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<p>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 <i>F</i> concentrated on [0,&#xa0;1] by means of random Bernstein polynomials, and for the derivatives of <i>F</i> by using random Bernstein–Kantorovich-type operators. In each case, the lengths of such bands and intervals depend upon the degree of smoothness of <i>F</i> 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.</p>

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Random positive linear operators and their applications to nonparametric statistics

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月4日

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  • José A. Adell,
  • J. T. Alcalá,
  • C. Sangüesa

摘要

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.