Random positive linear operators and their applications to nonparametric statistics
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911111195918336 |
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| author | Adell, José A. Alcalá, J. T. Sangüesa, C. |
| author_facet | Adell, José A. Alcalá, J. T. Sangüesa, C. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13931 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random positive linear operators and their applications to nonparametric statistics Adell, José A. Alcalá, J. T. Sangüesa, C. Statistics Theory Probability Primary: 62G05, 60E05, secondary: 41A25, 41A36 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. |
| title | Random positive linear operators and their applications to nonparametric statistics |
| topic | Statistics Theory Probability Primary: 62G05, 60E05, secondary: 41A25, 41A36 |
| url | https://arxiv.org/abs/2508.13931 |