Article

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

ABSTRACT

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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    • "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] "
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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. "
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