  We propose a novel semi-parametric classifier based on Mahalanobis distances
of an observation from the competing classes. Our tool is a generalized
additive model with the logistic link function that uses these distances as
features to estimate the posterior probabilities of the different classes.
While popular parametric classifiers like linear and quadratic discriminant
analyses are mainly motivated by the normality of the underlying distributions,
the proposed classifier is more flexible and free from such parametric
assumptions. Since the densities of elliptic distributions are functions of
Mahalanobis distances, this classifier works well when the competing classes
are (nearly) elliptic. In such cases, it often outperforms popular
nonparametric classifiers, especially when the sample size is small compared to
the dimension of the data. To cope with non-elliptic and possibly multimodal
distributions, we propose a local version of the Mahalanobis distance.
Subsequently, we propose another classifier based on a generalized additive
model that uses the local Mahalanobis distances as features. This nonparametric
classifier usually performs like the Mahalanobis distance based semiparametric
classifier when the underlying distributions are elliptic, but outperforms it
for several non-elliptic and multimodal distributions. We also investigate the
behaviour of these two classifiers in high dimension, low sample size
situations. A thorough numerical study involving several simulated and real
datasets demonstrate the usefulness of the proposed classifiers in comparison
to many state-of-the-art methods.
