Type Ia supernova parameter estimation: a comparison of two approaches using current datasets

We compare the traditional χ² and complete-likelihood approaches for determining parameter constraints from type Ia supernovae (SNe Ia) when the magnitude dispersion is to be estimated as well. The dataset we used was sample combination "e" from Kessler (2009, ApJS, 185, 32), comprising the first-year SDSS-II Supernova Survey together with ESSENCE, SNLS, HST, and a compilation of nearby SNe Ia. We considered cosmological constant + cold dark matter (ΛCDM) and spatially flat, constant w dark energy + cold dark matter (FwCDM) cosmological models and show that, for current data, there is a small difference in the best-fit values and a difference of about 30% in confidence contour areas if the MLCS2k2 light-curve fitter is adopted. For the SALT2 light-curve fitter the differences in area are less significant (≲13%). In both cases the likelihood approach gives more restrictive constraints. We argue for using the complete likelihood instead of the χ² approach when dealing with parameters in the expression for the variance.