Topp–Leone Family of Distributions: Some Properties and Application

This paper focuses on the application of Markov Chain Monte Carlo (MCMC) technique for estimating the parameters of log-logistic (LL) distribution which is dependent on a complete sample. To find Bayesian estimates for the parameters of the LL model OpenBUGS—established software for Bayesian analysis based on MCMC technique, is employed. It is presumed that samples for independent non informative set of priors for estimating LL parameters are drawn from posterior density function. A proposed module was developed and incorporated in OpenBUGS to estimate the Bayes estimators of the LL distribution. It is shown that statistically consistent parameter estimates and their respective credible intervals can be constructed through the use of OpenBUGS. Finally comparison of maximum likelihood estimate and Bayes estimates is carried out using three plots. Additively through this research it is established that computationally MCMC technique can be effortlessly put into practice. Elaborate procedure for applying MCMC, to estimate parameters of LL model, is demonstrated by making use of real survival data relating to bladder cancer patients.

In this study, we will look at estimating the parameters of the Gompertz distribution. We know that the maximum likelihood technique is the most often used method in the literature for parameter estimation. However, it is well known that the maximum likelihood estimators (MLEs) are biased for small sample sizes. As a result, we are motivated to produce nearly unbiased estimators for the parameters of this distribution. To be more specific, we concentrate on two bias-correction strategies (analytical and bootstrap approaches) to minimize MLE biases to the second order of magnitude. Monte Carlo simulations are used to compare the performances of these estimators. Finally, two real-data examples are offered to demonstrate the utility of our proposed estimators in small sample sizes.

In this paper, we discuss the parameters estimation of the Burr XII distribution. We know that the most popular method in the literature for parameter estimation is the maximum likelihood method. However, the maximum likelihood estimators (MLEs) are widely known to be biased for small sample sizes. Therefore, this motivate us to obtain approximately unbiased estimators for this distribution' parameters. Precisely, we focus on two bias-correction techniques (analytical and bootstrap approaches) to reduce the biases of the MLEs to the second order of magnitude. In order to compare the performance of these estimators, Monte Carlo simulations are conducted. Lastly, two real-data examples are provided to show the usefulness of these proposed estimators when sample sizes are small.