Berry-Esséen bound of sample quantiles for negatively associated sequence

Abstract

In this paper, we investigate the Berry-Esséen bound of the sample quantiles for the negatively associated random variables under some weak conditions. The rate of normal approximation is shown as O(n -1/9 ).

Uniformly asymptotic normality of sample quantiles estimator for linearly negative quadrant dependent samples

“…We obtain that the rate of normal approximation is under a LNQD sequence, which extends the result of Lemma 3.2 in Liu [ 2 ] and Lemma 2.1 in Yang [ 1 ] in some sense.…”

“…However, there are very few literature works on uniformly asymptotic normality of sample quantiles for a LNQD sequence which is weaker than a NA sequence. By using some inequalities for LNQD random variables, we investigate the uniformly asymptotic normality of the sample quantiles for a LNQD sequence under mild conditions and obtain the rate of normal approximation, the rate of convergence is near provided the third moment is finite, which extends and improves the corresponding results of Liu et al [ 2 ] and Yang et al [ 1 ] in some sense.…”

“…Assumptions (A2) and (A5) are used commonly in the literature. For example, Liu [ 2 ], Yang [ 1 , 13 , 14 ], Yang [ 16 ], Liang [ 19 ], and Li [ 20 ] used (A5) , (A2) and (A4) were used by Liu [ 2 ] and Yang [ 1 , 13 , 14 ], (A3) and (A6) were assumed in Liu [ 2 ]. Assumption (A5) is easily satisfied, for example, choosing , , .…”

“…Recently, Lahiri and Sun [ 12 ] obtained the Berry–Esséen bound of the sample quantiles for an α -mixing sequence. Yang et al [ 1 , 13 , 14 ] investigated the Berry–Esséen bound of the sample quantiles for a NA random sequence and a ϕ -mixing sequence, respectively, the convergence rate is . Considering other papers about Berry–Esséen bound, Cai and Roussas [ 15 ] studied the Berry–Esséen bound for the smooth estimator of a function under association sample.…”

Berry-Esseen bounds of weighted kernel estimator for a nonparametric regression model based on linear process errors under a LNQD sequence

“…Li et al [ 14 ] established the Berry-Esseen bounds of the wavelet estimator for a nonparametric regression model with linear process errors generated by φ -mixing sequences. Yang et al [ 15 ] investigated the Berry-Esseen bound of sample quantiles for NA random variables, the rate of normal approximation is shown as , etc.…”