Classification and discrimination problems with applications, part I

In Part I exact results for univariate (“p= 1”) two-group (“k = 2”) classification problems were derived assuming normality and equality of the variances. In Part IIa asymptotic results for multivariate (“p> I”) two-group classification and discrimination problems are based on the corresponding assumptions of multivariate normality and equality of the covariance matrices. The results (4.6.5), (4.6.6) and (4.6.7) are believed to be new. The asymptotic results in Section 4.6, together with results presented elsewhere in the literature, constitute the basis of various detailed proposals to deal with problems from actual statistical practice. Most of these proposals are modifications or specifications of existing ones. We shall pay some attention to (I) testing whether differences exist. But we are mainly interested in: (II) constructing a discriminant function, (III) assigning the individual under classification, and in (IV) constructing a confidence interval for “the” posterior probability that the individual under classification belongs to Population 2. An important part in our theory is played by various techniques for selecting variables in discriminant analysis. The need for such techniques follows from Section 4.10. The consequences of building-in a selection technique are discussed in Section 4.12. One of our proposals motivates the theory presented in Chapter 3 and is mentioned here for that reason: employ a large part of the data, say 70%, in order to construct a discriminant function (via a selection of variables); by applying this function to the rest of the data, the exact univariate theory of Part I becomes of application. Part IIb will contain a chapter on applications.