This paper presents adjusted profile likelihoods for α, the roughness parameter of the G 0 A (α, γ, L) distribution. This distribution has been widely used in the modeling, processing and analysis of data corrupted by speckle noise, e.g., synthetic aperture radar images. Specifically, we consider the following modified profile likelihoods: (i) the one proposed by Cox and Reid, and (ii) approximations to adjusted profile likelihood proposed by Barndorff-Nielsen, namely the approximations proposed by Severini and one based on results by Fraser, Reid and Wu. We focus on point estimation and on signalized likelihood ratio tests, the parameter of interest being the roughness parameter that indexes the distribution. As far as point estimation is concerned, the numerical evidence presented in the paper favors the Cox and Reid adjustment, and in what concerns signalized likelihood ratio tests, the results favor the approximation to Barndorff-Nielsen's adjustment based on the results by Fraser, Reid and Wu. An application to real synthetic aperture radar imagery is presented and discussed.

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Abstract. This paper presents several different adjusted profile likelihoods for the Weibull shape parameter. These adjustments aim at reducing the impact of the nuisance parameter on the likelihood-based inference regarding the parameter of interest. Both point estimation and hypothesis testing are considered. We also show that the ratio between the estimators and the shape parameter are pivotal quantities and that the size properties of the usual and adjusted profile likelihood ratio tests depend neither on the scale parameter nor on the value of the shape parameter set at the null hypothesis. The numerical results suggest that the adjustment obtained by Yang and Xie (2003) outperforms not only the profile likelihood inference but also inference based on competing adjusted profile likelihoods.

Abstract. We obtain adjustments to the profile likelihood function in Weibull regression models with and without censoring. Specifically, we consider two different modified profile likelihoods: (i) the one proposed by Cox and Reid (1987), and (ii) an approximation to the one proposed by Barndorff-Nielsen (1983), the approximation having been obtained using the results by Fraser and Reid (1995) and by Fraser et al. (1999). We focus on point estimation and likelihood ratio tests on the shape parameter in the class of Weibull regression models. We derive some distributional properties of the different maximum likelihood estimators and likelihood ratio tests. The numerical evidence presented in the paper favors the approximation to Barndorff-Nielsen's adjustment.

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