🤖 AI Summary
This study addresses the limited power of traditional goodness-of-fit tests in high-dimensional settings by proposing a novel approach based on the joint distribution of multiple samples. The method employs principal component analysis for dimensionality reduction and constructs confidence sets using k-nearest neighbor estimates of high-density regions, effectively integrating information from order statistics, empirical distribution function values, and moments. It is further extended to the two-sample case. To enhance test power, the approach incorporates non-uniform transformations and probability integral transforms, with inference carried out via permutation testing. Simulation studies demonstrate that the proposed method outperforms or matches existing classical and graphical tests across a range of alternative hypotheses, substantially improving both power and applicability in high-dimensional scenarios.
📝 Abstract
We propose goodness-of-fit tests based on simulated confidence sets for joint distributions of multiple sample statistics, focusing on absolutely continuous null distributions with known parameters. One class of tests uses hyperrectangular confidence sets for principal components of order statistics and related statistic vectors. Extending earlier work on horizontal and vertical confidence bands for cumulative distribution functions, these tests are compared with some classical, Zhang, and related graphical tests. Simulations show that the proposed procedures are competitive with, and often more powerful than, existing methods. We also study the geometry of principal-component-based statistics; under a normal null distribution, the first principal component corresponds to the sample mean, while the second is related to a linear analogue of variance.
A second class of tests uses confidence sets of arbitrary shape constructed through highest density regions. Unlike earlier kernel-density-based approaches, we use a k-nearest-neighbor method for detecting highest density regions, which is better suited to higher-dimensional statistic vectors. We study tests based on order statistics, empirical distribution function values, moments, and combinations of classical goodness-of-fit statistics. The resulting procedures are powerful against a wide range of alternatives.
We also outline a two-sample extension via permutation tests based on joint distributions of several statistics and compare moment-based versions with energy-distance permutation tests. Finally, we discuss transformations other than the probability integral transform, showing that mapping data to another target distribution, such as the standard normal, can be advantageous when powerful tests are available for that distribution.