## Abstract

Multivariate distributions are explored using the joint distributions of marginal sample quantiles. Limit theory for the mean of a function of order statistics is presented. The results include a multivariate central limit theorem and a strong law of large numbers. A result similar to Bahadur's representation of quantiles is established for the mean of a function of the marginal quantiles. In particular, it is shown that √ n(1/nσ ^{n}_{i=1}φ(X^{(1)}_{n : i}, ⋯ , X^{(d)}_{n : i}) - ȳ)=1/√nσ^{n} _{i=1} Z_{n,i} + oP (1) as n→ ∞, where ȳ is a constant and Z_{n,i} are i.i.d. random variables for each n. This leads to the central limit theorem. Weak convergence to a Gaussian process using equicontinuity of functions is indicated. The results are established under very general conditions. These conditions are shown to be satisfied in many commonly occurring situations.

Original language | English (US) |
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Pages (from-to) | 671-686 |

Number of pages | 16 |

Journal | Bernoulli |

Volume | 17 |

Issue number | 2 |

DOIs | |

State | Published - May 2011 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability