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tokens: we consider, the subhalo and two-halo term contributions are quite negligible. This is also the reason why we cannot place tight constraints on the subhalo fraction and subhalo mass. The contribution of the stellar mass of the BCG is only important at very small scales. Thus, the observed ESDs in this study can mainly provide us with the constraints on the host halo properties. Figure 6. The ESD measurements (shaded area) and the best-fit model predictions (black solid line) for groups in the four different mass bins. The dashed, dotted, dotted–dashed, and blue dashed lines are the contributions of an off-centered NFW profile, subhalo, stellar mass of the galaxy under consideration, and the two-halo term, respectively. Table 3 lists the best-fit parameters for groups in different mass bins. As we have seen from the likelihood distribution of parameters in Figure 5 , we can have a fairly good constraint on the halo mass of the group samples. However, if we compare the halo masses obtained from the ESDs with those obtained from the group catalog (c.f. Table 1 ), they are roughly underestimated by 0.1 to ∼0.2 dex. We will discuss this discrepancy in the following section. The overall off-center distances for our BCGs are quite small, assuring that BCGs are indeed good tracers of halo centers. On the other hand, the concentrations of the halos seem to be somewhat larger than the theoretical predictions (e.g., Zhao et al. 2009 ). However, since the concentration c and the off-center distance R sig are quite tightly correlated, if we adopt the lower value of R sig , the concentration c will drop significantly as well. Table 3 Posteriors of the Parameters We Fitted to Our Group Samples Listed in Table 4 are the best-fit parameters for the groups that are separated into the high and low X-ray luminosity subsamples. Although due to the smaller number of lens systems the error for each data point for our X-ray subsamples is somewhat larger and thus the constraints on the five parameters are somewhat weaker, we do see the prominent feature of the halo masses of the high X-ray luminosity subsamples being higher than their low luminosity counterparts in all mass bins. The difference is to 0.2 to ∼0.3 dex, which means that the high X-ray luminosity subsamples are nearly a factor of 2 more massive than their low X-ray luminosity counterparts. Table 4 Posteriors of the Parameters We Fitted to Our High and Low X-Ray Luminosity Subsamples 4. Eddington Bias of the Halo Mass Estimation Recent studies have shown that the combination of galaxy–galaxy lensing and the clustering of galaxies can be used to constrain cosmology (e.g., Cacciato et al. 2013 ; More et al. 2013 ; van den Bosch et al. 2013 ; Leauthaud et al. 2017 ), and that the halo masses estimated for groups using galaxy–galaxy lensing signals and hence the halo mass function can also be used to constrain the cosmological parameters. However, before doing so, one needs to make a careful study of the