Estimating smooth monotone functions

Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 1998 · Volume 60 · Issue 2

Abstract

Many situations call for a smooth strictly monotone functionfof arbitrary flexibility. The family of functions defined by the differential equationD2f=w Df, wherewis an unconstrained coefficient function comprises the strictly monotone twice differentiable functions. The solution to this equation isf = C0+C1D−1{exp(D−1w)}, whereC0andC1are arbitrary constants andD−1is the partial integration operator. A basis for expandingwis suggested that permits explicit integration in the expression off. In fitting data, it is also useful to regularizefby penalizing the integral ofw2since this is a measure of the relative curvature inf. Applications are discussed to monotone nonparametric regression, to the transformation of the dependent variable in non-linear regression and to density estimation.

Keywords

Convex functions, Density estimation, Generalized additive model, Linear differential equation, Monotonicity, Nonparametric regression, Regression spline, Spline smoothing

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