Random positive linear operators and their applications to nonparametric statistics

• José A. Adell ^[1] ; J. T. Alcalá ^[1] ; C. Sangüesa ^[1]
  1. [1] Universidad de Zaragoza

     Universidad de Zaragoza

     Zaragoza, España

     
• Localización: Test: An Official Journal of the Spanish Society of Statistics and Operations Research, ISSN-e 1863-8260, ISSN 1133-0686, Vol. 34, Nº. 4, 2025, págs. 981-1011
• Idioma: inglés
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• Resumen

  • 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.