# Bayesian Testing Procedure on the Lifetime Performance Index of Products Following Chen Lifetime Distribution Based on the Progressive Type-II Censored Sample

^{1}

^{2}

^{*}

## Abstract

**:**

_{L}is frequently used to monitor the larger-the-better lifetime performance of products. This research is related to the topic of asymmetrical probability distributions and applications across disciplines. Chen lifetime distribution with a bathtub shape or increasing failure rate function has many applications in the lifetime data analysis. We derived the uniformly minimum variance unbiased estimator (UMVUE) for C

_{L}, and we used this estimator to develop a hypothesis testing procedure of C

_{L}under a lower specification limit based on the progressive type-II censored sample. The Bayesian estimator for C

_{L}is also derived, and it is used to develop another hypothesis testing procedure. A simulation study is conducted to compare the average confidence levels for two procedures. Finally, one practical example is given to illustrate the implementation of our proposed non-Bayesian and Bayesian testing procedure.

## 1. Introduction

## 2. The Monotonic Relationship between the Lifetime Performance Index and the Conforming Rate

## 3. Results

#### 3.1. UMVUE for the Lifetime Performance Index and the Testing Procedure

_{1}is observed, then ${R}_{1}$ surviving units are randomly removed under the removal percentage ${p}_{1}$. When the ith failure time U

_{i}is observed, ${R}_{i}$ surviving units are randomly removed under the removal percentage ${p}_{i}$, i = 1,…, m − 1. When the mth failure time U

_{m}is observed, this experiment is terminated, and the remaining ${R}_{m}=n-{R}_{1}-\dots -{R}_{m-1}-m$ surviving units are all removed under the removal percentage ${p}_{m}$ = 1. Supposing that the failure times are following the Chen distribution, then ${U}_{1},\dots ,{U}_{m}$ is the progressive type-II censored sample under the censoring scheme ${R}_{1},\dots ,{R}_{m}$ with the removal percentages ${p}_{1},\dots ,{p}_{m}$. From Balakrishnan and Aggarwala [18], the likelihood function based on the progressive type-II censored sample ${U}_{1},\dots ,{U}_{m}$ is

#### 3.2. Bayesian Estimator for the Lifetime Performance Index and the Testing Procedure

#### 3.3. Simulation Study on Two Procedures

_{U}and then $L={e}^{{L}_{U}^{\beta}}-1$, α, where $n\le m$, a,b > 0

_{1}times. Then we have the estimated confidence level $1-{\widehat{\alpha}}_{1}=\frac{total\hspace{0.33em}count1}{{N}_{1}}$ and $1-{\widehat{\alpha}}_{2}=\frac{total\hspace{0.33em}count2}{{N}_{1}}$ for the first and the Bayesian credible intervals, respectively. Furthermore, we can obtain N

_{1}risk ${({C}_{L}-{\tilde{C}}_{L})}^{2}$ and ${({C}_{L}-{\ddot{C}}_{L})}^{2}$ for the UMVUE and Bayesian estimators.

_{2}times, we get N

_{2}estimated confidence levels ${(1-{\widehat{\alpha}}_{j})}_{1},\dots ,{(1-{\widehat{\alpha}}_{j})}_{{N}_{2}}$, j = 1 and 2 for the first and the Bayesian credible intervals, respectively. Furthermore, we can obtain the N

_{2}risks.

_{2}estimated confidence levels $1-{\tilde{\alpha}}_{j}={\displaystyle \sum _{i=1}^{1000}{(1-{\widehat{\alpha}}_{j})}_{i}/{N}_{2}}$, j = 1,2 as the average confidence level for two credible intervals. Take the average of N

_{2}risks to yield the estimated risks for the UMVUE estimator and Bayesian estimator.

_{1}= 100 and N

_{2}= 1000. The simulation results are reported in Table A2.

- Both credible intervals have average confidence levels very close to the nominal ones. Thus, the performance of both credible intervals is very satisfactory even for a small sample size n = 20 or larger sample size n = 30,100.
- The SMSEs for both credible intervals are about the same and very small in the scope of 0.000433 to 0.000523.
- The SMSEs for both credible intervals are decreasing when m is increasing for fixed n.
- The risk for the Bayesian estimator is smaller than the one for UMVUE. The discrepancy between the two estimators is decreasing when m is increasing for fixed n. The parameter (a,b) = (2,2) always has the smallest risk for both estimators. Generally speaking, the Bayesian estimator outperforms the UMVUE in terms of risk.

