Change-point Problems IMS Lecture Notes- Monograph Series (Volume 23, 1994) A NONPARAMETRIC TEST FOR HOMOGENEITY: APPLICATIONS TO PARAMETER ESTIMATION

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 em-pirical distribution function of the ith sample- and let the empirical distribution function of all the samples taken together be F. The problem is to test if these 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 o