# Statistical Analysis of Type-II Generalized Progressively Hybrid Alpha-PIE Censored Data and Applications in Electronic Tubes and Vinyl Chloride

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## Abstract

**:**

## 1. Introduction

- Derive the maximum likelihood estimators (MLEs) in addition to their two-sided approximate confidence intervals (ACIs), using observed Fisher’s information, of the alpha-PIE parameters $\alpha $ and $\mu $ or any associated function such as $R\left(t\right)$ and $h\left(t\right)$.
- Derive the Bayes’ estimators in addition to their two-sided highest posterior density (HPD) intervals, under independent gamma priors assumption, of $\alpha $, $\mu $, $R\left(t\right)$, and $h\left(t\right)$ using the squared error loss (SEL) function.
- To select the best progressive censoring patterns among various competing strategies, several criteria of optimality are proposed.
- Via extensive Monte Carlo simulations, on the basis of four accuracy criteria, namely, (i) root mean squared-errors, (ii) mean relative absolute biases, (iii) average confidence lengths, and (iv) coverage percentages, the performance of the acquired estimators is examined. Additionally, two real-word applications from the engineering and chemistry sectors, to evaluate how the offered approaches operate in practice and to choose the best censoring strategy, are examined.

## 2. Classical Inference

#### 2.1. Maximum Likelihood Estimators

#### 2.2. Asymptotic Interval Estimators

## 3. Bayes Inference

#### 3.1. Prior Functions

#### 3.2. Bayes Estimators

**Step****1:**- Set initial values ${\alpha}^{\left(0\right)}=\widehat{\alpha}$ and ${\mu}^{\left(0\right)}=\widehat{\mu}$.
**Step****2:**- Set $j=1$
**Step****3:**- Obtain ${\alpha}^{*}$ and ${\mu}^{*}$ from $N(\widehat{\alpha},{\widehat{v}}_{11})$ and $N(\widehat{\mu},{\widehat{v}}_{22})$, respectively.
**Step****4:**- Obtain ${\varphi}_{\alpha}=\frac{{\pi}_{\rho}^{\alpha}(\left.{\alpha}^{*}\right|{\mu}^{[j-1]},\mathbf{x})}{{\pi}_{\rho}^{\alpha}(\left.{\alpha}^{[j-1]}\right|{\mu}^{[j-1]},\mathbf{x})}$ and ${\varphi}_{\mu}=\frac{{\pi}_{\rho}^{\mu}\left(\left.{\mu}^{*}\right|{\alpha}^{\left[j\right]},\mathbf{x}\right)}{{\pi}_{\rho}^{\mu}\left(\left.{\mu}^{[j-1]}\right|{\alpha}^{\left[j\right]},\mathbf{x}\right)}$.
**Step****5:**- Obtain ${u}_{1}$ and ${u}_{2}$ from uniform $U(0,1)$ distribution.
**Step****6:**- If ${u}_{\alpha}\u2a7dmin\left\{1,{\varphi}_{\alpha}\right\}$ and ${u}_{\mu}\u2a7dmin\left\{1,{\varphi}_{\mu}\right\}$ set ${\alpha}^{\left[j\right]}={\alpha}^{*}$ and ${\mu}^{\left[j\right]}={\mu}^{*}$; else set ${\alpha}^{\left[j\right]}={\alpha}^{[j-1]}$ and ${\mu}^{\left[j\right]}={\mu}^{(j-1)}$, respectively.
**Step****7:**- Put $j=j+1$.
**Step****8:**- Redo Steps 3–7 for $\mathcal{B}$ times and obtain ${\alpha}^{\left[j\right]}$ and ${\mu}^{\left[j\right]}$ for $j=1,2,\dots ,\mathcal{B}$.
**Step****9:**- Obtain the RF (3) and HRF (4) using $({\alpha}^{\left[j\right]},{\mu}^{\left[j\right]}),\phantom{\rule{4pt}{0ex}}j=1,2,\dots ,\mathcal{B}$, at $t>0$, respectively as$${R}^{\left[j\right]}\left(t\right)=\frac{{\alpha}^{\left[j\right]}}{{\alpha}^{\left[j\right]}-1}\left(1-{{\alpha}^{\left[j\right]}}^{exp\left(-{\mu}^{\left[j\right]}{t}^{-1}\right)-1}\right),$$$${h}^{\left[j\right]}\left(t\right)=\frac{{\mu}^{\left[j\right]}log\left({\alpha}^{\left[j\right]}\right){t}^{-2}exp\left(-{\mu}^{\left[j\right]}{t}^{-1}\right){{\alpha}^{\left[j\right]}}^{exp\left(-{\mu}^{\left[j\right]}{t}^{-1}\right)-1}}{\left(1-{{\alpha}^{\left[j\right]}}^{exp\left(-{\mu}^{\left[j\right]}{t}^{-1}\right)-1}\right)}.$$
**Step****10:**- Obtain the Bayes estimate $\tilde{\pi}(\xb7)$ of $\pi (\xb7)$, after eliminating the first ${\mathcal{B}}^{*}$ samples as burn-in, as$$\begin{array}{c}\hfill \tilde{\pi}(\alpha ,\mu )=\frac{1}{\mathcal{B}-{\mathcal{B}}^{*}}\sum _{j={\mathcal{B}}^{*}+1}^{\mathcal{B}}{\pi}^{\left[j\right]}(\alpha ,\mu ).\end{array}$$
**Step****11:**- Obtain the HPD interval of $\pi (\xb7)$ via ordering ${\pi}^{\left[j\right]}(\alpha ,\mu )$ for $j={\mathcal{B}}^{*}+1,\dots ,\mathcal{B}$. Following Chen and Shao [28], the $(1-\u03f5)100\%$ HPD interval of $\pi (\alpha ,\mu )$ is given by$$\begin{array}{c}\hfill \left({\pi}^{\left({j}^{*}\right)},{\pi}^{\left({j}^{*}+\left(1-\u03f5\right)\left(\mathcal{B}-{\mathcal{B}}^{*}\right)\right)}\right),\end{array}$$$${\delta}^{\left({j}^{*}+\left[\left(1-\u03f5\right)\left(\mathcal{B}-{\mathcal{B}}^{*}\right)\right]\right)}-{\pi}^{\left({j}^{*}\right)}=\underset{1\u2a7dj\u2a7d\u03f5\left(\mathcal{B}-{\mathcal{B}}^{*}\right)}{min}\left({\pi}^{\left(j+\left[\left(1-\u03f5\right)\left(\mathcal{B}-{\mathcal{B}}^{*}\right)\right]\right)}-{\pi}^{\left[j\right]}\right),$$

