Nonparametric two-sample tests of high dimensional mean vectors via random integration.

Area of Science:

• Statistics
• Multivariate Analysis
• Hypothesis Testing

Background:

• Comparing means of two samples is a core statistical inference task.
• Existing methods (sum-of-squares, supremum statistics) have limitations and lack a unified approach.
• These methods often struggle in high-dimensional settings or with specific data structures.

Purpose of the Study:

• To develop a unified statistical framework for testing the equality of means in two samples.
• To extend existing methods, particularly for high-dimensional data, without restrictive assumptions.
• To provide a robust and adaptable testing procedure for diverse statistical scenarios.

Main Methods:

• Utilizing random integration of the difference to construct a new test statistic.
• Developing a general multivariate model to derive asymptotic properties of the test.
• Analyzing the test's performance without explicit constraints on covariance matrices or sparsity.

Main Results:

• The proposed framework unifies and extends numerous existing statistical tests.
• Asymptotic properties are derived under a general multivariate model, independent of dimension-sample size relationships.
• The test demonstrates high power (approaching 1) for weakly dense signals with similar signs and superior asymptotic relative Pitman efficiency.

Conclusions:

• The novel random integration framework offers a powerful and unified approach to testing mean equality.
• It provides flexibility and improved performance, especially in high-dimensional statistical inference.
• Numerical studies and real-world data confirm the practical utility and potential of the proposed method.