# Reliability Inference for the Multicomponent System Based on Progressively Type II Censored Samples from Generalized Pareto Distributions

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## Abstract

**:**

## 1. Introduction

#### Progressive Type II Censoring Scheme

## 2. The MSS System Based on Pareto Distribution

#### Reliability of MSS

## 3. Maximum Likelihood Estimation

**R**procedure uniroot to search the unit root for equation $g\left(\lambda \right)-\lambda =0$. The MLE of ${\delta}_{s,k}$, denoted as ${\widehat{\delta}}_{s,k}$, is then obtained by replacing ${\alpha}_{1}$ and ${\alpha}_{2}$ with their respective MLEs into Equation (5):

## 4. Confidence Intervals

#### 4.1. Confidence Interval Based on Fisher Information

#### 4.2. Bootstrap Confidence Interval

- Step 1
- Given a progressively type II censored sample $\{{x}_{\left(1\right)},{x}_{\left(2\right)},\cdots ,{x}_{\left({r}_{1}\right)}\}$ from the $GP({\alpha}_{1},\lambda )$ with the censoring scheme of ${R}_{x}=\{{R}_{x,1},{R}_{x,2},\dots ,{R}_{x,{r}_{1}}\}$ and a progressively type II censored sample $\{{y}_{\left(1\right)},{y}_{\left(2\right)},\cdots ,{y}_{\left({r}_{2}\right)}\}$ from the $GP({\alpha}_{2},\lambda )$ with the censoring scheme of ${R}_{y}=\{{R}_{y,1},{R}_{y,2},\dots ,{R}_{y,{r}_{2}}\}$. Obtain MLEs ${\widehat{\alpha}}_{1},{\widehat{\alpha}}_{2}$ and $\widehat{\lambda}$ using the procedure described in Section 3.
- Step 2
- A bootstrap progressively type II censored sample, denoted by $\{{x}_{\left(j\right)}^{*},j=1,2,\cdots ,{r}_{1}\}$, is generated from the $GP({\widehat{\alpha}}_{1},\widehat{\lambda})$ based on the censoring scheme of ${R}_{x}=\{{R}_{x,1},{R}_{x,2},\dots ,{R}_{x,{r}_{1}}\}$. A bootstrap progressively type II censored sample, denoted by $\{{y}_{\left(j\right)}^{*},j=1,2,\cdots ,{r}_{2}\}$, is generated from the $GP({\widehat{\alpha}}_{2},\widehat{\lambda})$ based on the censoring scheme of ${R}_{y}=\{{R}_{y,1},{R}_{y,2},\dots ,{R}_{y,{r}_{2}}\}$.
- Step 3
- The bootstrap estimates, ${\widehat{\alpha}}_{1}^{*},{\widehat{\alpha}}_{2}^{*}$ and ${\widehat{\lambda}}^{*}$, for ${\alpha}_{1},{\alpha}_{2}$ and $\lambda $ are obtained by using the procedure described in Section 3 with the generated bootstrap progressively type II censored samples in Step 2. Then, the bootstrap estimate of ${\delta}_{s,k}$ is obtained by using Equation (5) and replacing ${\alpha}_{1}$ and ${\alpha}_{2}$ by ${\widehat{\alpha}}_{1}^{*}$ and ${\widehat{\alpha}}_{2}^{*}$, respectively. Denote the obtained bootstrap estimate by ${\widehat{\delta}}_{s,k}^{*}$.
- Step 4
- Repeat steps 2 and 3 N times, where N is a given huge number. The bootstrap sample, $\{{\widehat{\delta}}_{{s,k}_{j}}^{*},j=1,2,\cdots ,N\}$, is collected.
- Step 5
- The empirical distribution function, denoted by $\widehat{G}$, based on the bootstrap sample, $\{{\widehat{\delta}}_{{s,k}_{j}}^{*}$, $j=1,2,\cdots ,N\}$, is obtained. Let ${\widehat{\delta}}_{{s,k}_{Bp}}\left(x\right)={\widehat{G}}^{-1}\left(x\right)$ for $0<x<1$. The $100(1-p)\%$ confidence interval of ${\delta}_{s,k}$ is given by$$({\widehat{\delta}}_{{s,k}_{Bp}}(p/2),{\widehat{\delta}}_{{s,k}_{Bp}}(1-p/2)).$$