## 4. Monte Carlo Simulations

- (a)
- For Set-1:
- Prior-1:$({a}_{1},{a}_{2})=(6,1)\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}{b}_{i}=5\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}i=1,2$;
- Prior-2:$({a}_{1},{a}_{2})=(12,2)\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}{b}_{i}=10\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}i=1,2$.

- (b)
- For Set-2:
- Prior-1:$({a}_{1},{a}_{2})=(7.5,2.5)\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}{b}_{i}=5\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}i=1,2$;
- Prior-2:$({a}_{1},{a}_{2})=(15,5)\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}{b}_{i}=10\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4pt}{0ex}}i=1,2$.

- All acquired point and interval estimates of $\alpha $, $\mu $, $R\left(t\right)$ or $h\left(t\right)$ have good behavior; this is a general note.
- As n(or FP%) increases, all results of all unknown parameters of life perform satisfactorily. A similar point is also true when the spacing between n and m is reduced.
- As ${T}_{i},\phantom{\rule{4pt}{0ex}}i=1,2$ grow, for both Sets 1 and 2, the RMSEs, MRABs, and ACLs of $\alpha $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ narrowed down, while their CPs increase.
- As anticipated, due to the gamma information, the Bayes point (or HPD interval) estimates of $\alpha $, $\mu $, $R\left(t\right)$, or $h\left(t\right)$ behave better compared to the others.
- All Bayesian computations performed based on Prior-2 provide more accurate results than those obtained based on Prior-1. This finding is due to the fact that the associated variance of Prior-2 is less than the associated variance of Prior-1.
- Comparing the suggested schemes 1, 2, and 3, for both Sets 1 and 2, it is seen that the point (or interval) estimates of $\delta $ have good results when all survival items $n-m$ are removed at the first stage (i.e., Scheme-1) and of $\mu $, $R\left(t\right)$, and $h\left(t\right)$ at the last stage (i.e., Scheme-3).
- In summary, it is advised to use MCMC samples to estimate the model parameters and reliability features of the alpha-PIE lifetime model when Type-II generalized progressively hybrid censored data are available.

## 5. Real Applications

#### 5.1. Electronic Tubes

#### 5.2. Vinyl Chloride

## 6. Optimal Progressive Designs

#### 6.1. Optimum from Electronic Tubes

- According to ${\mathcal{O}}_{i},\phantom{\rule{4pt}{0ex}}i=1,2,3$, the design of Sch[2] (in Sample ${\mathcal{S}}_{1}$) and the design of Sch[3] (in Samples ${\mathcal{S}}_{2}$ and ${\mathcal{S}}_{3}$) are the optimum censoring plans compared to others.
- According to ${\mathcal{O}}_{4}$, the design of Sch[1] (in Sample ${\mathcal{S}}_{1}$), the design of Sch[3] (in Sample ${\mathcal{S}}_{2}$) and the design of Sch[2] (in Sample ${\mathcal{S}}_{3}$) are the optimum censoring plans compared to others.

