Journal of Data Science 13(2015), 443-456 FUNCTIONAL VARYING COEFFICIENT MODEL WITH TIME- INDEPENDENT COVARIATE AND LONGITUDINAL RESPONSE Behdad Mostafaiy and Mohammad Reza Faridrohani Department of Statistics, Shahid Beheshti University Abstract: In this paper, we consider functional varying coefficient model in present of a time invariant covariate for sparse longitudinal data contaminated with some measurement errors. We propose a regularization method to estimate the slope function based on a reproducing kernel Hilbert space approach. As we will see, our procedure is easy to implement. Our simulation results show that the procedure performs well, especially when either sampling frequency or sample size increases. Applications of our method are illustrated in an analysis of a longitudinal CD4+ count dataset from an HIV study. Key words: CD4+ count, functional varying coefficient model, longitudinal data analysis, reproducing kernel Hilbert space, sparsity 1. Introduction In functional data, unlike multivariate data, the observations are naturally curves. In fact they are independent and identically distributed realizations of a stochastic process. See Ramsay and Silverman (2002, 2005) for an overview of methods and applications. See also Ferraty and Vieu (2006) and, Horváth and Kokoszka (2012). In many experiments realizations of involved trajectories are not directly observable. Instead, the observed data are obtained at discrete location points. These type of data are usually sparsely and irregularly sampled on random time points and are noise-contaminated. The aforementioned situation often occurs in many longitudinal experiments, for example in the most of biological, biomedical and medical studies. Varying coefficient models which are extension of parametric regression models, became popular after the works of Cleveland et al. (1991) and, Hastie and Tibshirani (1993). These models have been extensionally studied in the literature. Most of existing approach are based on polynomial spline, smoothing splines and local polynomial smoothing. See for example Hoover et al. (1998), Wu et al. (1998), Kauermann and Tutz (1999), Chiang et al. (2001), Wu and Chiang (2000) and Huang et al. (2002, 2004). See Fan and Zhang (2008) to review the collection of approaches and applications up-to that date. In the case of sparse designs and noisy measurement, we refer the readers to Sentürk and Müller (2008), Noh and Park (2010), Sentürk and Müller (2010), Sentürk and Nguyen (2011), Chiou et al. (2012), and Sentürk et al. (2013). In this paper, we consider the following functional varying coefficient model () =  0 () +  0 () + (), ∈ (1)