### 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 1 2011 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability

### Cite this

*Bernoulli*,

*17*(2), 671-686. https://doi.org/10.3150/10-BEJ287

}

*Bernoulli*, vol. 17, no. 2, pp. 671-686. https://doi.org/10.3150/10-BEJ287

**Limit theorems for functions of marginal quantiles.** / Babu, G. Jogesh; Bai, Zhidong; Choi, Kwok Pui; Mangalam, Vasudevan.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Limit theorems for functions of marginal quantiles

AU - Babu, G. Jogesh

AU - Bai, Zhidong

AU - Choi, Kwok Pui

AU - Mangalam, Vasudevan

PY - 2011/5/1

Y1 - 2011/5/1

N2 - 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 ni=1φ(X(1)n : i, ⋯ , X(d)n : i) - ȳ)=1/√nσn i=1 Zn,i + oP (1) as n→ ∞, where ȳ is a constant and Zn,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.

AB - 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 ni=1φ(X(1)n : i, ⋯ , X(d)n : i) - ȳ)=1/√nσn i=1 Zn,i + oP (1) as n→ ∞, where ȳ is a constant and Zn,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.

UR - http://www.scopus.com/inward/record.url?scp=79953856857&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79953856857&partnerID=8YFLogxK

U2 - 10.3150/10-BEJ287

DO - 10.3150/10-BEJ287

M3 - Article

AN - SCOPUS:79953856857

VL - 17

SP - 671

EP - 686

JO - Bernoulli

JF - Bernoulli

SN - 1350-7265

IS - 2

ER -