A Berry–Esseen theorem for sample quantiles under martingale difference sequences

Abstract

In this paper, we establish the uniformly asymptotic normality for sample quantiles based on martingale difference sequences under some suitable conditions. We obtain the rate of normality approximation of $O(n^{-1/4}\log n)$ by using some classical methods such as Bernstein type inequality, and so on. Finally, we verify asymptotic normality for the fixed quantile of the martingale difference sequences and present some numerical simulations to demonstrate the finite sample performances of the theoretical results.

Keywords:

• Berry–Esseen type bound
• sample quantile
• martingale difference sequences
• asymptotic normality
• Shapiro–Wilk test

MSC::

• 62G08
• 60G20

Acknowledgments

The authors are most grateful to the Editor and anonymous referee for carefully reading the manuscript and valuable suggestions which helped in improving an earlier version of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Supported by the the National Social Science Foundation of China (22BTJ059).