can possibly be due to the fact that $\sigma_{\text{int}}$ is obtained using only a nearby sample in the $\chi^2$ approach for MLCS2k2 while such distinction is not performed in the likelihood approach. This issue deserves further investigation and will be the subject of future work.

![Distributions for $\sigma_{\text{int}}$ for the $\Lambda$CDM model from the likelihood approach. We marginalized over all other parameters with flat prior. The horizontal lines depict the 68%, 95% and 99% confidence intervals. The dashed line is the result for the $\chi^2$ approach. *Left panel:* Results for the MLCS2k2 version of the SDSS compilation. *Right panel:* Results for the SALT2 version of the SDSS compilation. \[fig:SDSS\_sint\_dist\]](FIG4_a.eps "fig:"){width="49.00000%"} ![Distributions for $\sigma_{\text{int}}$ for the $\Lambda$CDM model from the likelihood approach. We marginalized over all other parameters with flat prior. The horizontal lines depict the 68%, 95% and 99% confidence intervals. The dashed line is the result for the $\chi^2$ approach. *Left panel:* Results for the MLCS2k2 version of the SDSS compilation. *Right panel:* Results for the SALT2 version of the SDSS compilation. \[fig:SDSS\_sint\_dist\]](FIG4_b.eps "fig:"){width="49.00000%"}

![image](FIG5_a.eps){width="49.00000%"} ![image](FIG5_b.eps){width="49.00000%"}\
![image](FIG5_c.eps){width="49.00000%"} ![image](FIG5_d.eps){width="49.00000%"}

We also allow for a possible variation of the parameters $\alpha$, $\beta$, $\mathcal{M}$ and $\sigma_\mathrm{int}$ with redshift, in the context of 