# Statistical Analysis and Theoretical Framework for a Partially Accelerated Life Test Model with Progressive First Failure Censoring Utilizing a Power Hazard Distribution

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## Abstract

**:**

## 1. Introduction

## 2. Model Descriptions and Assumptions

#### 2.1. Model Description

#### 2.2. Assumptions

- The lifetime of all units tested under various normal or accelerated conditions follows the PHFD.
- The lifetimes of test units are independent identically distributed random variables.
- The total number of units under test is $N={N}_{1}+{N}_{2}={n}_{1}{k}_{1}+{n}_{2}{k}_{2}$.
- Any unit has a lifetime of ${X}_{2}={\beta}^{-1}{X}_{1}$ under accelerated conditions.
- The lifetimes ${X}_{1i},i=1,2,\dots ,{m}_{1}$ of units assigned to the normal condition, while the lifetimes ${X}_{2i},i=1,2,\dots ,{m}_{2}$ of units assigned to the accelerated condition are independent of one another.

## 3. Maximum Likelihood Estimation

- (1)
- Use the method of moments or any other methods to estimate the parameters $\beta ,$ $\delta $ and $\rho $ as starting point of iteration, denote the estimates as $\left({\beta}_{0},{\delta}_{0},{\rho}_{0}\right)$, and set $l=0$.
- (2)
- Calculate ${\left(\frac{\partial \ell}{\partial \beta},\frac{\partial \ell}{\partial \delta},\frac{\partial \ell}{\partial \rho}\right)}_{\left({\beta}_{l},{\delta}_{l},{\rho}_{l}\right)}$ and the observed Fisher information matrix ${I}^{-1}\left(\beta ,\delta ,\rho \right),$ given in Section 3.
- (3)
- Update $\left(\beta ,\delta ,\rho \right)$ as$$\left({\beta}_{l+1},{\delta}_{l+1},{\rho}_{l+1}\right)=\left({\beta}_{l},{\delta}_{l},{\rho}_{l}\right)+{\left(\frac{\partial \ell}{\partial \beta},\frac{\partial \ell}{\partial \delta},\frac{\partial \ell}{\partial \rho}\right)}_{\left({\beta}_{l},{\delta}_{l},{\rho}_{l}\right)}\times {I}^{-1}\left(\beta ,\delta ,\rho \right).$$
- (4)
- Set $l=l+1$, and then go back to Step (1).
- (5)
- Continue the iterative steps until $\left|\left({\beta}_{l+1},{\delta}_{l+1},{\rho}_{l+1}\right)-\left({\beta}_{l},{\delta}_{l},{\rho}_{l}\right)\right|$ is smaller than a threshold value. The final estimates of $\beta ,$ $\delta $, and $\rho $ are the MLE of the parameters, denoted as $\widehat{\beta},$ $\widehat{\delta}$ and $\widehat{\rho}$.

#### 3.1. Consistent and Asymptotically Normal Estimators

#### 3.1.1. Consistency characteristic

**Theorem**

**1.**

**θ**and meets the subsequent criteria:

- 1.
- $\ell \left(\mathsf{\theta}\right|\underline{x})$ is differentiable in
**θ**for all x in the sample space. - 2.
- The expected value of the score function $\Omega \left(\underline{x},\mathsf{\theta}\right)={\displaystyle \frac{\partial \ell \left(\mathsf{\theta}\right|\underline{x})}{\partial \mathsf{\theta}}}$ is zero at the true parameter value, i.e., $E\left[\Omega \left(\underline{x},\mathsf{\theta}\right)\right]=0$ for $\mathsf{\theta}={\mathsf{\theta}}_{0}$.
- 3.
- The FIM ${I}^{-1}\left(\mathsf{\theta}\right)=E\left[\Omega \left(\underline{x},\mathsf{\theta}\right)\Omega {\left(\underline{x},\mathsf{\theta}\right)}^{T}\right]$ is positive definite at the true parameter value, i.e., ${I}^{-1}\left({\mathsf{\theta}}_{0}\right)>0$.

