  Observations of suspected coherent elastic neutrino-nucleus scatterings by
dark matter direct detection experiments highlight the need for an
investigation into the so-called "neutrino floor". We focus on the discovery
limit, a statistical concept to identify the neutrino floor, and analyze the
asymptotic behaviour of the profile binned likelihood ratio test statistic
where the likelihood is constructed by variate from events in each bin and pull
terms from neutrino fluxes. To achieve the asymptotic result, we propose two
novel methods: i) Asymptotic-Analytic method, which furnishes the analytic
result for large statistics, is applicable for more extra nuisance parameters,
and enables the identification of the most relevant parameters in the
statistical analysis; ii) Quasi-Asimov dataset, which is analogous to Asimov
dataset but with improved speed. Applying our methods to the neutrino floor, we
significantly accelerate the computation procedure compared to the previous
literature, and successfully address cases where Asimov dataset fails. Our
derivation on the asymptotic behavior of the test statistic not only
facilitates research into the impact of neutrinos on the search for dark
matter, but may also prove relevant in similar application scenarios.
