Testing for cubic smoothing splines under dependent data

In most research on smoothing splines the focus has been on estimation, while inference, especially hypothesis testing, has received less attention. By defining design matrices for fixed and random effects and the structure of the covariance matrices of random errors in an appropriate way, the cubic smoothing spline admits a mixed model formulation, which places this nonparametric smoother firmly in a parametric setting. Thus nonlinear curves can be included with random effects and random coefficients. The smoothing parameter is the ratio of the random-coefficient and error variances and tests for linear regression reduce to tests for zero random-coefficient variances. We propose an exact F-test for the situation and investigate its performance in a real pine stem data set and by simulation experiments. Under certain conditions the suggested methods can also be applied when the data are dependent.