**θ**.

**Proof.**

#### 3.1.2. Asymptotic Normality Characteristic

**Theorem**

**2.**

**Proof.**

## 4. Parametric Bootstrap

- Using the original PFFC sample as a foundation, ${x}_{j1:{m}_{j}:{n}_{j}:{k}_{j}}^{{R}_{j}},{x}_{j2:{m}_{j}:{n}_{j}:{k}_{j}}^{{R}_{j}},\cdots ,{x}_{j{m}_{j}:{m}_{j}:{n}_{j}:{k}_{j}}^{{R}_{j}}$ for $j=1,2$, obtain ${\widehat{\beta}}_{ML},$ ${\widehat{\delta}}_{ML}$ and ${\widehat{\rho}}_{ML}$.
- Employ the censoring plan $\left({n}_{j},{m}_{j},{k}_{j},{R}_{ji}\right)$ and $({\widehat{\beta}}_{ML},{\widehat{\delta}}_{ML},{\widehat{\rho}}_{ML})$ to generate a PFFC bootstrap sample ${x}_{j1:{m}_{j}:{n}_{j}:{k}_{j}}^{\ast {R}_{j}},{x}_{j2:{m}_{j}:{n}_{j}:{k}_{j}}^{\ast {R}_{j}},\cdots ,{x}_{j{m}_{j}:{m}_{j}:{n}_{j}:{k}_{j}}^{\ast {R}_{j}}$ for $j=1,2$.
- From ${x}_{j1:{m}_{j}:{n}_{j}:{k}_{j}}^{\ast {R}_{j}},{x}_{j2:{m}_{j}:{n}_{j}:{k}_{j}}^{\ast {R}_{j}},\cdots ,{x}_{j{m}_{j}:{m}_{j}:{n}_{j}:{k}_{j}}^{\ast {R}_{j}}$ calculate the bootstrap estimates, which are indicated by the symbol ${\widehat{\eta}}^{\ast},$ where ${\widehat{\eta}}^{\ast}={\widehat{\beta}}_{ML}^{\ast},$ ${\widehat{\delta}}_{ML}^{\ast}$ and ${\widehat{\rho}}_{ML}^{\ast}$.
- Steps 2 and 3 should be repeated $NB$ times to produce ${\widehat{\eta}}_{1}^{\ast},{\widehat{\eta}}_{2}^{\ast},\dots ,{\widehat{\eta}}_{NB}^{\ast}$.
- Sort ${\widehat{\eta}}_{j}^{\ast},$ $j=1,2,\dots ,NB,$ ascendingly as ${\widehat{\eta}}_{\left(j\right)}^{\ast},$ $j=1,2,\dots ,NB$.

#### 4.1. Parametric Boot-p

#### 4.2. Parametric Boot-t

## 5. Bayesian Estimation

- Start with an $(\beta ,\delta ,\rho )=({\widehat{\beta}}_{ML},{\widehat{\delta}}_{ML},{\widehat{\rho}}_{ML})$, and set $J=1$.
- Generate ${\beta}^{\left(J\right)}$ from $Gamma\left[{m}_{2},{\rho}^{\left(J-1\right)}{\Phi}_{2}\right]$.
- Generate ${\delta}^{\left(J\right)}$ according to the following:
- (a)
- Generate ${\delta}^{\ast}$ from normal distribution $N\left[{\delta}^{\left(J-1\right)},Var\left({\widehat{\delta}}_{ML}\right)\right]$ where $Var\left({\widehat{\delta}}_{ML}\right)$ the variance of $\delta $ given in (22).
- (b)
- Compute $r=min\left[1,\frac{{\pi}_{2}^{\ast}\left({\delta}^{\ast}\right|{\beta}^{\left(J\right)},{\rho}^{\left(J-1\right)},\underline{x})}{{\pi}_{2}^{\ast}\left({\delta}^{\left(J-1\right)}\right|{\beta}^{\left(J\right)},{\rho}^{\left(J-1\right)},\underline{x})}\right]$.
- (c)
- Generate a sample $\mu $ from the $U\left[0,1\right]$ distribution.
- (d)
- If $\mu \u2aafr$ set ${\delta}^{\left(J\right)}={\delta}^{\ast}$; otherwise, ${\delta}^{\left(J\right)}={\delta}^{\left(J-1\right)}$.

- Generate ${\rho}^{\left(J\right)}$ from $Gamma\left[{m}_{1}+{m}_{2}+{a}_{2},{\Phi}_{1}+{\beta}^{\left(J\right)}{\Phi}_{2}+{b}_{2}\right]$.
- Set $J=J+1$.
- To collect the required number of samples, repeat Steps 2–5 M times.

## 6. Simulation Study

- In every instance, as would be expected, the MSEs and AWs of all estimates decrease as sample sizes increase. It verifies the consistency features of each estimation method.
- With n and m keeping invariant, k increases both MSEs and AWs increase.
- In terms of decreased MSEs and AWs, the first scheme (I) performs the best when sample sizes are fixed and failures are observed.
- The MSE and AW both increase when removals are delayed.
- In terms of MSEs and AWs, Bayes estimation using MCMC performs better than the other approaches (ML, Boot-p, Boot-t).
- Due to having the smallest MSE and narrowest width, MCMC CRIs are, overall, the most satisfactory.
- Bootstrap methods outperform the ML approach in terms of MSEs and AWs. Furthermore, Boot-t performs better than Boot-p in terms of MSEs and AWs.
- The estimates produced by the ML, bootstrap, and Bayesian approaches are highly similar and have high CPs (around $0.95$).
- In spite of the fact that the Bayes estimators perform better than all other estimators, the simulation results show that all point and interval estimator approaches are efficient. The Bayes technique may be chosen if one has sufficient prior knowledge. If past knowledge of the topic being studied cannot be accessed, bootstrap approaches that primarily rely on MLEs are preferred.

