Limit theorems for functions of marginal quantiles

Details Limit theorems for functions of marginal quantiles Limit theorems for functions of marginal quantiles

2011-04-22

arxiv.org/abs/1104.4396v1

Bernoulli 2011, Vol. 17, No. 2, 671-686

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.3150/10-BEJ287 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/10-BEJ287

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 \[\sqrt{n}\Biggl(\frac{1}{n}\sum_{i=1}^n\phi\bigl(X_{n:i}^{(1)},...,X_{n:i}^{(d)}\bigr)-\bar{\gamma}\Biggr)=\frac{1}{\sqrt{n}}\sum_{i=1}^nZ_{n,i}+\mathrm{o}_P(1)\] as $n\rightarrow\infty$, where $\bar{\gamma}$ 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.

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