Bernoulli

  • Bernoulli
  • Volume 17, Number 2 (2011), 671-686.

Limit theorems for functions of marginal quantiles

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

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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 $$\sqrt{n}\Biggl(\frac{1}{n}\sum_{i=1}^{n}\phi\bigl(X_{n:i}^{(1)},\ldots,X_{n:i}^{(d)}\bigr)-\bar{\gamma}\Biggr)=\frac{1}{\sqrt{n}}\sum _{i=1}^{n}Z_{n,i}+\mathrm{o}_{P}(1)$$ as $n → ∞$, where $γ̄>$ 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.

Article information

Source
Bernoulli, Volume 17, Number 2 (2011), 671-686.

Dates
First available in Project Euclid: 5 April 2011

Permanent link to this document
https://projecteuclid.org/euclid.bj/1302009242

Digital Object Identifier
doi:10.3150/10-BEJ287

Mathematical Reviews number (MathSciNet)
MR2787610

Zentralblatt MATH identifier
1253.60024

Keywords
central limit theorem Cramér–Wold device lost association quantiles strong law of large numbers weak convergence of a process

Citation

Babu, G. Jogesh; Bai, Zhidong; Choi, Kwok Pui; Mangalam, Vasudevan. Limit theorems for functions of marginal quantiles. Bernoulli 17 (2011), no. 2, 671--686. doi:10.3150/10-BEJ287. https://projecteuclid.org/euclid.bj/1302009242


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References

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