[Paper Review] Robustness and accuracy of methods for high dimensional data analysis based on Student's t statistic

Aurore Delaigle, Peter Hall+1 more|arXiv (Cornell University)|Jan 22, 2010

Advanced Statistical Methods and Models8 citations

TL;DR

This paper establishes the robustness and second-order accuracy of Student's t-statistic and its bootstrap approximation in high-dimensional settings, particularly under heavy-tailed distributions and sparse signals. It demonstrates that the bootstrap effectively corrects skewness in extreme tails, outperforming normal and t-distribution approximations for tail probability estimation in p ≫ n problems.

ABSTRACT

Student's $t$ statistic is finding applications today that were never envisaged when it was introduced more than a century ago. Many of these applications rely on properties, for example robustness against heavy tailed sampling distributions, that were not explicitly considered until relatively recently. In this paper we explore these features of the $t$ statistic in the context of its application to very high dimensional problems, including feature selection and ranking, highly multiple hypothesis testing, and sparse, high dimensional signal detection. Robustness properties of the $t$-ratio are highlighted, and it is established that those properties are preserved under applications of the bootstrap. In particular, bootstrap methods correct for skewness, and therefore lead to second-order accuracy, even in the extreme tails. Indeed, it is shown that the bootstrap, and also the more popular but less accurate $t$-distribution and normal approximations, are more effective in the tails than towards the middle of the distribution. These properties motivate new methods, for example bootstrap-based techniques for signal detection, that confine attention to the significant tail of a statistic.

Motivation & Objective

• To investigate the robustness and accuracy of t-statistic-based methods in high-dimensional data with p ≫ n.
• To examine the performance of bootstrap, t-distribution, and normal approximations in estimating tail probabilities under heavy-tailed sampling distributions.
• To develop and validate bootstrap-based methods for signal detection and multiple hypothesis testing in sparse, high-dimensional settings.
• To quantify the convergence rates and higher-order accuracy of Studentized statistics under weak moment conditions.
• To establish theoretical guarantees for the higher criticism test using Studentized statistics in sparse signal detection.

Proposed method

• Analyzes moderate and large deviations of the Studentized t-statistic under weak moment assumptions, including only slightly finite second moments.
• Uses the bootstrap to approximate the sampling distribution of the t-statistic, showing it corrects skewness and achieves second-order accuracy in extreme tails.
• Applies large deviation and moderate deviation probability theory to derive bounds on tail probabilities for the t-statistic and its bootstrap approximation.
• Derives asymptotic expansions for the higher criticism test statistic under both null and alternative hypotheses using Studentized t-statistics.
• Employs Mill's ratio and extreme value approximations to analyze the behavior of tail probabilities in high-dimensional regimes.
• Re-parametrizes tail quantiles using s_n(q) = √(2q log p) to study the maximum signal detection power across different tail regions.

Experimental results

Research questions

• RQ1 How does the t-statistic perform in terms of robustness and accuracy when the underlying distribution has heavy tails and only a few finite moments?
• RQ2 To what extent does the bootstrap improve the accuracy of tail probability estimation for the t-statistic, especially in the extreme tails?
• RQ3 Can bootstrap-based methods achieve second-order accuracy in high-dimensional settings where classical approximations fail?
• RQ4 What is the detection boundary for sparse signals when using Studentized statistics in high-dimensional multiple testing?
• RQ5 How does the performance of higher criticism tests based on t-statistics compare to classical methods under heavy-tailed and sparse signal models?

Key findings

• The t-statistic exhibits robustness against heavy-tailed distributions even when only the second moment is finite, with convergence to normality faster than the unstandardized mean.
• Bootstrap approximations to the t-statistic achieve second-order accuracy, particularly in the tails, by correcting for skewness that plagues normal and t-distribution approximations.
• The bootstrap is more effective in estimating tail probabilities than normal or t-distribution approximations, especially when exceedance probabilities are exponentially or polynomially small.
• In sparse signal detection, the higher criticism test based on the t-statistic achieves optimal detection power when the signal strength and sparsity satisfy certain threshold conditions.
• The maximum signal detection power is determined by the function π(q, β, r), which governs the exponent of the tail probability, with optimal detection occurring at specific values of q depending on β and r.
• Under the alternative hypothesis, the higher criticism statistic ˜hc_n,α grows as L_p times a term involving the survival function of the standard normal, with the maximum over α in the tail region determining detection capability.