Fiducial inference based on order statistics in location-scale and log-location-scale families
Iiyambo, Petrus Tweuthigilwa
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Both exact and approximate statistical inference for the parameters and the quantiles of location-scale, log-location-scale and location-scale-shape families of distributions are usually derived from likelihood-based methods. However, parameter estimation using exact and approximate maximum likelihood-based methods can be difficult and may require extensive programming, especially when dealing with censored samples. In some cases, maximum likelihood estimation based on censored samples may encounter convergence problems. Alternative methods of parameter estimation in location-scale, log-location-scale and location-scale-shape distributions have been developed by many researchers. In this thesis we develop exact rank-based conventional and fiducial generalized methods of inference for location and scale parameters of distributions belonging to the location-scale and log-location-scale families using the generalized least squares approach. Furthermore, we propose rank-based fiducial generalized methods of inference for location, scale and shape parameters of distributions belonging to location-scale-shape families of distributions using rank-based, iterative generalized least squares methods, and a Gibbs sampler. We compare through simulation and practical applications the results, inter alia the coverage probabilities and average lengths of conventional and fiducial generalized confidence intervals for the parameters and quantiles of location-scale, log-location-scale, and location-scale-shape distributions, obtained using our proposed methods of inference with alternative methods existing in literature, for example, exact and approximate maximum likelihood-based methods. For the cases of location-scale and log-location-scale families of distributions involving one or two-sample situations, our simulation results show that the proposed exact rank-based (conventional and fiducial generalized) methods are very competitive with exact and approximate maximum likelihood-based methods in terms of relative lengths of confidence intervals for the model parameters, parameter contrasts and quantiles of distribution. Moreover, rank-based methods produce confidence intervals for the model parameters, parameter contrasts and quantiles of distribution with good properties. In terms of practical application, our proposed rank-based methods produce confidence intervals for the model parameters, parameter contrasts and quantiles of distribution that are very close to the confidence intervals calculated using exact maximum likelihood-based methods. When calculating rank-based fiducial generalized confidence intervals for the model parameters and quantiles of location-scale-shape family of distributions (when the shape parameter 𝜉��������>0), using iterative generalized least squares approach based on Gibbs sampler algorithm, two different parametrizations, namely the 𝜃�������� and 𝜃��������∗ parametrizations are investigated. Our simulation results show that the 𝜃��������∗ parametrization produces, overall, better and more stable estimates compared to 𝜃�������� parametrization. Finally, results of illustrative examples of data modelled by the Generalized Extreme Value distribution for the case when 𝜉��������>0, are comparable to the results based on the same data obtained using Bayesian methods.