## 5. Simulation Study

**R**procedure uniroot is used to search the solution, labeled by $\widehat{\lambda}$, of $\lambda $ to Equation (16). Then the MLEs, ${\widehat{\alpha}}_{1}$ and ${\widehat{\alpha}}_{2}$ can be obtained by plugging $\widehat{\lambda}$ into Equations (12) and (13), respectively. After the values of ${\widehat{\alpha}}_{1}$ and ${\widehat{\alpha}}_{2}$ are obtained, the MLE, ${\widehat{\delta}}_{s,k}$, of ${\delta}_{s,k}$ can be obtained by plugging ${\widehat{\alpha}}_{1}$ and ${\widehat{\alpha}}_{2}$ into Equation (5). Using the delta method with the Fisher information matrix described in Section 4.1, the CI-D of ${\delta}_{s,k}$ can be obtained for each combination of $(n,m)=(20,5)$, (30,15), (50,30) and (63,30) and different censoring schemes. The above procedure has been implemented for 10,000 simulation runs. The CP of the CI-D is obtained as the percentage of these 10,000 repetitions of simulated confidence intervals that cover the true ${\delta}_{s,k}$.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**The confidence interval of ${\delta}_{s,k}$ (CI-D) of ${\delta}_{s,k}$ and the coverage probability (CP) of the CI-D for $({\alpha}_{1},{\alpha}_{2},\lambda )=(2.5,2.5,1.0)$, $s=3$ and $k=5$.

n | m | Removal Scheme | CP * | Confidence Interval | |
---|---|---|---|---|---|

Average Lower Limit | Average Upper Limit | ||||

(0, 0, 0, 0, 15) | 0.9924 | 0.0091 | 0.9504 | ||

20 | 5 | (15, 0, 0, 0, 0) | 0.9931 | 0.0097 | 0.9479 |

(3, 3, 3, 3, 3) | 0.9927 | 0.0091 | 0.9503 | ||

(0, 0,…,15) | 0.9988 | 0.0364 | 0.9292 | ||

30 | 15 | (15, 0,…,0) | 0.9987 | 0.0389 | 0.9236 |

(3, 0,0,…,3,0, 0) | 0.9987 | 0.0376 | 0.9263 | ||

(0, 0, 0,…,0, 30) | 0.9335 | 0.1939 | 0.7815 | ||

50 | 30 | (30, 0, 0,…,0, 0) | 0.9995 | 0.0852 | 0.8832 |

(3, 0, 3,0,…,3, 0) | 0.9995 | 0.0832 | 0.8874 | ||

60 | 30 | (3,0,0,…,3, 0, 0) | 0.9995 | 0.0826 | 0.8884 |

**Table 2.**Estimation results of ${\delta}_{s,k}$ using the parametric percentile bootstrap method for $({\alpha}_{1},{\alpha}_{2},\lambda )=(2.5,2.5,1.0)$, $s=1$ and $k=1$.

n | m | Removal Scheme | Bias | MSE | CP * | Confidence Interval | |
---|---|---|---|---|---|---|---|

