Undercoverage of Wavelet-Based Resampling Confidence Intervals

Abstract

The decorrelating property of the discrete wavelet transformation (DWT) appears valuable because one can avoid estimating the correlation structure in the original data space by bootstrap resampling of the DWT. Several authors have shown that the wavestrap approximately retains the correlation structure of observations. However, simply retaining the same correlation structure of original observations does not guarantee enough variation for regression parameter estimators. Our simulation studies show that these wavestraps yield undercoverage of parameters for a simple linear regression for time series data of the type that arise in functional MRI experiments. It is disappointing that the wavestrap does not even provide valid resamples for both white noise sequences and fractional Brownian noise sequences. Thus, the wavestrap method is not completely valid in obtaining resamples related to linear regression analysis and should be used with caution for hypothesis testing as well. The reasons for these undercoverages are also discussed. A parametric bootstrap resampling in the wavelet domain is introduced to offer insight into these previously undiscovered defects in wavestrapping.

Keywords:

• Bootstrap
• Discrete wavelet transform (DWT)
• fMRI
• Inverse discrete wavelet transform (IDWT)
• Permutation tests
• Wavestrap

Mathematics Subject Classification:

• Primary 62G09
• Secondary 62P10

Acknowledgment

The authors would like to thank an associate editor and two referees for their constructive comments and suggestions. This study was supported by the U.S. Army Medical Research and Material Command numbers DAMD17-97-7-7024 and DAMD17-01-1-0741 through a consortium agreement with the University of Texas Southwestern Medical Center at Dallas, U.S. Public Health Service grant MO1-RR00633, and by a grant from the Perot Foundation, Dallas, Texas. The content of this paper does not necessarity reflect the position or the policy of the U.S. government, and no official endorsement should be inferred.