Estimating Smooth Monotone Functions

Summary Many situations call for a smooth strictly monotone function f of arbitrary flexibility. The family of functions defined by the differential equation D 2 f =w Df, where w is an unconstrained coefficient function comprises the strictly monotone twice differentiable functions. The solution to this equation is f = C 0 + C 1 D −1{exp(D −1 w)}, where C 0 and C 1 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 w 2 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.