Diagnostics can be summarized as follows. Let R(D,M) denote a result from the data D and the postulated model M, and let M(w) denote the perturbed model according to the perturbation w. The goal of assessing influence is to compare R(D,M) and (R,M(w)) as w varies through a set of relevant perturbations. The main issues in diagnostics are the choice of the perturbation scheme and the method of comparing R(D) and R(D,M(w)). The choice of R is at the discretion of the investigator. There have not been so many diagnostic methods in the field of multivariate statistical methods until recently except for regression and related works.
The objective of this thesis is to investigate the influence of observations on the estimators in multivariate statistical methods, and consequently one can identify the influential observations. The local influence method is adapted to principal component analysis, maximum likelihood factor analysis and linear discriminant analysis. The local influence method uses the curvatures of a perturbed surface as a way of comparing R(D) and R(D,M(w)). In this thesis, three simultaneous perturbations are considered, which are totally different from the individual perturbations used in the influence function method and the case deletion method. For summarizing the influence effects on a vector-valued estimate into a scalar measure, three local influence scalar-valued measures are proposed. It can be seen that the maximum slope direction vector and the empirical influence function give essentially the same influence information, and particularly this property is analytically proved in linear discriminant analysis. The local influence method is compared with the influence function method and the case deletion method in multivariate statistical methods through numerical examples. Those illustrate that the former method is more effective than the latter methods, and it avoids masking effects.