Comparative study to estimate the parameters of Failure probability Distributions and the Reliability using Maximum Likelihood and Bayes Methods

Abstract

The aim of this study is to apply lifetime models on the failure time of automatic teller machines (ATM) in Sudan and to estimate the reliability of machines, in order to compare between machines and methods of estimation, through use the failure models and the suitable statistical methods for failure data analysis. This research depends on Some hypothesis are that: Time to failure in ATM follows exponential distribution with one parameter, and decrease the reliability of the machines coincides with the increase of the machine's life and the increase of operating hours, the methods of statistical and mathematical approaches help in obtaining correct estimates of reliability if the data used in the estimations is correct and there is no significant differences between the mean of Maximum likelihood and Bayes parameter and the reliability estimators. Also the researcher presented the theoretical principles for Basic concepts and important measures for the reliability and the Lifetime-distributions and some methods of estimate the exponential distribution's lifetime models and Goodness of fit techniques. These it will be explained in chapter two and, three. The failure data has been taken from the Central Bank of Sudan, during the period of time (1/1/2017-31/5/2017). The statistical packages have been used for data analysis and models constricting are: Easy Fit, SPSS and Microsoft Excel. The comparison between 15 machines selected randomly (used simple random sample in selection) out of 28 machines from public banks in Sudan has done, through the lifetime models estimation used the Maximum likelihood and Bayes methods in estimate the parameters and reliability of exponential distribution. The comparison between methods of estimation has executed through mean square error values (MSE).And the most important results found out by researcher are: The failure-time of all machines follows exponential distribution with one-parameter and the hazard rate of machines is constant or independence of time. The mean square error values (MSE) of exponential parameter which is estimated by Maximum likelihood method is less than MSE of Bayes' estimator or the Maximum likelihood method is better than Bayes method in estimate the parameter and the reliability function for exponential distribution of automatic teller machines (ATM) in Sudan, The machines with high reliability have a low faults probability and hazard rate, while the machines with low reliability have a high fault probability and hazard vi rate, Whenever the mean time to failure for machine increases it indicates that the machine has high reliability and the reliability of the machines decreases when the working time of machines increases, the machines no (B28, B5 and B35) have high reliability compared to the other machines and the machines (B23and B33) have low reliability, and lastly there is no significant differences between the mean of Maximum likelihood and Bayes parameter and the reliability estimators at 𝛼 and 95% Confidence Interval of the Difference. Ultimately, as the researcher suggests that to emphasize the importance of the subject of reliability in the studies and evaluations the machines or the differentiation between one system and another. And it recommends that possibly, to depend on Maximum likelihood method to estimate the parameter of the Exponential distribution and the reliability of the ATM machines in Sudan. Furthermore, the researcher suggests that to extend the study span to include all types of ATM faults (out of cash and out of serves).Also the researcher recommends that when expanding or adding a new machine, it is preferable to buy the machine with high reliability. In addition to that, the researcher recommends that it is better to follow the remedial policy of maintenance for the machines which says that maintenance is made when a defect occurs and the precautionary policy should be replaced because the great percentage of failure will occur shortly after the operation of precautionary maintenance and finally the researcher suggests that to accurate recording of failures which occur in all machines that leads to determine the time, interval, type of failure as well as the cause of failure.