Fast and Accurate Inference for the Smoothing Parameter in Semiparametric Models

We adapt the method developed in Paige, Trindade, and Fernando (2009) in order to make approximate inference on optimal smoothing parameters for penalized spline, and partially linear models. The method is akin to a parametric bootstrap where Monte Carlo simulation is replaced by saddlepoint approximation, and is applicable whenever the underlying estimator can be expressed as the root of an estimating equation that is a quadratic form in normal random variables. This is the case under a variety of common optimality criteria such as ML, REML, GCV, and AIC. We apply the method to some well-known datasets in the literature, and find that under the ML and REML criteria it delivers a performance that is nearly exact, with computational speeds that are at least an order of magnitude faster than exact methods. Perhaps most importantly, the proposed method also offers a computationally feasible alternative where no known exact methods exist, e.g. GCV and AIC.