PROBABILISTIC APPROACH TO THE SCHOENBERG SPLINE OPERATOR AND UNIMODAL DENSITY ESTIMATOR

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, re ning the knots  while keeping the degree xed of B-splines yields better estimates. We also give a numerical example to  verify our results.