Estimation of Monotone Functions via P-Splines: A Constrained Dynamical Optimization Approach

Estimation of monotone functions has broad applications in statistics, engineering, and science. This paper addresses asymptotic behaviors of monotone penalized spline estimators using constrained dynamical optimization techniques. The underlying regression function is approximated by a B-spline of an arbitrary degree subject to the first-order difference penalty. The optimality conditions for spline coefficients give rise to a size-dependent complementarity problem. As a key technical result of the paper, the uniform Lipschitz property of optimal spline coefficients is established by exploiting piecewise linear and polyhedral theory. This property forms a cornerstone for stochastic boundedness, uniform convergence, and boundary consistency of the monotone estimator. The estimator is then approximated by a solution of a differential equation subject to boundary conditions. This allows the estimator to be represented by a kernel regression estimator defined by a related Green’s function of an ODE. The asymptotic normality is established at interior points via the Green’s function. The convergence rate is shown to be independent of spline degrees, and the number of knots does not affect asymptotic distribution, provided that it tends to infinity fast enough.