PROBABILISTIC APPROACH TO THE SCHOENBERG SPLINE OPERATOR AND UNIMODAL DENSITY ESTIMATOR

Abstract

Using Chebyshev's inequality, we provide a probabilistic proof of the uniform convergence for continuous functions on a closed interval by Schoenberg's variation diminishing spline operator. Furthermore, we introduce a unimodal density estimator based on this spline operator and thus generalize that of Bernstein polynomials and beta density. The advantage of this method is the local property. That is, rening the knots while keeping the degree xed of B-splines yields better estimates. We also give a numerical example to verify our results.