Confidence Intervals for Nonparametric Curve Estimates

Abstract

Numerous nonparametric regression methods exist that yield consistent estimators of function curves. Often, one is also interested in constructing confidence intervals for the unknown function. When a function estimate is based on a single global smoothing parameter the resulting confidence intervals may hold their desired confidence level 1 – α on average but because bias in nonparametric estimation is not uniform, they do not hold the desired level uniformly at all design points. Most research in this area has focused on mean squared error properties of the estimator, for example MISE, itself a global measure. In addition, measures like MISE are one step removed from the practical issue of coverage probability. Recent work that focuses on coverage probability has considered only coverage in an average sense, ignoring the important issue of uniformity of coverage across the design space. To deal with this problem, a new estimator is developed which uses a local cross-validation criterion (LCV) to determine a separate smoothing parameter for each design point. The local smoothing parameters are then used to compute the point estimators of the regression curve and the corresponding pointwise confidence intervals. Incorporation of local information through the new method is shown, via Monte Carlo simulation, to yield more uniformly valid pointwise confidence intervals for nonparametric regression curves. Diagnostic plots are developed (Breakout Plots) to visually inspect the degree of uniformity of coverage of the confidence intervals. The approach, here applied to cubic smoothing splines, easily generalizes to many other 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 improved coverage probabilities through reliable pointwise confidence intervals.

KEY WORDS:

• Coverage probability
• Local cross-validation
• Nonparametric regression
• Pointwise confidence intervals
• Smoothing splines