Bayesian Estimation of the Shape Parameter of Generalized Rayleigh Distribution Under Symmetric and Asymmetric Loss Functions Chapter One Aim and Objectives of the Study

 

1. By estimating the shape parameter (α) when the scale parameter (λ) is known using both informative and non-informative priors under symmetric lossfunction
2. By estimating the shape parameter (α) when the scale parameter (λ) is known using both informative and non-informative priors under asymmetric lossfunction
3. To compare the performances of the proposed estimators with that of Maximum Likelihood Estimators in terms of Mean SquareError

CHAPTER TWO:

LITERATURE REVIEW

Kundu and Raqab (2005) considered different estimation procedures (maximum likelihood estimation, modified moment estimator, least square estimator, weighted least square estimator, percentile-based estimator and modified L-moment estimator) to estimate the unknown parameter(s) of a GRD. The study compared the performance of the different estimators using Monte-Carlo simulations mainly with respect to their biases and mean square errors for different sample sizes and different parameter values. (Kundu and Raqab, 2005) showed that, when the sample size is small (say, n=10), the performance of most of  the methods are quite bad. In particular, the estimation of the shape parameter (α) becomes very difficult for small sample sizes. The biases of all the methods are  quite severe for   small and moderate sample sizes (n ≤ 20). The study also showed that, the least square estimate performs quite well for n≤20 and for n ≥ 30, the weighted least square estimates outperforms the least square estimate marginally. If mean square error is considered, the weighted least square performs better than the rest in most of the cases considered.

Different methods of estimating R = P (Y < X ) when Y and X both follow GRD with different shape parameters but the same scale parameter were compared by (Raqab and Kundu, 2005). When the scale parameter is unknown, it is observed that the MLEs of the three unknown parameters can be obtained by solving one non-linear equation. An iterative procedure for computing the MLEs of the unknown parameters and the MLE of R was developed. The asymptotic distribution of R was obtained and this was used to compute the asymptotic confidence intervals. It was observed that, even when the sample size is quite small, the asymptotic confidence intervals work quite well. Two bootstrap confidence intervals were also proposed and their performances were also quite satisfactory. On the other hand, when the scale parameter is known they compare MLE and UMVUE with different Bayes estimators. It was observed that the Bayes estimators with non-informative priors behave quite similarly with the MLEs

In similar studies, Mahdi (2006), estimated the parameters of a Rayleigh distribution using five different estimation techniques namely: maximum likelihood, method of moment, probability weighted moment method, least square method and least absolute deviation method. He proposed the modified maximum likelihood estimation method for the parameters and compares it with the above methods. In his comparison, all the methods performed reasonably well except the method of moments. On the other hand, the modified maximum likelihood method provides better estimates  for  the  parameters  when  the sample sizes are not small (n≥10), while in the case of small samples, the probability weighted moment method outperforms the maximum likelihood method for the  estimation of the threshold parameter and performs almost as good as the  maximum  likelihood  method for the estimation of the scale parameter. Hence, Mahdi (2006) recommended the  use of modified maximum likelihood method for the parameter estimation of the Rayleigh distribution if the sample size is large.

A conventional method for estimating the two parameters of generalized Rayleigh distribution for different sample sizes (small, medium and large) was suggested by (AL- Naqeeb and Hamed, 2009). The estimators were compared by using mean square error (MSE) as a performance measure based on the simulated data used in the study.

Abdel-Hady (2013) extended the Marshall and Olkin’s bivariate exponential model to the Generalized Bivariate Rayleigh (GBR) Distribution. The CDF, pdf and the conditional distribution of the GBR distribution were computed. The maximum likelihood estimation procedure for the estimation of the GBR parameters when all parameters are unknown and the observed Fisher Information Matrix were also derived.

Al – Kanani and Abbas, (2014) studied the Bayesian and non-bayesian methods in estimating the parameters of the GRD. The non-bayesian m

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