Estimating Smooth Monotone Functions
- 1 July 1998
- journal article
- research article
- Published by Oxford University Press (OUP) in Journal of the Royal Statistical Society Series B: Statistical Methodology
- Vol. 60 (2) , 365-375
- https://doi.org/10.1111/1467-9868.00130
Abstract
Many situations call for a smooth strictly monotone function f of arbitrary flexibility. The family of functions defined by the differential equation D 2f =w Df, where w is an unconstrained coefficient function comprises the strictly monotone twice differentiable functions. The solution to this equation is f = C0 + C1 D−1{exp(D−1w)}, where C0 and C1 are arbitrary constants and D−1 is the partial integration operator. A basis for expanding w is suggested that permits explicit integration in the expression of f. In fitting data, it is also useful to regularize f by penalizing the integral of w2 since this is a measure of the relative curvature in f. Applications are discussed to monotone nonparametric regression, to the transformation of the dependent variable in non-linear regression and to density estimation.This publication has 1 reference indexed in Scilit:
- Monotone Regression Splines in ActionStatistical Science, 1988