## Abstract

Let {X_{n}} be a strictly stationary φ-mixing process with Σ_{j=1}^{∞} φ^{ 1 2}(j) < ∞. It is shown in the paper that if X_{1} is uniformly distributed on the unit interval, then, for any t ∈ [0, 1], |F_{n}^{-1}(t) - t + F_{n}(t) - t| = O(n^{- 3 4}(log log n)^{ 3 4}) a.s. and sup_{0≤t≤1} |F_{n}^{-1}(t) - t + F_{n}(t) - t| = (O(n^{- 3 4}(log n)^{ 1 2}(log log n)^{ 1 4}) a.s., where F_{n} and F_{n}^{-1}(t) denote the sample distribution function and tth sample quantile, respectively. In case {X_{n}} is strong mixing with exponentially decaying mixing coefficients, it is shown that, for any t ∈ [0, 1], |F_{n}^{-1}(t) - t + F_{n}(t) - t| = O(n^{- 3 4}(log n)^{ 1 2}(log log n)^{ 3 4}) a.s. and sup_{0≤t≤1} |F_{n}^{-1}(t) - t + F_{n}(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 language | English (US) |
---|---|

Pages (from-to) | 532-549 |

Number of pages | 18 |

Journal | Journal of Multivariate Analysis |

Volume | 8 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1978 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Numerical Analysis
- Statistics, Probability and Uncertainty