Discriminant analysis through a semiparametric model

We consider a semiparametric generalisation of normal-theory discriminant analysis. The semiparametric model assumes that, after unspecified univariate monotone transformations, the class distributions are multivariate normal. We introduce an estimation procedure based on the distribution quantiles, in which the parameters of the semiparametric model are estimated directly without estimating the nonparametric transformations. The procedure is computationally fast and the estimation accuracy is shown to have the usual parametric rate. The relationship between the method and more general nonparametric discriminant analysis is discussed. The semiparametric specification of the class densities is a submodel of the nonparametric log density functional analysis of variance model in which the main effects are completely nonparametric but the interaction terms are specified semiparametrically. Simulations and real examples are used to illustrate the procedure.