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Bahadur-type representation of sample conditional quantiles based on weakly dependent data. (English) Zbl 0798.62067

Let \(\{(X_ i,Z_ i)\}\) be a strictly stationary sequence of two- dimensional random vectors defined on a probability space \((\Omega, {\mathcal F}, P)\). For \(0<p<1\), let \(\theta_ p (x)\) denote the \(p\)-quantile of \(Z\) given \(X=x\). P. K. Bhattacharya and A. K. Gangopadhyay [Ann. Stat. 18, No. 3, 1400-1415 (1990; Zbl 0706.62040)] obtained Bahadur type representations of nearest neighbor and kernel estimators of \(\theta_ p(x)\) when \(\{(X_ i,Z_ i)\}\) are independent and identically distributed random vectors. In this paper the author considers the analogous problem when \(\{(X_ i,Z_ i)\}\) satisfies some mixing condition.

MSC:

62G20 Asymptotic properties of nonparametric inference
62G05 Nonparametric estimation
62G30 Order statistics; empirical distribution functions
60F15 Strong limit theorems

Citations:

Zbl 0706.62040
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