Robust Bayesian Estimation

Abstract

Bayes methods in statistical inference are one of the important methods, and most of the research and messages tend to use the Bayes method in the estimation process. The regular Bayes method does not meet this problem, so in this thesis it is possible to verify the existence of prior data conflict by modeling the parameters of the prior distribution and then comparing the standard deviation of the prior distribution with the standard deviation of the posterior distribution, if the value of the standard deviation of the prior distribution is greater than the deviation. The standard distribution for the posterior distribution, it means that there is a problem of prior data conflict. Then we used an approach to solve this problem through a set of prior distributions called this approach by the robust Bayesian method, to identify the behavior of the estimators, two types of failure models were used, the first Weibull distribution to match it with continuous data. The second is a (Binomial) distribution to match the discrete data, the regular Bayes method is compared with the robust Bayesian method by using integrated mean square error (IMSE). In the Weibull distribution, the scale parameter (θ) and the survival function were estimated for two simulation experiments, the first was in the case of prior data unconflict the second was in the case of prior data conflict, so the simulation results showed that the robust Bayes method is the best by using the comparison criterion integrated mean square error (IMSE). On the practical side, real data were collected from Al-Manathira Hospital of the Najaf Health Department for the deaths of heart attack patients for 2018, the time of admission of the patient to the hospital until death was recorded, which is the time Exit where a sample of (15) patients was collected and the test of goodness of fit showed that the data follow a Weibull distribution with two parameters, the robust Bayes method was used to estimate the scale parameter and the survival function. As for the Binomial distribution, the parameter (P) and survival function were estimated for two experiments from the first simulation, which was in the case of prior data unconflict, as for the second experiment, it was in the case of prior data conflict. The simulation results showed that the robust Bayes method is the best by using the comparison criterion (IMSE). On the practical side, real data were collected from Yarmouk Teaching Hospital on breast cancer patients’ mortality from 2010 to 2017, and the test of goodness of fit showed that the data follow a Binomial distribution, the robust Bayes method was used to estimate the parameter (P) and survival function.