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On the Bahadur Representation of Quantiles for a Sample from ρ*-Mixing Structure Population

Dudek Dagmara, Kuczmaszewska Anna

Advances in Science and Technology. Research Journal

2022

Vol. 16, no 3

316--330

In this paper, we establish the strong consistency and the Bahadur representation of sample quantiles for ρ*-mixing random variables. Additionally, the asymptotic normality and the Berry-Esseen bound of sample quantiles for ρ*-mixing random variables are presented. Additionally, we provide the rate of convergence of sample quantiles to population counterparts. Moreover, numerical simulation is presented to ilustrate and verify obtained results.

On non-uniform Berry-Esseen bounds for time series

Jirak M.

Probability and Mathematical Statistics

2015

Vol. 35, Fasc. 1

1--14

Given a stationary sequence {Xk}k ϵ Z, non-uniform bounds for the normal approximation in the Kolmogorov metric are established. The underlying weak dependence assumption includes many popular linear and nonlinear time series from the literature, such as ARMA or GARCH models. Depending on the number of moments p, typical bounds in this context are of the size O(mp−1 n−p/2+1), where we often find that m = mn = log n. In our setup, we can essentially improve upon this rate by the factor m−p/2, yielding a bound of O(mp/2−1 n−p/2+1). Among other things, this allows us to recover a result from the literature, which is due to Ibragimov.