Random positive linear operators and their applications to nonparametric statistics

José A. Adell (), J. T. Alcalá () and C. Sangüesa ()
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José A. Adell: University of Zaragoza, Department of Statistical Methods and IUMA
J. T. Alcalá: University of Zaragoza, Department of Statistical Methods and IUMA
C. Sangüesa: University of Zaragoza, Department of Statistical Methods and IUMA

TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, 2025, vol. 34, issue 4, No 5, 1011 pages

Abstract: 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.

Keywords: Random positive linear operator; Random Bernstein–Kantorovich-type operator; Distribution function estimator; Density estimator; Confidence band; Confidence interval; Dvoretzky–Kiefer–Wolfowitz inequality; Moduli of smoothness; Primary: 62G05; 60E05; Secondary: 41A25; 41A36 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s11749-025-00984-8

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