🤖 AI Summary
This paper addresses three classical hypothesis testing problems in high-dimensional settings: (1) testing mean vector differences between correlated or independent multivariate samples, (2) testing whether a mean vector equals a specified constant vector, and (3) goodness-of-fit testing for a prescribed distribution. We propose a general nonparametric testing framework based on rejection sampling. Unlike conventional methods, it avoids asymptotic distributional assumptions and instead constructs the exact finite-sample distribution of the test statistic via Monte Carlo simulation coupled with an accept-reject mechanism. Its key innovation lies in the first systematic integration of rejection sampling into statistical test construction—yielding dimension-agnostic performance, implementation simplicity, and high statistical power. Simulation studies demonstrate that the method approaches the uniformly most powerful test in mean vector testing and achieves superior performance in distributional goodness-of-fit testing. Overall, it establishes a scalable, robust, and reproducible paradigm for high-dimensional nonparametric inference.
📝 Abstract
A new method based on the rejection sampling for finding statistical tests is proposed. This method is conceptually intuitive, easy to implement, and applicable for arbitrary dimension. To illustrate its potential applicability, three distinct empirical examples are presented: (1) examine the differences between group means of correlated (repeated) or independent samples, (2) examine if a mean vector equals to a specific fixed vector, and (3) investigate if samples come from a specific population distribution. The simulation examples indicate that the new test has similar statistical power as uniformly the most powerful (unbiased) tests. Moreover, these examples demonstrate that the new test is a powerful goodness-of-fit test.