Goodness of Fit of a Discriminant Function from the Vector Space of Dummy Variables

Abstract

Summary The problem of discrimination among several groups can be looked upon as the problem of “association” between the vector of variables under consideration and a vector of “dummy” variables representing the contrasts among the groups. In certain situations, observations on one or more of these dummy variables are explicitly available and tests have been derived by Bartlett and Williams, for the adequacy of a set of assigned discriminant functions from among these dummy variables. In this paper, the statistical distributions of these test criteria are derived analytically, assuming the null hypothesis to be true.

Elliptically contoured model and factorization of wllks1A

“…Anderson, Fang, and Hsu (1986) derive several null robust Wilks' A tests for the ECM (also see Gupta and Varga, 1993). The present paper shows that Kshirsagar's (1971) normal theory null densities hold for the ECM. The derivations of the densities of A1, A2, A3, Aj, and A5 are recorded in the next section.…”

“…The Bartlett decomposition of the underlying multivariate beta distribution into independent factors determines the distribution of these factors. The present paper generalizes Kshirsagar's (1971) normal theory to the elliptically contoured model, and shows that his results are null robust for the elliptically contoured model. …”

“…Using the above Bartlett decompositions, Kshirsagar (1971) records the densities of A2, Ag, and 4 to be respectively A(n -g,g,q -g), A(n -2g,p -g,q -g), and A(n -g,p -g,q -g) where A(n -q,p,q -g) denotes the density of M (p x p) given by where the letter K generically denotes the normalizing constants of density functions in this paper. Now an n component vector y is said to have an n-variate elliptically contoured model (ECM) density if its characteristic function @(t) is of the form Thus e.g., has the characteristic function…”

“…Following Kshirsagar (1971) we assume the context of multivariate normal linear regression model to be…”

“…

In a series of papers, Kshirsagar (1964Kshirsagar ( , 1971 and McHenry and Kshirsagar (1977), factorize Wilks' A into a number of factors and find the independent null multivariate beta densities of these factors. These factors are the likelihood ratio test criteria for testing the goodness of fit of certain assigned discriminant functions or canonical variables either in the space of independent or dependent variables.

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