Modern biomedical datasets are often high-dimensional at multiple levels of biological organization. Practitioners must therefore grapple with data to estimate sparse or low-rank structures so as to adhere to the principle of parsimony. Further complicating matters is the presence of groups in data, each of which may have distinct associations with explanatory variables or be characterized by fundamentally different covariates. These themes in data analysis are explored in the context of classification. Vertex Discriminant Analysis (VDA) offers flexible linear and nonlinear models for classification that generalize the advantages of support vector machines to data with multiple classes. The proximal distance principle, which leverages projection and proximal operators in the design of practical algorithms, handily facilitates variable selection in VDA via nonconvex distance-to-set penalties directly controlling the number of active variables. Two flavors of sparse VDA are developed to address data in which instances may be homogeneous or heterogeneous with respect to predictors characterizing classes. Empirical studies illustrate how VDA is adapted to class-specific variable selection on simulated and real datasets, with an emphasis on applications to cancer classification via gene expression patterns.

Mahalanobis distance may be used as a measure of the disparity between an individual’s profile of scores and the average profile of a population of controls. The degree to which the individual’s profile is unusual can then be equated to the proportion of the population who would have a larger Mahalanobis distance than the individual. Several estimators of this proportion are examined. These include plug-in maximum likelihood estimators, medians, the posterior mean from a Bayesian probability matching prior, an estimator derived from a Taylor expansion, and two forms of polynomial approximation, one based on Bernstein polynomial and one on a quadrature method. Simulations show that some estimators, including the commonly-used plug-in maximum likelihood estimators, can have substantial bias for small or moderate sample sizes. The polynomial approximations yield estimators that have low bias, with the quadrature method marginally to be preferred over Bernstein polynomials. However, the polynomial estimators sometimes yield infeasible estimates that are outside the 0–1 range. While none of the estimators are perfectly unbiased, the median estimators match their definition; in simulations their estimates of the proportion have a median error close to zero. The standard median estimator can give unrealistically small estimates (including 0) and an adjustment is proposed that ensures estimates are always credible. This latter estimator has much to recommend it when unbiasedness is not of paramount importance, while the quadrature method is recommended when bias is the dominant issue.