), thereby offering algorithmic and modeling advantages. It is also a good alternative to ordinary least squares (OLS) regression. The approach is based on the \emph{Risk Quadrangle} (RQ) framework, which links four stochastic functionals – error, regret, risk, and deviation – through a statistic. For the biased mean quadrangle, the statistic is the biased mean. We study properties of the new quadrangle, such as \emph{subregularity}, and establish its relationship to the quantile quadrangle. Numerical experiments confirm the theoretical statements and illustrate the practical implications.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 4.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a strong theoretical framework extending Rockafellar’s Risk Quadrangle to biased mean regression and superexpectation error, offering attractive LP formulations. However, it focuses primarily on mathematical derivations and equivalence proofs, with relatively limited empirical backtesting compared to its rich theoretical development.

Research Flowchart

  flowchart TD
    A[Research Goal: Introduce Biased Mean Regression E[Y] + x] --> B[Framework: Apply Risk Quadrangle Theory]
    B --> C[Methodology: Minimize Superexpectation Error]
    D[Inputs: Portfolio Loss & Environmental Data] --> E[Computation: Solve Linear Programming LP Problem]
    C --> E
    E --> F[Theoretical Outcomes: Prove Equivalence to Quantile Regression, OLS x=0 & CVaR]
    F --> G[Practical Findings: Computationally Efficient Alternative to OLS & Quantile Methods]

Cited References (60)

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2. C onditional V alue-at- R isk for G eneral L oss D istributions (2002)
3. R isk T uning with G eneralized L inear R egression (2008)
4. CV a R N orm and A pplications in O ptimization (2014)
5. The F undamental R isk Q uadrangle in R isk M anagement, O ptimization and S tatistical E stimation (2013)
6. Convex Optimization (2004)
7. Support V ector R egression M achines (1996)
8. Support V ector M ethod for F unction A pproximation, R egression E stimation and S ignal P rocessing (1996)

1 of 60 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.