**Alex Trindade **

Texas Tech University

A fast and accurate method of confidence interval construction for the smoothing parameter in penalized spline and partially linear models is proposed. The method is akin to a parametric percentile bootstrap where Monte-Carlo simulation is replaced by saddlepoint approximation, and can therefore be viewed as an approximate bootstrap. It is applicable in a quite general setting, requiring only that the underlying estimator be the root of an estimating equation that is a quadratic form in normal random variables. This is the case under a variety of optimality criteria such as those commonly denoted by ML, REML, GCV, and AIC. Simulations studies reveal that under the ML and REML criteria, the method delivers a near-exact performance with computational speeds that are an order of magnitude faster than existing exact methods, and two orders of magnitude faster than a classical bootstrap. Perhaps most importantly, the proposed method also offers a computationally feasible alternative when no known exact or asymptotic methods exist, e.g. GCV and AIC. An application is illustrated by applying the methodology to the well-known fossil data. Giving a range of plausible smooths in this instance can help answer questions about the statistical significance of apparent features in the data.