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

TL;DR

This paper introduces a high-dimensional mean testing framework for shrinking random variables, enhancing backtesting of value-at-risk by overcoming limitations of traditional methods in high-dimensional settings.

Contribution

It develops a novel pooling-based test statistic with theoretical guarantees and a bootstrap procedure, improving validation in high-dimensional financial risk models.

Findings

Superior performance in simulations for high-dimensional data

Effective in backtesting value-at-risk with real data

Theoretically valid even when marginal normality fails

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.