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

Huihui Chen, Darinka Dentcheva, Yang Lin, Gregory J. Stock

TL;DR

This work derives central limit theorems for vector-valued, nested composite risk functionals when using mixed estimators that combine empirical and smoothing components. It builds a general framework employing Hadamard directional differentiability, strong approximate identities, and kernel smoothing to obtain CLTs for both scalar- and vector-valued functionals, with explicit Brownian-limit representations and covariances. The results apply to risk-measure optimization, comparisons of risk measures, and systemic risk estimation, providing verifiable conditions and bandwidth guidelines for kernel estimators. Simulations indicate that smoothing reduces estimator bias and improves finite-sample accuracy in both univariate and multivariate settings, enabling reliable high-dimensional risk analysis.

Abstract

This paper focuses on vector-valued composite functionals, which may be nonlinear in probability. Our primary goal is to establish central limit theorems for these functionals when mixed estimators are employed. Our study is relevant to the evaluation and comparison of risk in decision-making contexts and extends to functionals that arise in machine learning methods. A generalized family of composite risk functionals is presented, which encompasses most of the known coherent risk measures including systemic measures of risk. The paper makes two main contributions. First, we analyze vector-valued functionals, providing a framework for evaluating high-dimensional risks. This framework facilitates the comparison of multiple risk measures, as well as the estimation and asymptotic analysis of systemic risk and its optimal value in decision-making problems. Second, we derive novel central limit theorems for optimized composite functionals when mixed types of estimators: empirical and smoothed estimators are used. We provide verifiable sufficient conditions for the central limit formulae and show their applicability to several popular measures of risk.

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

TL;DR

This work derives central limit theorems for vector-valued, nested composite risk functionals when using mixed estimators that combine empirical and smoothing components. It builds a general framework employing Hadamard directional differentiability, strong approximate identities, and kernel smoothing to obtain CLTs for both scalar- and vector-valued functionals, with explicit Brownian-limit representations and covariances. The results apply to risk-measure optimization, comparisons of risk measures, and systemic risk estimation, providing verifiable conditions and bandwidth guidelines for kernel estimators. Simulations indicate that smoothing reduces estimator bias and improves finite-sample accuracy in both univariate and multivariate settings, enabling reliable high-dimensional risk analysis.

Abstract

This paper focuses on vector-valued composite functionals, which may be nonlinear in probability. Our primary goal is to establish central limit theorems for these functionals when mixed estimators are employed. Our study is relevant to the evaluation and comparison of risk in decision-making contexts and extends to functionals that arise in machine learning methods. A generalized family of composite risk functionals is presented, which encompasses most of the known coherent risk measures including systemic measures of risk. The paper makes two main contributions. First, we analyze vector-valued functionals, providing a framework for evaluating high-dimensional risks. This framework facilitates the comparison of multiple risk measures, as well as the estimation and asymptotic analysis of systemic risk and its optimal value in decision-making problems. Second, we derive novel central limit theorems for optimized composite functionals when mixed types of estimators: empirical and smoothed estimators are used. We provide verifiable sufficient conditions for the central limit formulae and show their applicability to several popular measures of risk.
Paper Structure (11 sections, 11 theorems, 89 equations, 3 figures)

This paper contains 11 sections, 11 theorems, 89 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Framework and Preliminaries
  3. Central limit theorems for mixed smoothed and empirical estimators
  4. Scalar-valued composite risk functionals
  5. Central limit theorem for vector-valued composite functionals
  6. Central limit theorem for sample-based stochastic optimization
  7. Application
  8. Risk measures representable as optimal value of an optimization problem
  9. Comparison of two risk measures
  10. Central Limit Theorem for Systemic Risk
  11. Conclusions

Key Result

Theorem 3.1

Suppose an index set $J\subseteq\{1, \ldots, k+1\}$ is fixed, $m_0 = 1$, and the following conditions are satisfied: Then ${\sqrt[]{n}( \varrho_{\mu}^{(n,J)}[X]-\varrho[X]) \xrightarrow{d} \xi_1(G)}$, where $G(\cdot)=(G_1(\cdot),\ldots,G_k(\cdot),G_{k+1})$ is zero-mean Brownian process on $I$. Here $G_j(\cdot)$ is a Brownian process of dimension $m_{j-1}$ on $I_j$, $j=1,\ldots,k$, and $G_{k+1}$ i

Figures (3)

  • Figure 1: Density histogram of the distribution of the estimator $\varrho^{(n)}_K$ and $\varrho^{(n)}$ with sample sizes of 30, 50, 100, and 200 (arranged clockwise).
  • Figure 2: Density histogram of the distribution of the estimator $\varrho^{(n)}_K$ and $\varrho^{(n)}$ with sample sizes of 30, 50, 100, and 200 (arranged clockwise).
  • Figure 3: Density histogram of the distribution of the systemic risk $\varrho^{(K,n)}_{\rm sys}[X]$ and $\varrho^{(n)}_{\rm sys}[X]$ with sample sizes of 30, 50, 100, and 200 (arranged clockwise).

Theorems & Definitions (21)

  • Definition 2.1
  • Theorem 3.1
  • proof
  • Definition 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 4.1
  • proof
  • ...and 11 more