# Statistical Inference and Optimal Design of Accelerated Life Testing for the Chen Distribution under Progressive Type-II Censoring

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Chen Distribution

## 3. Model Assumptions

- (1)
- Under stress level ${s}_{i}$, $i=1,2,\cdots ,k$, the lifetimes of the units follow$$\begin{array}{c}\hfill {f}_{i}\left(x;{\alpha}_{i},\beta \right)={\alpha}_{i}\beta {x}^{\beta -1}exp\left\{{\alpha}_{i}\left(1-{e}^{{x}^{\beta}}\right)+{x}^{\beta}\right\},\phantom{\rule{2.em}{0ex}}x>0,{\alpha}_{i}>0,\beta >0.\end{array}$$
- (2)
- The relationship between the stress level ${s}_{i}$ and the lifetime distribution parameter ${\alpha}_{i}$ is$$\begin{array}{c}\hfill ln\left({\alpha}_{i}\right)={b}_{0}+b{\phi}_{i},\phantom{\rule{1.em}{0ex}}i=0,1,\cdots ,k,\end{array}$$

## 4. Maximum Likelihood Estimation

## 5. Fisher Information Matrix

## 6. Parametric Bootstrap Intervals

#### 6.1. The Bootstrap Percentile Confidence Interval

- -
- Step 1: Utilize the progressive censored sample to compute the MLE $\hat{\mathrm{\Theta}}=\left({\hat{\alpha}}_{0},\hat{\lambda},\hat{\beta}\right)$.
- -
- Step 2: Generate a progressive censored sample by regarding $\hat{\mathrm{\Theta}}$ as the parameter values for the Chen distribution.
- -
- Step 3: Compute the MLE ${\hat{\mathrm{\Theta}}}^{*}=\left({\hat{\alpha}}_{0}^{*},{\hat{\lambda}}^{*},{\hat{\beta}}^{*}\right)$ using the samples from Step 2.
- -
- Step 4: Repeat Step 2 and 3, B times.
- -
- Step 5: Estimate the cdf of $\hat{{\theta}^{*}}$ by $\hat{{F}_{1}}\left(x\right)=P\left(\hat{{\theta}^{*}}\le x\right)$ where $\theta $ is ${\alpha}_{0},\lambda $ or $\beta $ and let ${\hat{\theta}}_{Boot-p}={\hat{{F}_{1}}}^{-1}\left(x\right)$ be the inverse of $\hat{{F}_{1}}$ for a given x $\left(0<x<1\right)$.

#### 6.2. The Bootstrap-t Confidence Interval

- -
- Steps 1 and 2 are the same as those in the algorithm of the bootstrap percentile confidence interval.
- -
- Step 3: Compute the MLE $\hat{{\theta}^{*}}$ using the progressive censored samples from Step 2 where $\theta $ is ${\alpha}_{0},\lambda $ or $\beta $. Then, obtain the statistic$${T}^{*}=\frac{\hat{{\theta}^{*}}-\hat{\theta}}{\sqrt{V\hat{a}r\left(\hat{{\theta}^{*}}\right)}}.$$
- -
- Step 4: Repeat Step 2 and 3, B times.
- -
- Step 5: Estimate the cdf of ${T}^{*}$ by $\hat{{F}_{2}}\left(x\right)=P\left({T}^{*}\le x\right)$ and let ${\hat{\theta}}_{Boot-t}=\hat{\theta}+\sqrt{V\hat{a}r\left(\hat{{\theta}^{*}}\right)}$ ${\hat{{F}_{2}}}^{-1}\left(x\right)$ for a given x $\left(0<x<1\right)$.

