Inference using shape-restricted regression splines

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.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS167 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

International income poverty measurement: which way now?

“…See Meyer (2008) for a discussion of monotone splines. 28 Using those curves, the splines, the estimated elasticities are also increasing with higher mean consumption which is consistent with the arguments made by Ravallion and Chen (2011).…”

Estimation procedures for exchangeable Marshall copulas with hydrological application

“…It can be also derived from Meyer (2008) that at each knot exactly one basis function has a positive second derivative, so that for our convex splines the proper set of basis functions is up to order four. In the same work, it is also shown that for the unrestricted case and bounded mesh ratio the asymptotically optimal number of knots is l ≈ n 1/(2p+1) with p being the order of the polynomial pieces.…”

“…For our simulation purposes we used the code provided by Meyer (2008) in the supplementary material. As this estimate returns convex and monotone result, only the boundary condition is not necessarily satisfied.…”

Checking unimodality using isotonic regression: an application to breast cancer mortality rates

“…Several authors have dealt with related problems using the isotonic regression for monotone regression models or for alternative shape constraints. Some representative references are Brunk (1970), Dykstra (1983), Hastie and Tibshirani (1986), Bachetti (1989), Huang (2002), Andersson et al (2004), Meyer (2008), Shively et al (2011) and Rueda and Lombardía (2012) among many others. There has been also considerable previous work on procedures testing homogeneity against monotonicity or unimodality (see for example Basso and Salmaso, 2011, Shi, 1988and Wolfe, 2006, but the problem of considering testing monotonicity against unimodality has not been studied yet.…”