Please use this identifier to cite or link to this item: https://doi.org/10.3150/10-BEJ287
Title: Limit theorems for functions of marginal quantiles
Authors: Babu, G.J.
Bai, Z. 
Choi, K.P. 
Mangalam, V.
Keywords: Central limit theorem
Cramér-wold device
Lost association
Quantiles
Strong law of large numbers
Weak convergence of a process
Issue Date: May-2011
Citation: Babu, G.J., Bai, Z., Choi, K.P., Mangalam, V. (2011-05). Limit theorems for functions of marginal quantiles. Bernoulli 17 (2) : 671-686. ScholarBank@NUS Repository. https://doi.org/10.3150/10-BEJ287
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=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. © 2011 ISI/BS.
Source Title: Bernoulli
URI: http://scholarbank.nus.edu.sg/handle/10635/105196
ISSN: 13507265
DOI: 10.3150/10-BEJ287
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