model. Thus, in the framework of a spatially flat DGP model, we can directly rewrite the above equation and obtain the angular diameter distance between redshifts $z_1$ and $z_2$ $$\begin{aligned}
D_A^{DGP}(z_1,z_2)&=&\frac{c}{H_0 (1+z_2)}\times\nonumber\\
\null&\null&\hskip-0.3in
\int^{z_2}_{z_1}[\sqrt{\Omega_m(1+z)^3+\Omega_{r_c}}+\sqrt{\Omega_{r_c}}]^{-1}
dz.\nonumber\end{aligned}$$

Now we introduce a new statistic, the “Area Minimization Statistic" to constrain cosmological parameters, which has been recently proposed to test the evolution of the Universe, and then applied to the investigation of dynamical properties of dark energy [@Yennapureddy17; @Melia18c]. It should be noted that our reconstructed distance ratio $\mathcal{D}_{\rm obs}$, with the corresponding theoretical model value $\mathcal{D}^{\rm \Lambda{\rm CDM}}$ or $\mathcal{D}^{\rm DGP}$, is an continuous function. Therefore, the discrete sampling statistics (such as the $\chi^2$ statistic) is not sufficient enough to provide an effective way to make a comparison between different models, because the sampling at random points to obtain the squares of differences between model and reconstructed curve would lose information between these points.

![[*Top panel:*]{} The distribution of frequency versus area differential $\Delta A$ for a mock sample with lens bin $0.3<z_l<0.32$; [*Bottom panel:*]{} its corresponding cumulative probability distribution.](fig4_1.eps "fig:"){width="0.9\linewidth"}\
0.2in ![[*Top panel:*]{} The distribution of frequency versus area differential $\Delta A$ for a mock sample with lens bin $0.3<z_l<0.32$; [*Bottom panel:*]{} 