Abstract
Under normality, Flury and Schmid [Quadratic discriminant functions with constraints on the covariances matrices: some asymptotic results, J. Multivariate Anal. 40 (1992) 244–261] investigated the asymptotic properties of the quadratic discrimination procedure under hierarchical models for the scatter matrices, that is: (i) arbitrary scatter matrices, (ii) common principal components, (iii) proportional scatter matrices and (iv) identical matrices. In this paper, we study the properties of robust quadratic discrimination rules based on robust estimates of the involved parameters. Our analysis is based on the partial influence functions of the functionals related to these parameters and allows to derive the asymptotic variances of the estimated coefficients under models (i)–(iv). From them, we conclude that the asymptotic variances verify the same order relations as those obtained by Flury and Schmid [Quadratic discriminant functions with constraints on the covariances matrices: some asymptotic results, J. Multivariate Anal. 40 (1992) 244–261] for the classical estimators. We also perform a Monte Carlo study for different sample sizes and different hierarchies which shows the advantage of using robust procedures over classical ones, when anomalous data are present. It also confirms that better rates of misclassification can be achieved if a more parsimonious model among all the correct ones is used instead of the standard quadratic discrimination.
Highlights
Assume that we are dealing with independent observations from two independent samples in Rp with location parameter μi and dispersion/covariance matrix i, i = 1, 2
The matrices satisfy a common principal component (CPC) model, i.e., i = i T, i = 1, 2, where = 1, . . . , p is the orthogonal matrix of the common eigenvectors and i = diag( i1, . . . , ip) are diagonal matrices containing the eigenvalues for each population
1 j ni,1 distributed i 2 are independent observations from within each sample, following a general multivariate location–dispersion distribution functionals such that mi (Fi) with location parameter μi and scatter matrix i that do not need to be equal to the population mean and covariance matrix, since we do not assume the existence of second moments as in the classical setting
Summary
When x has an elliptically symmetric distribution F, i.e, when the distributions of the neighborhood are restricted to be spherically symmetric, the robust location functional equals μ while the robust scatter functional is proportional to These scatter functionals are calibrated so that under the central normal model they provide Fisher–consistent estimators of the covariance matrix. 1 j ni ,1 distributed i 2 are independent observations from within each sample, following a general multivariate location–dispersion distribution Fi with location parameter μi and scatter matrix i that do not need to be equal to the population mean and covariance matrix, since we do not assume the existence of second moments as in the classical setting. Some proofs are given in the Appendix while the others can be found in Bianco et al [1]
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