Geodesic
Some asymptotic properties between smooth empirical and quantile processes for dependent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 3, pp. 565-580 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\widehat F_n$ be the smooth empirical estimator obtained by integrating a kernel type density estimator based on a random sample of size $n$ from continuous distribution function $F$. The almost sure deviation between smooth empirical and smooth quantile processes is investigated under $\phi$-mixing and strong mixing conditions. We derive a pointwise as well as a uniform Bahadur–Kieffer type representation for smooth quantiles under cases of $\phi$-mixing and strong mixing. These results extend those of Babu and Singh [J. Multivariate Anal., 8 (1978), pp. 532–549] and Ralescu [J. Statist. Plann. Inference, 32 (1992), pp. 243–258].
Keywords: kernel density estimator, almost sure deviation, smooth empirical process, smooth quantile process. 
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S. Sun; W. Zhu. Some asymptotic properties between smooth empirical and quantile processes for dependent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 3, pp. 565-580. http://geodesic.mathdoc.fr/item/TVP_2021_66_3_a8/

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