Limit theorems for functions of marginal quantiles

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

Research output: Contribution to journalArticlepeer-review

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 2011

All Science Journal Classification (ASJC) codes

  • Statistics and Probability

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