LOCAL ASYMPTOTICS FOR POLYNOMIAL SPLINE REGRESSION

Jianhua Z.\ Huang

Abstract: In this paper we develop a general theory of local asymptotics for 
least squares estimates over polynomial spline spaces in a regression problem.
The polynomial spline spaces we consider include univariate splines, tensor 
product splines, and bivariate or multivariate splines on triangulations.
We establish asymptotic normality of the estimate and study the magnitude of 
the bias due to spline approximation. 
The asymptotic normality holds uniformly over the points
where the regression function is to be estimated and uniformly over a broad
class of design densities, error distributions and regression functions. 
The bias is controlled by the minimum $L_\infty$ norm of the error when the 
target regression function is approximated by a function in the polynomial 
spline space that is used to define the estimate. The control of bias relies
on the stability in $L_\infty$ norm of $L_2$ projections onto polynomial 
spline spaces. Asymptotic normality of least squares estimates over
polynomial or trigonometric polynomial spaces is also treated by the general
theory. In addition, a preliminary analysis of additive models is provided.