  We develop optimised estimators of two sorts of power spectra for fields
defined on the sky, in the presence of partial sky coverage. The first is the
cross-power spectrum of two fields on the sky; the second is the skew spectrum
of three fields. These can probe the Integrated Sachs Wolfe Effect (ISW) at
large angular scales and the Sunyaev Z\'eldovich (SZ) effect from hot gas in
clusters at small angular scales. The skew spectrum, recently introduced by
Munshi & Heavens (2009), is an optimised statistic which can be tuned to study
a particular form of non-Gaussianity, such as may arise in the early Universe,
but which retains information on the nature of non-Gaussianity. In this paper
we develop the mathematical formalism for the skew spectrum of 3 different
fields. When applied to the CMB, this allows us to explore the contamination of
the skew spectrum by secondary sources of CMB fluctuations. Considering the
three-point function, the study of the bispectrum provides valuable information
regarding cross-correlation of secondaries with lensing of CMB with much higher
significance compared to just the study involving CMB sky alone. We use our
analytical models to study specific cases of cosmological interest which
include cross-correlating CMB with various large scale tracers to probe ISW and
SZ effects for cross spectral analysis and use the formalism to study the
signal-to-noise ratio for detection of the weak lensing of the CMB by
cross-correlating it with different tracers as well as point sources for CMB
experiments such as Planck (abridged).
