Estimating smooth monotone functions

Abstract

Many situations call for a smooth strictly monotone function f of arbitrary flexibility. The family of functions defined by the differential equation D2f= 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. © 1998 Royal Statistical Society.