### A nonparametric test for homogeneity: applications to parameter estimation [chapter]

K. Ghoudi, D. McDonald
1994 Lecture Notes-Monograph Series
Testing for homogeneity has many applications in statistical analysis. For example, regression analysis may be viewed as determining the set of parameters that makes the residuals homogeneous. Assume for each i = 1, --.-, q a small sample n ( i ) observations is collected. Let N = x : = l n ( i ) , let F' be the empirical distribution function of the ith sampleand let the empirical distribution function of all the samples taken together be F . The problem is to test if these samples are
more » ... samples are homogeneous. Lehmann (1951) considered the problem of testing the equality of the distributions of q samples. He proposed the statistic The asymptotic properties of Crimer-Von Mises statistics like the above were studied by Kiefer (1959) . He considered the case where n ( i ) + oo while q stayed fixed. McDonald (1991) considered the situation where q + oo and n ( i ) stays fixed for univariate observations for a more general family of statistics called mndomness statistics. In case of multivariate observations similar asymptotics are discussed in Ghoudi (1992). Here we present an application of the above statistics to the estimation of the parameters of regression models with independent additive errors. The main novelty of our approach is the use of blocking to contrast the empirical distribution of the residuals of observations whose independent variables are in the same block with the empirical distribution of all the residuals taken together. Our estimated regression surface is the one whose residuals minimize a randomness statistic like Lehmann's. Confidence intervals and a test for the model follow without assumptions on the error distribution. Introduction. Consider a multivariate linear regression model. Observations of a dependent vector y = ( y l , y2, . . . , yd) and independent variables x = ( x l , 2 2 , -, x p ) are indexed by t = 1, ---, N and are governed by the model