## 7. Practical Analysis of Engineering Data

- Normal use condition: 0.18, 0.19, 0.19, 0.34, 0.36, 0.40, 0.44, 0.44, 0.45, 0.46, 0.47,0.53, 0.57, 0.57, 0.63, 0.65, 0.70, 0.71, 0.71, 0.75, 0.76, 0.76, 0.79,0.80, 0.85, 0.98, 1.01, 1.07, 1.12, 1.14, 1.15, 1.17, 1.20, 1.23, 1.24,1.25, 1.26, 1.32, 1.33, 1.33, 1.39, 1.42, 1.50, 1.55, 1.58, 1.59, 1.62, 1.68, 1.70, 1.79, 2.00, 2.01, 2.04, 2.54, 3.61, 3.76, 4.65, 8.97.
- Accelerated stress condition: 0.13, 0.16, 0.20, 0.20, 0.21, 0.25, 0.26, 0.28, 0.28, 0.30, 0.31, 0.33, 0.35, 0.35, 0.35, 0.39, 0.50, 0.52, 0.58, 0.60, 0.60, 0.62, 0.63, 0.67, 0.71, 0.73, 0.75, 0.75, 0.78, 0.80, 0.80, 0.86, 0.90, 0.91, 0.93, 0.93, 0.94, 0.98, 0.99, 1.01, 1.03, 1.06, 1.06, 1.10, 1.22, 1.22, 1.24, 1.28, 1.39, 1.39, 1.46, 1.48, 1.52, 1.74, 1.95, 2.46, 3.02, 5.16.

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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$\mathit{\beta}$ | $\mathit{\delta}$ | ||||||||
---|---|---|---|---|---|---|---|---|---|

$(\mathit{k},\mathit{n},{\mathit{m}}_{\mathbf{1}},{\mathit{m}}_{\mathbf{2}})$ | CS | ML | Boot-p | Boot-t | Bayes | ML | Boot-p | Boot-t | Bayes |

