Inference on lifetime performance index of two-parameter exponential distribution: Censored and record data

Author
Sayyed Heidar Jafari
shahrood university of technology
Abstract
Introduction

Improving the quality, compliance rate and product life is a priority in manufacturing industries. If the performance of a process is evaluated by the lifetime of a product, it is clear that a larger lifetime indicates better product quality and higher process capability, and this process is accepted. For instance, the lifetime of electronic components exhibits a larger-the-better type of quality characteristic. The process capability index is one of the indicators for evaluating the capability of a process. To measure the larger-the-better quality characteristic, a process capability index has been defined as , where and are the mean and standard deviation of the process, respectively, and is the lower specification limit.

In various manufacturing and service processes, the assumption of normality is often not valid. The two-parameter exponential distribution has many applications in engineering, biological studies, epidemiology, and medical sciences. In engineering analysis, the location parameter is called the threshold value or warranty time and the scale parameter is the expected average as well as warranty life. If the lifetime of a manufactured product follows a two-parameter exponential distribution with location parameter and scale parameter , then the capability index becomes .



Material and methods

Since the performance capability index has been used in industries, point estimation, construction of confidence intervals, and testing the hypothesis about this index have been considered. However, the classical inference cannot be applied to . Therefore, we utilized the concepts of generalized pivotal quantity and generalized test variable for inference on this parameter. Here, we suppose that different kinds of schemes such as double type II censoring, progressive censoring, record values, and k-record values. In each case, the maximum likelihood and uniformly minimum variance unbiased estimators are derived. Then, a generalized confidence interval and a generalized p-value are suggested. The generalized confidence interval is evaluated by Monte Carlo simulation, and the presented approaches are illustrated using some real examples.



Results and discussion

We study the coverage probability and expected length of the proposed generalized confidence interval for under a progressive censoring scheme. We found that the coverage probability is close to the confidence coefficient. Also, it does not depend on the values of parameters and sample size. Besides, the expected length decreases when decreases (or the sample size increases). The shortest expected length of the generalized confidence interval is when the value of is close to the location parameter .



Conclusion

The following conclusions were drawn from this research.

The minimum variance unbiased estimator for has a closed-form under all considered schemes.
The proposed generalized inference for is a satisfactory and easy approach.
This article contains all results in other papers such as Lee et al. (2011) and Gunasekera, and Wijekularathna (2019).
Different indicators have been introduced to evaluate the process’s capability. Some of the most important ones that are widely used in manufacturing industries have been introduced by Wooluru et al. (2014). The methods introduced in this paper, can be generalized and used for these process capability indicators.
Keywords
power graph
automorphism group
abelian group
nilpotent
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