Distribution function estimation by constrained polynomial spline regression

Abstract

A smooth monotone polynomial spline (PS) estimator is proposed for the cumulative distribution function. The proposed method applies a constrained PS regression to smooth the empirical distribution function, while simultaneously ensures monotonicity by imposing a set of linear constraints on the coefficients of the PS functions. This feature is not shared by its kernel counterpart in [Cheng, M.Y., and Peng, L. (2002), ‘Regression Modeling for Nonparametric Estimation of Distribution and Quantile Functions’, Statistica Sinica, 12, 1043–1060], as the kernel estimator is not necessarily monotone. Under mild assumptions, both L 2 and uniform convergence rates are obtained. Our simulation studies show that the proposed estimator has better finite sample performance than the simple empirical distribution function. We also illustrate the use of the proposed method by analysing two real data examples.