On almost sure convergence rates for the kernel estimator of a covariance operator under negative association

Document Type : Special Issue Dedicated to memory of Prof. Mahbanoo Tata

Author

Department of Statistics, Ordered Data, Reliability and Dependency Center of Excellence, Ferdowsi University of Mashhad, Mashhad, Iran

Abstract

    It is suppose that $\{X_n,~n\geq 1\}$ is a strictly stationary sequence of negatively associated random variables with continuous distribution function F. The aim of this paper is to estimate the distribution of $(X_1,X_{k+1})$ for $k\in I\!\!N_0$ using kernel type estimators. We also estimate the covariance function of the limit empirical process induced by the sequence $\{X_n,~n\geq 1\}$. Then, we obtain uniform strong convergence rates for the kernel estimator of the distribution function of $(X_1,X_{k+1})$. These rates, which do not require any condition on the covariance structure of the variables, were not already found. Furthermore, we show that the covariance function of the limit empirical process based on kernel type estimators has uniform strong convergence rates assuming a convenient decrease rate of covariances $Cov(X_1,X_{n+1}),~n\geq 1$. Finally, the convergence rates obtained here are empirically compared with corresponding results already achieved by some authors.

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References

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Journal of Mahani Mathematical Research
Volume 13, Issue 3 - Serial Number 28
Special Issue Dedicated to Memory of Professor Mahbanoo Tata
August 2024
Pages 91-104

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History

  • Receive Date: 30 December 2023
  • Revise Date: 15 March 2024
  • Accept Date: 12 July 2024

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