doi:10.1007/s11749-025-00984-8engAdell, José A.Alcalá, J. T.Sangüesa, C.Random positive linear operators and their applications to nonparametric statisticsART-2025-145452We 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.2025http://zaguan.unizar.es/record/16293210.1007/s11749-025-00984-8http://zaguan.unizar.es/record/162932oai:zaguan.unizar.es:162932info:eu-repo/grantAgreement/ES/DGA/E48-23Rinfo:eu-repo/grantAgreement/ES/DGA/S41-23Rinfo:eu-repo/grantAgreement/ES/MINECO-FEDER/PID2021-123737NB-I00Test (2025), [31 pp.]byhttps://creativecommons.org/licenses/by/4.0/deed.esinfo:eu-repo/semantics/openAccess