Statistical Inference on the Modified Gumbel Distribution Parameters

The work in this thesis is concerned with the progress and development of the Gumbel distribution by the introduction of a new parameter namely, the shape parameter. Generalization of the Gumbel distribution is established. The work is also concerned with the investigation of the finite sample performance of asymptotic inference procedures using the likelihood function based on the modified distributions. The study includes investigating the adequacy of asymptotic inferential procedures in small samples. The maximum likelihood estimator of the parameters of modified distributions is not available in closed form. Thus a simulation study is conducted to investigate the bias, asymptotic variance (ASV), finite sample variance (FSV), and the mean square error (MSE) of the maximum likelihood estimator of the parameters of the modified distribution. Exact testing hypothesis procedures for the modified distribution are intractable. Therefore three standard large sample statistics based on maximum likelihood estimator were considered, which are the likelihood ratio, the Wald, and the Rao statistics. Their performances in finite samples in terms of their sizes and powers are investigated and compared. Confidence intervals based on the likelihood ratio, the Wald, and the Rao statistics were studied. The performances in terms of the attainment of the nominal error probability and symmetry of lower and upper probabilities were investigated and compared. The main findings of the simulation studies of the inference procedures for the parameters of the modified Gumbel distribution indicate that the estimate of the shape parameter is nearly unbiased, while estimates of the location and scale parameters tend to be slightly biased for small sample size of the univariate distribution, while for bivariate models, estimate of the scale and shape parameters performance are satisfactory in terms of bias and variance in all the situations considered. In the hypothesis testing o f the m odified d istribution, the 1 ikelihood ratio statistic appears to perform better than the Wald and the Rao statistics. Interval estimates for the scale parameter based on Wald and Rao statistics are highly symmetric and tend to be slightly anticonservative, while intervals based on the likelihood ratio statistics are in general symmetric and attain the nominal error probability. For the shape parameter, all intervals tend to be symmetric in the lower and upper error probabilities. Results of the simulations also indicate that the modified extreme value models can contribute meaningfully in solving several problems of the environmental data, particularly the air pollution data.

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