  In this article, we study the binomial mixture model under the regime that
the binomial size $m$ can be relatively large compared to the sample size $n$.
This project is motivated by the GeneFishing method (Liu et al., 2019), whose
output is a combination of the parameter of interest and the subsampling noise.
To tackle the noise in the output, we utilize the observation that the density
of the output has a U shape and model the output with the binomial mixture
model under a U shape constraint. We first analyze the estimation of the
underlying distribution F in the binomial mixture model under various
conditions for F. Equipped with these theoretical understandings, we propose a
simple method Ucut to identify the cutoffs of the U shape and recover the
underlying distribution based on the Grenander estimator (Grenander, 1956). It
has been shown that when $m = {\Omega}(n^{2/3})$, the identified cutoffs
converge at the rate $O(n^{-1/3})$. The $L_1$ distance between the recovered
distribution and the true one decreases at the same rate. To demonstrate the
performance, we apply our method to varieties of simulation studies, a GTEX
dataset used in (Liu et al., 2019) and a single cell dataset from Tabula Muris.
