One of the primary goals in analyzing fMRI data is to estimate the Hemodynamic Response Function (HRF), which is a large-dimensional parameter vector possessing some form of sparsity. This paper introduces a varyingdimensional model for the HRF, and develops novel regularization methods for estimating the HRF from fMRI time series via incorporating the sparsity feature. Particularly, we present three types of penalty choice methods: the Lasso, the adaptive Lasso and the SCAD. Simulation studies

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... demonstrate the advantages of regularization methods, in terms of sparsity recovery, over conventional non-regularized approaches which restrict the HRF to be fixed low dimensional without capturing the sparsity structure. We illustrate the regularized methods for estimating the HRF using a real fMRI data set and compare with results offered by a popular imaging analysis tool AFNI.