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Uniform strong consistency of kernel density estimators under dependence. (English) Zbl 0843.62041
Summary: It is shown that the kernel density estimators converge a.s. uniformly on compact subsets of the variable under $\alpha$-mixing. In particular, the rates of convergence for the estimators will be investigated to analyze dependency effects.

##### MSC:
 62G07 Density estimation 62G20 Nonparametric asymptotic efficiency
Full Text:
##### References:
 [1] Cai; Z.; Roussas, G.: Uniform strong estimation under ${\alpha}$-mixing, with rates. Statist. probab. Lett. 15, 47-55 (1992) · Zbl 0757.62024 [2] Cox; D.; Kim, T. Y.: Moment bounds for mixing random variables useful in nonparametric function estimation. Stochastic process appl. 56, 151-158 (1995) · Zbl 0817.62027 [3] Rosenblatt, M.: Density estimates and Markov sequences. Nonparametric techniques in statistical inference, 199-210 (1970) [4] Roussas, G.: Nonparametric estimation in Markov processes. Ann. inst. Statist. math. 21, 73-87 (1969) · Zbl 0181.45804 [5] Roussas, G.: Nonparametric estimation in mixing sequences of random variables. J. statist. Planning inference 18, 135-149 (1988) · Zbl 0658.62048 [6] Silverman, B.W.: Density estimation for statistics and data analysis. (1986) · Zbl 0617.62042 [7] Yakowitz, S.: Nonparametric density estimation, prediction and regression for Markov sequences. J. amer. Statist. assoc. 80, 215-221 (1985) · Zbl 0566.62029 [8] Yu, B.: Density estimation in the L$\infty$norm for dependent data with applications to the Gibbs sampler. Ann. statist. 21, 711-735 (1993) · Zbl 0792.62035