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

Xueping Hu,Keming Yu,Rong Jiang,Tong Zhang

doi:10.1186/s13660-018-1788-6

Abstract

In the present article, by utilizing some inequalities for linearly negative quadrant dependent random variables, we discuss the uniformly asymptotic normality of sample quantiles for linearly negative quadrant dependent samples under mild conditions. The rate of uniform asymptotic normality is presented and the rate of convergence is near O(n^{-1/4} log n) when the third moment is finite, which extends and improves the corresponding results of Yang et al. (J. Inequal. Appl. 2011:83, 2011) and Liu et al. (J. Inequal. Appl. 2014:79, 2014) under negatively associated random samples in some sense.

Highlights

1 Introduction We first recall the definition of negative (NA, for short), negative quadrant dependent (NQD, for short), and linearly negative quadrant dependent (LNQD, for short) sequences
Liu et al [2] presented the Berry–Esséen bound of the sample quantiles for a NA sequence as follows
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 O(n–1/4 log n) 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

Summary

Introduction

We first recall the definition of negative (NA, for short), negative quadrant dependent (NQD, for short), and linearly negative quadrant dependent (LNQD, for short) sequences.Definition 1.1 (Joag-Dev and Proschan [3]) Random variables {Xi}1≤i≤n are said to be NA if, for every pair of disjoint subsets A, B ⊂ {1, 2, . . . , n}, Cov f (Xi, i ∈ A), g(Xj, j ∈ B) ≤ 0, where f and g are real coordinate-wise nondecreasing functions provided the covariance exists. Theorem A Let p ∈ (0, 1) and {Xn}n≥1 be a second-order stationary NA sequence with a common marginal distribution function F and EXn = 0, n ≥ 1. 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 O(n–1/6 log n log log n).

Results

Conclusion