A Sieve Bootstrap Two-Sample t-Test Under Serial Correlation

Abstract

The classical two-sample t-test assumes that observations are independent. A violation of this assumption could lead to unreliable or even erroneous conclusions. However, in many biological studies, data are recorded over time and hence exhibit serial correlation. In order to take such temporal dependence into account, we suggest applying the sieve bootstrap method to generate replications of the observed data and then using these proxy-dependent processes to construct the empirical distribution for the t-statistic. The proposed method is fast, distribution-free, and well approximates the nominal significance level. We illustrate our approach in application to detection problem of brain activity in functional magnetic resonance imaging (fMRI) and a longitudinal study of weight growth in rats.

Key Words:

• Dependence effect
• Hypothesis testing
• Resampling
• Serial correlation
• Sieve bootstrap
• Two-sample t-test

ACKNOWLEDGMENTS

We thank Drs. Dietmar Cordes, Joel Dubin, Joseph Gastwirth, Frank Konietschke, Mary Thompson, and Nikita Khromov-Borisov for providing the data and stimulating discussions and advice. The research of Yulia R. Gel was supported by grants from the National Science and Engineering Research Council of Canada. Bei Chen was partially supported by the WatRisk and MITACS scholarships. This work was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET: http://www.sharcnet.ca).

Notes

The NIG distribution is solely determined by its four parameters α, β, μ, and δ, corresponding to shape, skewness, location, and scale. The mean, variance, skewness, and kurtosis of NIG are respectively defined by μ + βδ/γ, δα²/γ³, , and 3(1 + 4β²/α²)/δγ, where . Various combinations of these four parameters cover a broad range of continuous distribution of different shapes. Due to its flexibility, NIG is often used in modeling heavy-tailed financial processes (for more detailed discussion see, e.g., Barndorff-Nielsen, 1997; and references therein).

The Tukey family of CN distribution has the cumulative distribution function (cdf) F _{CN λ, α} (x) = (1 − α)Φ(x) + αΦ(x/λ), where Φ(x) is the cdf of standard normal distribution, 0 ≤ α ≤1 and λ > 0 are constants (Gleason, 1993; Tukey, 1960).

Note: A nominal level is α = 0.05. Number of Monte Carlo simulations is 1000 and number of bootstrap replications of each sample is 1000.

Note: A nominal level is α = 0.05. Number of Monte Carlo simulations is 1000 and number of bootstrap replications of each sample is 1000.

Note: A nominal level is α = 0.05. Number of Monte Carlo simulations is 1000 and number of bootstrap replications of each sample is 1000.

The data are kindly provided by Professor Dietmar Cordes, School of Medicine, University of Colorado, Denver.