Local influence in testing some hypotheses about covariance matrices

Abstract

An assessment of the influence of observations under minor perturbation of the statistical model is important to evaluate the inexactness of the model. Much attention has been given to problems of detecting observations which individually or jointly influence on the statistical model and to problems of assessing the influence of such observations. Most approaches have not overcome the masking effect or swamping effect caused by multiple outliers so well. The main purpose of this thesis is to detect observations which have large local influence in testing the equality of two covariance matrices and blockwise independence of covariance matrix by adaptation of the local influence method introduced by Cook (1986) and extended by Wu and Luo (1993a). We derive the following procedures to apply the local influence method to the maximum likelihood statistic for the tests. First, we obtain a procedure for computing the curvatures and their associated direction vectors for the surface of maximum likelihood statistic which is perturbed by putting a weight on the covariance matrix for each observation. This result is illustrated with the plot of the direction vectors corresponding to the largest curvature and the second largest one. Hence we can locate the influential observations which are separated from the rest of data. Second, we get direction vector corresponding to the maximum slope for the surface of maximum likelihood statistic at the point denoting no perturbation. This approach is the by-product of the above procedure and the result obtained by this method is complementary with the above procedure. Third, the case-deletion method, which consumes large amount of computing time, is applied to the tests and its result is compared with that of local influence method to prove the effect of the local influence method. Analyzing the results obtained from the above procedures, it is evident that the local influence method provides very informative, simple and unified app...