On likelihood ratio testing for penalized splines

Penalized spline regression using a mixed effects representation is one of the most popular nonparametric regression tools to estimate an unknown regression function $f(\cdot )$ . In this context testing for polynomial regression against a general alternative is equivalent to testing for a zero variance component. In this paper, we fill the gap between different published null distributions of the corresponding restricted likelihood ratio test under different assumptions. We show that: (1) the asymptotic scenario is determined by the choice of the penalty and not by the choice of the spline basis or number of knots; (2) non-standard asymptotic results correspond to common penalized spline penalties on derivatives of $f(\cdot )$ , which ensure good power properties; and (3) standard asymptotic results correspond to penalized spline penalties on $f(\cdot )$ itself, which lead to sizeable power losses under smooth alternatives. We provide simple and easy to use guidelines for the restricted likelihood ratio test in this context.