Outliers In Discriminant Analysis (influence Function, Contaminant Observation, Divergence Value).

Abstract

Outliers, from a subjective point of view, are observations which are discordant from the other remaining observations. The influence function is an appealing and intuitive qualitative method to evaluate the effect per observation on a statistic with a contaminated model relative to a basic model. Initially, the influence function was designed to improve estimators from a robustness point of view, but it also can be used as a tool to detect outliers. Campbell (1978), in his paper The influence function as an aid in outlier detection in discriminant analysis, was the first researcher to study this concept in a between-group study, in particular in a discriminant analysis. Outliers in discriminant analysis are often taken to be indicative of an incorrect measurement or an incorrect allocation of the observation. Campbell's work assumes multivariate normal distribution in each of the two populations but restricts it to a common covariance matrix. The parameters that describe important features of a discriminant analysis and whose influence functions Campbell evaluated are the Mahalanobis distance, which give an idea of the degree of group separation, and the coefficients vector of the linear discriminant function which give the relative order of importance of the discriminating variables. In this research, new results are presented. First Campbell tried to approximate the distribution of the norm for the coefficient vector of the linear discriminant function by a suitable distribution with the same first two moments. The second moment, which was not solved by him, is evaluated here and consequently the approximated distribution can be obtained. Second, a generalization, without assumption of equality of covariance matrices has been done in this research. Influence function for the mean as well as for the variance of the general quadratic discriminant function in each group have been evaluated. Third, Campbell's work has been generalized with unequal covariance matrices but with the discriminant function restricted to be linear. The parameter value called divergence, defined in information theory, for this linear discriminant function evaluated by Kullback has been used in this research. This parameter reflects group separation and the Mahalanobis distance can be obtained from it assuming equal covariance matrices. This work shows the evaluation of the influence function for this divergence value. This function is unbounded and its distribution has been approximated by a suitable distribution with the same first two moments. Fourth, the theoretical results are extended to the replacement of the parameter values by their estimates for practical applications. Fifth, we compute asymptotic variances for the extended sample influence functions.