## 7. Bayesian Estimation

#### 7.1. Tierney and Kadane Technique

#### 7.2. Lindley’s Approximation

## 8. Optimal Design

#### 8.1. D-Optimality

#### 8.2. A-Optimality

## 9. Simulation Study

- (1)
- According to Table 1, Table 2, Table 3, Table 4, Table 5 and Table 6, the parameter estimation under different progressive censoring schemes all roughly follows the pattern of better performance with increasing sample size. Different censoring schemes have different performance in different estimation methods. As the censoring scheme of withdrawing one surviving unit for each unit observed to fail from the beginning of the test (for example, $\left(1*15\right)$), the performance in point estimation is poor, especially when the sample size is relatively small and the estimation bias is relatively large. However, this censoring scheme has no obvious difference from other schemes in interval estimation.
- (2)
- According to the results of the point estimates in Table 1, Table 2 and Table 3, the estimated values are all close to the true values and, roughly, the MSE decreases as the sample size increases, so all the point estimation methods mentioned are valid. Based on the MSEs of the parameter estimators, the three methods perform similarly when the progressive censoring scheme is determined. For the estimation of ${\alpha}_{0}$, the BE obtained from the Lindley’s approximation performs better than BE obtained from the Tierney and Kadane technique. Additionally, BE obtained from the Tierney and Kadane technique performs slightly better than MLE. For $\lambda $ estimation, BE obtained from the Lindley’s approximation performs best, and the BE obtained from the Tierney and Kadane technique and MLE perform similarly. For $\beta $ estimation, BE obtained from the Lindley’s approximation performs best, followed by BE obtained from Lindley’s approximation and MLE. Overall, BE obtained from the Lindley’s approximation performs well for all three parameters.
- (3)
- According to the results of the interval estimation in Table 4, Table 5 and Table 6, generally speaking, the asymptotic confidence intervals perform best. The bootstrap percentile confidence intervals and the bootstrap-t confidence intervals perform similarly when the progressive censoring scheme is determined. For all three interval estimation methods, the AIL of $\lambda $ is longer than those of ${\alpha}_{0}$ and $\beta $. The COVP of the intervals of ${\alpha}_{0}$ outperforms those of both $\lambda $ and $\beta $ in parametric bootstrap intervals.
- (4)
- According to Table 7 and Table 8 for the optimal transformed stress level, the values based on A-optimality are smaller than those based on D-optimality. For a similar censoring scheme, the results based on D-optimality become larger as the sample size increases. However, the results based on A-optimality are the opposite. In general, the optimal transformed stress levels do not differ significantly in each of the optimal criteria.

## 10. Real Data Analysis

- (1)
- The point estimates of ${\alpha}_{0}$ and $\beta $ obtained from the three different point estimation methods are similar, but the point estimates of $\lambda $ are slightly different. For $\lambda $ estimation, the MLE is maximal, followed by the BE obtained from the Lindley’s approximation and the BE obtained from the Tierney and Kadane technique;
- (2)
- In terms of AIL, the results obtained by the three interval estimation methods do not differ significantly. The bootstrap-t confidence intervals outperform slightly the asymptotic confidence intervals and the bootstrap percentile confidence intervals. This is consistent with the pattern that parametric bootstrap intervals outperform asymptotic intervals when the sample size is small.
- (3)
- For the bounds of the interval estimates of ${\alpha}_{0}$, the bootstrap percentile confidence intervals and the bootstrap-t confidence intervals are similar. For the interval estimates of $\lambda $, the upper and lower bounds of the asymptotic confidence intervals are significantly larger than those of the bootstrap percentile confidence intervals and the bootstrap-t confidence intervals. The bounds of the bootstrap-t confidence intervals are slightly larger than those of the bootstrap percentile confidence intervals. For the bounds on the interval estimates of $\beta $, the results obtained by the three methods are similar.

## 11. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 8.**For ${n}_{i}$ assigned units, progressive Type-II censoring scheme $\left({R}_{i1},{R}_{i2},\cdots ,{R}_{i{m}_{i}}\right)$ at stress level ${s}_{i}$, $i=1,2,\cdots k$, in constant-stress accelerated life testing.

**Figure 9.**The empirical cumulative distribution function and the fitted cumulative distribution function for failure data under stress levels ${s}_{1}=30$ kV (Left, data 1) and ${s}_{2}=36$ kV (Right, data 2).

**Table 1.**The expected values (EV) and mean square errors (MSE) of the maximum likelihood estimates for different sample sizes $\left({n}_{i},{m}_{i}\right)$ and different censoring schemes ${R}_{i}$.

