Some Asymptotic Results on Generalized Penalized Spline Smoothing

The paper discusses asymptotic properties of penalized spline smoothing if the spline basis increases with the sample size. The proof is provided in a generalized smoothing model allowing for non-normal responses. The results are extended in two ways. First, assuming the spline coefficients to be a priori normally distributed links the smoothing framework to generalized linear mixed models (GLMM). We consider the asymptotic rates such that Laplace approximation is justified and the resulting fits in the mixed model correspond to penalized spline estimates. Secondly, we make use of a fully Bayesian viewpoint by imposing a priori distribution on all parameters and coefficients. We argue that with the postulated rates at which the spline basis dimension increases with the sample size the posterior distribution of the spline coefficients is approximately normal. The validity of this result is investigated in finite samples by comparing Markov Chain Monte Carlo (MCMC) results with their asymptotic approximation in a simulation study.