On likelihood ratio testing for penalized splines

摘要

Penalized spline regression using a mixed effects representation is one of the most popular nonparametric regression tools to estimate an unknown regression function f(•). 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(•), which ensure good power properties; and (3) standard asymptotic results correspond to penalized spline penalties on f(•) 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.