  This paper proposes improved methods for the maximum likelihood (ML)
estimation of the equivalent number of looks $L$. This parameter has a
meaningful interpretation in the context of polarimetric synthetic aperture
radar (PolSAR) images. Due to the presence of coherent illumination in their
processing, PolSAR systems generate images which present a granular noise
called speckle. As a potential solution for reducing such interference, the
parameter $L$ controls the signal-noise ratio. Thus, the proposal of efficient
estimation methodologies for $L$ has been sought. To that end, we consider
firstly that a PolSAR image is well described by the scaled complex Wishart
distribution. In recent years, Anfinsen et al. derived and analyzed estimation
methods based on the ML and on trace statistical moments for obtaining the
parameter $L$ of the unscaled version of such probability law. This paper
generalizes that approach. We present the second-order bias expression proposed
by Cox and Snell for the ML estimator of this parameter. Moreover, the formula
of the profile likelihood modified by Barndorff-Nielsen in terms of $L$ is
discussed. Such derivations yield two new ML estimators for the parameter $L$,
which are compared to the estimators proposed by Anfinsen et al. The
performance of these estimators is assessed by means of Monte Carlo
experiments, adopting three statistical measures as comparison criterion: the
mean square error, the bias, and the coefficient of variation. Equivalently to
the simulation study, an application to actual PolSAR data concludes that the
proposed estimators outperform all the others in homogeneous scenarios.