#### 3.4. Example

## 4. Discussion

_{L}to develop a hypothesis testing procedure for the lifetime performance index based on the complete sample for an exponential distribution lifetime. In practice, we cannot observe all lifetimes of products due to the restrictive resources or some experimental factors. In this case, we can only observe type I or type II censored data. Integrating the progressive censoring, which allows the removal of units progressively (accidental breakage of units) at some time points, including the final termination point, the progressive censored data is collected. Wu [7] derived the maximum likelihood estimator (MLE) for the lifetime performance index based on the progressive type-I interval censored sample and built a testing procedure about C

_{L}when the lifetime of products follows a Chen lifetime distribution. Wu and Chang [8] derived the MLE for the lifetime performance index based on the progressive type-I interval censored sample and built a testing procedure about C

_{L}with the lifetime of products following an exponentiated Frech’et distribution. Wu and Hsieh [9] found the MLE for the lifetime performance index based on the progressive type-I interval censored sample and built a testing procedure about C

_{L}when the lifetime of products follows the Gompertz distribution. Progressive type-I interval censoring has the advantage of the convenience of collecting data for experimenters. However, experimenters can only observe the number of failure units at each inspection time, not the failure time for each experimental unit. Under the progressive type-II censoring, the experiment terminates when the mth failure time is observed and the failure times for the first m units, excluding the progressive censored units, are collected. Laumen and Cramer [10] derived the MLE for the lifetime performance index from gamma distributions and built a testing procedure under the same censoring. For the exponential lifetime model, Lee et al. [11] derived the UMVUE for the lifetime performance index and utilized it to build a hypothesis testing procedure for C

_{L}. Wu et al. [12] considered two Bayesian tests based on two Bayesian estimators and made simulation comparisons on the test power for two procedures. For the Burr XII model, Lee et al. [13] constructed the UMVUE for the lifetime performance index and utilized it to build a hypothesis testing procedure for C

_{L}. Wu et al. [14] consider another lifetime performance index and utilize its MLE to develop the testing procedure about C

_{L}. Lee [15] assessed the lifetime performance index of Rayleigh products based on the Bayesian estimation and used it to build a testing procedure for C

_{L}. Our theoretical contribution for this paper is to find the UMVUE for the lifetime performance index based on the MLE for Chen lifetime products and prove this result. We also find the Bayesian estimator for the lifetime performance index. We develop two testing procedures about C

_{L}based on the UMVUE and Bayesian estimators. We also make the simulation comparison for these two tests and these two estimators. The Chen distribution is a two-parameter lifetime distribution with a bathtub shape or increasing failure rate function (see Chen [16]). The property of the failure rate function is illustrated in Figure 1. The practical implications of our research are to provide two assessment testing procedures for products following lifetime distributions with a bathtub shape or increasing failure rate function. The practical application of our research is illustrated by the example of 18 failure times (days) of electronic devices (see Xie and Lai [21]) given in Section 3.4. There is not any research on the evaluation of lifetime performance index for products from a Chen distribution based on progressive type-II censored sample in the literature. Our research goal is to expand the field of the assessment on the lifetime performance index from an exponential distribution, gamma distribution, Rayleigh distribution, and Burr XII distribution to include Chen distribution based on the progressive type-II censored sample. Our research is needed to help engineers to manage the reliability of their high-quality products.

## 5. Conclusions

#### 5.1. Summary

#### 5.2. Limitations and Future Research Directions

_{1}= … = R

_{m}= 0, the censored sample is reduced to the complete sample. When R

_{1}= … = R

_{m−}

_{1}= 0 and R

_{m}$\ne $ 0, the censored sample is reduced to the right type II censored sample. When R

_{1}$\ne $ 0 and R

_{2}… = R

_{m}= 0, the censored sample is reduced to the left type II censored sample. Therefore, the progressive type-II censored sample covers the cases of right type II censored sample, left type II censored sample, and complete sample. In the future, we can extend the research to other censoring schemes, for example, progressive type-I interval censoring, hybrid type II censoring. We can also extend the lifetime distribution to other kinds, for example, exponentiated Frech’et, exponentiated Weibull, exponentiated extreme value.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**The lifetime performance index ${C}_{L}$ and its corresponding conforming rates ${P}_{r}$.