$(2,40,15,15)$ | I | 0.22546 | 0.23547 | 0.20417 | 0.18652 | 0.33457 | 0.32485 | 0.30635 | 0.28965 |

II | 0.24687 | 0.25473 | 0.22563 | 0.19968 | 0.35647 | 0.34783 | 0.32968 | 0.30124 | |

III | 0.27365 | 0.26475 | 0.23984 | 0.22364 | 0.37856 | 0.36451 | 0.34789 | 0.32658 | |

$(2,40,20,25)$ | I | 0.17635 | 0.17335 | 0.15364 | 0.13478 | 0.29365 | 0.28647 | 0.26455 | 0.24365 |

II | 0.19635 | 0.19124 | 0.17365 | 0.15879 | 0.31254 | 0.30654 | 0.28574 | 0.26458 | |

III | 0.23345 | 0.22654 | 0.20658 | 0.18657 | 0.35647 | 0.34998 | 0.32456 | 0.29571 | |

$(2,60,30,30)$ | I | 0.14532 | 0.14110 | 0.12365 | 0.10859 | 0.25648 | 0.24782 | 0.23011 | 0.21109 |

II | 0.15998 | 0.15366 | 0.13948 | 0.11932 | 0.27457 | 0.26475 | 0.24986 | 0.22145 | |

III | 0.16997 | 0.16984 | 0.15621 | 0.13265 | 0.30564 | 0.29658 | 0.27694 | 0.24362 | |

$(2,60,35,40)$ | I | 0.11997 | 0.10999 | 0.09587 | 0.91124 | 0.21456 | 0.20548 | 0.18325 | 0.15964 |

II | 0.12658 | 0.11965 | 0.10689 | 0.09463 | 0.23547 | 0.22457 | 0.19875 | 0.17145 | |

III | 0.13475 | 0.12968 | 0.11654 | 0.10556 | 0.25639 | 0.24573 | 0.22143 | 0.20325 | |

$(2,90,45,45)$ | I | 0.09967 | 0.09745 | 0.08869 | 0.08234 | 0.18635 | 0.18002 | 0.15968 | 0.13124 |

II | 0.10568 | 0.11002 | 0.09378 | 0.08968 | 0.21245 | 0.20321 | 0.17663 | 0.15347 | |

III | 0.12065 | 0.11890 | 0.10554 | 0.99976 | 0.23475 | 0.22657 | 0.19661 | 0.16999 | |

$(2,90,60,75)$ | I | 0.09345 | 0.09164 | 0.08345 | 0.07965 | 0.15635 | 0.14658 | 0.13001 | 0.11475 |

II | 0.97367 | 0.09535 | 0.09128 | 0.08467 | 0.17658 | 0.16584 | 0.15348 | 0.13554 | |

III | 0.10369 | 0.10024 | 0.09786 | 0.09164 | 0.20214 | 0.19648 | 0.17653 | 0.14587 | |

$(4,40,15,15)$ | I | 0.26548 | 0.25545 | 0.22416 | 0.20653 | 0.35456 | 0.34487 | 0.32636 | 0.29999 |

II | 0.27655 | 0.26547 | 0.24635 | 0.21455 | 0.37365 | 0.36784 | 0.34621 | 0.32632 | |

III | 0.29365 | 0.28456 | 0.26459 | 0.23587 | 0.39652 | 0.38843 | 0.36427 | 0.33994 | |

$(4,40,20,25)$ | I | 0.19639 | 0.19339 | 0.17368 | 0.15479 | 0.33364 | 0.32646 | 0.29454 | 0.27361 |

II | 0.21356 | 0.21012 | 0.19875 | 0.17348 | 0.35652 | 0.34721 | 0.31265 | 0.29478 | |

III | 0.23854 | 0.23187 | 0.21365 | 0.19597 | 0.37452 | 0.36543 | 0.34652 | 0.31247 | |

$(4,60,30,30)$ | I | 0.17534 | 0.17113 | 0.15364 | 0.13852 | 0.28646 | 0.27783 | 0.25024 | 0.23128 |

II | 0.19658 | 0.18635 | 0.16996 | 0.15023 | 0.30245 | 0.29654 | 0.27683 | 0.25362 | |

III | 0.21345 | 0.20689 | 0.18965 | 0.17647 | 0.32654 | 0.31475 | 0.29012 | 0.26948 | |

$(4,60,35,40)$ | I | 0.13998 | 0.12996 | 0.11581 | 0.10125 | 0.24453 | 0.23547 | 0.20326 | 0.17965 |

II | 0.15234 | 0.14687 | 0.12897 | 0.11012 | 0.26481 | 0.25644 | 0.22345 | 0.19634 | |

III | 0.17124 | 0.16589 | 0.14658 | 0.13011 | 0.28635 | 0.27461 | 0.24867 | 0.21543 | |

