Finite sample t-tests for high-dimensional means.

Area of Science:

• Statistics
• Statistical Inference
• High-Dimensional Data Analysis

Background:

• Asymptotic statistical tests struggle with accurate Type I error rates when sample sizes are small.
• Traditional methods require diverging sample sizes, limiting their application in fields with limited data.

Purpose of the Study:

• To develop novel statistical tests for mean vectors in high-dimensional settings with very small sample sizes.
• To establish tests that maintain accurate Type I error rates under these challenging conditions.

Main Methods:

• Proposed one-sample, two-sample, and ANOVA tests for mean vectors.
• Established asymptotic -distributions for proposed -statistics, requiring only diverging dimensionality and fixed, small sample sizes (≥3).
• Developed nonparametric tests applicable to both normally distributed and heavy-tailed data.

Main Results:

• The proposed tests maintain accurate Type I error rates across various sample sizes and data dimensionalities.
• Theoretical results are confirmed through simulation studies.
• Demonstrated practical application on an fMRI dataset.

Conclusions:

• The developed nonparametric tests offer a robust solution for hypothesis testing with high-dimensional, small-sample data.
• These methods overcome limitations of traditional asymptotic tests, expanding their applicability.
• The tests provide reliable inference in scenarios previously considered intractable.