  Motivated by applications in high-dimensional data analysis where strong
signals often stand out easily and weak ones may be indistinguishable from the
noise, we develop a statistical framework to provide a novel categorization of
the data into the signal, noise, and indistinguishable subsets. The
three-subset categorization is especially relevant under high-dimensionality as
a large proportion of signals can be obscured by the large amount of noise.
Understanding the three-subset phenomenon is important for the researchers in
real applications to design efficient follow-up studies. %For example,
candidates belonging to the signal subset may have priority for more focused
study, while those in the noise subset can be removed; and, for candidates in
the indistinguishable subset, additional data may be collected to further
separate weak signals from the noise. We develop an efficient data-driven
procedure to identify the three subsets. Theoretical study shows that, under
certain conditions, only signals are included in the identified signal subset
while the remaining signals are included in the identified indistinguishable
subsets with high probability. Moreover, the proposed procedure adapts to the
unknown signal intensity, so that the identified indistinguishable subset
shrinks with the true indistinguishable subset when signals become stronger.
The procedure is examined and compared with methods based on FDR control using
Monte Carlo simulation. Further, it is applied successfully in a real-data
application to identify genomic variants having different signal intensity.
