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

TL;DR

This paper establishes central limit theorems for vector-valued composite functionals, including risk measures in decision-making and machine learning, using mixed estimators like empirical and smoothed types.

Contribution

It introduces a framework for high-dimensional risk evaluation and derives new CLTs for optimized composite functionals with mixed estimators.

Findings

Central limit theorems for composite functionals with mixed estimators

Framework for high-dimensional risk comparison and systemic risk analysis

Applicability to popular risk measures

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.