$\left({\mathit{n}}_{\mathit{i}},{\mathit{m}}_{\mathit{i}}\right)$ | ${\mathit{R}}_{\mathit{i}}$ | EV${}_{{\hat{\mathit{\alpha}}}_{0}}$ | MSE${}_{{\hat{\mathit{\alpha}}}_{0}}$ | EV${}_{\hat{\mathit{\lambda}}}$ | MSE${}_{\hat{\mathit{\lambda}}}$ | EV${}_{\hat{\mathit{\beta}}}$ | MSE${}_{\hat{\mathit{\beta}}}$ |
---|---|---|---|---|---|---|---|

$\left(30,15\right)$ | ($0\ast 7$, 15, $0\ast 7$) | $0.4933$ | $0.0321$ | $2.1014$ | $0.0736$ | $0.7474$ | $0.0142$ |

($0\ast 5$, $3\ast 5$, $0\ast 5$) | $0.4946$ | $0.0324$ | $2.1050$ | $0.0786$ | $0.7484$ | $0.0151$ | |

($1\ast 15$) | $0.4967$ | $0.0367$ | $2.1238$ | $0.0932$ | $0.7567$ | $0.0182$ | |

$\left(45,25\right)$ | ($0\ast 12$, 20, $0\ast 12$) | $0.4728$ | $0.0144$ | $2.0573$ | $0.0330$ | $0.7281$ | $0.0071$ |

($0\ast 10$, $4\ast 5$, $0\ast 10$) | $0.4729$ | $0.0146$ | $2.0583$ | $0.0368$ | $0.7281$ | $0.0076$ | |

($1\ast 20$, $0\ast 5$) | $0.4703$ | $0.0139$ | $2.0595$ | $0.0355$ | $0.7281$ | $0.0074$ | |

$\left(60,35\right)$ | ($0\ast 17$, 25, $0\ast 17$) | $0.4643$ | $0.0096$ | $2.0456$ | $0.0235$ | $0.7217$ | $0.0049$ |

($0\ast 15$, $5\ast 5$, $0\ast 15$) | $0.4671$ | $0.0100$ | $2.0409$ | $0.0224$ | $0.7205$ | $0.0048$ | |

($1\ast 25$, $0\ast 10$) | $0.4650$ | $0.0098$ | $2.0402$ | $0.0228$ | $0.7199$ | $0.0050$ |

**Table 2.**The expected values (EV) and mean square errors (MSE) of the Bayes estimates using the Tierney and Kadane technique for different sample sizes $\left({n}_{i},{m}_{i}\right)$ and different censoring schemes ${R}_{i}$.

$\left({\mathit{n}}_{\mathit{i}},{\mathit{m}}_{\mathit{i}}\right)$ | ${\mathit{R}}_{\mathit{i}}$ | EV${}_{{\hat{\mathit{\alpha}}}_{0}}$ | MSE${}_{{\hat{\mathit{\alpha}}}_{0}}$ | EV${}_{\hat{\mathit{\lambda}}}$ | MSE${}_{\hat{\mathit{\lambda}}}$ | EV${}_{\hat{\mathit{\beta}}}$ | MSE${}_{\hat{\mathit{\beta}}}$ |
---|---|---|---|---|---|---|---|

$\left(30,15\right)$ | ($0\ast 7$, 15, $0\ast 7$) | 0.4957 | 0.0291 | 2.0937 | 0.0715 | 0.7386 | 0.0130 |

($0\ast 5$, $3\ast 5$, $0\ast 5$) | 0.4957 | 0.0299 | 2.1003 | 0.0779 | 0.7397 | 0.0139 | |

($1\ast 15$) | 0.4953 | 0.0315 | 2.1187 | 0.0902 | 0.7476 | 0.0166 | |

$\left(45,25\right)$ | ($0\ast 12$, 20, $0\ast 12$) | 0.4744 | 0.0147 | 2.0541 | 0.0330 | 0.7229 | 0.0067 |

($0\ast 10$, $4\ast 5$, $0\ast 10$) | 0.4747 | 0.0150 | 2.0563 | 0.0350 | 0.7236 | 0.0069 | |

($1\ast 20$, $0\ast 5$) | 0.4750 | 0.0142 | 2.0556 | 0.0345 | 0.7244 | 0.0071 | |

$\left(60,35\right)$ | ($0\ast 17$, 25, $0\ast 17$) | 0.4669 | 0.0093 | 2.0368 | 0.0220 | 0.7157 | 0.0047 |

($0\ast 15$, $5\ast 5$, $0\ast 15$) | 0.4664 | 0.0096 | 2.0364 | 0.0223 | 0.7154 | 0.0046 | |

($1\ast 25$, $0\ast 10$) | 0.4664 | 0.0098 | 2.0401 | 0.0235 | 0.7172 | 0.0049 |

**Table 3.**The expected values (EV) and mean square errors (MSE) of the Bayes estimates using the Lindley’s approximation for different sample sizes $\left({n}_{i},{m}_{i}\right)$ and different censoring schemes ${R}_{i}$.

