  A classifier for two or more samples is proposed when the data are
high-dimensional and the underlying distributions may be non-normal. The
classifier is constructed as a linear combination of two easily computable and
interpretable components, the $U$-component and the $P$-component. The
$U$-component is a linear combination of $U$-statistics which are averages of
bilinear forms of pairwise distinct vectors from two independent samples. The
$P$-component is the discriminant score and is a function of the projection of
the $U$-component on the observation to be classified. Combined, the two
components constitute an inherently bias-adjusted classifier valid for
high-dimensional data. The simplicity of the classifier helps conveniently
study its properties, including its asymptotic normal limit, and extend it to
multi-sample case. The classifier is linear but its linearity does not rest on
the assumption of homoscedasticity. Probabilities of misclassification and
asymptotic properties of their empirical versions are discussed in detail.
Simulation results are used to show the accuracy of the proposed classifier for
sample sizes as small as 5 or 7 and any large dimensions. Applications on real
data sets are also demonstrated.
