Limit theorems for functions of marginal quantiles

G. Jogesh Babu, Zhidong Bai, Kwok Pui Choi, Vasudevan Mangalam

Research output: Contribution to journalArticle

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 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.

Original languageEnglish (US)
Pages (from-to)671-686
Number of pages16
JournalBernoulli
Volume17
Issue number2
DOIs
StatePublished - May 1 2011

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Quantile
Limit Theorems
Central limit theorem
Bahadur Representation
Equicontinuity
Sample Quantiles
I.i.d. Random Variables
Strong law of large numbers
Multivariate Distribution
Weak Convergence
Order Statistics
Gaussian Process
Joint Distribution

All Science Journal Classification (ASJC) codes

  • Statistics and Probability

Cite this

Babu, G. Jogesh ; Bai, Zhidong ; Choi, Kwok Pui ; Mangalam, Vasudevan. / Limit theorems for functions of marginal quantiles. In: Bernoulli. 2011 ; Vol. 17, No. 2. pp. 671-686.
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Babu, GJ, Bai, Z, Choi, KP & Mangalam, V 2011, 'Limit theorems for functions of marginal quantiles', 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.

In: Bernoulli, Vol. 17, No. 2, 01.05.2011, p. 671-686.

Research output: Contribution to journalArticle

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