Penalized partially linear models using orthonormal wavelet bases with an application to fMRI time series

In this paper, we consider modeling the nonparametric component in partially linear models (PLM) using orthogonal wavelet expansions. We introduce a regularized estimator of the nonparametric part in the wavelet domain. The key innovation here is that the nonparametric part can be efficiently estimated by choosing an appropriate penalty function for which the hard and soft thresholding estimators are particular cases. This avoids excessive bias in estimating the parametric component. We give an efficient estimation algorithm. A large scale simulation study is also conducted to illustrate the finite sample properties of the estimator. The estimator is finally applied to real neurophysiological functional MRI data sets that are suspected to contain both smooth and transient drift features.