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Moderate deviation principles for nonparametric recursive distribution estimators using Bernstein polynomials

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Abstract

In this paper we prove moderate deviations principles for the recursive estimators of a distribution function defined by the stochastic approximation algorithm based on Bernstein polynomials introduced by Jmaei el al. (J Nonparametr Stat 29:792–805, 2017). We show that the considered estimator gives the same pointwise moderate deviations principle (MDP) as the recursive kernel distribution estimator proposed in  Slaoui (Math Methods Stat 23(4):306–325, 2014b) and whose large and moderate deviation principles were established by  Slaoui (Stat Interface 12(3):439–455, 2009).

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References

  1. Babu, G.J., Canty, A.J., Chaubey, Y.P.: Application of Bernstein polynomials for smooth estimation of a distribution and density function. J. Statist. Plan. Inference 105, 377–392 (2002)

    Article  MathSciNet  Google Scholar 

  2. Bojanic, R., Seneta, E.: A unified theory of regularly varying sequences. Math. Z. 134(2), 91–106 (1973)

    Article  MathSciNet  Google Scholar 

  3. Galambos, J., Seneta, E.: Regularly varying sequences. Proc. Am. Math. Soc. 41(1), 110–116 (1973)

    Article  MathSciNet  Google Scholar 

  4. Guan, Z.: Efficient and robust density estimation using Bernstein type polynomials. J. Nonparameter Stat. 28(2), 250–271 (2016)

    Article  MathSciNet  Google Scholar 

  5. Jmaei, A., Slaoui, Y., Dellagi, W.: Recursive distribution estimators defined by stochastic approximation method using Bernstein polynomials. J. Nonparameter Stat. 29, 792–805 (2017)

    Article  MathSciNet  Google Scholar 

  6. Klebaner, F., Liptser, R.: Large deviations for past-dependent recursions. Probl. Inf. Transm. 32(4), 23–34 (1996)

    MATH  Google Scholar 

  7. Leblanc, A.: On estimationg distribution function using Bernstein polynomials. Ann. Inst. Stat. Math. 64, 919–943 (2012)

    Article  Google Scholar 

  8. Mokkadem, A., Pelletier, M.: A companion for the Kiefer–Wolfowitz–Blum stochastic approximation algorithm. Ann. Stat. 35(4), 1749–1772 (2007)

    Article  MathSciNet  Google Scholar 

  9. Mokkadem, A., Pelletier, M., Slaoui, Y.: The stochastic approximation method for the estimation of a multivariate probability density. J. Stat. Plan. Inference 139(7), 2459–2478 (2009)

    Article  MathSciNet  Google Scholar 

  10. Puhalskii, A.A.: The method of stochastic exponentials for large deviations. Stochastic Process. Appl. 54(1), 45–70 (1994)

    Article  MathSciNet  Google Scholar 

  11. Puhalskii,A. A.: Large deviations for stochastic processes. LMS/EPSRC Short Course: Stochastic Stability, Large Deviations and Coupling Methods. Heriot-Watt University, Edinburgh (2006)

  12. Slaoui, Y.: The stochastic approximation method for the estimation of a distribution function. Math. Methods Stat. 23(4), 306–325 (2014b)

    Article  MathSciNet  Google Scholar 

  13. Slaoui, Y.: Large and moderate deviation principles for recursive kernel distribution estimators defined by stochastic approximation method. Opuscula Mathematica 39(5), 733–746 (2019)

    Article  MathSciNet  Google Scholar 

  14. Slaoui, Y., Jmaei, A.: Recursive density estimators based on Robbins–Monro’s scheme and using Bernstein polynomials. Stat. Interface 12(3), 439–455 (2019)

    Article  MathSciNet  Google Scholar 

  15. Varadhan, S.R.S.: Large Deviations. Ann. Probab. 36(2), 397–419 (2008)

    Article  MathSciNet  Google Scholar 

  16. Vitale, R.A.: A Bernstein polynomial approach to density function estimation. Stat. Inference Rel. Top. 2, 87–99 (1975)

    Article  MathSciNet  Google Scholar 

  17. Wang, T., Guan, Z.: Bernstein polynomial model for nonparametric multivariate density. Statistics 53(2), 321–338 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author would like to thank the Editor Editor-in-Chief Prof. Marco Castrillón López of Revista Matemática Complutense and the referees for their very helpful comments, which led to considerable improvement of the original version of the paper and a more sharply focused presentation.

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Correspondence to Yousri Slaoui.

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Slaoui, Y. Moderate deviation principles for nonparametric recursive distribution estimators using Bernstein polynomials. Rev Mat Complut 35, 147–158 (2022). https://doi.org/10.1007/s13163-021-00384-0

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