#### 6.2. Optimum from Vinyl Chloride

- According to ${\mathcal{O}}_{1}$, the design of Sch[1] (in Sample ${\mathcal{S}}_{1}$) and the design of Sch[2] (in Samples ${\mathcal{S}}_{i},\phantom{\rule{4pt}{0ex}}i=2,3$) are the optimum censoring plans compared to others.
- According to ${\mathcal{O}}_{i},\phantom{\rule{4pt}{0ex}}i=2,3$, the design of Sch[1] (in Samples ${\mathcal{S}}_{i},\phantom{\rule{4pt}{0ex}}i=1,3$) are the optimum censoring plans compared to others.
- According to ${\mathcal{O}}_{i},\phantom{\rule{4pt}{0ex}}i=2,3$, the designs Sch[3] and Sch[2] (in Sample ${\mathcal{S}}_{2}$), respectively, are the optimum censoring plans compared to others.
- According to ${\mathcal{O}}_{4}$, the design of Sch[3] (in Samples ${\mathcal{S}}_{i},\phantom{\rule{4pt}{0ex}}i=1,2$) and the design of Sch[2] (in Samples ${\mathcal{S}}_{3}$) are the optimum censoring plans compared to others.

## 7. Concluding Remarks

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 3.**Autocorrelation (

**top**) and trace (

**bottom**) plots for MCMC draws of $\alpha $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$.

**Figure 9.**Fitted PDFs (

**a**); fitted RFs (

**b**); scaled TTT (

**c**); contour (

**d**); plots from electronic tubes data.

**Figure 10.**Density (

**left**) and trace (

**right**) plots of $\alpha $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ from electronic tubes data.

**Figure 11.**The PP plots of the alpha-PIE and its competitive distributions from vinyl chloride data.

**Figure 12.**Fitted PDFs (

**a**); fitted RFs (

**b**); scaled-TTT (

**c**); contour (

**d**) plots from vinyl chloride data.

**Figure 13.**Density (

**left**) and Trace (

**right**) plots of $\alpha $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ from vinyl chloride data.

q | ${\mathit{K}}_{\mathit{q}}$ | ${\mathit{D}}_{\mathit{q}}$ | ${\mathit{\Re}}_{\mathit{q}}({\mathit{T}}_{\mathit{\tau}};\mathit{\vartheta})$ | ${\mathit{R}}_{{\mathit{d}}_{\mathit{\tau}}+1}^{*}$ |
---|---|---|---|---|

1 | ${\Pi}_{i=1}^{{d}_{1}}{\sum}_{k=i}^{m}\left({R}_{k}+1\right)$ | ${d}_{1}$ | ${\left[1-F\left({T}_{1}\right)\right]}^{{R}_{{d}_{1}+1}^{*}}$ | $n-{d}_{1}-{\sum}_{k=1}^{m-1}{R}_{k}$ |

2 | ${\Pi}_{i=1}^{m}{\sum}_{k=i}^{m}\left({R}_{k}+1\right)$ | m | 1 | 0 |

3 | ${\Pi}_{i=1}^{{d}_{2}}{\sum}_{k=i}^{m}\left({R}_{k}+1\right)$ | ${d}_{2}$ | ${\left[1-F\left({T}_{2}\right)\right]}^{{R}_{{d}_{2}+1}^{*}}$ | $n-{d}_{2}-{\sum}_{k=1}^{{d}_{2}}{R}_{k}$ |

Plan | Author(s) | Setting |
---|---|---|

Type-I PHC | Kundu and Joarder [7] | ${T}_{1}\to 0$ |

Type-II PHC | Childs et al. [6] | ${T}_{2}\to \infty $ |

Type-I Hybrid | Epstein [8] | ${T}_{1}\to 0$, ${R}_{i}=0,\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,m-1,$ and ${R}_{m}=n-m$ |

Type-II Hybrid | Childs et al. [9] | ${T}_{2}\to \infty $, ${R}_{i}=0,\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,m-1,$ and ${R}_{m}=n-m$ |

Type-I censoring | Epstein and Sobel [10] | ${T}_{1}=0$, $m=n$, ${R}_{i}=0,\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,m-1,$ and ${R}_{m}=n-m$ |

