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Central limit theorems for vector-valued composite functionals with smoothing and applications

Author

Listed:
  • Huihui Chen

    (Stevens Institute of Technology)

  • Darinka Dentcheva

    (Stevens Institute of Technology)

  • Yang Lin

    (Hartford Steam Boiler)

  • Gregory J. Stock

    (Raritan Valley Community College)

Abstract

This paper focuses on vector-valued composite functionals, which may be nonlinear in probability. Our goal is establishing central limit theorems for these functionals when employed by mixed estimators. Our study is relevant to the evaluation and comparison of risk in decision-making contexts and extends to functionals that arise in machine learning. A generalized family of composite risk functionals is presented, which encompasses coherent risk measures, including systemic risk. The paper makes two main contributions. First, we analyze vector-valued functionals and provide a framework for evaluating high-dimensional risks. This enables comparison of multiple risk measures and supports estimation and asymptotic analysis of systemic risk and its optimal value in decision-making. Second, we derive new central limit theorems for optimized composite functionals using mixed estimators, including empirical and smoothed types. We give verifiable conditions for central limit formulae and demonstrate their applicability to several risk measures.

Suggested Citation

  • Huihui Chen & Darinka Dentcheva & Yang Lin & Gregory J. Stock, 2025. "Central limit theorems for vector-valued composite functionals with smoothing and applications," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 77(5), pages 821-852, October.
    • Handle: RePEc:spr:aistmt:v:77:y:2025:i:5:d:10.1007_s10463-025-00934-z
    DOI: 10.1007/s10463-025-00934-z
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    References listed on IDEAS

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    1. PAVLO A. Krokhmal, 2007. "Higher moment coherent risk measures," Quantitative Finance, Taylor & Francis Journals, vol. 7(4), pages 373-387.
    2. Aray Almen & Darinka Dentcheva, 2024. "On Risk Evaluation and Control of Distributed Multi-agent Systems," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 2025-2054, November.
    3. Radulovic, Dragan & Wegkamp, Marten, 2003. "Necessary and sufficient conditions for weak convergence of smoothed empirical processes," Statistics & Probability Letters, Elsevier, vol. 61(3), pages 321-336, February.
    4. Darinka Dentcheva & Yang Lin & Spiridon Penev, 2023. "Stability and Sample-Based Approximations of Composite Stochastic Optimization Problems," Operations Research, INFORMS, vol. 71(5), pages 1871-1888, September.
    5. Darinka Dentcheva & Spiridon Penev & Andrzej Ruszczyński, 2017. "Statistical estimation of composite risk functionals and risk optimization problems," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(4), pages 737-760, August.
    6. Rockafellar, R. Tyrrell & Uryasev, Stanislav, 2002. "Conditional value-at-risk for general loss distributions," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1443-1471, July.
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