title: Stochastic dynamic models and Chebyshev splines.
abstract: In this article, we establish a connection between a stochastic dynamic model (SDM) driven by a linear stochastic differential equation (SDE) and a Chebyshev spline, which enables researchers to borrow strength across fields both theoretically and numerically. We construct a differential operator for the penalty function and develop a reproducing kernel Hilbert space (RKHS) induced by the SDM and the Chebyshev spline. The general form of the linear SDE allows us to extend the well-known connection between an integrated Brownian motion and a polynomial spline to a connection between more complex diffusion processes and Chebyshev splines. One interesting special case is connection between an integrated Ornstein-Uhlenbeck process and an exponential spline. We use two real data sets to illustrate the integrated Ornstein-Uhlenbeck process model and exponential spline model and show their estimates are almost identical.
authors.affiliation: Biostatistics and Bioinformatics Branch, Division of Intramural Population Health Research, Eunice Kennedy Shriver National Institute of Child Health and Human Development, National Institutes of Health, Rockville, MD 20852, U.S.A.
authors.forename: Ruzong
authors.initials: R
authors.lastname: Fan
authors.affiliation: Biostatistics Branch, Division of Cancer Epidemiology and Genetics, National Cancer Institute, National Institutes of Health, Rockville, MD 20850, U.S.A.
authors.forename: Bin
authors.initials: B
authors.lastname: Zhu
authors.affiliation: Department of Statistics & Applied Probability, University of California-Santa Barbara, Santa Barbara, CA 93106, U.S.A.
authors.forename: Yuedong
authors.initials: Y
authors.lastname: Wang
keywords: Brownian motion; Ornstein–Uhlenbeck process; reproducing kernel Hilbert space; smoothing splines; stochastic differential equations