$(4,90,45,45)$ | I | 0.11968 | 0.10744 | 0.09866 | 0.09233 | 0.21636 | 0.20003 | 0.17969 | 0.14125 |

II | 0.13546 | 0.12896 | 0.10554 | 0.09989 | 0.23154 | 0.22547 | 0.19568 | 0.16532 | |

III | 0.14896 | 0.13997 | 0.11856 | 0.10743 | 0.25473 | 0.24516 | 0.22341 | 0.18678 | |

$(4,90,60,75)$ | I | 0.10346 | 0.10005 | 0.09344 | 0.08966 | 0.18634 | 0.17657 | 0.15002 | 0.12671 |

II | 0.11597 | 0.11063 | 0.10557 | 0.09764 | 0.21548 | 0.20362 | 0.17695 | 0.14635 | |

III | 0.13124 | 0.12897 | 0.11869 | 0.10323 | 0.23684 | 0.22457 | 0.19632 | 0.17021 |

$(\mathit{k},\mathit{n},{\mathit{m}}_{1},{\mathit{m}}_{2})$ | CS | ML | Boot-p | Boot-t | Bayes |
---|---|---|---|---|---|

$(2,40,15,15)$ | I | 0.52634 | 0.51635 | 0.47654 | 0.42658 |

II | 0.53642 | 0.52369 | 0.49652 | 0.45234 | |

III | 0.56471 | 0.55632 | 0.52362 | 0.47685 | |

$(2,40,20,25)$ | I | 0.49632 | 0.48657 | 0.42364 | 0.39874 |

II | 0.51243 | 0.50247 | 0.45632 | 0.41867 | |

III | 0.53624 | 0.52463 | 0.47562 | 0.43869 | |

$(2,60,30,30)$ | I | 0.45783 | 0.44568 | 0.38745 | 0.35476 |

II | 0.47695 | 0.46357 | 0.40693 | 0.37985 | |

III | 0.49863 | 0.48655 | 0.43675 | 0.40127 | |

$(2,60,35,40)$ | I | 0.41236 | 0.40238 | 0.35968 | 0.32154 |

II | 0.43658 | 0.42563 | 0.37454 | 0.34578 | |

III | 0.46112 | 0.45027 | 0.40321 | 0.36942 | |

$(2,90,45,45)$ | I | 0.37695 | 0.36546 | 0.31258 | 0.28994 |

II | 0.39542 | 0.38456 | 0.34127 | 0.31253 | |

III | 0.41258 | 0.40357 | 0.37124 | 0.34624 | |

$(2,90,60,75)$ | I | 0.33642 | 0.32145 | 0.28635 | 0.24751 |

II | 0.35628 | 0.34658 | 0.30547 | 0.27136 | |

III | 0.37564 | 0.36472 | 0.33453 | 0.29954 | |

$(4,40,15,15)$ | I | 0.54635 | 0.53636 | 0.49655 | 0.44657 |

II | 0.56243 | 0.55364 | 0.51247 | 0.47635 | |

III | 0.58672 | 0.57463 | 0.53241 | 0.49356 | |

$(4,40,20,25)$ | I | 0.51634 | 0.50655 | 0.44366 | 0.42875 |

II | 0.53624 | 0.52471 | 0.46572 | 0.44632 | |

III | 0.56328 | 0.55473 | 0.48652 | 0.46211 | |

$(4,60,30,30)$ | I | 0.47782 | 0.46569 | 0.40746 | 0.37475 |

II | 0.49363 | 0.48657 | 0.42869 | 0.39674 | |

III | 0.51364 | 0.50472 | 0.45362 | 0.42578 | |

$(4,60,35,40)$ | I | 0.43237 | 0.42239 | 0.37967 | 0.34155 |

II | 0.45362 | 0.44572 | 0.39452 | 0.36973 | |

III | 0.48965 | 0.47658 | 0.42583 | 0.39112 | |

$(4,90,45,45)$ | I | 0.39696 | 0.38545 | 0.33257 | 0.30995 |

II | 0.41283 | 0.40324 | 0.36254 | 0.32164 | |

III | 0.44658 | 0.43657 | 0.39655 | 0.36442 | |

$(4,90,60,75)$ | I | 0.35642 | 0.34145 | 0.30635 | 0.26751 |

II | 0.37625 | 0.37001 | 0.32487 | 0.28974 | |

III | 0.41235 | 0.40586 | 0.35646 | 0.31587 |

$(\mathit{k},\mathit{n},{\mathit{m}}_{1},{\mathit{m}}_{2})$ | CS | MLE | Boot-p | Boot-t | Bayes | ||||
---|---|---|---|---|---|---|---|---|---|

