Statistica Sinica 19 (2009), 325-342

POLYNOMIAL SPLINE CONFIDENCE BANDS FOR

REGRESSION CURVES

Jing Wang and Lijian Yang

University of Illinois at Chicago and Michigan State University

Abstract: Asymptotically exact and conservative confidence bands are obtained for a nonparametric regression function, using piecewise constant and piecewise linear spline estimation, respectively. Compared to the pointwise confidence interval of Huang (2003), the confidence bands are inflated by a factor proportional to $\left\{ \log \left( n\right) \right\} ^{1/2}$, with the same width order as the Nadaraya-Watson bands of Härdle (1989), and the local polynomial bands of Xia (1998) and Claeskens and Van Keilegom (2003). Simulation experiments corroborate the asymptotic theory. The linear spline band has been used to identify an appropriate polynomial trend for fossil data.

Key words and phrases: Brownian bridge, B spline, knots, nonparametric regression, quantile transformation.