Kernel density estimates are frequently used, based on a second order kernel. Thus, the bias inherent to the estimates has an order of O(<i>h</i>²_<i>n</i>). In this note, a method of correcting the bias in the kernel density estimates is provided, which reduces the bias to a smaller order. Effectively, this method produces a higher order kernel based on a second order kernel. For a kernel function <i>K</i>, the functions W_k(x)=^k-1₁₌₀(^k_l+1)x^lK^(l)(x)/l! and [1/<i>K</i>^{(<i>k</i> 1)}(<i>x</i>)/<i>x</i> d <i>x</i>]<i>K</i>^{(<i>k</i> 1)}(<i>x</i>)/<i>x</i> are kernels of order <i>k</i>, under some mild conditions.

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In the event-related functional magnetic resonance imaging (fMRI) data analysis, there is an extensive interest in accurately and robustly estimating the hemodynamic response function (HRF) and its associated statistics (eg, the magnitude and duration of the activation). Most methods to date are developed in the time domain and they have utilized almost exclusively the temporal information of fMRI data without accounting for the spatial information. The aim of this paper is to develop a multiscale adaptive smoothing model (MASM) in the frequency domain by integrating the spatial and temporal information to adaptively and accurately estimate HRFs pertaining to each stimulus sequence across all voxels in a three-dimensional (3D) volume. We use two sets of simulation studies and a real data set to examine the finite sample performance of MASM in estimating HRFs. Our real and simulated data analyses confirm

Semiparametric regression models are very useful for longitudinal data analysis. The complexity of semiparametric models and the structure of longitudinal data pose new challenges to parametric inferences and model selection that frequently arise from longitudinal data analysis. In this article, two new approaches are proposed for estimating the regression coefficients in a semiparametric model. The asymptotic normality of the resulting estimators is established. An innovative class of variable selection procedures is proposed to select significant variables in the semiparametric models. The proposed procedures are distinguished from others in that they simultaneously select significant variables and estimate unknown parameters. Rates of convergence of the resulting estimators are established. With a proper choice of regularization parameters and penalty functions, the proposed variable selection procedures