to minimizing $\mathcal{L}$. However, this is not the case for current SNe Ia observations and neglecting the last but one term in Eq. (\[loglike\]) would, in principle, lead to a biased result. Our proposal is minimizing the following functions for each case discussed in the previous section $$\label{like_salt}
\mathcal{L}_{\mathrm{SALT2}}(\boldsymbol{\theta}, \boldsymbol{\delta}, {\cal M},
\sigma_\mathrm{int})=\chi^2_{\mathrm{SALT2}}(\boldsymbol{\theta},
\boldsymbol{\delta}, {\cal M}, \sigma_\mathrm{int})
+\sum_i^N\ln(\sigma_i^2(\boldsymbol{\delta}) + \sigma_\mathrm{int}^ 2)$$ and $$\label{like_mlcs}
\mathcal{L}_{\mathrm{MLCS2k2}}(\boldsymbol{\theta}, h,
\sigma_\mathrm{int})=\chi^2_{\mathrm{MLCS2k2}}(\boldsymbol{\theta}, h,
\sigma_\mathrm{int})
+\sum_i^N\ln(\sigma_i^2 +\sigma_\mathrm{int}^ 2),$$ where we neglected parameter-independent terms. $\chi^2_{\mathrm{SALT2}}(\boldsymbol{\theta}, \boldsymbol{\delta}, {\cal M},
\sigma_\mathrm{int})$ and $\chi^2_{\mathrm{MLCS2k2}}(\boldsymbol{\theta}, h, \sigma_\mathrm{int})$ are given, respectively, by Eqs. (\[chi2\_salt2\]) and (\[chi2\_mlcs2k2\]) now considering $\sigma_\mathrm{int}$ *also as a free parameter*. With this procedure, we can obtain directly unbiased probability distributions functions for all parameters, including $\sigma_\mathrm{int}$ and $\boldsymbol{\delta}$.

Results {#sec:results}
=======

In this section we compare the results obtained from the $\chi^2$ and the likelihood approaches, as described in Sections \[sec:least\_squares\] and \[sec:likelihood\], using real data from the SDSS first year compilation [@Kessler2009].

In order to perform the comparison, we considered the following cosmological models:

-   $\Lambda$CDM, in which we can write the Friedmann equation, in terms of the parameters $\boldsymbol{\theta}=(\Omega_{m0},\Omega_{k0})$, as $$E^2(z; \boldsymbol{\theta})= \Omega_{m0}(1+z)^3 +\Omega_{k0}(1+z)^2
    +(1-\Omega_{m0}-\Omega_{k0}).$$

-   F$w$CDM, described by $\boldsymbol{\theta}=(\Omega_{m0},w)$ and $$E^2(z; \boldsymbol{\theta})= \Omega_{m0}(1+z)^3 +
    %  \Omega_{k0}(1+z)^2 