  The trispectrum of the cosmic microwave background can be used to assess the
level of non-Gaussianity on cosmological scales. It probes the fourth order
moment, as a function of angular scale, of the probability distribution
function of fluctuations and has been shown to be sensitive to primordial
non-gaussianity, secondary anisotropies (such as the Ostriker-Vishniac effect)
and systematic effects (such as astrophysical foregrounds). In this paper we
develop a formalism for estimating the trispectrum from high resolution sky
maps which incorporates the impact of finite sky coverage. This leads to a
series of operations applied to the data set to minimize the effects of
contamination due to the Gaussian component and correlations between estimates
at different scales. To illustrate the effect of the estimation process, we
apply our procedure to the BOOMERanG data set and show that it is consistent
with Gaussianity. This work presents the first estimation of the CMB
trispectrum on sub-degree scales.