Average | Average | ||||||

Lower Limit | Upper Limit | ||||||

(0, 0, 0, 0, 15) | −0.0037 | 0.0072 | 0.9635 | 0.3205 | 0.6588 | ||

20 | 5 | (15, 0, 0, 0, 0) | −0.0201 | 0.0197 | 0.9618 | 0.2045 | 0.7334 |

(3, 3, 3, 3, 3) | −0.0091 | 0.0211 | 0.9573 | 0.2240 | 0.7498 | ||

(0, 0,…,15) | −0.0068 | 0.0215 | 0.9551 | 0.2285 | 0.7541 | ||

30 | 15 | (15, 0,…,0) | −0.0122 | 0.0067 | 0.9663 | 0.3055 | 0.6482 |

(3,0,0,…,3,0, 0) | −0.0078 | 0.0070 | 0.9651 | 0.3136 | 0.6542 | ||

(0, 0, 0,…,0, 30) | −0.0041 | 0.0055 | 0.9584 | 0.3433 | 0.6400 | ||

50 | 20 | (30, 0, 0,…,0, 0) | −0.0131 | 0.0050 | 0.9585 | 0.3274 | 0.6285 |

(3, 0, 3,0,…,3, 0) | −0.0071 | 0.0053 | 0.9602 | 0.3379 | 0.6364 | ||

(0, 0, 0, …, 0, 20) | −0.0042 | 0.0035 | 0.9611 | 0.3677 | 0.6139 | ||

30 | (20, 0, 0, 0,…,0) | −0.0113 | 0.0034 | 0.9567 | 0.3574 | 0.6062 | |

(2, 0, 0,…,2, 0,0) | −0.0078 | 0.0035 | 0.9595 | 0.3628 | 0.6105 | ||

60 | 30 | (3,0,0,…,3, 0, 0) | −0.0068 | 0.0035 | 0.9598 | 0.3644 | 0.6118 |

**Table 3.**Estimation results of ${\delta}_{s,k}$ using the parametric percentile bootstrap method for $({\alpha}_{1},{\alpha}_{2},\lambda )=(2.5,2.5,1.0)$, $s=1$ and $k=2$.

n | m | Removal Scheme | Bias | MSE | CP * | Confidence Interval | ||
---|---|---|---|---|---|---|---|---|

Average | Average | |||||||

Lower Limit | Upper Limit | |||||||

20 | 5 | (0, 0, 0, 0, 15) | −0.0205 | 0.0267 | 0.9551 | 0.3242 | 0.8912 | |

(15, 0, 0, 0, 0) | −0.0338 | 0.0255 | 0.9618 | 0.2922 | 0.8772 | |||

(3, 3, 3, 3, 3) | −0.0227 | 0.0264 | 0.9573 | 0.3184 | 0.8884 | |||

30 | 15 | (0, 0,…,15) | −0.0084 | 0.0090 | 0.9635 | 0.4477 | 0.8216 | |

(15, 0,…,0) | −0.0174 | 0.0086 | 0.9663 | 0.4285 | 0.8124 | |||

(3, 0,0,…,3,0, 0) | −0.0128 | 0.0089 | 0.9651 | 0.4389 | 0.8177 | |||

50 | 20 | (0, 0, 0,…,0, 30) | −0.0078 | 0.0069 | 0.9584 | 0.4773 | 0.8057 | |

(30, 0, 0,…,0, 0) | −0.0175 | 0.0065 | 0.9585 | 0.4572 | 0.7953 | |||

(3, 0, 3,0,…,3, 0) | −0.0110 | 0.0067 | 0.9602 | 0.4705 | 0.8025 | |||

30 | (0, 0, 0,…,0, 20) | −0.0068 | 0.0044 | 0.9611 | 0.5087 | 0.7825 | ||

(20, 0, 0, 0,…,0) | −0.0145 | 0.0043 | 0.9567 | 0.4959 | 0.7750 | |||

(2, 0, 0,…,2, 0,0) | −0.0107 | 0.0044 | 0.9595 | 0.5025 | 0.7791 | |||

60 | 30 | (3,0,0,…,3, 0, 0) | −0.0096 | 0.0045 | 0.9598 | 0.5046 | 0.7804 |

**Table 4.**Estimation results of ${\delta}_{s,k}$ using the parametric percentile bootstrap method for $({\alpha}_{1},{\alpha}_{2},\lambda )=(2.5,2.5,1.0)$, $s=3$ and $k=3$.

n | m | Removal Scheme | Bias | MSE | CP * | Confidence Interval | ||
---|---|---|---|---|---|---|---|---|

