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

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.