${\mathit{C}}_{\mathit{L}}$ | ${\mathit{P}}_{\mathit{r}}$ | ${\mathit{C}}_{\mathit{L}}$ | ${\mathit{P}}_{\mathit{r}}$ | ${\mathit{C}}_{\mathit{L}}$ | ${\mathit{P}}_{\mathit{r}}$ |
---|---|---|---|---|---|

$-\infty $ | 0.000000 | −0.125 | 0.324652 | 0.550 | 0.637628 |

−3.000 | 0.018316 | 0.000 | 0.367879 | 0.575 | 0.653770 |

−2.750 | 0.023518 | 0.125 | 0.416862 | 0.600 | 0.670320 |

−2.500 | 0.030197 | 0.150 | 0.427415 | 0.625 | 0.687289 |

−2.250 | 0.038774 | 0.175 | 0.438235 | 0.650 | 0.704688 |

−2.125 | 0.043937 | 0.200 | 0.449329 | 0.675 | 0.722527 |

−2.000 | 0.049787 | 0.225 | 0.460704 | 0.700 | 0.740818 |

−1.750 | 0.063928 | 0.250 | 0.472367 | 0.725 | 0.759572 |

−1.500 | 0.082085 | 0.275 | 0.484325 | 0.750 | 0.778801 |

−1.250 | 0.105399 | 0.300 | 0.496585 | 0.775 | 0.798516 |

−1.125 | 0.119433 | 0.325 | 0.509156 | 0.800 | 0.818731 |

−1.000 | 0.135335 | 0.350 | 0.522046 | 0.825 | 0.839457 |

−0.750 | 0.173774 | 0.375 | 0.535261 | 0.850 | 0.860708 |

−0.500 | 0.223130 | 0.400 | 0.548812 | 0.875 | 0.882497 |

−0.250 | 0.286505 | 0.425 | 0.562705 | 0.900 | 0.904837 |

−0.225 | 0.293758 | 0.450 | 0.576950 | 0.925 | 0.927743 |

−0.200 | 0.301194 | 0.475 | 0.591555 | 0.950 | 0.951229 |

−0.175 | 0.308819 | 0.500 | 0.606531 | 0.975 | 0.975310 |

−0.15 | 0.316637 | 0.525 | 0.621885 | 1.000 | 1.000000 |

**Table A2.**Average confidence level for ${C}_{L}$, SMSE (in the first parentheses) and risk (in the second parentheses) under L

_{U}= 0.1 and $1-\alpha $ = 0.95.

(a,b) = (2,2) | (a,b) = (2,5) | (a,b) = (5,2) | ||||||
---|---|---|---|---|---|---|---|---|

n | m | $({\mathit{R}}_{1},\dots ,{\mathit{R}}_{\mathit{m}})$ | UMVUE | Bayes | UMVUE | Bayes | UMVUE | Bayes |

20 | 10 | $(5,4,1,0\times 7)$ | 0.95046 | 0.95049 | 0.95062 | 0.95039 | 0.95025 | 0.95018 |

(0.000459) | (0.000454) | (0.000485) | (0.000466) | (0.000439) | (0.00046) | |||

(0.032338) | (0.020448) | (0.206719) | (0.129118) | (0.166737) | (0.082863) | |||

$(0\times 3,2,3,3,2,0\times 3)$ | 0.94954 | 0.94965 | 0.94927 | 0.94990 | 0.94871 | 0.94895 | ||

(0.000458) | (0.000475) | (0.000478) | (0.000491) | (0.000493) | (0.000496) | |||

(0.033270) | (0.020028) | (0.202740) | (0.126668) | (0.172086) | (0.083376) | |||

$(0\times 7,1,4,5)$ | 0.94997 | 0.94944 | 0.94962 | 0.94882 | 0.95002 | 0.94905 | ||

(0.000479) | (0.000513) | (0.000476) | (0.000490) | (0.000458) | (0.000464) | |||

(0.033348) | (0.020464) | (0.210638) | (0.129526) | (0.167786) | (0.083427) | |||

15 | $(4,1,0\times 13)$ | 0.94989 | 0.95060 | 0.95011 | 0.94988 | 0.94956 | 0.94983 | |

(0.000435) | (0.000442) | (0.000435) | (0.000452) | (0.000471) | (0.000501) | |||

(0.019462) | (0.014550) | (0.128238) | (0.092888) | (0.101481) | (0.063165) | |||

$(0\times 6,1,3,1,0\times 6)$ | 0.95047 | 0.95079 | 0.95005 | 0.95018 | 0.94850 | 0.94915 | ||

(0.000481) | (0.000465) | (0.000473) | (0.000457) | (0.000510) | (0.000511) | |||

(0.020197) | (0.014692) | (0.127588) | (0.091372) | (0.103466) | (0.063199) | |||

$(0\times 13,1,4)$ | 0.94974 | 0.95031 | 0.95007 | 0.94971 | 0.94926 | 0.94936 | ||

(0.000439) | (0.000463) | (0.000509) | (0.000516) | (0.000488) | (0.000507) | |||

(0.020154) | (0.014541) | (0.130222) | (0.093632) | (0.100478) | (0.062896) | |||

