High Dimensional Mean Test for Shrinking Random Variables with Applications to Backtesting

2026-02-10

AI Summary

This study addresses the poor performance of conventional mean tests in high-dimensional settings, where issues such as mean cancellation or failure of marginal asymptotic normality arise. To overcome these limitations, the authors propose a novel testing framework that aggregates observations over overlapping subsets of dimensions, estimates subset-specific means, and constructs a test statistic based on the maximum absolute subset mean. The null distribution of this statistic is approximated using a multiplier bootstrap procedure. This approach effectively mitigates mean cancellation induced by simple averaging, retains desirable statistical properties even under nonstandard asymptotic conditions, and naturally extends to Value-at-Risk (VaR) backtesting. Extensive simulations and empirical analyses demonstrate that the proposed method achieves superior testing power and practical utility in high-dimensional scenarios.

Technology Category

• Machine Learning: Ensemble Methods
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

We propose a high dimensional mean test framework for shrinking random variables, where the underlying random variables shrink to zero as the sample size increases. By pooling observations across overlapping subsets of dimensions, we estimate subsets means and test whether the maximum absolute mean deviates from zero. This approach overcomes cancellations that occur in simple averaging and remains valid even when marginal asymptotic normality fails. We establish theoretical properties of the test statistic and develop a multiplier bootstrap procedure to approximate its distribution. The method provides a flexible and powerful tool for the validation and comparative backtesting of value-at-risk. Simulations show superior performance in high-dimensional settings, and a real-data application demonstrates its practical effectiveness in backtesting.

Problem

• high dimensional mean test
• shrinking random variables
• backtesting
• value-at-risk
• mean deviation

Innovation

• high-dimensional mean test
• shrinking random variables
• multiplier bootstrap
• backtesting
• overlapping subsets