Type-II censoring | Epstein and Sobel [10] | ${T}_{1}=0$, ${T}_{2}\to \infty $, ${R}_{i}=0,\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,m-1,$ and ${R}_{m}=n-m$ |

0.1415 | 0.3484 | 0.3994 | 0.4174 | 0.5937 | 1.1045 | 1.7323 | 1.8348 | 2.3467 | 2.4651 |

2.6155 | 2.7425 | 3.1356 | 3.2259 | 3.4177 | 3.5551 | 3.5681 | 3.7287 | 9.2817 | 9.3208 |

Model | MLE(St.Er) | N-L | A | B | C-A | H-Q | K-S (p-Value) | |
---|---|---|---|---|---|---|---|---|

$\mathsf{\alpha}$ | $\mathsf{\mu}$ | |||||||

APIE | 21.826 (55.135) | 0.4149 (0.2867) | 43.3798 | 90.7596 | 92.7511 | 91.4655 | 91.1484 | 0.1962 (0.375) |

IE | - | 0.9052 (0.2024) | 45.0282 | 92.0564 | 93.0522 | 92.2787 | 92.2508 | 0.2930 (0.051) |

IL | - | 1.2990 (0.2240) | 46.4118 | 94.8236 | 95.8194 | 95.0459 | 95.0180 | 0.3265 (0.0212) |

IW | 0.8531 (0.1361) | 1.0019 (0.2383) | 44.4847 | 92.9694 | 94.9609 | 93.6753 | 93.3581 | 0.2342 (0.1897) |

IG | 0.8692 (0.2390) | 0.7869 (0.2871) | 44.8932 | 93.7863 | 95.7778 | 94.4922 | 94.1751 | 0.2677 (0.0935) |

INH | 0.5622 (0.1660) | 2.7258 (1.7357) | 43.5167 | 91.0335 | 93.0249 | 91.7393 | 91.4222 | 0.2033 (0.3340) |

Scheme | Sample | ${\mathit{T}}_{1}\left({\mathit{d}}_{1}\right)$ | ${\mathit{T}}_{2}\left({\mathit{d}}_{2}\right)$ | Generated Data | ${\mathit{R}}^{*}$ | ${{\mathcal{T}}^{*}}^{*}$ |
---|---|---|---|---|---|---|

Sch[1] | ${\mathcal{S}}_{1}$ | 3.8 (10) | 4.5 (10) | 0.1415, 0.3994, 0.4174, 0.5937, 1.1045, 1.8348, 2.4651, 2.7425, 3.2259, 3.4177 | 1 | 3.8 |

${\mathcal{S}}_{2}$ | 2.5 (6) | 3.8 (10) | 0.1415, 0.4174, 0.5937, 1.7323, 2.3467, 2.4651, 2.6155, 3.2259, 3.5551, 3.5681 | 0 | 3.5681 | |

${\mathcal{S}}_{3}$ | 1.5 (4) | 3.3 (9) | 0.1415, 0.4174, 0.5937, 1.1045, 1.7323, 1.8348, 2.6155, 2.7425, 3.2259 | 2 | 3.3 | |

Sch[2] | ${\mathcal{S}}_{1}$ | 2.8 (10) | 3.5 (10) | 0.1415, 0.4174, 0.5937, 1.1045, 1.7323, 1.8348, 2.3467, 2.4651, 2.6155, 2.7425 | 2 | 2.8 |

${\mathcal{S}}_{2}$ | 1.8 (5) | 3.6 (10) | 0.1415, 0.3484, 0.5937, 1.1045, 1.7323, 2.3467, 2.4651, 2.6155, 3.1356, 3.5551 | 0 | 3.5551 | |

${\mathcal{S}}_{3}$ | 1.4 (4) | 3.6 (9) | 0.1415, 0.3994, 0.4174, 1.1045, 1.8348, 2.3467, 2.4651, 3.4177, 3.5551 | 3 | 3.6 | |

Sch[3] | ${\mathcal{S}}_{1}$ | 2.8 (10) | 3.2 (10) | 0.1415, 0.3994, 0.4174, 0.5937, 1.1045, 1.7323, 1.8348, 2.3467, 2.6155, 2.7425 | 2 | 2.8 |

${\mathcal{S}}_{2}$ | 1.2 (6) | 3.4 (10) | 0.1415, 0.3484, 0.3994, 0.4174, 0.5937, 1.1045, 1.7323, 1.8348, 2.6155, 3.2259 | 0 | 3.2259 | |

${\mathcal{S}}_{3}$ | 1.9 (6) | 3.5 (9) | 0.1415, 0.4174, 0.5937, 1.1045, 1.7323, 1.8348, 2.3467, 2.7425, 3.4177 | 3 | 3.5 |

**Table 6.**Bayesian and classical estimates of $\alpha $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ from electronic tube data.