30 | 15 | $(7,5,3,0\times 12)$ | 0.95063 | 0.95028 | 0.95007 | 0.95010 | 0.95138 | 0.94991 |

(0.000501) | (0.000488) | (0.000488) | (0.000531) | (0.000468) | (0.000457) | |||

(0.020698) | (0.014852) | (0.125239) | (0.092391) | (0.103473) | (0.063256) | |||

$(0\times 6,4,7,4,0\times 6)$ | 0.94924 | 0.94998 | 0.95129 | 0.95087 | 0.94943 | 0.94928 | ||

(0.000467) | (0.000472) | (0.000447) | (0.000459) | (0.000475) | (0.000488) | |||

(0.020231) | (0.014722) | (0.128099) | (0.092867) | (0.102906) | (0.063306) | |||

$(0\times 12,3,5,7)$ | 0.94985 | 0.95003 | 0.95074 | 0.94997 | 0.95084 | 0.95019 | ||

(0.000464) | (0.000454) | (0.000475) | (0.000464) | (0.000483) | (0.000480) | |||

(0.020609) | (0.014676) | (0.126370) | (0.093766) | (0.103612) | (0.063977) | |||

20 | $(5,4,1,0\times 17)$ | 0.94953 | 0.94969 | 0.95032 | 0.95063 | 0.94956 | 0.95000 | |

(0.000484) | (0.000483) | (0.000474) | (0.000454) | (0.000472) | (0.000475) | |||

(0.014779) | (0.011676) | (0.089691) | (0.071110) | (0.074365) | (0.051170) | |||

$(0\times 8,2,3,3,2,0\times 8)$ | 0.95019 | 0.94991 | 0.95013 | 0.94941 | 0.94926 | 0.94932 | ||

(0.000491) | (0.000517) | (0.000050) | (0.000502) | (0.000434) | (0.000481) | |||

(0.014722) | (0.011651) | (0.093602) | (0.072828) | (0.074644) | (0.051560) | |||

$(0\times 17,1,4,5)$ | 0.94885 | 0.94927 | 0.95023 | 0.95008 | 0.95031 | 0.95006 | ||

(0.000503) | (0.000523) | (0.000484) | (0.000491) | (0.000457) | (0.000451) | |||

(0.014757) | (0.011515) | (0.090619) | (0.072060) | (0.073405) | (0.051221) | |||

100 | 20 | $(60,20,0\times 18)$ | 0.94984 | 0.94966 | 0.94989 | 0.94950 | 0.95015 | 0.95040 |

(0.000491) | (0.000487) | (0.000508) | (0.000506) | (0.000492) | (0.000499) | |||

(0.014944) | (0.011491) | (0.093359) | (0.072604) | (0.073616) | (0.051114) | |||

$(0\times 8,20,20,20,20,0\times 8)$ | 0.94895 | 0.94883 | 0.94985 | 0.95042 | 0.94994 | 0.94926 | ||

(0.000449) | (0.000475) | (0.000466) | (0.000489) | (0.000465) | (0.000471) | |||

(0.014812) | (0.011674) | (0.08942) | (0.071394) | (0.074201) | (0.051082) | |||

$(0\times 18,20,60)$ | 0.95057 | 0.95005 | 0.95010 | 0.94935 | 0.95048 | 0.94950 | ||

(0.000458) | (0.000463) | (0.000439) | (0.000444) | (0.000446) | (0.000485) | |||

(0.014641) | (0.011405) | (0.093107) | (0.071695) | (0.073540) | (0.051245) |

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**Figure 3.**Ecdf-plot with the transformed data in the example and the black dotted line is the cdf from the exponential distribution.

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Wu, S.-F.; Chang, W.-T. Bayesian Testing Procedure on the Lifetime Performance Index of Products Following Chen Lifetime Distribution Based on the Progressive Type-II Censored Sample. *Symmetry* **2021**, *13*, 1322.
https://doi.org/10.3390/sym13081322

**AMA Style**

Wu S-F, Chang W-T. Bayesian Testing Procedure on the Lifetime Performance Index of Products Following Chen Lifetime Distribution Based on the Progressive Type-II Censored Sample. *Symmetry*. 2021; 13(8):1322.
https://doi.org/10.3390/sym13081322

**Chicago/Turabian Style**

Wu, Shu-Fei, and Wei-Tsung Chang. 2021. "Bayesian Testing Procedure on the Lifetime Performance Index of Products Following Chen Lifetime Distribution Based on the Progressive Type-II Censored Sample" *Symmetry* 13, no. 8: 1322.
https://doi.org/10.3390/sym13081322