A Modified chi^2-Test for CMB Analyses

Abstract

We present a new general procedure for determining a given set of quantities. To this end, we define certain statistic, that we call 'modified $\chi^2$' ($\chi^2_M$), because of its similarity with the standard $\chi^2$. The terms of this $\chi^2_M$ are made up of the fluctuations of an unbiased estimator of some statistical quantities, and certain weights. Only the diagonal terms of the covariance matrix explicitly appear in our statistic, while the full covariance matrix (and not its inverse) is implicitly included in the calculation of the weights. Choosing these weights we may obtain, through minimising the $\chi^2_M$, the estimator that provides the minimum RMS, either for those quantities or for the parameters on which these quantities depend. In this paper, we describe our method in the context of Cosmic Microwave Background experiments, in order to obtain either the statistical properties of the maps, or the cosmological parameters. The test here is constructed out of some estimator of the two-point correlation function at different angles. For the problem of one parameter estimation, we show that our method has the same power as the maximum likelihood method. We have also applied this method to Monte Carlo simulations of the COBE-DMR data, as well as to the actual 4-year data, obtaining consistent results with previous analyses. We also provide a very good analytical approximation to the distribution function of our statistic, which could also be useful in other contexts.