On deviations between empirical and quantile processes for mixing random variables

Indian Statistical Institute, Calcutta, India
Journal of Multivariate Analysis (Impact Factor: 0.93). 12/1978; 8(4):532-549. DOI: 10.1016/0047-259X(78)90031-3
Source: RePEc


Let {Xn} be a strictly stationary φ-mixing process with . It is shown in the paper that if X1 is uniformly distributed on the unit interval, then, for any t ∈ [0, 1], a.s. and a.s., where Fn and Fn−1(t) denote the sample distribution function and tth sample quantile, respectively. In case {Xn} is strong mixing with exponentially decaying mixing coefficients, it is shown that, for any t ∈ [0, 1], a.s. and sup0≤t≤1 |Fn−1(t) − t + Fn(t) − t| = a.s. The results are further extended to general distributions, including some nonregular cases, when the underlying distribution function is not differentiable. The results for φ-mixing processes give the sharpest possible orders in view of the corresponding results of Kiefer for independent random variables.

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    • "random variables to the many dependent cases of random variables. For example, Sen [14], Babu and Singh [2], Yoshihara [32], Sun [20], Wang et al. [25] and Zhang et al. [33] studied the Bahadur representation under the cases of ϕ-mixing sequences or strong mixing (α-mixing) sequences. Wendler [28] investigated the Bahadur representation for U-quantiles of α-mixing sequences and functionals of absolutely regular sequences. "
    [Show abstract] [Hide abstract] ABSTRACT: It can be found that widely orthant dependent (WOD) random variables are weaker than extended negatively orthant dependent (END) random variables, while END random variables are weaker than negatively orthant dependent (NOD) and negatively associated (NA) random variables. In this paper, we investigate the Bahadur representation of sample quantiles based on WOD sequences. Our results extend the corresponding ones of Ling [N.X. Ling, The Bahadur representation for sample quantiles under negatively associated sequence, Statistics and Probability Letters 78(16) (2008), 2660–2663], Xu et al. [S.F. Xu, L. Ge, Y. Miao, On the Bahadur representation of sample quantiles and order statistics for NA sequences, Journal of the Korean Statistical Society 42(1) (2013), 1–7] and Li et al. [X.Q. Li, W.Z. Yang, S.H. Hu, X.J. Wang, The Bahadur representation for sample quantile under NOD sequence, Journal of Nonparametric Statistics 23(1) (2011), 59–65] for the case of NA sequences or NOD sequences.
    Preview · Article · Jan 2014 · Filomat
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    • "However, it can easily be extended to plug-in estimators of more general L-functionals L K with dK having compact support strictly within (0, 1). Under the stronger mixing conditions α(n) ≤ Ke −εn , ε > 0, and α(n) ≤ Kn −8 the result of Theorem 3.6 is basically already known from [4] and [36] "
    [Show abstract] [Hide abstract] ABSTRACT: Both Marcinkiewicz-Zygmund strong laws of large numbers (MZ-SLLNs) and ordinary strong laws of large numbers (SLLNs) for plug-in estimators of general statistical functionals are derived. It is used that if a statistical functional is "sufficiently regular", then a (MZ-) SLLN for the estimator of the unknown distribution function yields a (MZ-) SLLN for the corresponding plug-in estimator. It is in particular shown that many L-, V- and risk functionals are "sufficiently regular", and that known results on the strong convergence of the empirical process of \alpha-mixing random variables can be improved. The presented approach does not only cover some known results but also provides some new strong laws for plug-in estimators of particular statistical functionals.
    Preview · Article · Jan 2013 · Statistics: A Journal of Theoretical and Applied Statistics
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    • "Using this representation, one can express asymptotically sample quantiles as an average of i.i.d random variables and can obtain limiting properties of the sample quantiles. Among others, Sen (1972), Babu and Singh (1978) and Yoshihara (1995) gave the Bahadur representation of sample quantiles for some dependent sequences, such as φ-mixing random variable sequences and strongly mixing random variable sequences respectively. "
    [Show abstract] [Hide abstract] ABSTRACT: In this article, we investigate a Bahadur representation of sample quantiles based on negatively associated (NA) sequence. Our results in this note extend Sun's results [Sun, S.X., 2006. The Bahadur representation of sample quantile under week dependence. Statist. Probab. Lett. 76, 1238-1244] which are obtained under other weak dependence.
    Preview · Article · Nov 2008 · Statistics [?] Probability Letters
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