  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.
