Thesis (M.Sc.) - Cairo University - Faculty of Graduate Studies for Statistical Research - Department of Mathematical Statistics

In many statistical situations, classical distributions do not provide appropriate fits to real data. Recently, attempts have been made to define new families of probability distributions that extend well-known distributions and at the same time provide great flexibility in modeling data in practice. In this thesis, a three-parameter continuous distribution is constructed called power transmuted inverse Rayleigh using a power transformation methodology. A comprehensive account of the statistical properties is discussed including; quantile function, moments, order statistics, incomplete moments, mean residual life function and Rényi entropy. Furthermore, estimation by methods of maximum likelihood, least squares and percentiles are discussed. A simulation study is implemented to compare the performance of different estimates. Finally, a real data application is used to illustrate the usefulness of the proposed distribution in modelling real data. Furthermore, Bayesian estimation is used to estimate the population parameters of the newproposed distribution based on informative and non- informative priors represented by gamma and Jeffery{u2019}s priors respectively.The Bayesian estimators are motivated by four loss functions which are minimum expected loss function, squared error loss function, precautionary lossfunctionandlinear-exponential loss function. Markov Chain Monte Carlo method is implemented for investigating the accuracy of estimates for different sample sizes. Numerical study is performed based on relative absolute biases and estimated risk in order to examine and compare the behavior of the parameters{u2019} Bayesian estimates

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