$\left({\mathit{n}}_{\mathit{i}},{\mathit{m}}_{\mathit{i}}\right)$ | ${\mathit{R}}_{\mathit{i}}$ | EV${\hat{\mathit{\alpha}}}_{0}$ | MSE${\hat{\mathit{\alpha}}}_{0}$ | EV${}_{\hat{\mathit{\lambda}}}$ | MSE${}_{\hat{\mathit{\lambda}}}$ | EV${}_{\hat{\mathit{\beta}}}$ | MSE${}_{\hat{\mathit{\beta}}}$ |
---|---|---|---|---|---|---|---|

$\left(30,15\right)$ | ($0\ast 7$, 15, $0\ast 7$) | 0.4869 | 0.0270 | 2.0424 | 0.0491 | 0.7117 | 0.0098 |

($0\ast 5$, $3\ast 5$, $0\ast 5$) | 0.4825 | 0.0265 | 2.0425 | 0.0514 | 0.7099 | 0.0105 | |

($1\ast 15$) | 0.4826 | 0.0258 | 2.0462 | 0.0593 | 0.7119 | 0.0125 | |

$\left(45,25\right)$ | ($0\ast 12$, 20, $0\ast 12$) | 0.4699 | 0.0126 | 2.0290 | 0.0269 | 0.7088 | 0.0058 |

($0\ast 10$, $4\ast 5$, $0\ast 10$) | 0.4704 | 0.0138 | 2.0290 | 0.0285 | 0.7089 | 0.0060 | |

($1\ast 20$, $0\ast 5$) | 0.4708 | 0.0133 | 2.0286 | 0.0304 | 0.7085 | 0.0065 | |

$\left(60,35\right)$ | ($0\ast 17$, 25, $0\ast 17$) | 0.4632 | 0.0088 | 2.0190 | 0.0192 | 0.7059 | 0.0042 |

($0\ast 15$, $5\ast 5$, $0\ast 15$) | 0.4648 | 0.0087 | 2.0189 | 0.0196 | 0.7054 | 0.0042 | |

($1\ast 25$, $0\ast 10$) | 0.4660 | 0.0096 | 2.0171 | 0.0194 | 0.7051 | 0.0042 |

**Table 4.**The lower bounds (LB), upper bounds (UB), average interval lengths (AIL) and coverage probabilities (COVP) of the $95\%$ asymptotic confidence intervals for different sample sizes $\left({n}_{i},{m}_{i}\right)$ and different censoring schemes ${R}_{i}$.

$\left({\mathit{n}}_{\mathit{i}},{\mathit{m}}_{\mathit{i}}\right)$ | ${\mathit{R}}_{\mathit{i}}$ | Parameter | LB | UB | AIL | COVP |
---|---|---|---|---|---|---|

$\left(30,15\right)$ | ($0\ast 7$, 15, $0\ast 7$) | ${\alpha}_{0}$ | 0.2015 | 0.7859 | 0.5844 | 0.9422 |

$\lambda $ | 1.6609 | 2.5389 | 0.8779 | 0.9556 | ||

$\beta $ | 0.5503 | 0.9425 | 0.3922 | 0.9403 | ||

($0\ast 5$, $3\ast 5$, $0\ast 5$) | ${\alpha}_{0}$ | 0.2020 | 0.7876 | 0.5856 | 0.9451 | |

