The Bahadur representation for sample quantiles under negatively associated sequence

A remark on the Bahadur representation of sample quantiles for negatively associated sequences

“…This motives us to investigate further the Bahadur representation of sample quantiles for negatively associated random variables. In this paper, under mild conditions, the Bahadur representation of sample quantiles under negatively associated sequences is obtained, whose convergence rate improves the corresponding t n obtained by Ling (2008) and reaches t n n −1/4 under the same conditions as the ones in Theorem 2 of that paper.…”

“…Sun (2006) established the Bahadur representation for sample quantiles under the strongly mixing sequences with polynomially decaying rate. Recently, Ling (2008) investigated the Bahadur representation for sample quantiles under negative associated sequences. Assuming that {t n , n ≥ 1} is a positive real number sequence satisfying t n → 0 and √ nt 2 n / log n → ∞, Ling (2008) obtained the convergence rate t n of the Bahadur representation for sample quantiles under the assumption of negative association in Theorem 2, which is obviously slower than the rate of convergence t n n −1/4 obtained in Theorem 1 by the factor n −1/4 .…”

“…Recently, Ling (2008) investigated the Bahadur representation for sample quantiles under negative associated sequences. Assuming that {t n , n ≥ 1} is a positive real number sequence satisfying t n → 0 and √ nt 2 n / log n → ∞, Ling (2008) obtained the convergence rate t n of the Bahadur representation for sample quantiles under the assumption of negative association in Theorem 2, which is obviously slower than the rate of convergence t n n −1/4 obtained in Theorem 1 by the factor n −1/4 . However, the convergence rate of the Bahadur representation for sample quantiles under the assumption of φ-mixing or α-mixing dependence obtained in Sun (2006) and Yoshihara (1995) is equal to the convergence one obtained in the relevant auxiliary theorem.…”

Berry-Esséen bound of sample quantiles for negatively associated sequence

“…Sun [5] established the Bahadur representation for the sample quantiles under a-mixing sequence with polynomially decaying rate. Ling [6] investigated the Bahadur representation for the sample quantiles under NA sequence. Li et al [7] investigated the Bahadur representation of the sample quantile based on negatively orthant-dependent (NOD) sequence, which is weaker than NA sequence.…”

Moment inequalities for m-negatively associated random variables and their applications

“…As pointed out and proved by Joag-Dev and Proschan (1983), a number of well-known multivariate distributions possess the NA property, such as multinomial, convolution of unlike multionmial, multivariate hypergeometric, Dirichlet, permutation distribution, negatively correlated normal distribution, random sampling without replacement and joint distribution of ranks. For more details about NA random variables, one can refer to Matula (1992), Shao (2000), Chen et al (2008), Ling (2008), Liang and Zhang (2010), Sung (2011), Zarei and Jabbari (2011), Hu (2012, 2014), Wang et al (2011Wang et al ( , 2014a, and so on.…”