  We consider the problem of estimating the direction of arrival of a signal
embedded in $K$-distributed noise, when secondary data which contains noise
only are assumed to be available. Based upon a recent formula of the Fisher
information matrix (FIM) for complex elliptically distributed data, we provide
a simple expression of the FIM with the two data sets framework. In the
specific case of $K$-distributed noise, we show that, under certain conditions,
the FIM for the deterministic part of the model can be unbounded, while the FIM
for the covariance part of the model is always bounded. In the general case of
elliptical distributions, we provide a sufficient condition for unboundedness
of the FIM. Accurate approximations of the FIM for $K$-distributed noise are
also derived when it is bounded. Additionally, the maximum likelihood estimator
of the signal DoA and an approximated version are derived, assuming known
covariance matrix: the latter is then estimated from secondary data using a
conventional regularization technique. When the FIM is unbounded, an analysis
of the estimators reveals a rate of convergence much faster than the usual
$T^{-1}$. Simulations illustrate the different behaviors of the estimators,
depending on the FIM being bounded or not.
