its corresponding cumulative probability distribution.](fig4_2.eps "fig:"){width="0.9\linewidth"}

The most important assumption of “Area Minimization Statistic" is that the measurement errors are Gaussian, which should be satisfied by the mock sample with GP reconstructed curves and possible variation of $\mathcal{D}$ away from $\mathcal{D}_{\rm obs}$. More specifically, such statistic is realized by a Gaussian randomized value $$\mathcal{D}_{i,\,{\rm mock}}(\langle z_l\rangle, z_s)=\mathcal{D}_{i,\,{\rm obs}}(\langle z_l\rangle, z_s)+r
\sigma_{\mathcal{D}_{i,obs}}\;,$$ where $r$ is characterized by a Gaussian distribution $r=0.0\pm1.0$, and $\mathcal{D}_{i,\,{\rm obs}}(\langle z_l\rangle, z_s)$ represents the actual measurement at source redshift $z_s$, with 1$\sigma$ error denoted by $\sigma_{\mathcal{D}_{i,obs}}$. Therefore, the function $\mathcal{D}_{\rm mock}(\langle z_l\rangle,
z_s)$ corresponding to mock sample could be straightforwardly obtained. Finally, a normalized absolute area difference between $\mathcal{D}_{\rm mock}(\langle z_l\rangle, z_s)$ and the GP reconstructed function of the actual data can be defined as $$\Delta A= \int_{z_{\rm min}}^{z_{\rm
max}}dz_s\bigg(\frac{\big|\mathcal{D}_{\rm mock}(\langle z_l\rangle,
z_s)-
        \mathcal{D}_{\rm obs}(\langle z_l\rangle, z_s)\big|}{\sigma_{\mathcal{D}}}\bigg)\;,$$ where $z_{\rm min}$ and $z_{\rm max}$ are the minimum and maximum redshifts of the mock sample. This process is repeated 10000 times in order to guarantee unbiased final results, from which one could derive a distribution of frequency versus area differential $\Delta
A$ and the cumulative probability distribution. The results are shown in Fig. 4, in which one can clearly 