Scheme | Sample | Par. | MLE | MCMC | ACI | HPD | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Est. | St.Er | Est. | St.Er | Lower | Upper | IW | Lower | Upper | IW | |||

Sch[1] | ${\mathcal{S}}_{1}$ | $\alpha $ | 28.054 | 19.395 | 27.953 | 0.1417 | 0.0000 | 66.067 | 66.067 | 27.760 | 28.149 | 0.3884 |

$\mu $ | 0.4391 | 0.1520 | 0.3742 | 0.0971 | 0.1411 | 0.7370 | 0.5960 | 0.2349 | 0.5154 | 0.2805 | ||

$R\left(0.5\right)$ | 0.8892 | 0.0542 | 0.8500 | 0.0589 | 0.7831 | 0.9954 | 0.2123 | 0.7599 | 0.9247 | 0.1649 | ||

$h\left(0.5\right)$ | 0.4043 | 0.1578 | 0.5098 | 0.1566 | 0.0949 | 0.7137 | 0.6187 | 0.3098 | 0.7522 | 0.4424 | ||

${\mathcal{S}}_{2}$ | $\alpha $ | 55.496 | 17.196 | 55.395 | 0.1431 | 21.792 | 89.200 | 67.408 | 55.193 | 55.588 | 0.3945 | |

$\mu $ | 0.5042 | 0.1509 | 0.4346 | 0.1038 | 0.2083 | 0.8000 | 0.5917 | 0.2847 | 0.5846 | 0.2999 | ||

$R\left(0.5\right)$ | 0.9389 | 0.0344 | 0.9138 | 0.0383 | 0.8716 | 0.9923 | 0.1207 | 0.8589 | 0.9617 | 0.1029 | ||

$h\left(0.5\right)$ | 0.2500 | 0.1179 | 0.3295 | 0.1197 | 0.0190 | 0.4810 | 0.4620 | 0.1770 | 0.5037 | 0.3267 | ||

${\mathcal{S}}_{3}$ | $\alpha $ | 48.976 | 17.349 | 48.875 | 0.1419 | 14.973 | 82.979 | 68.005 | 48.682 | 49.073 | 0.3905 | |

$\mu $ | 0.4676 | 0.1425 | 0.3995 | 0.1007 | 0.1884 | 0.7468 | 0.5584 | 0.2612 | 0.5477 | 0.2865 | ||

$R\left(0.5\right)$ | 0.9248 | 0.0405 | 0.8942 | 0.0460 | 0.8454 | 0.9947 | 0.1493 | 0.8253 | 0.9521 | 0.1267 | ||

$h\left(0.5\right)$ | 0.2965 | 0.1324 | 0.3888 | 0.1367 | 0.0371 | 0.5559 | 0.5188 | 0.2138 | 0.5946 | 0.3808 | ||

Sch[2] | ${\mathcal{S}}_{1}$ | $\alpha $ | 45.398 | 17.543 | 45.297 | 0.1430 | 11.015 | 79.782 | 68.767 | 45.095 | 45.488 | 0.3926 |

$\mu $ | 0.3808 | 0.1221 | 0.3254 | 0.0877 | 0.1416 | 0.6200 | 0.4784 | 0.2005 | 0.4610 | 0.2605 | ||

$R\left(0.5\right)$ | 0.8887 | 0.0557 | 0.8483 | 0.0630 | 0.7796 | 0.9979 | 0.2184 | 0.7560 | 0.9328 | 0.1768 | ||

$h\left(0.5\right)$ | 0.4084 | 0.1657 | 0.5178 | 0.1689 | 0.0837 | 0.7332 | 0.6494 | 0.2937 | 0.7786 | 0.4849 | ||

${\mathcal{S}}_{2}$ | $\alpha $ | 36.008 | 17.983 | 35.909 | 0.1399 | 0.7610 | 71.254 | 70.493 | 35.712 | 36.105 | 0.3924 | |

$\mu $ | 0.4145 | 0.1378 | 0.3519 | 0.0950 | 0.1445 | 0.6845 | 0.5400 | 0.2173 | 0.4902 | 0.2729 | ||

$R\left(0.5\right)$ | 0.8920 | 0.0551 | 0.8516 | 0.0613 | 0.7841 | 0.9989 | 0.2148 | 0.7588 | 0.9269 | 0.1681 | ||

$h\left(0.5\right)$ | 0.3969 | 0.1633 | 0.5069 | 0.1649 | 0.0768 | 0.7171 | 0.6402 | 0.2890 | 0.7462 | 0.4572 | ||