ACIs | ACIs | ACIs | CRIs | ||||||

AWs | CPs | AWs | CPs | AWs | CPs | AWs | CPs | ||

$(2,40,15,15)$ | I | 3.2536 | 0.941 | 3.1562 | 0.951 | 2.9974 | 0.951 | 2.9265 | 0.959 |

II | 3.2785 | 0.939 | 3.1847 | 0.941 | 3.1047 | 0.954 | 2.9847 | 0.954 | |

III | 3.3246 | 0.938 | 3.2354 | 0.945 | 3.1648 | 0.949 | 3.1025 | 0.951 | |

$(2,40,20,25)$ | I | 3.1346 | 0.941 | 3.0994 | 0.943 | 2.8575 | 0.947 | 2.7996 | 0.961 |

II | 3.1954 | 0.942 | 3.1168 | 0.950 | 2.9347 | 0.948 | 2.8465 | 0.974 | |

III | 3.2246 | 0.929 | 3.1648 | 0.939 | 2.9877 | 0.951 | 2.9364 | 0.963 | |

$(2,60,30,30)$ | I | 3.0258 | 0.938 | 2.9578 | 0.954 | 2.7754 | 0.950 | 2.6789 | 0.955 |

II | 3.1045 | 0.937 | 3.0987 | 0.941 | 2.8346 | 0.955 | 2.7245 | 0.958 | |

III | 3.1567 | 0.941 | 3.1011 | 0.939 | 2.9124 | 0.954 | 2.8654 | 0.963 | |

$(2,60,35,40)$ | I | 2.9567 | 0.938 | 2.8836 | 0.938 | 2.6648 | 0.949 | 2.5763 | 0.964 |

II | 2.9997 | 0.937 | 2.9475 | 0.941 | 2.7135 | 0.942 | 2.6345 | 0.954 | |

III | 3.1245 | 0.941 | 3.0899 | 0.942 | 2.8366 | 0.947 | 2.7541 | 0.971 | |

$(2,90,45,45)$ | I | 2.8746 | 0.943 | 2.7986 | 0.947 | 2.5564 | 0.951 | 2.4975 | 0.972 |

II | 2.9257 | 0.939 | 2.8345 | 0.951 | 2.5987 | 0.946 | 2.5563 | 0.966 | |

III | 2.9765 | 0.937 | 2.8841 | 0.946 | 2.6634 | 0.955 | 2.5946 | 0.967 | |

$(2,90,60,75)$ | I | 2.7568 | 0.951 | 2.6899 | 0.942 | 2.4757 | 0.953 | 2.3999 | 0.958 |

II | 2.8364 | 0.948 | 2.7246 | 0.943 | 2.5376 | 0.955 | 2.4462 | 0.955 | |

III | 2.8822 | 0.943 | 2.7864 | 0.941 | 2.5947 | 0.951 | 2.5146 | 0.957 | |

$(4,40,15,15)$ | I | 3.3536 | 0.952 | 3.2562 | 0.951 | 3.1974 | 0.949 | 3.0926 | 0.962 |

II | 3.4125 | 0.954 | 3.3154 | 0.952 | 3.2246 | 0.948 | 3.1124 | 0.958 | |

III | 3.4855 | 0.947 | 3.3698 | 0.944 | 3.2997 | 0.951 | 3.1994 | 0.962 | |

$(4,40,20,25)$ | I | 3.2347 | 0.946 | 3.1993 | 0.937 | 2.9576 | 0.953 | 2.8395 | 0.958 |

II | 3.3145 | 0.937 | 3.2564 | 0.951 | 3.1045 | 0.952 | 2.9457 | 0.961 | |

III | 3.3765 | 0.938 | 3.3145 | 0.947 | 3.2247 | 0.956 | 3.1046 | 0.962 | |

$(4,60,30,30)$ | I | 3.1254 | 0.941 | 3.0579 | 0.939 | 2.8757 | 0.957 | 2.7786 | 0.958 |

II | 3.2547 | 0.944 | 3.1147 | 0.938 | 2.9456 | 0.955 | 2.8359 | 0.956 | |

III | 3.3254 | 0.940 | 3.2169 | 0.941 | 3.1456 | 0.954 | 2.9248 | 0.955 | |

$(4,60,35,40)$ | I | 3.0766 | 0.951 | 2.9837 | 0.936 | 2.7649 | 0.953 | 2.6462 | 0.961 |

II | 3.1365 | 0.949 | 3.0689 | 0.941 | 2.8365 | 0.949 | 2.7154 | 0.960 | |

III | 3.2355 | 0.950 | 3.1347 | 0.945 | 2.9446 | 0.955 | 2.8122 | 0.958 | |

$(4,90,45,45)$ | I | 2.9745 | 0.929 | 2.8985 | 0.943 | 2.6565 | 0.948 | 2.5976 | 0.957 |

II | 3.0997 | 0.955 | 2.9648 | 0.933 | 2.7253 | 0.959 | 2.6541 | 0.964 | |

III | 3.1605 | 0.941 | 3.0765 | 0.941 | 2.8223 | 0.958 | 2.7446 | 0.955 | |

$(4,90,60,75)$ | I | 2.8567 | 0.949 | 2.7898 | 0.951 | 2.5758 | 0.954 | 2.4995 | 0.974 |

II | 2.9124 | 0.934 | 2.8247 | 0.952 | 2.6647 | 0.953 | 2.5731 | 0.966 | |

III | 2.9999 | 0.941 | 2.8976 | 0.949 | 2.7764 | 0.955 | 2.6474 | 0.962 |

$(\mathit{k},\mathit{n},{\mathit{m}}_{1},{\mathit{m}}_{2})$ | CS | MLE | Boot-p | Boot-t | Bayes | ||||
---|---|---|---|---|---|---|---|---|---|

