$M_0(t, \lambda)$, $M_1(t,
    \lambda)$ are the zeroth and the first moments of the distribution of training sample SEDs. One might consider adding moments of higher order to Eq. (\[salt2restflux\]).

    To compare with photometric SNe Ia data, the (unredshifted) observer-frame flux in passband $Y$ is calculated as $$F^Y(p(1+z))=(1+z)\int d\lambda'\left[\lambda'
    \phi(p,\lambda')T^Y(\lambda' (1+z))\right],
    \label{salt2obsflux}$$ where $T^Y(\lambda)$ defines the transmission curve of the observer-frame filter $Y$, and $z$ is the redshift.

    As called attention to above, each SN Ia light curve is fitted separately using Eqs. (\[salt2restflux\]) and (\[salt2obsflux\]) to determine the parameters $x_0$, $x_1$, and $c$ with corresponding errors. However, the SALT2 light-curve fit does not yield an independent distance modulus estimate for each SN Ia. As we will see in the next section, the distance moduli are estimated as part of a global parameter fit to an ensemble of SN Ia light curves in which cosmological parameters and global SN Ia properties are also obtained.

In the next section we will discuss how to obtain constraints on cosmological parameters using MLCS2k2 and SALT2 output quantities as our data.

The traditional $\chi^2$ approach {#sec:least_squares}
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The prevailing SNe Ia cosmological analysis is based on the $\chi^2$ function: $$\label{chi2_base}
\chi ^2:=\boldsymbol{X}^T \boldsymbol{\Sigma}^{-1}\boldsymbol{X}$$ where $\boldsymbol{X}:=(\boldsymbol\mu-\boldsymbol{\mu}_{th}(\boldsymbol{z},