$\lambda $ | 1.6553 | 2.5524 | 0.8971 | 0.9595 | ||

$\beta $ | 0.5470 | 0.9495 | 0.4025 | 0.9436 | ||

($1\ast 15$) | ${\alpha}_{0}$ | 0.2018 | 0.7889 | 0.5871 | 0.9476 | |

$\lambda $ | 1.6240 | 2.6262 | 1.0022 | 0.9649 | ||

$\beta $ | 0.5285 | 0.9863 | 0.4578 | 0.9404 | ||

$\left(45,25\right)$ | ($0\ast 12$, 20, $0\ast 12$) | ${\alpha}_{0}$ | 0.2565 | 0.6871 | 0.4306 | 0.9467 |

$\lambda $ | 1.7324 | 2.3861 | 0.6538 | 0.9535 | ||

$\beta $ | 0.5784 | 0.8780 | 0.2995 | 0.9465 | ||

($0\ast 10$, $4\ast 5$, $0\ast 10$) | ${\alpha}_{0}$ | 0.2569 | 0.6878 | 0.4309 | 0.9461 | |

$\lambda $ | 1.7296 | 2.3883 | 0.6587 | 0.9546 | ||

$\beta $ | 0.5768 | 0.8793 | 0.3026 | 0.9456 | ||

($1\ast 20$, $0\ast 5$) | ${\alpha}_{0}$ | 0.2553 | 0.6840 | 0.4287 | 0.9446 | |

$\lambda $ | 1.7270 | 2.3995 | 0.6725 | 0.9546 | ||

$\beta $ | 0.5742 | 0.8847 | 0.3106 | 0.9470 | ||

$\left(60,35\right)$ | ($0\ast 17$, 25, $0\ast 17$) | ${\alpha}_{0}$ | 0.2860 | 0.6448 | 0.3588 | 0.9461 |

$\lambda $ | 1.7700 | 2.3133 | 0.5433 | 0.9535 | ||

$\beta $ | 0.5940 | 0.8454 | 0.2514 | 0.9474 | ||

($0\ast 15$, $5\ast 5$, $0\ast 15$) | ${\alpha}_{0}$ | 0.2856 | 0.6437 | 0.3581 | 0.9444 | |

$\lambda $ | 1.7683 | 2.3144 | 0.5461 | 0.9523 | ||

$\beta $ | 0.5935 | 0.8465 | 0.2531 | 0.9486 | ||

($1\ast 25$, $0\ast 10$) | ${\alpha}_{0}$ | 0.2853 | 0.6447 | 0.3594 | 0.9422 | |

$\lambda $ | 1.7675 | 2.3144 | 0.5468 | 0.9562 | ||

$\beta $ | 0.5927 | 0.8471 | 0.2544 | 0.9491 |

**Table 5.**The lower bounds (LB), upper bounds (UB), average interval lengths (AIL) and coverage probabilities (COVP) of the $95\%$ bootstrap percentile confidence interval for different sample sizes $\left({n}_{i},{m}_{i}\right)$ and different censoring schemes ${R}_{i}$.

$\left({\mathit{n}}_{\mathit{i}},{\mathit{m}}_{\mathit{i}}\right)$ | ${\mathit{R}}_{\mathit{i}}$ | Parameter | LB | UB | AIL | COVP |
---|---|---|---|---|---|---|

$\left(30,15\right)$ | ($0\ast 7$, 15, $0\ast 7$) | ${\alpha}_{0}$ | 0.2748 | 1.0015 | 0.7267 | 0.9180 |

$\lambda $ | 1.8041 | 2.9166 | 1.1125 | 0.8520 | ||

$\beta $ | 0.6093 | 1.0612 | 0.4519 | 0.8560 | ||

($0\ast 5$, $3\ast 5$, $0\ast 5$) | ${\alpha}_{0}$ | 0.2762 | 1.0105 | 0.7343 | 0.9320 | |

$\lambda $ | 1.7954 | 2.9324 | 1.1370 | 0.8680 | ||

$\beta $ | 0.6050 | 1.0685 | 0.4635 | 0.8540 | ||

($1\ast 15$) | ${\alpha}_{0}$ | 0.2760 | 1.0237 | 0.7476 | 0.9260 | |

$\lambda $ | 1.7939 | 3.0883 | 1.2945 | 0.8800 | ||

$\beta $ | 0.5961 | 1.1200 | 0.5238 | 0.8800 | ||

$\left(45,25\right)$ | ($0\ast 12$, 20, $0\ast 12$) | ${\alpha}_{0}$ | 0.3040 | 0.7959 | 0.4919 | 0.9260 |

