The Annals of Applied Statistics

Inference using shape-restricted regression splines

Mary C. Meyer

Full-text: Open access

Abstract

Regression splines are smooth, flexible, and parsimonious nonparametric function estimators. They are known to be sensitive to knot number and placement, but if assumptions such as monotonicity or convexity may be imposed on the regression function, the shape-restricted regression splines are robust to knot choices. Monotone regression splines were introduced by Ramsay [Statist. Sci. 3 (1998) 425–461], but were limited to quadratic and lower order. In this paper an algorithm for the cubic monotone case is proposed, and the method is extended to convex constraints and variants such as increasing-concave. The restricted versions have smaller squared error loss than the unrestricted splines, although they have the same convergence rates. The relatively small degrees of freedom of the model and the insensitivity of the fits to the knot choices allow for practical inference methods; the computational efficiency allows for back-fitting of additive models. Tests of constant versus increasing and linear versus convex regression function, when implemented with shape-restricted regression splines, have higher power than the standard version using ordinary shape-restricted regression.

Article information

Source
Ann. Appl. Stat. Volume 2, Number 3 (2008), 1013-1033.

Dates
First available: 13 October 2008

Permanent link to this document
http://projecteuclid.org/euclid.aoas/1223908050

Digital Object Identifier
doi:10.1214/08-AOAS167

Zentralblatt MATH identifier
1149.62033

Mathematical Reviews number (MathSciNet)
MR2516802

Citation

Meyer, Mary C. Inference using shape-restricted regression splines. The Annals of Applied Statistics 2 (2008), no. 3, 1013--1033. doi:10.1214/08-AOAS167. http://projecteuclid.org/euclid.aoas/1223908050.


Export citation

References

  • Delecroix, M., Simioni, M. and Thomas-Agnan, C. (1995). A shape constrained smoother: simulation study. Comput. Statist. 10 155–175.
  • Eilers, P. H. C. and Marx, B. D. (1996). Flexible smoothing with B-splines and penalties. Statist. Sci. 11 89–121.
  • Fredenhagen, S., Oberle, H. J. and Opfer, G. (1999). On the construction of optimal monotone cubic spline interpolations. J. Approx. Theory 96 182–201.
  • Fraser, D. A. S. and Massam, H. (1989). A mixed primal-dual bases algorithm for regression under inequality constraints. Application to convex regression. Scand. J. Statist. 16 65–74.
  • Friedman, J. H. and Silverman, B. W. (1989). Flexible parsimonious smoothing and additive modeling. Technometrics 31 3–21.
  • Fritsch, F. N. and Carlson, R. E. (1980). Monotone piecewise cubic interpolation. SIAM J. Numer. Anal. 17 238–246.
  • Hastie, T. J. and Tibshirani, R. J. (1990). Generalized Additive Models. Chapman and Hall/CRC, London.
  • Huang, J. Z. and Stone, C. J. (2002). Extended linear modeling with splines. In Nonlinear Estimation and Classification (D. D. Dension, M. H. Hansen, C. C. Holmes, B. Malick, B. Yu, eds.) 213–234. Springer, New York.
  • Mammen, E. (1991). Estimating a smooth monotone regression function. Ann. Statist. 19 724–740.
  • Mammen, E. and Thomas-Agnan, C. (1999). Smoothing splines and shape restrictions. Scand. J. Statist. 26 239–252.
  • Meyer, M. C. (1996). Shape restricted inference with applications to nonparametric regression, smooth nonparametric function estimation, and density estimation. Dissertation, Univ. Michigan.
  • Meyer, M. C. (1999). An extension of the mixed primal-dual bases algorithm to the case of more constraints than dimensions. J. Statist. Plann. Inference 81 13–31.
  • Meyer, M. C. (2008). Supplements to “Inference using shape restricted regression splines.” DOI: 10.1214/08-AOAS167SUPPA, DOI: 10.1214/08-AOAS167SUPPB, DOI: 10.1214/08-AOAS167SUPPC, DOI: 10.1214/08-AOAS167SUPPD.
  • Meyer, M. and Woodroofe, M. (2000). On the degrees of freedom in shape-restricted regression. Ann. Statist. 28 1083–1104.
  • Meyer M. C. (2003). A test for linear versus convex regression function using shape-restricted regression. Biometrika 90 223–232.
  • Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1992). Numerical Recipes in FORTRAN 77, 2nd ed. Cambridge Univ. Press.
  • Ramsay, J. O. (1988). Monotone regression splines in action. Statist. Sci. 3 425–461.
  • Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference. Wiley, Chichester.
  • Ruppert, D., Wand, M. P. and Carroll, R. J. (2003). Semiparametric Regression. Cambridge Univ. Press.
  • Tantiyaswasdikul, C. and Woodroofe, M. B. (1994). Isotonic smoothing splines under sequential designs. J. Statist. Plann. Inference 38 75–87.