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arXiv · 2508.13931

Random positive linear operators and their applications to nonparametric statistics

José A. Adell·J. T. Alcalá·C. Sangüesa
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Abstract

We outline a general procedure on how to apply random positive linear operators in nonparametric estimation. As a consequence, we give explicit confidence bands and intervals for a distribution function $F$ concentrated on $[0,1]$ by means of random Bernstein polynomials, and for the derivatives of $F$ by using random Bernstein-Kantorovich type operators. In each case, the lengths of such bands and intervals depend upon the degree of smoothness of $F$ or its corresponding derivatives, measured in terms of appropriate moduli of smoothness. In particular, we estimate the uniform distribution function by means of a random polynomial of second order. This estimator is much simpler and performs better than the classical uniform empirical process used in the celebrated Dvoretzky-Kiefer-Wolfowitz inequality.

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José A. Adell, J. T. Alcalá, C. Sangüesa. 2025-08-19. Random positive linear operators and their applications to nonparametric statistics. https://arxiv.org/abs/2508.13931

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