$\lambda $ | 1.8200 | 2.5674 | 0.7474 | 0.8980 | ||

$\beta $ | 0.6171 | 0.9435 | 0.3265 | 0.8900 | ||

($0\ast 10$, $4\ast 5$, $0\ast 10$) | ${\alpha}_{0}$ | 0.3056 | 0.8003 | 0.4947 | 0.9220 | |

$\lambda $ | 1.8125 | 2.5607 | 0.7482 | 0.9040 | ||

$\beta $ | 0.6144 | 0.9442 | 0.3298 | 0.9000 | ||

($1\ast 20$, $0\ast 5$) | ${\alpha}_{0}$ | 0.3134 | 0.8193 | 0.5059 | 0.9120 | |

$\lambda $ | 1.7963 | 2.5448 | 0.7484 | 0.9180 | ||

$\beta $ | 0.6068 | 0.9414 | 0.3346 | 0.8960 | ||

$\left(60,35\right)$ | ($0\ast 17$, 25, $0\ast 17$) | ${\alpha}_{0}$ | 0.3188 | 0.7094 | 0.3906 | 0.9460 |

$\lambda $ | 1.8287 | 2.4219 | 0.5932 | 0.9180 | ||

$\beta $ | 0.6188 | 0.8854 | 0.2666 | 0.9060 | ||

($0\ast 15$, $5\ast 5$, $0\ast 15$) | ${\alpha}_{0}$ | 0.3184 | 0.7082 | 0.3898 | 0.9420 | |

$\lambda $ | 1.8368 | 2.4402 | 0.6035 | 0.9040 | ||

$\beta $ | 0.6251 | 0.8957 | 0.2706 | 0.8820 | ||

($1\ast 25$, $0\ast 10$) | ${\alpha}_{0}$ | 0.3144 | 0.7022 | 0.3878 | 0.9200 | |

$\lambda $ | 1.8367 | 2.4400 | 0.6032 | 0.9020 | ||

$\beta $ | 0.6219 | 0.8917 | 0.2699 | 0.8880 |

**Table 6.**The lower bounds (LB), upper bounds (UB), average interval lengths (AIL) and coverage probabilities (COVP) of the $95\%$ bootstrap-t confidence interval for different sample sizes $\left({n}_{i},{m}_{i}\right)$ and different censoring schemes ${R}_{i}$.

$\left({\mathit{n}}_{\mathit{i}},{\mathit{m}}_{\mathit{i}}\right)$ | ${\mathit{R}}_{\mathit{i}}$ | Parameter | LB | UB | AIL | COVP |
---|---|---|---|---|---|---|

$\left(30,15\right)$ | ($0\ast 7$, 15, $0\ast 7$) | ${\alpha}_{0}$ | 0.2779 | 1.0088 | 0.7309 | 0.9180 |

$\lambda $ | 1.8004 | 2.8922 | 1.0919 | 0.8660 | ||

$\beta $ | 0.6098 | 1.0624 | 0.4526 | 0.8460 | ||

($0\ast 5$, $3\ast 5$, $0\ast 5$) | ${\alpha}_{0}$ | 0.2803 | 1.0228 | 0.7425 | 0.9140 | |

$\lambda $ | 1.7986 | 2.9278 | 1.1293 | 0.8640 | ||

$\beta $ | 0.6091 | 1.0729 | 0.4639 | 0.8580 | ||

($1\ast 15$) | ${\alpha}_{0}$ | 0.2862 | 1.0585 | 0.7723 | 0.9260 | |

$\lambda $ | 1.7877 | 3.0774 | 1.2897 | 0.8680 | ||

$\beta $ | 0.5981 | 1.1247 | 0.5266 | 0.8540 | ||

$\left(45,25\right)$ | ($0\ast 12$, 20, $0\ast 12$) | ${\alpha}_{0}$ | 0.2991 | 0.7843 | 0.4851 | 0.9300 |

$\lambda $ | 1.8251 | 2.5750 | 0.7499 | 0.8900 | ||

$\beta $ | 0.6203 | 0.9461 | 0.3257 | 0.8760 | ||

($0\ast 10$, $4\ast 5$, $0\ast 10$) | ${\alpha}_{0}$ | 0.3063 | 0.8031 | 0.4968 | 0.9240 | |

