Confidence intervals for nonparametric curve estimates: Toward more uniform pointwise coverage

Numerous nonparametric regression methods exist that yield consistent estim ators of function curves. Often, one is also interested in constructing con fidence intervals for the unknown function. When a function estimate is bas ed on a single global smoothing parameter the resulting confidence interval s may hold their desired confidence level 1 - alpha on average but because bias in nonparametric estimation is not uniform, they do not hold the desir ed level uniformly at all design points. Most research in this area has foc used on mean squared error properties of the estimator, for example MISE, i tself a global measure. In addition, measures like MISE are one step remove d from the practical issue of coverage probability. Recent work that focuse s on coverage probability has considered only coverage in an average sense, ignoring the important issue of uniformity of coverage across the design s pace. To deal with this problem, a new estimator is developed which uses a local cross-validation criterion (LCV) to determine a separate smoothing pa rameter for each design point. The local smoothing parameters are then used to compute the point estimators of the regression curve and the correspond ing pointwise confidence intervals. Incorporation of local information thro ugh the new method is shown, via Monte Carlo simulation, to yield more unif ormly valid pointwise confidence intervals for nonparametric regression cur ves. Diagnostic plots are developed (Breakout Plots) to visually inspect th e degree of uniformity of coverage of the confidence intervals. The approac h, here applied to cubic smoothing splines, easily generalizes to many othe r nonparametric regression estimators. The improved curve estimation is not a solely theoretical improvement such as providing an estimator that has a faster EASE convergence rate but shows its worth empirically by yielding i mproved coverage probabilities through reliable pointwise confidence interv als.