A modified χ2-test for cosmic microwave background analyses

Rubiño-Martín, J. A.; Betancort-Rijo, J. E.
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Monthly Notices of the Royal Astronomical Society, Volume 345, Issue 1, pp. 221-232.

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We present a new general procedure for determining a given set of quantities. To this end, we define a certain statistic, which we call `modified χ2' (χ2M), because of its similarity to the standard χ2. The terms of this χ2M 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 appear explicitly in our statistic, while the full covariance matrix (and not its inverse) is included implicitly in the calculation of the weights. Choosing these weights, we may obtain, through minimizing χ2M, 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-yr 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.