${\mathcal{S}}_{3}$ | $\alpha $ | 30.437 | 18.638 | 30.333 | 0.1447 | 0.0000 | 66.966 | 66.966 | 30.125 | 30.523 | 0.3977 | |

$\mu $ | 0.4378 | 0.1532 | 0.3714 | 0.0989 | 0.1377 | 0.7380 | 0.6004 | 0.2271 | 0.5176 | 0.2905 | ||

$R\left(0.5\right)$ | 0.8930 | 0.0555 | 0.8531 | 0.0600 | 0.7842 | 0.9919 | 0.2077 | 0.7596 | 0.9274 | 0.1678 | ||

$h\left(0.5\right)$ | 0.3933 | 0.1638 | 0.5016 | 0.1608 | 0.0723 | 0.7144 | 0.6421 | 0.2955 | 0.7480 | 0.4525 | ||

Sch[3] | ${\mathcal{S}}_{1}$ | $\alpha $ | 34.013 | 18.493 | 33.913 | 0.1396 | 0.0000 | 70.259 | 70.259 | 33.728 | 34.109 | 0.3815 |

$\mu $ | 0.4838 | 0.1496 | 0.4185 | 0.0987 | 0.1906 | 0.7770 | 0.5865 | 0.2725 | 0.5621 | 0.2896 | ||

$R\left(0.5\right)$ | 0.9146 | 0.0423 | 0.8845 | 0.0459 | 0.8316 | 0.9975 | 0.1659 | 0.8152 | 0.9430 | 0.1278 | ||

$h\left(0.5\right)$ | 0.3281 | 0.1321 | 0.4151 | 0.1312 | 0.0691 | 0.5871 | 0.5179 | 0.2417 | 0.6126 | 0.3709 | ||

${\mathcal{S}}_{2}$ | $\alpha $ | 17.027 | 11.353 | 16.926 | 0.1426 | 0.0000 | 39.279 | 39.279 | 16.7161 | 17.108 | 0.3920 | |

$\mu $ | 0.4427 | 0.1532 | 0.3787 | 0.0969 | 0.1425 | 0.7429 | 0.6004 | 0.2442 | 0.5259 | 0.2817 | ||

$R\left(0.5\right)$ | 0.8615 | 0.0616 | 0.8184 | 0.0649 | 0.7406 | 0.9823 | 0.2417 | 0.7235 | 0.9072 | 0.1838 | ||

$h\left(0.5\right)$ | 0.4831 | 0.1668 | 0.5907 | 0.1608 | 0.1562 | 0.8099 | 0.6537 | 0.3756 | 0.8365 | 0.4609 | ||

${\mathcal{S}}_{3}$ | $\alpha $ | 49.677 | 12.061 | 49.578 | 0.1383 | 26.038 | 73.315 | 47.277 | 49.392 | 49.767 | 0.3754 | |

$\mu $ | 0.5061 | 0.1499 | 0.4349 | 0.1053 | 0.2124 | 0.7999 | 0.5876 | 0.2904 | 0.5913 | 0.3009 | ||

$R\left(0.5\right)$ | 0.9356 | 0.0356 | 0.9092 | 0.0400 | 0.8659 | 0.9967 | 0.1308 | 0.8499 | 0.9583 | 0.1084 | ||

$h\left(0.5\right)$ | 0.2608 | 0.1201 | 0.3435 | 0.1233 | 0.0254 | 0.4963 | 0.4709 | 0.1896 | 0.5270 | 0.3374 |

0.1 | 0.1 | 0.2 | 0.2 | 0.4 | 0.4 | 0.4 | 0.5 | 0.5 | 0.5 | 0.6 | 0.6 | 0.8 | 0.9 | 0.9 | 1.0 | 1.1 |

1.2 | 1.2 | 1.3 | 1.8 | 2.0 | 2.0 | 2.3 | 2.4 | 2.5 | 2.7 | 2.9 | 3.2 | 4.0 | 5.1 | 5.3 | 6.8 | 8.0 |

Model | MLE(St.Er) | N-L | A | B | C-A | H-Q | K-S (p-Value) | |
---|---|---|---|---|---|---|---|---|

