Robust Data Depth Procedures Based Weighted Estimator with Application in Multivariate Discriminant Analysis

Data depth concept used to measure the deepness of a given point in the entire multivariate data cloud. It leads to center-outward ordering of sample points used rather than usual smallest to largest rank. The ordering starts from middle and moves in all directions. Multivariate location and scatter can be computed by using the depth value of each data point. Various depth procedures have been established by many authors. In this paper, a new depth procedure is proposed, namely Modified Mahalanobis Depth (MMD), which calculates depth based on robust distance with Minimum Covariance Determinant (MCD) approach and a weight function is established to determine the location and scale. The superiority of the proposed depth based procedure over existing depth procedures has been studied in simulated environment using R software with respect to application in discriminant analysis. In order to study the superiority of the proposed data depth procedure (MMD) it has been applied in discriminant analysis by comparing the Apparent Error Rate (AER) in the context of classification problems. From the experiment, through real and simulation studies, it reveals that, the AER of proposed data depth procedure is almost similar to existing depth procedures in case of less contamination level. But, when the contamination level, sample size and the number of dimension increases, the AER of the proposed data depth procedure (MMD) is less compared with other existing depth procedures. The proposed depth procedure performs well when compared with the existing procedures even with higher contamination levels and larger sample sizes. Further, it is concluded that the proposed procedure gives more accuracy in the context of classifying the objects when compared with the existing procedures. The proposed procedure is most suitable to the research communities who are performing statistical data analysis techniques by computing the measure of location and scatter. The data depth procedure introduced in this thesis can be beneficial to researchers, who work on machine learning techniques by considering the factors such as noise, computational time, ease algorithm approach and high dimensionality.