$\lambda $ | 1.8170 | 2.5678 | 0.7508 | 0.9180 | ||

$\beta $ | 0.6151 | 0.9446 | 0.3295 | 0.9020 | ||

($1\ast 20$, $0\ast 5$) | ${\alpha}_{0}$ | 0.3135 | 0.8194 | 0.5059 | 0.9180 | |

$\lambda $ | 1.8088 | 2.5659 | 0.7571 | 0.9100 | ||

$\beta $ | 0.6123 | 0.9485 | 0.3362 | 0.8860 | ||

$\left(60,35\right)$ | ($0\ast 17$, 25, $0\ast 17$) | ${\alpha}_{0}$ | 0.3255 | 0.7213 | 0.3957 | 0.9280 |

$\lambda $ | 1.8254 | 2.3876 | 0.5623 | 0.9060 | ||

$\beta $ | 0.6194 | 0.8699 | 0.2505 | 0.9120 | ||

($0\ast 15$, $5\ast 5$, $0\ast 15$) | ${\alpha}_{0}$ | 0.3182 | 0.7099 | 0.3917 | 0.9460 | |

$\lambda $ | 1.8306 | 2.4291 | 0.5984 | 0.9020 | ||

$\beta $ | 0.6199 | 0.8886 | 0.2687 | 0.9040 | ||

($1\ast 25$, $0\ast 10$) | ${\alpha}_{0}$ | 0.3173 | 0.7077 | 0.3904 | 0.9260 | |

$\lambda $ | 1.8339 | 2.4316 | 0.5977 | 0.9160 | ||

$\beta $ | 0.6227 | 0.8927 | 0.2700 | 0.9120 |

**Table 7.**The average optimal transformed stress level ${h}_{k}^{\ast}$ using D-optimality for different sample sizes $\left({n}_{i},{m}_{i}\right)$ and different censoring schemes ${R}_{i}$.

$\left({\mathit{n}}_{\mathit{i}},{\mathit{m}}_{\mathit{i}}\right)$ | ${\mathit{R}}_{\mathit{i}}$ | ${\mathit{h}}_{\mathit{k}}^{\ast}$ | ${\mathit{R}}_{\mathit{i}}$ | ${\mathit{h}}_{\mathit{k}}^{\ast}$ | ${\mathit{R}}_{\mathit{i}}$ | ${\mathit{h}}_{\mathit{k}}^{\ast}$ |
---|---|---|---|---|---|---|

$\left(30,15\right)$ | ($0\ast 7$, 15, $0\ast 7$) | 8.2396 | ($0\ast 5$, $3\ast 5$, $0\ast 5$) | 8.1901 | ($1\ast 15$) | 7.7286 |

$\left(45,25\right)$ | ($0\ast 12$, 20, $0\ast 12$) | 8.6145 | ($0\ast 10$, $4\ast 5$, $0\ast 10$) | 8.5932 | ($1\ast 20$, $0\ast 5$) | 8.5267 |

$\left(60,35\right)$ | ($0\ast 17$, 25, $0\ast 17$) | 8.7662 | ($0\ast 15$, $5\ast 5$, $0\ast 15$) | 8.7874 | ($1\ast 25$, $0\ast 10$) | 8.7212 |

**Table 8.**The average optimal transformed stress level ${h}_{k}^{\ast}$ using A-optimality for different sample sizes $\left({n}_{i},{m}_{i}\right)$ and different censoring schemes ${R}_{i}$.

$\left({\mathit{n}}_{\mathit{i}},{\mathit{m}}_{\mathit{i}}\right)$ | ${\mathit{R}}_{\mathit{i}}$ | ${\mathit{h}}_{\mathit{k}}^{\ast}$ | ${\mathit{R}}_{\mathit{i}}$ | ${\mathit{h}}_{\mathit{k}}^{\ast}$ | ${\mathit{R}}_{\mathit{i}}$ | ${\mathit{h}}_{\mathit{k}}^{\ast}$ |
---|---|---|---|---|---|---|

$\left(30,15\right)$ | ($0\ast 7$, 15, $0\ast 7$) | 6.7736 | ($0\ast 5$, $3\ast 5$, $0\ast 5$) | 6.7567 | ($1\ast 15$) | 6.7330 |