Average | Average | |||||||

Lower Limit | Upper Limit | |||||||

(0, 0, 0, 0, 15) | 0.0115 | 0.0137 | 0.9551 | 0.0938 | 0.5257 | |||

20 | 5 | (15, 0, 0, 0, 0) | −0.0005 | 0.0112 | 0.9618 | 0.0818 | 0.4962 | |

(3, 3, 3, 3, 3) | 0.0093 | 0.0131 | 0.9573 | 0.0915 | 0.5192 | |||

(0, 0,…,15) | 0.0271 | 0.0042 | 0.9635 | 0.1386 | 0.3988 | |||

30 | 15 | (15, 0,…,0) | −0.0042 | 0.0036 | 0.9663 | 0.1303 | 0.3873 | |

(3, 0,0,…,3,0, 0) | −0.0006 | 0.0039 | 0.9651 | 0.1348 | 0.3938 | |||

(0, 0, 0,…,0, 30) | 0.0010 | 0.0031 | 0.9584 | 0.1507 | 0.3776 | |||

50 | 20 | (30, 0, 0,…,0, 0) | −0.0062 | 0.0027 | 0.9585 | 0.1417 | 0.3658 | |

(3, 0, 3,0,…,3, 0) | −0.0013 | 0.0030 | 0.9602 | 0.1470 | 0.3738 | |||

(0, 0, 0,…,0, 20) | −0.0005 | 0.0020 | 0.9611 | 0.1641 | 0.3500 | |||

30 | (20, 0, 0, 0,…,0) | −0.0060 | 0.0018 | 0.9567 | 0.1580 | 0.3427 | ||

(2, 0, 0,…,2, 0,0) | −0.0032 | 0.0019 | 0.9595 | 0.1611 | 0.3467 | |||

60 | 30 | (3,0,0,…,3, 0, 0) | −0.0024 | 0.0019 | 0.9598 | 0.1621 | 0.3480 |

**Table 5.**Estimation results of ${\delta}_{s,k}$ using the parametric percentile bootstrap method for $({\alpha}_{1},{\alpha}_{2},\lambda )=(2.5,2.5,1.0)$, $s=2$ and $k=4$.

n | m | Removal Scheme | Bias | MSE | CP * | Confidence Interval | |
---|---|---|---|---|---|---|---|

Average | Average | ||||||

Lower Limit | Upper Limit | ||||||

(0, 0, 0, 0, 15) | −0.0125 | 0.0315 | 0.9551 | 0.2637 | 0.8750 | ||

20 | 5 | (15, 0, 0, 0, 0) | −0.0279 | 0.0294 | 0.9618 | 0.2347 | 0.8575 |

(3, 3, 3, 3, 3) | −0.0151 | 0.0310 | 0.9573 | 0.258 | 0.8715 | ||

(0, 0,…,15) | −0.0057 | 0.0111 | 0.9635 | 0.3754 | 0.786 | ||

30 | 15 | (15, 0,…,0) | −0.0161 | 0.0104 | 0.9663 | 0.3569 | 0.7754 |

(3, 0,0,…,3,0, 0) | −0.0108 | 0.0108 | 0.9651 | 0.3669 | 0.7819 | ||

(0, 0, 0,…,0, 30) | −0.0059 | 0.0085 | 0.9584 | 0.4035 | 0.7668 | ||

50 | 20 | (30, 0, 0,…,0, 0) | −0.0171 | 0.0078 | 0.9585 | 0.3838 | 0.7539 |

(3, 0, 3,0,…,3, 0) | −0.0096 | 0.0083 | 0.9602 | 0.3968 | 0.7628 | ||

(0, 0,0,…,0,20) | −0.0058 | 0.0055 | 0.9611 | 0.4338 | 0.7377 | ||

30 | (20, 0,0,0,…,0) | −0.0146 | 0.0053 | 0.9567 | 0.4209 | 0.7287 | |

(2,0,0,…,2,0,0) | −0.0102 | 0.0054 | 0.9595 | 0.4276 | 0.7337 | ||

60 | 30 | (3,0,0,…,3,0,0) | −0.0090 | 0.0055 | 0.9598 | 0.4297 | 0.7352 |

**Table 6.**Estimation results of ${\delta}_{s,k}$ using the parametric percentile bootstrap method for $({\alpha}_{1},{\alpha}_{2},\lambda )=(2.5,2.5,1.0)$, $s=4$ and $k=4$.

n | m | Removal Scheme | Bias | MSE | CP * | Confidence Interval | |
---|---|---|---|---|---|---|---|

