Statistical Analysis of Diffusion Tensors in Diffusion-weighted Magnetic Resonance Image Data

In this technical report, we give detailed information about how to establish asymptotic theory for one-step weighted least-squares estimates of tensors, estimated eigenvalues and eigenvectors, and pseudo-likelihood ratio statistics. We establish the strong convergence rate and asymptotic normality for the one-step weighted least-squares estimates of tensors. We derive the first-order and second-order expansions of the eigenvalues and eigenvectors of the estimated diffusion tensors. We also derive the asymptotic distributions of pseudolikelihood ratio statistics under the null hypotheses to classify tensor morphologies. ∗H. Zhu is Associate Professor of Biostatistics (E-mail: [email protected]), Department of Biostatistics and Biomedical Research Imaging Center, and J. G. Ibrahim is Alumni Distinguished Professor of Biostatistics (E-mail: [email protected]), Department of Biostatistics, University of North Carolina at Chapel Hill, NC 27599-7420. H. Zhang is Professor of Biostatistics (E-mail: [email protected]), Department of Epidemiology and Public Health, Yale University School of Medicine, New Haven, CT 06520-8034. B. Peterson is Professor of Psychiatry (E-mail: [email protected]), Department of Psychiatry, Columbia University Medical Center and the New York State Psychiatric Institute. We thank the Editor, the Associate Editor, and two anonymous referees for valuable suggestions, which greatly helped to improve our presentation. Thanks to Dr. Jason Royal for his invaluable editorial assistance. This work was supported in part by NSF grant SES-06-43663 to Dr. Zhu, NIDA grants DA016750 and DA017713 to Dr. Zhang, NIDA grant DA017820 and NIMH grants MH068318 and K02-74677 to Dr. Peterson, NIH grants GM 70335 and CA 74015 to Dr. Ibrahim, as well as by the Suzanne Crosby Murphy Endowment at Columbia University Medical Center, and by the Thomas D. Klingenstein and Nancy D. Perlman Family Fund.