Uniform Approximations for Distributions of Continuous Random Variables with Application in Dual Statis Method

Abstract

The matrix S =  tr(WiQWjQ)  i,j=1,...,k where Q is a symmetric positive definite matrix and Wi = X′ iDiXi , i = 1, ..., k is formed by data tables Xi and diagonal matrices of weights Di , plays a central role in dual STATIS method. In this paper, we approximate the distribution function of the entries of S, assuming data tables Xi given by Ui + Ei , i = 1, ..., k with independent random matrices Ei representing errors, in order to obtain (approximately) the distribution of Sv, where v is the orthonormal eigenvector of S associated to the largest eigenvalue. To achieve this goal, we approximate uniformly the distribution of each entry of S. In general, our technique consists in to approximate uniformly the distribution sequence  g(V n + µn), n > 1 , where g is some smooth function of several variables, {V n, n > 1} is a sequence of identically distributed random vectors of continuous type and {µn} is a non-random vector sequence.