Confidence Intervals for the Estimated Smoothing Parameter in Non Parametric Regression

Most nonparametric regression estimates smooth the observed data in the process of generating a suitable estimate. The resulting curve estimate is sensitive to the amount of smoothing introduced and it is important to examine the estimate over a range of smoothing choices. 'One way of specifying such a range is to estimate the optimal value of the smoothing parameter (bandwidth) with respect to some loss function and then report a confidence interval for this parameter estimate. This article describes two strategies for constructing such confidence intervals using asymptotic approximations and simulation techniques. Suppose that data is observed from the model Yj;= f(t,l:) + ej; l::;k::;n where f is a function that is twice differentiable and {e,l:} are mean zero, independent, random variables. A cubic smoothing spline estimate of f is considered where the smoothing parameter is chosen using generalized cross-validation. Confidence intervals are constructed for the smoothing parameter that minimizes average squared error using the asymptotic distribution of the cross-validation function and by a version of the bootstrap. Although this bootstrap method involves more computation, it yields confidence intervals that tend to have a shorter width. In general, this second method is easier to implement since it avoids the problem of deriving the complicated formulas for the asymptotic variance. Also for spline estimates one can significantly reduce the amount of computation through a useful orthogonal decomposition of the problem. Although this paper considers cross-validation for choosing the smoothing parameter, one could use these same techniques to construct intervals for other bandwidth selection methods. Subject Classification: (Amos-mos): Primary 62G05 Secondary 62G15