ACIs | ACIs | ACIs | CRIs | ||||||

AWs | CPs | AWs | CPs | AWs | CPs | AWs | CPs | ||

$(2,40,15,15)$ | I | 2.0456 | 0.939 | 1.8947 | 0.941 | 1.7458 | 0.941 | 1.6978 | 0.958 |

II | 2.1354 | 0.937 | 1.9365 | 0.945 | 1.8657 | 0.945 | 1.7245 | 0.956 | |

III | 2.3654 | 0.951 | 2.0584 | 0.943 | 1.9658 | 0.948 | 1.8654 | 0.955 | |

$(2,40,20,25)$ | I | 1.8654 | 0.948 | 1.7548 | 0.950 | 1.6645 | 0.946 | 1.5471 | 0.961 |

II | 1.9345 | 0.943 | 1.8673 | 0.939 | 1.7654 | 0.944 | 1.6745 | 0.960 | |

III | 2.1346 | 0.952 | 1.9694 | 0.954 | 1.8568 | 0.952 | 1.7589 | 0.958 | |

$(2,60,30,30)$ | I | 1.6547 | 0.954 | 1.5694 | 0.941 | 1.4573 | 0.951 | 1.3365 | 0.957 |

II | 1.7548 | 0.939 | 1.6947 | 0.939 | 1.5576 | 0.952 | 1.4289 | 0.964 | |

III | 1.8345 | 0.937 | 1.7784 | 0.941 | 1.6532 | 0.954 | 1.5364 | 0.955 | |

$(2,60,35,40)$ | I | 1.4698 | 0.951 | 1.3687 | 0.945 | 1.3001 | 0.951 | 1.2874 | 0.974 |

II | 1.5643 | 0.948 | 1.4568 | 0.943 | 1.4999 | 0.954 | 1.3568 | 0.958 | |

III | 1.6637 | 0.943 | 1.5587 | 0.950 | 1.5003 | 0.949 | 1.4346 | 0.956 | |

$(2,90,45,45)$ | I | 1.2654 | 0.952 | 1.1547 | 0.939 | 1.1136 | 0.947 | 1.0996 | 0.955 |

II | 1.3654 | 0.954 | 1.2756 | 0.954 | 1.2136 | 0.948 | 1.1564 | 0.961 | |

III | 1.4587 | 0.939 | 1.3654 | 0.941 | 1.3122 | 0.951 | 1.2456 | 0.960 | |

$(2,90,60,75)$ | I | 1.1769 | 0.937 | 1.0987 | 0.939 | 1.0778 | 0.950 | 1.0658 | 0.958 |

II | 1.2365 | 0.951 | 1.1647 | 0.941 | 1.1034 | 0.955 | 1.1001 | 0.957 | |

III | 1.3124 | 0.948 | 1.2546 | 0.945 | 1.1865 | 0.954 | 1.1236 | 0.964 | |

$(4,40,15,15)$ | I | 2.1454 | 0.943 | 1.9948 | 0.943 | 1.8457 | 0.951 | 1.7979 | 0.955 |

II | 2.2365 | 0.952 | 2.1457 | 0.950 | 1.9658 | 0.954 | 1.8547 | 0.974 | |

III | 2.3691 | 0.954 | 2.2574 | 0.939 | 2.1365 | 0.949 | 1.9432 | 0.958 | |

$(4,40,20,25)$ | I | 1.9655 | 0.939 | 1.8549 | 0.954 | 1.7644 | 0.947 | 1.6472 | 0.956 |

II | 2.0547 | 0.937 | 1.9358 | 0.941 | 1.8576 | 0.948 | 1.7569 | 0.955 | |

III | 2.1369 | 0.951 | 2.0965 | 0.939 | 1.9658 | 0.951 | 1.8694 | 0.961 | |

$(4,60,30,30)$ | I | 1.7548 | 0.948 | 1.6695 | 0.941 | 1.5574 | 0.950 | 1.4366 | 0.960 |

II | 1.8476 | 0.943 | 1.7764 | 0.945 | 1.6547 | 0.955 | 1.5467 | 0.958 | |

III | 1.9568 | 0.952 | 1.8649 | 0.943 | 1.7466 | 0.954 | 1.6573 | 0.957 | |

$(4,60,35,40)$ | I | 1.5697 | 0.954 | 1.4686 | 0.950 | 1.4002 | 0.951 | 1.3875 | 0.964 |

II | 1.6694 | 0.941 | 1.5473 | 0.939 | 1.5012 | 0.954 | 1.4768 | 0.955 | |

III | 1.7589 | 0.939 | 1.6377 | 0.954 | 1.5999 | 0.949 | 1.5152 | 0.974 | |

$(4,90,45,45)$ | I | 1.3652 | 0.937 | 1.2548 | 0.941 | 1.2137 | 0.947 | 1.1593 | 0.961 |

II | 1.4586 | 0.951 | 1.3475 | 0.939 | 1.3104 | 0.948 | 1.2468 | 0.960 | |

III | 1.5624 | 0.948 | 1.4586 | 0.929 | 1.4007 | 0.951 | 1.3567 | 0.959 | |

$(4,90,60,75)$ | I | 1.2767 | 0.943 | 1.1988 | 0.941 | 1.1576 | 0.950 | 1.1051 | 0.953 |

II | 1.3654 | 0.952 | 1.2689 | 0.939 | 1.2145 | 0.955 | 1.1698 | 0.961 | |

III | 1.4652 | 0.954 | 1.3584 | 0.941 | 1.2997 | 0.954 | 1.2563 | 0.959 |

$(\mathit{k},\mathit{n},{\mathit{m}}_{1},{\mathit{m}}_{2})$ | CS | MLE | Boot-p | Boot-t | Bayes | ||||
---|---|---|---|---|---|---|---|---|---|

