Statistica Sinica 25 (2015),
Abstract: In statistical analysis of functional magnetic resonance imaging
(fMRI), dealing with the temporal correlation is a major challenge in assessing
changes within voxels. This paper aims to address this issue by considering
a semiparametric model for single-voxel fMRI. For the error process in
the semiparametric model with autocorrelation matrix R, we adopt the
difference-based method to construct a banded estimate of R, and propose
a refined estimate
*-1 of R-1. Under mild regularity conditions, we
establish consistency of
and
* with explicit convergence rates. We also
demonstrate convergence of
*-1 in mean square under the L∞ norm, though
this convergence property does not hold for
-1. Data-driven procedures for
choosing the banding parameter and refining the estimate are developed, and
simulation studies reveal their satisfactory performance. Numerical results
suggest that
*-1 performs well when applied to the semiparametric test
statistics for detecting brain activity.
Key words and phrases: Autocorrelation matrix, difference-based method, fMRI, inverse, semiparametric model.