The Annals of Statistics

Local asymptotics for polynomial spline regression

Jianhua Z. Huang

Full-text: Open access

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.

Article information

Source
Ann. Statist., Volume 31, Number 5 (2003), 1600-1635.

Dates
First available in Project Euclid: 9 October 2003

Permanent link to this document
https://projecteuclid.org/euclid.aos/1065705120

Digital Object Identifier
doi:10.1214/aos/1065705120

Mathematical Reviews number (MathSciNet)
MR2012827

Zentralblatt MATH identifier
1042.62035

Subjects
Primary: 62G07: Density estimation
Secondary: 62G20: Asymptotic properties

Keywords
Asymptotic normality least squares nonparametric regression polynomial regression regression spline

Citation

Huang, Jianhua Z. Local asymptotics for polynomial spline regression. Ann. Statist. 31 (2003), no. 5, 1600--1635. doi:10.1214/aos/1065705120. https://projecteuclid.org/euclid.aos/1065705120


Export citation

References

  • Barrow, D. L. and Smith, P. W. (1978). Asymptotic properties of best $L_2[0,1]$ approximation by splines with variable knots. Quart. Appl. Math. 36 293--304.
  • Chen, Z. (1991). Interaction spline models and their convergence rates. Ann. Statist. 19 1855--1868.
  • Chui, C. K. (1988). Multivariate Splines. SIAM, Philadelphia.
  • de Boor, C. (1976). A bound on the $L_\infty$-norm of $L_2$-approximation by splines in terms of a global mesh ratio. Math. Comp. 30 765--771.
  • de Boor, C., Höllig, K. and Riemenschneider, S. (1993). Box Splines. Springer, New York.
  • Descloux, J. (1972). On finite element matrices. SIAM J. Numer. Anal. 9 260--265.
  • DeVore, R. A. and Lorentz, G. G. (1993). Constructive Approximation. Springer, Berlin.
  • Douglas, J., Dupont, T. and Wahlbin, L. (1975). Optimal $L_\infty$ error estimates for Galerkin approximations to solutions of two-point boundary value problems. Math. Comp. 29 475--483.
  • Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Chapman and Hall, London.
  • Gasser, T., Sroka, L. and Jennen-Steinmetz, C. (1986). Residual variance and residual pattern in nonlinear regression. Biometrika 73 625--633.
  • Hall, P., Kay, J. W. and Titterington, D. M. (1990). Asymptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika 77 521--528.
  • Hansen, M. (1994). Extended linear models, multivariate splines, and ANOVA. Ph.D. dissertation, Dept. Statistics, Univ. California, Berkeley.
  • Hansen, M., Kooperberg, C. and Sardy, S. (1998). Triogram models. J. Amer. Statist. Assoc. 93 101--119.
  • Härdle, W. (1990). Applied Nonparametric Regression. Cambridge Univ. Press.
  • Hart, J. D. (1997). Nonparametric Smoothing and Lack-of-Fit Tests. Springer, New York.
  • Huang, J. Z. (1998a). Projection estimation for multiple regression with application to functional ANOVA models. Ann. Statist. 26 242--272.
  • Huang, J. Z. (1998b). Functional ANOVA models for generalized regression. J. Multivariate Anal. 67 49--71.
  • Huang, J. Z. (1999). Asymptotics for polynomial spline regression under weak conditions. Unpublished manuscript.
  • Huang, J. Z. (2001). Concave extended linear modeling: A theoretical synthesis. Statist. Sinica 11 173--197.
  • Huang, J. Z., Kooperberg, C., Stone, C. J. and Truong, Y. K. (2000). Functional ANOVA modeling for proportional hazards regression. Ann. Statist. 28 961--999.
  • Huang, J. Z. and Stone, C. J. (1998). The $L_2$ rate of convergence for event history regression with time-dependent covariates. Scand. J. Statist. 25 603--620.
  • Huang, J. Z., Wu, C. O. and Zhou, L. (2000). Polynomial spline estimation and inference for varying coefficient models with longitudinal data. Statist. Sinica. To appear.
  • Kooperberg, C., Stone, C. J. and Truong, Y. K. (1995a). The $L_2$ rate of convergence for hazard regression. Scand. J. Statist. 22 143--157.
  • Kooperberg, C., Stone, C. J. and Truong, Y. K. (1995b). Rate of convergence for logspline spectral density estimation. J. Time Ser. Anal. 16 389--401.
  • Lehmann, E. L. (1999). Elements of Large-Sample Theory. Springer, New York.
  • Lehmann, E. L. and Loh, W.-Y. (1990). Pointwise versus uniform robustness of some large-sample tests and confidence intervals. Scand. J. Statist. 17 177--187.
  • Oswald, P. (1994). Multilevel Finite Element Approximations: Theory and Applications. Teubner, Stuttgart.
  • Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York.
  • Ruppert, D. and Wand, M. P. (1994). Multivariate locally weighted least squares regression. Ann. Statist. 22 1346--1370.
  • Rice, J. (1984). Bandwidth choice for nonparametric regression. Ann. Statist. 12 1215--1230.
  • Schumaker, L. (1981). Spline Functions: Basic Theory. Wiley, New York.
  • Stone, C. J. (1982). Optimal global rates of convergence for nonparametric regression. Ann. Statist. 10 1040--1053.
  • Stone, C. J. (1985). Additive regression and other nonparametric models. Ann. Statist. 13 689--705.
  • Stone, C. J. (1986). The dimensionality reduction principle for generalized additive models. Ann. Statist. 14 590--606.
  • Stone, C. J. (1989). Uniform error bounds involving logspline models. In Probability, Statistics and Mathematics: Papers in Honor of Samuel Karlin (T. W. Anderson, K. B. Athreya and D. L. Iglehart, eds.) 335--355. Academic Press, New York.
  • Stone, C. J. (1990). Large-sample inference for logspline models. Ann. Statist. 18 717--741.
  • Stone, C. J. (1991). Asymptotics for doubly flexible logspline response models. Ann. Statist. 19 1832--1854.
  • Stone, C. J. (1994). The use of polynomial splines and their tensor products in multivariate function estimation (with discussion). Ann. Statist. 22 118--184.
  • Stone, C. J., Hansen, M., Kooperberg, C. and Truong, Y. (1997). Polynomial splines and their tensor products in extended linear modeling (with discussion). Ann. Statist. 25 1371--1470.
  • Szegö, G. (1975). Orthogonal Polynomials, 4th ed. Amer. Math. Soc., Providence, RI.
  • Zhou, S., Shen, X. and Wolfe, D. A. (1998). Local asymptotics for regression splines and confidence regions. Ann. Statist. 26 1760--1782.