$\mathsf{\alpha}$ | $\mathsf{\mu}$ | |||||||

APIE | 21.364 (52.796) | 0.2524 (0.1725) | 57.2457 | 118.4915 | 121.5442 | 118.8786 | 119.5325 | 0.0952 (0.918) |

IE | - | 0.5725 (0.0982) | 59.1930 | 120.3860 | 121.9124 | 120.5110 | 120.9066 | 0.1470 (0.454) |

IL | - | 0.8774 (0.1127) | 61.8136 | 125.6272 | 127.1535 | 125.7522 | 126.1477 | 0.1908 (0.168) |

IW | 0.8805 (0.1093) | 0.6539 (0.1347) | 58.6266 | 121.2532 | 124.3059 | 121.6403 | 122.2942 | 0.1134 (0.774) |

IG | 0.9002 (0.1904) | 0.5154 (0.1434) | 59.0659 | 122.1319 | 125.1846 | 122.5190 | 123.1729 | 0.1310 (0.604) |

INH | 0.6089 (0.1518) | 1.4368 (0.7440) | 57.5539 | 119.1079 | 122.1606 | 119.4950 | 120.1490 | 0.1004 (0.883) |

Scheme | Sample | ${\mathit{T}}_{1}\left({\mathit{d}}_{1}\right)$ | ${\mathit{T}}_{2}\left({\mathit{d}}_{2}\right)$ | Generated Data | ${\mathit{R}}^{*}$ | ${{\mathcal{T}}^{*}}^{*}$ |
---|---|---|---|---|---|---|

Sch[1] | ${\mathcal{S}}_{1}$ | 4.2 (17) | 4.8 (17) | 0.1, 0.2, 0.4, 0.4, 0.5, 0.6, 0.8, 0.9, 0.9, 1.0, 1.1, 1.2, 1.3, 1.8, 2.3, 2.4, 4.0 | 1 | 4.2 |

${\mathcal{S}}_{2}$ | 1.5 (12) | 5.5 (17) | 0.1, 0.2, 0.4, 0.4, 0.4, 0.5, 0.6, 0.6, 0.9, 1.0, 1.1, 1.2, 2.0, 2.0, 2.5, 2.7, 5.1 | 0 | 5.1 | |

${\mathcal{S}}_{3}$ | 1.1 (12) | 1.3 (15) | 0.1, 0.2, 0.4, 0.5, 0.5, 0.5, 0.6, 0.6, 0.8, 0.9, 0.9, 1.0, 1.1, 1.2, 1.2 | 4 | 1.3 | |

Sch[2] | ${\mathcal{S}}_{1}$ | 2.2 (11) | 6.9 (17) | 0.1, 0.2, 0.4, 0.5, 0.5, 0.6, 0.8, 0.9, 1.2, 1.3, 2.0, 2.3, 2.7, 4.0, 5.1, 5.3, 6.8 | 1 | 6.9 |

${\mathcal{S}}_{2}$ | 1.7 (13) | 5.4 (17) | 0.1, 0.2, 0.5, 0.5, 0.6, 0.6, 0.8, 0.9, 0.9, 1.0, 1.1, 1.2, 1.2, 2.3, 2.4, 4.0, 5.3 | 0 | 5.3 | |

${\mathcal{S}}_{3}$ | 1.1 (12) | 1.3 (14) | 0.1, 0.2, 0.4, 0.4, 0.4, 0.5, 0.6, 0.6, 0.8, 0.9, 0.9, 1.0, 1.1, 1.2 | 4 | 1.3 | |

Sch[3] | ${\mathcal{S}}_{1}$ | 2.6 (17) | 2.8 (17) | 0.1, 0.2, 0.2, 0.4, 0.4, 0.4, 0.5, 0.5, 0.5, 0.6,0.9, 0.9, 1.2, 1.2, 1.3, 2.0, 2.5 | 2 | 2.6 |

${\mathcal{S}}_{2}$ | 1.4 (13) | 4.4 (17) | 0.1, 0.2, 0.2, 0.4, 0.4, 0.4, 0.5, 0.5, 0.5, 0.6, 0.9, 1.1, 1.3, 1.8, 2.4, 2.9, 4.0 | 0 | 4.0 | |

${\mathcal{S}}_{3}$ | 0.8 (10) | 1.4 (16) | 0.1, 0.2, 0.2, 0.4, 0.4, 0.4, 0.5, 0.5, 0.5, 0.6, 0.9, 1.0, 1.1, 1.2, 1.2, 1.3 | 3 | 1.4 |

**Table 10.**Bayesian and classical estimates of $\alpha $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ from vinyl chloride data.

