A robust estimation method of location and scale with application in monitoring process variability

Abstract

This thesis consists of two parts; theoretical and application. The first part proposes the development of a new method for robust estimation of location and scale, in data concentration step (C-step), of the most widely used method known as fast minimum covariance determinant (FMCD). This new method is as effective as FMCD and minimum vector variance (MVV) but with lower computational complexity. In FMCD, the optimality criterion of C-step is still quite cumbersome if the number of variables p is large because of the computation of sample generalized variance. This is the reason why MVV has been introduced. The computational complexity of the C-step in FMCD is of order O(p3 ) while MVV is O(p2 ). This is a significant improvement especially for the case when p is large. In this case, although MVV is faster than FMCD, it is still time consuming. Thus, this is the principal motivation of this thesis, that is, to find another optimal criterion which is of far higher computational efficiency. In this study, two other different optimal criteria which will be able to reduce the running time of C-step is proposed. These criteria are (i) the covariance matrix equality and (ii) index set equality. Both criteria do not require any statistical computations, including the generalized variance in FMCD and vector variance in MVV. Since only a logical test is needed, the computational complexities of the C-step are of order O(p ln p) . The second part is the application of the proposed criteria in robust Phase I operation of multivariate process variability based on individual observations. Besides that, to construct a more sensitive Phase II operation, both Wilks’ W statistic and Djauhari’s F statistic are used. Both statistics have different distributions and is used to measure the effect of an additional observation on covariance structure.