Average | Average | ||||||

Lower Limit | Upper Limit | ||||||

(0, 0, 0, 0, 15) | 0.0131 | 0.0107 | 0.9551 | 0.0726 | 0.4595 | ||

20 | 5 | (15, 0, 0, 0, 0) | 0.0021 | 0.0085 | 0.9618 | 0.0630 | 0.4295 |

(3, 3, 3, 3, 3) | 0.0100 | 0.0102 | 0.9573 | 0.0707 | 0.4529 | ||

(0, 0,…,15) | 0.0032 | 0.0031 | 0.9635 | 0.1080 | 0.3339 | ||

30 | 15 | (15, 0,…,0) | −0.0027 | 0.0026 | 0.9663 | 0.1013 | 0.3231 |

(3, 0,0,…,3,0, 0) | 0.0003 | 0.0029 | 0.9651 | 0.1050 | 0.3292 | ||

(0, 0, 0,…,0, 30) | 0.0016 | 0.0023 | 0.9584 | 0.1178 | 0.3139 | ||

50 | 20 | (30, 0, 0,…,0, 0) | −0.0046 | 0.0019 | 0.9585 | 0.1104 | 0.3030 |

(3, 0, 3,0,…,3, 0) | −0.0004 | 0.0022 | 0.9602 | 0.1153 | 0.3105 | ||

(0, 0, 0,…,0, 20) | <0.0001 | 0.0014 | 0.9611 | 0.1286 | 0.2884 | ||

30 | (20, 0, 0, 0,…,0) | −0.0047 | 0.0013 | 0.9567 | 0.1236 | 0.2818 | |

(2, 0, 0,…,2, 0,0) | −0.0023 | 0.0014 | 0.9595 | 0.1262 | 0.2855 | ||

60 | 30 | (3,0,0,…,3, 0, 0) | −0.0016 | 0.0014 | 0.9598 | 0.1270 | 0.2866 |

**Table 7.**Estimation results of ${\delta}_{s,k}$ using the parametric percentile bootstrap method for $({\alpha}_{1},{\alpha}_{2},\lambda )=(2.5,2.5,1.0)$, $s=1$ and $k=5$.

n | m | Removal Scheme | Bias | MSE | CP * | Confidence Interval | |
---|---|---|---|---|---|---|---|

Average | Average | ||||||

Lower Limit | Upper Limit | ||||||

(0, 0, 0, 0, 15) | −0.0387 | 0.0242 | 0.9551 | 0.4494 | 0.9689 | ||

20 | 5 | (15, 0, 0, 0, 0) | −0.0488 | 0.0244 | 0.9618 | 0.4097 | 0.9635 |

(3, 3, 3, 3, 3) | −0.0403 | 0.0241 | 0.9573 | 0.4423 | 0.9679 | ||

(0, 0,…,15) | −0.0145 | 0.0074 | 0.9635 | 0.6034 | 0.9404 | ||

30 | 15 | (15, 0,…,0) | −0.0220 | 0.00742 | 0.966 | 0.5814 | 0.9352 |

(3, 0,0,…,3,0, 0) | −0.0182 | 0.0074 | 0.9651 | 0.5934 | 0.9382 | ||

(0, 0, 0,…,0, 30) | −0.0124 | 0.0056 | 0.9584 | 0.6382 | 0.9321 | ||

50 | 20 | 30, 0, 0,…,0, 0) | −0.0204 | 0.0055 | 0.9585 | 0.6158 | 0.9258 |

(3, 0, 3,0,…,3, 0) | −0.0150 | 0.0056 | 0.9602 | 0.6308 | 0.9302 | ||

(0, 0, 0,…,0, 20) | −0.0095 | 0.0036 | 0.9611 | 0.6744 | 0.9188 | ||

50 | 30 | (20, 0, 0, 0,…,0) | −0.0161 | 0.0036 | 0.9567 | 0.6605 | 0.9138 |

(2, 0, 0,…,2, 0,0) | −0.0129 | 0.0036 | 0.9595 | 0.6677 | 0.9166 | ||

60 | 30 | (3,0,0,…,3, 0, 0) | −0.0120 | 0.0036 | 0.9598 | 0.6700 | 0.9174 |

**Table 8.**Estimation results of ${\delta}_{s,k}$ using the parametric percentile bootstrap method for $({\alpha}_{1},{\alpha}_{2},\lambda )=(2.5,2.5,1.0)$, $s=3$ and $k=5$.

n | m | Removal Scheme | Bias | MSE | CP * | Confidence Interval | |
---|---|---|---|---|---|---|---|

