  The Neyman-Pearson (NP) paradigm in binary classification seeks classifiers
that achieve a minimal type II error while enforcing the prioritized type I
error controlled under some user-specified level $\alpha$. This paradigm serves
naturally in applications such as severe disease diagnosis and spam detection,
where people have clear priorities among the two error types. Recently, Tong,
Feng and Li (2018) proposed a nonparametric umbrella algorithm that adapts all
scoring-type classification methods (e.g., logistic regression, support vector
machines, random forest) to respect the given type I error upper bound $\alpha$
with high probability, without specific distributional assumptions on the
features and the responses. Universal the umbrella algorithm is, it demands an
explicit minimum sample size requirement on class $0$, which is often the more
scarce class, such as in rare disease diagnosis applications. In this work, we
employ the parametric linear discriminant analysis (LDA) model and propose a
new parametric thresholding algorithm, which does not need the minimum sample
size requirements on class $0$ observations and thus is suitable for small
sample applications such as rare disease diagnosis. Leveraging both the
existing nonparametric and the newly proposed parametric thresholding rules, we
propose four LDA-based NP classifiers, for both low- and high-dimensional
settings. On the theoretical front, we prove NP oracle inequalities for one
proposed classifier, where the rate for excess type II error benefits from the
explicit parametric model assumption. Furthermore, as NP classifiers involve a
sample splitting step of class $0$ observations, we construct a new adaptive
sample splitting scheme that can be applied universally to NP classifiers, and
this adaptive strategy reduces the type II error of these classifiers.
