Article
On deviations between empirical and quantile processes for mixing random variables
ABSTRACT
Let {Xn} be a strictly stationary φmixing 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 φmixing processes give the sharpest possible orders in view of the corresponding results of Kiefer for independent random variables.

Source Available from: export.arxiv.org
 "However, it can easily be extended to plugin estimators of more general Lfunctionals 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 MarcinkiewiczZygmund strong laws of large numbers (MZSLLNs) and
ordinary strong laws of large numbers (SLLNs) for plugin 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
plugin 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 \alphamixing random variables can be
improved. The presented approach does not only cover some known results but
also provides some new strong laws for plugin estimators of particular
statistical functionals. Statistics: A Journal of Theoretical and Applied Statistics 01/2013; 48(5). DOI:10.1080/02331888.2013.800075 · 0.53 Impact Factor

Source Available from: sciencedirect.com
 "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 φmixing 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, 12381244] which are obtained under other weak dependence. Statistics [?] Probability Letters 11/2008; 78(16):26602663. DOI:10.1016/j.spl.2008.03.026 · 0.60 Impact Factor

Source Available from: journals.cambridge.org
 "As can be seen in the proof of Lemma 3.5 of Babu and Singh [1], the existence of the term log log log M is due to the second moment bound for the sum of amixing random variables. One can prove this lemma easily by following the same arguments used by Babu and Singh [1] and using the fact that F(X) follows a uniform (0, 1) distribution for any continuous random variable X with smooth underlying distribution function F. We omit the proof here. "
[Show abstract] [Hide abstract]
ABSTRACT: In this paper, we examine the rate of convergence of moving block bootstrap (MBB) approximations to the distributions of normalized sample quantiles based on strongly mixing observations. Under suitable smoothness and regularity conditions on the onedimensional marginal distribution function, the rate of convergence of the MBB approximations to distributions of centered and scaled sample quantiles is of order O(n−1¼ log logn). Journal of the Australian Mathematical Society 03/2007; 82(02):263  282. DOI:10.1017/S1446788700016074 · 0.14 Impact Factor
Show more Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.