Using the Bayes estimator for Weibull parameters estimation taking into account left-truncated and rightcensored data

Today, when operation of commercial plants is organised, they are expected to comply to constantly increasing requirements for safety, dependability and efficiency of operation. The methods and procedures that are employed for the purpose of improving the safety and dependability of commercial plants are based on the information on the dependability of components, systems and equipment. In order to identify the objective dependability characteristics of such facilities, their behaviour in operation is monitored. In the course of facility operation monitoring, periods of continuous fault-free operation, periods of downtime, causes of downtime, failures, defects and malfunctions of items, frequency and depth of preventive maintenance of elements and systems, as well as other information are recorded. It should be noted that elements and systems of today’s industrial facilities, such as nuclear power plants, petrochemical complexes, etc., are classified as highly dependable equipment. Failures of such equipment are rare. The number of same-type facilities is extremely small.

Aim. Given the above, the problem arises of developing methods for reliable estimation if item dependability characteristics on the basis of limited statistical information.

Method. The paper examines a method for calculating facility dependability indicators on the basis of statistical information obtained in operation, i.e. a method for minimising the risk function while taking into account left-truncated and right-censored data for the purpose of Weibull distribution parameter estimation.

Conclusions. By way of example, the authors refer to a method for evaluating dependability indicators based on complete, right-censored and left-truncated operation times, as, in practice, such combination is quite common. The form of likelihood functions for the Weibull distribution is given. A test case is examined, whereas, using the risk function minimisation, estimates of the Weibull distribution parameters are obtained for a sample that contains full, left-truncated and right-censored data. The authors examined the variation in the Weibull distribution values and their accuracy depending on the proportion of truncated and censored data.

1. Antonov A.V. [System Analysis: A Study Guide. 4-th edition, revised and extended]. Moscow: INFRA-M; 2017. (in Russ.)

2. Antonov A.V., Yershov A.N., Chepurko V.A. [Estimating the parameters of the distribution law in cases of incomplete failure information]. In: [Diagnostics and prediction of the state of complex systems. Proceedings no. 17 of the ACS Department]. Obninsk: OINPE;2007:16-21. (in Russ.)

3. Cox D.R., Oakes D. Analysis of Survival Data. Moscow: Finansy i statistika; 1988.

4. Balakrishnan N., Debanjan M. Likelihood inference for left truncated and right censored data. Computational Statistics and Data Analysis; 2011. P. 58.

5. Hong Y.Q., Meeker W.Q., McCalley J.D. Prediction of remaining life of power transformers based on left truncated and right censored lifetime data. The Annals of Applied Statistics 2009;3(2):857-879.