Estimating smooth monotone functions

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.