Average | Average | ||||||

Lower Limit | Upper Limit | ||||||

20 | 5 | (0, 0, 0, 0, 15) | −0.0006 | 0.0309 | 0.9551 | 0.2045 | 0.8179 |

(15, 0, 0, 0, 0) | −0.0171 | 0.0277 | 0.9618 | 0.1804 | 0.7945 | ||

(3, 3, 3, 3, 3) | −0.0035 | 0.03024 | 0.9573 | 0.2000 | 0.8131 | ||

30 | 15 | (0, 0,…,15) | −0.0016 | 0.0108 | 0.9635 | 0.2965 | 0.7036 |

(15, 0,…,0) | −0.0122 | 0.0098 | 0.9663 | 0.2805 | 0.6900 | ||

(3, 0, 0,…,3, 0,0) | −0.0067 | 0.0104 | 0.9651 | 0.2891 | 0.6978 | ||

50 | 20 | (0, 0, 0,…,0, 30) | −0.0028 | 0.0083 | 0.9584 | 0.3204 | 0.6794 |

(30, 0, 0, 0,…,0) | −0.0141 | 0.0074 | 0.9585 | 0.3032 | 0.6647 | ||

(3, 0, 3, 0,…,3, 0) | −0.0065 | 0.0079 | 0.9602 | 0.3146 | 0.6748 | ||

50 | 30 | (0, 0, 0,…,0, 20) | −0.0037 | 0.0053 | 0.9611 | 0.3465 | 0.6457 |

(20, 0, 0, 0,…,0) | −0.0125 | 0.0050 | 0.9567 | 0.3351 | 0.6358 | ||

(2, 0, 0,…,2, 0, 0) | −0.0085 | 0.0052 | 0.9595 | 0.3410 | 0.6413 | ||

60 | 30 | (3, 0, 0,…,3, 0, 0) | −0.0069 | 0.00532 | 0.9598 | 0.3429 | 0.6430 |

**Table 9.**Estimation results of ${\delta}_{s,k}$ using the parametric percentile bootstrap method for $({\alpha}_{1},{\alpha}_{2},\lambda )=(2.5,2.5,1.0)$, $s=5$ and $k=5$.

n | m | Removal Scheme | Bias | MSE | CP * | Confidence Interval | |
---|---|---|---|---|---|---|---|

Average | Average | ||||||

Lower Limit | Upper Limit | ||||||

20 | 5 | (0, 0, 0, 0, 15) | 0.0134 | 0.0086 | 0.9551 | 0.0592 | 0.4089 |

(15, 0, 0, 0, 0) | 0.0034 | 0.0067 | 0.9618 | 0.05128 | 0.3793 | ||

(3, 3, 3, 3, 3) | 0.0115 | 0.0081 | 0.9573 | 0.05770 | 0.4023 | ||

30 | 15 | (0, 0,…,15) | 0.0034 | 0.0024 | 0.9635 | 0.0885 | 0.2873 |

(15, 0,…,0) | −0.0019 | 0.0020 | 0.9663 | 0.0829 | 0.2773 | ||

(3, 0,0,…,3,0, 0) | 0.0008 | 0.0022 | 0.9651 | 0.0859 | 0.2830 | ||

50 | 20 | (0, 0, 0,…,0, 30) | 0.0018 | 0.0017 | 0.9584 | 0.0967 | 0.2687 |

(30, 0, 0,…,0, 0) | −0.0036 | 0.0014 | 0.9585 | 0.0905 | 0.2588 | ||

(3, 0, 3,0,…,3, 0) | <0.0001 | 0.0016 | 0.9602 | 0.0946 | 0.2656 | ||

30 | (0, 0, 0,…,0, 20) | 0.0002 | 0.0011 | 0.9611 | 0.1057 | 0.2453 | |

(20, 0, 0, 0,…,0) | −0.0038 | 0.0010 | 0.9567 | 0.1015 | 0.23942 | ||

(2, 0, 0,…,2, 0,0) | −0.0017 | 0.0010 | 0.9595 | 0.1037 | 0.2427 | ||

60 | 30 | (3,0,0,…,3, 0, 0) | −0.0011 | 0.0010 | 0.9598 | 0.1043 | 0.2437 |

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**MDPI and ACS Style**

Sauer, L.; Lio, Y.; Tsai, T.-R.
Reliability Inference for the Multicomponent System Based on Progressively Type II Censored Samples from Generalized Pareto Distributions. *Mathematics* **2020**, *8*, 1176.
https://doi.org/10.3390/math8071176

**AMA Style**

Sauer L, Lio Y, Tsai T-R.
Reliability Inference for the Multicomponent System Based on Progressively Type II Censored Samples from Generalized Pareto Distributions. *Mathematics*. 2020; 8(7):1176.
https://doi.org/10.3390/math8071176

**Chicago/Turabian Style**

Sauer, Lauren, Yuhlong Lio, and Tzong-Ru Tsai.
2020. "Reliability Inference for the Multicomponent System Based on Progressively Type II Censored Samples from Generalized Pareto Distributions" *Mathematics* 8, no. 7: 1176.
https://doi.org/10.3390/math8071176