$\left(45,25\right)$ | ($0\ast 12$, 20, $0\ast 12$) | 6.7087 | ($0\ast 10$, $4\ast 5$, $0\ast 10$) | 6.7015 | ($1\ast 20$, $0\ast 5$) | 6.6857 |

$\left(60,35\right)$ | ($0\ast 17$, 25, $0\ast 17$) | 6.6858 | ($0\ast 15$, $5\ast 5$, $0\ast 15$) | 6.6738 | ($1\ast 25$, $0\ast 10$) | 6.6786 |

Stress Level | Data Set | K-S Statistic | p-Value |
---|---|---|---|

30 kV | 7.74, 17.05, 20.46, 21.02, 22.66, 43.40, 47.30, 139.07, 144.12, 175.88, 194.90 | 0.2203 | 0.5858 |

36 kV | 0.35, 0.59, 0.96, 0.99, 1.69, 1.97, 2.07, 2.58, 2.71, 2.90, 3.67, 3.99, 5.35, 13.77, 25.50 | 0.2179 | 0.4154 |

Stress Level | ${\mathit{h}}_{\mathit{i}}$ | $\left({\mathit{n}}_{\mathit{i}},{\mathit{m}}_{\mathit{i}}\right)$ | ${\mathit{R}}_{\mathit{i}}$ | Censored Data |
---|---|---|---|---|

30 kV | 1 | $\left(11,10\right)$ | ($0\ast 4$, 1, $0\ast 5$) | 7.74, 17.05, 20.46, 21.02, 22.66, 47.30, 139.07, 144.12, 175.88, 194.90 |

36 kV | 1.44966 | $\left(15,14\right)$ | ($0\ast 6$, 1, $0\ast 7$) | 0.35, 0.59, 0.96, 0.99, 1.69, 1.97, 2.07, 2.58, 2.90, 3.67, 3.99, 5.35, 13.77, 25.50 |

**Table 11.**The maximum likelihood estimates (MLE) and the Bayes estimates using the Tierney and Kadane technique ($B{E}_{TK}$) and Lindley’s approximation ($B{E}_{LA}$).

Parameter | MLE | ${\mathit{BE}}_{\mathit{TK}}$ | ${\mathit{BE}}_{\mathit{LA}}$ |
---|---|---|---|

${\alpha}_{0}$ | 0.0025 | 0.0032 | 0.0026 |

$\lambda $ | 22.8063 | 20.4310 | 21.3963 |

$\beta $ | 0.2639 | 0.2605 | 0.2673 |

**Table 12.**The 95% asymptotic confidence intervals (ACI), the 95% bootstrap percentile confidence intervals (Boot-p) and the 95% bootstrap-t confidence intervals (Boot-t).

Parameter | ACI | Boot-p | Boot-t |
---|---|---|---|

${\alpha}_{0}$ | (0.0012, 0.0039) 0.0027 | (0.0025, 0.0218) 0.0193 | (0.0025, 0.0043) 0.0018 |

$\lambda $ | (9.9884, 35.6241) 25.6356 | (4.8530, 28.3640) 23.5111 | (6.0894, 28.2857) 22.1964 |

$\beta $ | (0.2292, 0.2986) 0.0694 | (0.2162, 0.2848) 0.0686 | (0.2259, 0.2812) 0.0553 |

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**MDPI and ACS Style**

Zhang, W.; Gui, W.
Statistical Inference and Optimal Design of Accelerated Life Testing for the Chen Distribution under Progressive Type-II Censoring. *Mathematics* **2022**, *10*, 1609.
https://doi.org/10.3390/math10091609

**AMA Style**

Zhang W, Gui W.
Statistical Inference and Optimal Design of Accelerated Life Testing for the Chen Distribution under Progressive Type-II Censoring. *Mathematics*. 2022; 10(9):1609.
https://doi.org/10.3390/math10091609

**Chicago/Turabian Style**

Zhang, Wenjie, and Wenhao Gui.
2022. "Statistical Inference and Optimal Design of Accelerated Life Testing for the Chen Distribution under Progressive Type-II Censoring" *Mathematics* 10, no. 9: 1609.
https://doi.org/10.3390/math10091609