Partial Informative Normal and Bayesian Smoothing Splines

• Activity Number: 681
• Type: Topic Contributed
• Date/Time: Thursday, August 8, 2013 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Sifan Liu*+ and Dongchu Sun
• Companies: University of Missouri - Columbia and University of Missouri
• Keywords: Partial Informative Normal ; Spline smoothing ; Linear mixed model ; Bayesian estimator ; Non-parametric regression ; Semi-parametric model

Abstract

There is a well-known Bayesian interpretation for function estimation by spline smoothing using a limit of proper normal priors. This limiting prior has the same form with Partial Informative Normal (PIN), which was introduced in Sun et al. (1999). In this talk, we show that, under certain conditions, the linear transformation of PIN random variable, the linear combination of independent PIN random variables and the sum of a PIN random variable and a linear transformation of a random variable with constant prior, all follow some PIN distribution. We show that a normal semi-parametric model can be expressed in a normal linear mixed model associated with the third case of PIN's by adding a linear component with constant prior to the common smoothing spline problem. We derive necessary and sufficient conditions for the propriety of the posterior for both the linear parametric component and the nonparametric smooth component, while assuming objective priors on the variance of noise and the noise-signal ratio. Numerical examples are given for illustration.