Test of Significance Based on Wavelet Thresholding and Neyman's Truncation

Abstract

Traditional nonparametric tests, such as the Kolmogorov—Smirnov test and the Cramér—Von Mises test, are based on the empirical distribution functions. Although these tests possess root-n consistency, they effectively use only information contained in the low frequencies. This leads to low power in detecting fine features such as sharp and short aberrants as well as global features such as high-frequency alternations. The drawback can be repaired via smoothing-based test statistics. In this article we propose two such kind of test statistics based on the wavelet thresholding and the Neyman truncation. We provide extensive evidence to demonstrate that the proposed tests have higher power in detecting sharp peaks and high frequency alternations, while maintaining the same capability in detecting smooth alternative densities as the traditional tests. Similar conclusions can be made for two-sample nonparametric tests of distribution functions. In that case, the traditional linear rank tests such as the Wilcoxon test and the Fisher—Yates test have low power in detecting two nearby densities where one has local features or contains high-frequency components, because these procedures are essentially testing the uniform distribution based on the sample mean of rank statistics. In contrast, the proposed tests use more fully the sampling information and have better ability in detecting subtle features.

Key Words:

• Adaptive Neyman test
• Goodness-of-fit
• Hard-thresholding parameter
• Soft-thresholding parameter
• Two-sample test
• Wavelet thresholding