Coverage errors for Student's t confidence intervals comparable to those in Hall (1988) (2501.07645v1)

Abstract: Table 1 of Hall (1988) contains asymptotic coverage error formulas for some nonparametric approximate 95% confidence intervals for the mean based on $n$ IID samples. The table includes an entry for an interval based on the central limit theorem using Gaussian quantiles and the Gaussian maximum likelihood variance estimate. It is missing an entry for the very widely used Student $t$ confidence intervals. This note makes a mild numerical correction for the Gaussian entry and provides an entry for the Student $t$ intervals. For skewness $\gamma$ and kurtosis $\kappa$, the corrected Gaussian formula is $0.14\kappa -2.16\gamma^2-3.42$ and the formula for the $t$ intervals is $0.14\kappa -2.16\gamma^2$. The impetus to revisit this estimate arose from the surprisingly robust performance of Student's t statistic in randomized quasi-Monte Carlo sampling.

• The paper introduces a novel formula for Student's t interval coverage errors, computed as 0.14κ - 2.16γ².
• It employs asymptotic expansions and numerical refinements to extend Hall’s work, particularly in randomized quasi-Monte Carlo sampling.
• The findings guide researchers in selecting robust confidence intervals for nonparametric, high-dimensional, and skewed data analyses.

Analysis of "Coverage Errors for Student's $t$ Confidence Intervals Comparable to Those in Hall (1988)"

The paper "Coverage errors for Student's $t$ confidence intervals comparable to those in Hall (1988)" by Art B. Owen provides a critical exploration of coverage error formulas within the context of nonparametric confidence interval methods for estimating the mean of a random variable. Specifically, this paper revisits the well-recognized findings of Hall (1988), augmenting prior work with pertinent details regarding the widely applied Student's $t$ confidence intervals.

Overview and Key Contributions

The paper is structured to fill a notable gap in the original Hall's table of asymptotic coverage error formulas for 95% confidence intervals. Hall's work, while extensive, lacked coverage on Student's $t$ confidence intervals, despite their prominent usage in statistical analyses. In addition, this paper provides a numerical refinement for the Gaussian entry present in Hall's table.

Central to this exploration is the development of a formula for the Student's $t$ intervals, shown to be $0.14\kappa - 2.16\gamma^2$, where $\kappa$ represents kurtosis and $\gamma$ denotes skewness. This contrasts with the corrected Gaussian formula $0.14\kappa - 2.16\gamma^2 - 3.42$, providing a comprehensive understanding of the statistical properties underlying these methodologies.

Methodological Insights

The research draws its impetus from recent findings on the robustness of Student's $t$ statistic, particularly in scenarios involving randomized quasi-Monte Carlo (RQMC) sampling. The surprising performance of the Student's $t$ intervals in these contexts serves as a compelling case for revisiting coverage error estimations. The paper critically employs asymptotic expansions and derives the necessary coverage error formulas, employing a modest correction to Hall's Gaussian interval formula along the way.

Implications and Future Prospects

The inclusion of Student's $t$ intervals alongside Gaussian and bootstrap intervals in the context of Hall's earlier work paves the way for a more nuanced understanding of interval estimation approaches. By elucidating the statistical underpinnings that dictate coverage accuracy, this paper offers practitioners the necessary tools to select the most appropriate confidence interval method in empirical research.

From a theoretical standpoint, these results reinforce the reliability of traditional studentized approaches in contexts where data deviate from normative assumptions, such as in RQMC sampling. As future advancements in statistical methodologies unfold, the foundational work presented here can inform further refinement of interval estimation techniques, particularly in addressing anomalies encountered in high-dimensional modeling or datasets with complex dependency structures.

In summary, this paper contributes to statistical literature by conclusively extending Hall's 1988 formational insights with a comprehensive coverage analysis of Student's $t$ intervals, reaffirming the method's efficacy through robust mathematical exposition and numerical validation. As such, this work stands to significantly aid statisticians in confidently employing these methodologies across various application domains.