LIMIT THEOREMS FOR FUNCTIONS OF MARGINAL QUANTILES AND ITS APPLICATION.

Abstract

A broken sample problem has been studied by statistician,which is random sample observed for a low-component random variable X and Y,however,the link (or correspondences information ) between the X-component and the Y-components are broken ( or even missing ). A method for re-pairing the broken sample is proposed as well as making statistical inference. Meanwhile,multivariate data ordering schemes has a successful application in the color image processing. So in this paper,we extended the broken sample formulation to study the limit theorem for functions of marginal quantiles.We mainly studied how to explore multivariate distribution using the joint distribution of marginal quantiles. Limit theory for the mean of functions of order statistics is presented. The result include multivariate central theorem and strong law of large numbers.This leads to the central limit theorem. A weak convergence to a Gaussian process using equicontinuity of functions is indicated. The conditions ,under which these results are established.Simulation results of the Marshall-Olkin bivariate exponential distribution and the Farlie-Gumbel-Morgenstern family of copulas are demonstrated to show our two main theoretical results satisfy in many examples that include several commonly occuring situations.