On deviations between empirical and quantile processes for mixing random variables

Gutti Jogesh Babu, Kesar Singh

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49 Scopus citations

Abstract

Let {Xn} be a strictly stationary φ-mixing process with Σj=1 φ 1 2(j) < ∞. It is shown in the paper that if X1 is uniformly distributed on the unit interval, then, for any t ∈ [0, 1], |Fn-1(t) - t + Fn(t) - t| = O(n- 3 4(log log n) 3 4) a.s. and sup0≤t≤1 |Fn-1(t) - t + Fn(t) - t| = (O(n- 3 4(log n) 1 2(log log n) 1 4) 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], |Fn-1(t) - t + Fn(t) - t| = O(n- 3 4(log n) 1 2(log log n) 3 4) a.s. and sup0≤t≤1 |Fn-1(t) - t + Fn(t) - t| = O(n- 3 4(log n)(log log n) 1 4) 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.

Original languageEnglish (US)
Pages (from-to)532-549
Number of pages18
JournalJournal of Multivariate Analysis
Volume8
Issue number4
DOIs
StatePublished - Dec 1978

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Numerical Analysis
  • Statistics, Probability and Uncertainty

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