  We discuss the derivation of the analytic properties of the cross-power
spectrum estimator from multi-detector CMB anisotropy maps. The method is
computationally convenient and it provides unbiased estimates under very broad
assumptions. We also propose a new procedure for testing for the presence of
residual bias due to inappropriate noise subtraction in pseudo-$C_{\ell}$
estimates. We derive the analytic behavior of this procedure under the null
hypothesis, and use Monte Carlo simulations to investigate its efficiency
properties, which appear very promising. For instance, for full sky maps with
isotropic white noise, the test is able to identify an error of 1% on the noise
amplitude estimate.
