  We propose a computationally feasible estimator for the needlet trispectrum,
which develops earlier work on the bispectrum by Donzelli et al. (2012). Our
proposal seems to enjoy a number of useful properties, in particular a) the
construction exploits the localization properties of the needlet system, and
hence it automatically handles masked regions; b) the procedure incorporates a
quadratic correction term to correct for the presence of instrumental noise and
sky-cuts; c) it is possible to provide analytic results on its statistical
properties, which can serve as a guidance for simulations. The needlet
trispectrum we present here provides the natural building blocks for the
efficient estimation of nonlinearity parameters on CMB data, and in particular
for the third order constants $g_{NL}$ and $\tau_{NL}$.
