High-dimensional bootstrap and asymptotic expansion (2404.05006v3)

Abstract: The recent seminal work of Chernozhukov, Chetverikov and Kato has shown that bootstrap approximation for the maximum of a sum of independent random vectors is justified even when the dimension is much larger than the sample size. In this context, numerical experiments suggest that third-moment match bootstrap approximations would outperform normal approximation even without studentization, but the existing theoretical results cannot explain this phenomenon. In this paper, we first show that Edgeworth expansion, if justified, can give an explanation for this phenomenon. Second, we obtain valid Edgeworth expansions in the high-dimensional setting when the random vectors have Stein kernels. Finally, we prove the second-order accuracy of a double wild bootstrap method in this setting. As a byproduct, we find an interesting blessing of dimensionality phenomenon: The single third-moment match wild bootstrap is already second-order accurate in high-dimensions if the covariance matrix has identical diagonal entries and bounded eigenvalues.

• The paper shows that Edgeworth expansions justify bootstrap approximations that outperform normal methods in high-dimensional settings.
• The paper validates Edgeworth expansions in non-studentized frameworks using Stein kernels to achieve second-order accuracy.
• The paper introduces an innovative double wild bootstrap method that leverages the blessing of dimensionality under specific covariance conditions.

High-Dimensional Bootstrap and Asymptotic Expansion

The paper "High-dimensional bootstrap and asymptotic expansion" by Yuta Koike investigates the problem of bootstrap approximation accuracy in high-dimensional statistical settings, specifically when the dimension of the data is considerably larger than the sample size. The paper builds on the foundational work by Chernozhukov, Chetverikov, and Kato, which demonstrated the validity of Gaussian-type approximations for the maximum of a sum of independent random vectors in high-dimensional contexts.

Key Contributions

1. Edgeworth Expansion Justification: The paper illustrates that if Edgeworth expansions are applicable, they can elucidate the phenomenon where third-moment match bootstrap approximations surpass normal approximations in terms of performance. This is particularly significant in non-studentized scenarios.
2. Validation of Edgeworth Expansions in High Dimensions: The paper establishes valid Edgeworth expansions for high-dimensional settings by leveraging Stein kernels. This is achieved when analyzing random vectors with these kernels, which serve as a pivotal tool for expansions in such settings.
3. Accurate Double Wild Bootstrap Method: An innovative double wild bootstrap method is proposed and shown to possess second-order accuracy in high-dimensional arrangements. This method is crucial for achieving high approximation accuracy without requiring stringent pivot conditions.
4. Blessing of Dimensionality Phenomenon: A surprising finding is that a single third-moment match wild bootstrap can be second-order accurate if the covariance matrix possesses identical diagonal entries and limited eigenvalue bounds.

Implications of the Research

The research has profound implications for both theoretical and practical dimensions of high-dimensional statistics:

• Theoretical Implications: The results highlight the significance of Stein's method and Edgeworth expansions in advancing our understanding of statistical properties in high dimensions. These methodologies offer a more nuanced understanding of coverage probability errors and allow for refined predictions of statistical behavior in complex dimensional setups.
• Practical Implications: Practitioners dealing with high-dimensional data can adopt these results to improve the performance of bootstrap methods without studentization. This enhances the accuracy of inferential procedures in various applications, including genomics, finance, and machine learning, where dimensionality often surpasses sample size.

Future Directions

The paper opens several avenues for future research, including:

• Extending the Edgeworth expansion framework to other types of statistics beyond maxima and exploring its potential in other non-Gaussian settings.
• Investigating the bounds and conditions under which the blessing of dimensionality can be harnessed for other statistical problems.
• Developing computationally efficient algorithms that leverage the theoretical advancements of Stein kernels and Edgeworth expansions in practical applications.

In summary, Koike's work on high-dimensional bootstrap and asymptotic expansion provides substantial progress in the field of high-dimensional statistics, offering new insights and methodologies for addressing the challenges posed by large-dimensional datasets.