ACIs | ACIs | ACIs | CRIs | ||||||

AWs | CPs | AWs | CPs | AWs | CPs | AWs | CPs | ||

$(2,40,15,15)$ | I | 4.0564 | 0.929 | 3.8654 | 0.939 | 3.6548 | 0.947 | 2.9584 | 0.951 |

II | 4.2658 | 0.955 | 4.0569 | 0.954 | 3.8614 | 0.948 | 3.1653 | 0.958 | |

III | 4.4635 | 0.941 | 4.2355 | 0.941 | 4.0563 | 0.951 | 3.4658 | 0.957 | |

$(2,40,20,25)$ | I | 3.7365 | 0.949 | 3.5468 | 0.939 | 3.3659 | 0.950 | 3.0125 | 0.964 |

II | 3.9652 | 0.934 | 3.7659 | 0.941 | 3.5476 | 0.955 | 3.2486 | 0.955 | |

III | 4.1365 | 0.929 | 3.9254 | 0.939 | 3.7463 | 0.954 | 3.3192 | 0.974 | |

$(2,60,30,30)$ | I | 3.5564 | 0.955 | 3.3684 | 0.954 | 3.1468 | 0.947 | 2.8567 | 0.958 |

II | 3.7685 | 0.941 | 3.5477 | 0.941 | 3.3698 | 0.948 | 3.0568 | 0.957 | |

III | 3.9476 | 0.949 | 3.7666 | 0.939 | 3.5574 | 0.951 | 3.2695 | 0.964 | |

$(2,60,35,40)$ | I | 3.3659 | 0.934 | 3.1457 | 0.941 | 2.8745 | 0.950 | 2.6547 | 0.955 |

II | 3.5687 | 0.929 | 3.3394 | 0.939 | 3.1692 | 0.955 | 2.7954 | 0.974 | |

III | 3.6985 | 0.955 | 3.5147 | 0.954 | 3.2998 | 0.954 | 3.0045 | 0.958 | |

$(2,90,45,45)$ | I | 3.1254 | 0.941 | 2.7994 | 0.941 | 2.5998 | 0.947 | 2.4577 | 0.957 |

II | 3.3258 | 0.949 | 2.9957 | 0.939 | 2.7984 | 0.948 | 2.5969 | 0.964 | |

III | 3.5462 | 0.934 | 3.2598 | 0.941 | 3.0243 | 0.951 | 2.8635 | 0.955 | |

$(2,90,60,75)$ | I | 2.9965 | 0.929 | 2.6954 | 0.939 | 2.3874 | 0.950 | 2.2466 | 0.974 |

II | 3.1564 | 0.955 | 2.8547 | 0.954 | 2.5146 | 0.955 | 2.3721 | 0.958 | |

III | 3.3467 | 0.941 | 3.0119 | 0.941 | 2.7568 | 0.954 | 2.5599 | 0.957 | |

$(4,40,15,15)$ | I | 4.2564 | 0.949 | 4.0654 | 0.939 | 3.8548 | 0.947 | 3.1584 | 0.964 |

II | 4.4689 | 0.934 | 4.2658 | 0.941 | 4.0654 | 0.948 | 3.2581 | 0.955 | |

III | 4.5956 | 0.929 | 4.4365 | 0.939 | 4.1997 | 0.951 | 3.5876 | 0.974 | |

$(4,40,20,25)$ | I | 3.9365 | 0.955 | 3.7468 | 0.954 | 3.5659 | 0.950 | 3.2125 | 0.958 |

II | 4.1358 | 0.941 | 3.9524 | 0.941 | 3.7456 | 0.955 | 3.4658 | 0.957 | |

III | 4.3692 | 0.949 | 4.1365 | 0.939 | 3.9647 | 0.954 | 3.6524 | 0.964 | |

$(4,60,30,30)$ | I | 3.7563 | 0.934 | 3.5685 | 0.941 | 3.2467 | 0.947 | 3.0568 | 0.955 |

II | 3.9658 | 0.929 | 3.7458 | 0.939 | 3.4578 | 0.948 | 3.1995 | 0.974 | |

III | 4.1568 | 0.955 | 3.9651 | 0.954 | 3.6654 | 0.951 | 3.4554 | 0.958 | |

$(4,60,35,40)$ | I | 3.5657 | 0.941 | 3.3454 | 0.941 | 3.0746 | 0.950 | 2.8548 | 0.957 |

II | 3.7441 | 0.949 | 3.5662 | 0.939 | 3.2313 | 0.955 | 3.0533 | 0.964 | |

III | 3.8954 | 0.934 | 3.7441 | 0.941 | 3.4225 | 0.954 | 3.1899 | 0.955 | |

$(4,90,45,45)$ | I | 3.3256 | 0.929 | 2.9895 | 0.939 | 2.7999 | 0.947 | 2.6578 | 0.974 |

II | 3.4996 | 0.955 | 3.1645 | 0.954 | 2.8974 | 0.948 | 2.8557 | 0.958 | |

III | 3.7154 | 0.941 | 3.4571 | 0.941 | 3.1824 | 0.951 | 3.0079 | 0.957 | |

$(4,90,60,75)$ | I | 3.1063 | 0.949 | 2.8956 | 0.939 | 2.5875 | 0.950 | 2.4467 | 0.964 |

II | 3.3651 | 0.938 | 3.1587 | 0.943 | 2.7458 | 0.955 | 2.6552 | 0.955 | |

III | 3.4985 | 0.397 | 3.3334 | 0.942 | 2.8965 | 0.954 | 2.8324 | 0.974 |

Normal use condition: $({k}_{1},{n}_{1},{m}_{1})=(2,29,15).$ |

${R}_{1}$ = (3, 1, 1, 2, 0, 1, 2, 1, 0, 2, 0, 1, 0, 0, 0). |

0.18, 0.19, 0.36, 0.45, 0.47, 0.57, 0.63, 0.70, 0.71, 0.76, 0.79, 0.85, 1.01, 3.76, 8.97. |

Accelerated stress condition: $({k}_{2},{n}_{2},{m}_{2})=(2,29,18).$ |

${R}_{2}$ = (1, 1, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 0, 1, 0, 0, 0, 0). |

0.13, 0.20, 0.21, 0.26, 0.31, 0.35, 0.50, 0.58, 0.60, 0.63, 0.75, 0.78, 0.80, 0.94, 1.22, 1.95, |

2.46, 3.02. |

**Table 7.**Estimates of $\left(\beta ,\delta ,\rho \right)$ and its corresponding $95\%$ CI using PFFC under CS-PALT.

Parameter | ${(.)}_{\mathit{ML}}$ | ${(.)}_{\mathit{Boot}-\mathit{p}}$ | ${(.)}_{\mathit{Boot}-\mathit{t}}$ | ${(.)}_{\mathit{BS}}$ | |
---|---|---|---|---|---|

$\beta $ | Estimate | 1.70985 | 1.74865 | 1.66473 | 1.54899 |

$95\%$ CI | (0.8371, 3.4926) | (0.9436, 3.3772) | (0.8945, 2.9246) | (0.9932, 2.6745) | |

$\delta $ | Estimate | 0.15323 | 0.16225 | 0.13482 | 0.12557 |

$95\%$ CI | (0.0295, 0.7953) | (0.0365, 0.8766) | (0.0334, 0.7452) | (0.0215, 0.6935) | |

$\rho $ | Estimate | 0.28209 | 0.26942 | 0.21641 | 0.19984 |

$95\%$ CI | (0.1701, 0.4679) | (0.1432, 0.5211) | (0.1165, 0.4263) | (0.0989, 0.3762) |

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© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Abd-El-Monem, A.; Eliwa, M.S.; El-Morshedy, M.; Al-Bossly, A.; EL-Sagheer, R.M.
Statistical Analysis and Theoretical Framework for a Partially Accelerated Life Test Model with Progressive First Failure Censoring Utilizing a Power Hazard Distribution. *Mathematics* **2023**, *11*, 4323.
https://doi.org/10.3390/math11204323

**AMA Style**

Abd-El-Monem A, Eliwa MS, El-Morshedy M, Al-Bossly A, EL-Sagheer RM.
Statistical Analysis and Theoretical Framework for a Partially Accelerated Life Test Model with Progressive First Failure Censoring Utilizing a Power Hazard Distribution. *Mathematics*. 2023; 11(20):4323.
https://doi.org/10.3390/math11204323

**Chicago/Turabian Style**

Abd-El-Monem, Amel, Mohamed S. Eliwa, Mahmoud El-Morshedy, Afrah Al-Bossly, and Rashad M. EL-Sagheer.
2023. "Statistical Analysis and Theoretical Framework for a Partially Accelerated Life Test Model with Progressive First Failure Censoring Utilizing a Power Hazard Distribution" *Mathematics* 11, no. 20: 4323.
https://doi.org/10.3390/math11204323