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Inference using shape-restricted regression splines. (English) Zbl 1149.62033

Summary: 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 J. O. Ramsay [Stat. Sci. 3, 425–461 (1998)], 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 functions, when implemented with shape-restricted regression splines, have higher power than the standard version using ordinary shape-restricted regression.

MSC:

62G08 Nonparametric regression and quantile regression
65C60 Computational problems in statistics (MSC2010)

Software:

pchip; SemiPar
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References:

[1] Delecroix, M., Simioni, M. and Thomas-Agnan, C. (1995). A shape constrained smoother: simulation study. Comput. Statist. 10 155-175. · Zbl 0936.62047
[2] Eilers, P. H. C. and Marx, B. D. (1996). Flexible smoothing with B -splines and penalties. Statist. Sci. 11 89-121. · Zbl 0955.62562
[3] Fredenhagen, S., Oberle, H. J. and Opfer, G. (1999). On the construction of optimal monotone cubic spline interpolations. J. Approx. Theory 96 182-201. · Zbl 0934.41009
[4] 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. · Zbl 0672.62077
[5] Friedman, J. H. and Silverman, B. W. (1989). Flexible parsimonious smoothing and additive modeling. Technometrics 31 3-21. · Zbl 0672.65119
[6] Fritsch, F. N. and Carlson, R. E. (1980). Monotone piecewise cubic interpolation. SIAM J. Numer. Anal. 17 238-246. · Zbl 0423.65011
[7] Hastie, T. J. and Tibshirani, R. J. (1990). Generalized Additive Models . Chapman and Hall/CRC, London. · Zbl 0747.62061
[8] 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. · Zbl 1142.62379
[9] Mammen, E. (1991). Estimating a smooth monotone regression function. Ann. Statist. 19 724-740. · Zbl 0737.62038
[10] Mammen, E. and Thomas-Agnan, C. (1999). Smoothing splines and shape restrictions. Scand. J. Statist. 26 239-252. · Zbl 0932.62051
[11] Meyer, M. C. (1996). Shape restricted inference with applications to nonparametric regression, smooth nonparametric function estimation, and density estimation. Dissertation, Univ. Michigan.
[12] 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. · Zbl 1057.62510
[13] 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. · Zbl 1149.62033
[14] Meyer, M. and Woodroofe, M. (2000). On the degrees of freedom in shape-restricted regression. Ann. Statist. 28 1083-1104. · Zbl 1105.62340
[15] Meyer M. C. (2003). A test for linear versus convex regression function using shape-restricted regression. Biometrika 90 223-232. · Zbl 1034.62057
[16] Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1992). Numerical Recipes in FORTRAN 77 , 2nd ed. Cambridge Univ. Press. · Zbl 0778.65002
[17] Ramsay, J. O. (1988). Monotone regression splines in action. Statist. Sci. 3 425-461.
[18] Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference . Wiley, Chichester. · Zbl 0645.62028
[19] Ruppert, D., Wand, M. P. and Carroll, R. J. (2003). Semiparametric Regression . Cambridge Univ. Press. · Zbl 1038.62042
[20] Tantiyaswasdikul, C. and Woodroofe, M. B. (1994). Isotonic smoothing splines under sequential designs. J. Statist. Plann. Inference 38 75-87. · Zbl 0814.62042
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