Statistical Issues on the Neutrino Mass Hierarchy with $\Delta \chi^{2}$

The Neutrino Mass Hierarchy Determination ($\nu$ MHD) is one of the main goals of the major current and future neutrino experiments. The statistical analysis usually proceeds from a standard method, a single dimensional estimator $(1D-\Delta \chi^{2})$ that shows some draw-backs and concerns, together with a debatable strategy. The draw-backs and considerations of the standard method will be explianed through the following three main issues. First issue is the limited power of the standard method. The $\Delta \chi^{2}$ estimator provides us with different results when different simulation procedures were used. Second issue, when $\chi^{2}_{min(NH)}$ and $\chi^{2}_{min(IH)}$ are drawn in a $2D$ map, their strong positive correlation manifests $\chi^{2}$ as a bi-dimensional instead of single dimensional estimator. The overlapping between the $\chi^{2}$ distributions of the two hypotheses leads to the experiment sensitivity reduction. Third issue is the robustness of the standard method. When the JUNO sensitivity is obtained using different procedures, $\Delta \chi^{2}$ as one dimensional and $\chi^{2}$ as two dimensional estimator, the experimental sensitivity varies with the different values of the atmospheric mass, the input parameter. We computed the oscillation of $\vert\overline{\Delta \chi^{2}} \vert$ with the input parameter values, $\vert\Delta m^{2} \vert_{input}$. The MH significance using the standard method, $\Delta\chi^{2}$, strongly depends on the values of the parameter $\vert\Delta m^{2} \vert_{input}$. Consequently, the experiment sensitivity depends on the precision of the atmospheric mass. This evaluation of the standard method confirms the draw-backs.