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.