Likelihood and Bayesian Inference in the Lomax Distribution under Progressive Censoring
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The Lomax distribution has been used as a statistical model in several fields, especially for business failure data and reliability engineering. Accurate parameter estimation is very important because it is the base for most inferences from this model. In this paper, we shall study this problem in detail. We developed several points and interval estimators for the parameters of this model assuming the data are type II progressively censored. Specifically, we derive the maximum likelihood estimator and the associated Wald interval. Bayesian point and interval estimators were considered. Since they can't be obtained in a closed form, we used a Markov chains Monte Carlo technique, the so called the Metropolis – Hastings algorithm to obtain approximate Bayes estimators and credible intervals. The asymptotic approximation of Lindley to the Bayes estimator is obtained for the present problem. Moreover, we obtained the least squares and the weighted least squares estimators for the parameters of the Lomax model. Simulation techniques were used to investigate and compare the performance of the various estimators and intervals developed in this paper. We found that the Lindley's approximation to the Bayes estimator has the least mean squared error among all estimators and that the Bayes interval obtained using the Metropolis – Hastings to have better overall performance than the Wald intervals in terms of coverage probabilities and expected interval lengths. Therefore, Bayesian techniques are recommended for inference in this model. An example of real data on total rain volume is given to illustrate the application of the methods developed in this paper.
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