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Kernel estimates under association: strong uniform consistency

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  • Roussas, George G.

Abstract

Let X1, X2,... be associated random variables forming a strictly stationary sequence, and let f be the probability density function of X1. For r [greater-or-equal, slanted] 0 integer, let f(r) be the rth order derivative of f. Under suitable regularity conditions on a kernel function K, a sequence of bandwidths {hn}, the derivatives f(s), s = 0, 1,..., r, and the covariances Cov(X1, Xi), i [greater-or-equal, slanted] 2, the usual kernel estimate of f(r)(x) is shown to be strongly consistent, uniformly in x. An application is also presented in the estimation of the hazard rate. Finally, certain covergence rates are also discussed.

Suggested Citation

  • Roussas, George G., 1991. "Kernel estimates under association: strong uniform consistency," Statistics & Probability Letters, Elsevier, vol. 12(5), pages 393-403, November.
    • Handle: RePEc:eee:stapro:v:12:y:1991:i:5:p:393-403
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    Citations

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    Cited by:

    1. Ioannides, D. A. & Roussas, G. G., 1999. "Exponential inequality for associated random variables," Statistics & Probability Letters, Elsevier, vol. 42(4), pages 423-431, May.
    2. Roussas, George G., 1995. "Asymptotic normality of a smooth estimate of a random field distribution function under association," Statistics & Probability Letters, Elsevier, vol. 24(1), pages 77-90, July.
    3. Cai, Zongwu & Roussas, George G., 1998. "Kaplan-Meier Estimator under Association," Journal of Multivariate Analysis, Elsevier, vol. 67(2), pages 318-348, November.
    4. Pierre Jacob & Paulo Oliveira, 1999. "Histograms and Associated Point Processes," Statistical Inference for Stochastic Processes, Springer, vol. 2(3), pages 227-251, October.
    5. Cai, Zongwu & Roussas, George G., 1997. "Smooth estimate of quantiles under association," Statistics & Probability Letters, Elsevier, vol. 36(3), pages 275-287, December.
    6. Igor S. Borisov & Yuliana Yu. Linke & Pavel S. Ruzankin, 2021. "Universal weighted kernel-type estimators for some class of regression models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(2), pages 141-166, February.
    7. Guessoum, Zohra & Ould Saïd, Elias & Sadki, Ourida & Tatachak, Abdelkader, 2012. "A note on the Lynden-Bell estimator under association," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 1994-2000.
    8. Roussas, George G., 2001. "An Esséen-type inequality for probability density functions, with an application," Statistics & Probability Letters, Elsevier, vol. 51(4), pages 397-408, February.
    9. Huang, Wen-Tao & Xu, Bing, 2002. "Some maximal inequalities and complete convergences of negatively associated random sequences," Statistics & Probability Letters, Elsevier, vol. 57(2), pages 183-191, April.
    10. Li, Yongming & Yang, Shanchao & Wei, Chengdong, 2011. "Some inequalities for strong mixing random variables with applications to density estimation," Statistics & Probability Letters, Elsevier, vol. 81(2), pages 250-258, February.
    11. Masry, Elias, 2002. "Multivariate probability density estimation for associated processes: strong consistency and rates," Statistics & Probability Letters, Elsevier, vol. 58(2), pages 205-219, June.
    12. Guodong Xing & Shanchao Yang, 2010. "Some Exponential Inequalities for Positively Associated Random Variables and Rates of Convergence of the Strong Law of Large Numbers," Journal of Theoretical Probability, Springer, vol. 23(1), pages 169-192, March.

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