2009/12/09 by Xiao Wang, Wang, Xiao, Jinglai Shen +3
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Image and Signal Denoising Methods #Statistics Theory (math.ST) #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.0912.1824
arxiv created 2009/12/09 · openalex publication_date 2009/12/09 · arxiv updated 2010/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper addresses asymptotic properties of general penalized spline estimators with an arbitrary B-spline degree and an arbitrary order difference penalty. The estimator is approximated by a solution of a linear differential equation subject to suitable boundary conditions. It is shown that, in certain sense, the penalized smoothing corresponds approximately to smoothing by the kernel method. The equivalent kernels for both inner points and boundary points are obtained with the help of Green's functions of the differential equation. Further, the asymptotic normality is established for the estimator at interior points. It is shown that the convergence rate is independent of the degree of the splines, and the number of knots does not affect the asymptotic distribution, provided that it tends to infinity fast enough.