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

Liujun Chen, Chen Zhou

TL;DR

The paper addresses high-dimensional mean testing in settings where individual means shrink to zero as the sample size grows, a scenario that breaks standard marginal normality. It introduces a subsets-based pooling approach that constructs $d$ overlapping subsets of size $q$ and uses a max-type statistic $\mathcal{M}$ together with multiplier bootstrap to recover the full null $\mu_j=0$ for all $j$ while accommodating dependence across dimensions. Theoretical results establish size control and power under rho-mixing and Lyapunov-type conditions, and the framework is applied to VaR backtesting via both validation and comparative tests, with simulations and a real-data study demonstrating improved performance in high dimensions. The method provides a flexible, robust tool for tail-risk validation across many assets or risk factors, enabling more reliable backtest-based judgments in finance and climate contexts.

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.

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

TL;DR

The paper addresses high-dimensional mean testing in settings where individual means shrink to zero as the sample size grows, a scenario that breaks standard marginal normality. It introduces a subsets-based pooling approach that constructs overlapping subsets of size and uses a max-type statistic together with multiplier bootstrap to recover the full null for all while accommodating dependence across dimensions. Theoretical results establish size control and power under rho-mixing and Lyapunov-type conditions, and the framework is applied to VaR backtesting via both validation and comparative tests, with simulations and a real-data study demonstrating improved performance in high dimensions. The method provides a flexible, robust tool for tail-risk validation across many assets or risk factors, enabling more reliable backtest-based judgments in finance and climate contexts.

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.
Paper Structure (14 sections, 8 theorems, 140 equations, 3 figures, 1 table)

This paper contains 14 sections, 8 theorems, 140 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methodology
  3. A naive test
  4. The subsets-based test
  5. Application: Backtesting for VaR
  6. Validation Backtest
  7. Comparative Backtest
  8. Simulation
  9. Real data Application
  10. Proofs
  11. Proof of Theorem \ref{['Theorem:naive']}
  12. Proofs of Theorem \ref{['theorem:size:improved']}
  13. Proof of Theorem \ref{['theorem:power:improved']}
  14. Proofs of Other results

Key Result

Theorem 1

Assume that Conditions assum:mixing, assum:var:low:bound and assum:lyaponov hold. Then, under $H_0$, as $n\to\infty$,

Figures (3)

  • Figure 1: The empirical size (left) and power (right) of subsets-based pooling test, marginal test, and naive test over different levels of $q$.
  • Figure 2: The empirical size (left) and power (right) of subsets-based pooling test, marginal test, and naive test over different levels of $d$.
  • Figure 3: Heatmap of the upper tail dependence coefficients for the filtered residuals in the first training period for the S&P constituents. Each row and column corresponds to a stock, and stocks are grouped by the Global Industry Classification Standard (GICS) employed by Standard and Poor's: Financials (FIN), Energy (ENE), Utilities (UTIL), Information Technology (IT), Health Care (HC), Consumer Discretionary (CD), Consumer Staples (CS), Industrials (IND), Communication Services (COM), and Real Estate (RE).

Theorems & Definitions (16)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • proof : Proof of Theorem \ref{['Theorem:naive']}
  • Lemma S2
  • proof : Proof of Lemma \ref{['lemma:upper:bound:Y']}
  • Lemma S3
  • proof : Proof of Lemma \ref{['lemma:fourth:moment']}
  • Lemma S4
  • ...and 6 more