Dealing with monotone likelihood in a model for speckled data

In this paper we study maximum likelihood estimation (MLE) of the roughness parameter of the distribution for speckled imagery (Frery et al., 1997). We discover that when a certain criterion is satisfied by the sample moments, the likelihood function is monotone and MLE estimates are infinite, implying an extremely homogeneous region. We implement three corrected estimators in an attempt to obtain finite parameter estimates. Two of the estimators are taken from the literature on monotone likelihood ([Firth, 1993] and [Jeffreys, 1946]) and one, based on resampling, is proposed by the authors. We perform Monte Carlo experiments to compare the three estimators. We find the estimator based on the Jeffreys prior to be the worst. The choice between Firth's estimator and the Bootstrap estimator depends on the value of the number of looks (which is given before estimation) and the specific needs of the user. We also apply the estimators to real data obtained from synthetic aperture radar (SAR). These results corroborate the Monte Carlo findings. Further clarification of the choice between the Firth and Bootstrap estimators will be obtained through future studies of the classification properties of these estimators.