The Crossing Statistic: Dealing with Unknown Errors in the Dispersion of Type Ia Supernovae

TL;DR

The Crossing Statistic is a new method designed to compare models with data when the intrinsic errors are poorly known, especially useful for cosmological supernova data, and it generalizes the traditional chi-squared approach.

Contribution

We introduce the Crossing Statistic, a novel goodness-of-fit measure less sensitive to unknown errors, improving model discrimination in cosmology.

Findings

The Crossing Statistic can distinguish models where chi-squared fails.

It reduces to chi-squared in the last mode, acting as a generalization.

Applicable to various data sets, especially supernova cosmology.

Abstract

We propose a new statistic that has been designed to be used in situations where the intrinsic dispersion of a data set is not well known: The Crossing Statistic. This statistic is in general less sensitive than `chi^2' to the intrinsic dispersion of the data, and hence allows us to make progress in distinguishing between different models using goodness of fit to the data even when the errors involved are poorly understood. The proposed statistic makes use of the shape and trends of a model's predictions in a quantifiable manner. It is applicable to a variety of circumstances, although we consider it to be especially well suited to the task of distinguishing between different cosmological models using type Ia supernovae. We show that this statistic can easily distinguish between different models in cases where the `chi^2' statistic fails. We also show that the last mode of the Crossing Statistic is identical to `chi^2', so that it can be considered as a generalization of `chi^2'.