Scheme | Sample | Par. | MLE | MCMC | ACI | HPD | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Est. | St.Er | Est. | St.Er | Lower | Upper | IW | Lower | Upper | IW | |||

Sch[1] | ${\mathcal{S}}_{1}$ | $\alpha $ | 60.396 | 8.489 | 60.349 | 0.0832 | 43.757 | 77.034 | 33.278 | 60.218 | 60.485 | 0.2670 |

$\mu $ | 0.2862 | 0.0652 | 0.2641 | 0.0496 | 0.1584 | 0.4140 | 0.2556 | 0.1775 | 0.3515 | 0.1741 | ||

$R\left(0.5\right)$ | 0.8466 | 0.0508 | 0.8216 | 0.0491 | 0.7471 | 0.9461 | 0.1990 | 0.7374 | 0.9003 | 0.1629 | ||

$h\left(0.5\right)$ | 0.5326 | 0.1395 | 0.5956 | 0.1259 | 0.2592 | 0.8060 | 0.5467 | 0.3965 | 0.8226 | 0.4262 | ||

${\mathcal{S}}_{2}$ | $\alpha $ | 41.368 | 12.570 | 41.320 | 0.0844 | 16.730 | 66.006 | 49.275 | 41.188 | 41.459 | 0.2718 | |

$\mu $ | 0.3018 | 0.0729 | 0.2777 | 0.0521 | 0.1589 | 0.4447 | 0.2859 | 0.1888 | 0.3673 | 0.1785 | ||

$R\left(0.5\right)$ | 0.8351 | 0.0532 | 0.8089 | 0.0502 | 0.7309 | 0.9393 | 0.2084 | 0.7238 | 0.8858 | 0.1620 | ||

$h\left(0.5\right)$ | 0.5582 | 0.1415 | 0.6224 | 0.1253 | 0.2809 | 0.8354 | 0.5545 | 0.4174 | 0.8301 | 0.4127 | ||

${\mathcal{S}}_{3}$ | $\alpha $ | 183.76 | 6.8550 | 183.74 | 0.0554 | 170.33 | 197.20 | 26.871 | 183.64 | 183.83 | 0.1951 | |

$\mu $ | 0.1897 | 0.0421 | 0.1798 | 0.0321 | 0.1072 | 0.2721 | 0.1649 | 0.1232 | 0.2413 | 0.1181 | ||

$R\left(0.5\right)$ | 0.8116 | 0.0581 | 0.7907 | 0.0521 | 0.6976 | 0.9255 | 0.2279 | 0.6979 | 0.8776 | 0.1797 | ||

$h\left(0.5\right)$ | 0.6467 | 0.1513 | 0.6950 | 0.1274 | 0.3501 | 0.9433 | 0.5932 | 0.4717 | 0.9238 | 0.4521 | ||

Sch[2] | ${\mathcal{S}}_{1}$ | $\alpha $ | 50.611 | 17.847 | 50.562 | 0.0856 | 15.632 | 85.589 | 69.957 | 50.427 | 50.701 | 0.2738 |

$\mu $ | 0.3078 | 0.0780 | 0.2810 | 0.0548 | 0.1549 | 0.4607 | 0.3058 | 0.1894 | 0.3741 | 0.1847 | ||

$R\left(0.5\right)$ | 0.8522 | 0.0523 | 0.8250 | 0.0501 | 0.7497 | 0.9547 | 0.2050 | 0.7412 | 0.9015 | 0.1602 | ||

$h\left(0.5\right)$ | 0.5146 | 0.1444 | 0.5837 | 0.1291 | 0.2315 | 0.7977 | 0.5662 | 0.3762 | 0.7973 | 0.4211 | ||

${\mathcal{S}}_{2}$ | $\alpha $ | 104.59 | 11.9356 | 104.55 | 0.0705 | 81.194 | 127.98 | 46.787 | 104.43 | 104.67 | 0.2364 | |

$\mu $ | 0.2352 | 0.0547 | 0.2195 | 0.0408 | 0.1280 | 0.3423 | 0.2143 | 0.1482 | 0.2958 | 0.1476 | ||

$R\left(0.5\right)$ | 0.8333 | 0.0557 | 0.8096 | 0.0511 | 0.7241 | 0.9424 | 0.2184 | 0.7153 | 0.8896 | 0.1743 | ||

$h\left(0.5\right)$ | 0.5785 | 0.1506 | 0.6362 | 0.1289 | 0.2834 | 0.8736 | 0.5902 | 0.4276 | 0.8757 | 0.4481 | ||

${\mathcal{S}}_{3}$ | $\alpha $ | 671.16 | 11.8650 | 670.21 | 663.50 | 647.91 | 694.42 | 46.510 | 664.45 | 675.78 | 11.3371 | |

$\mu $ | 0.1277 | 0.0286 | 0.1815 | 0.0584 | 0.0717 | 0.1837 | 0.1120 | 0.1410 | 0.2273 | 0.0863 | ||

$R\left(0.5\right)$ | 0.7706 | 0.0665 | 0.8597 | 0.0938 | 0.6402 | 0.9010 | 0.2608 | 0.8009 | 0.9101 | 0.1092 | ||

$h\left(0.5\right)$ | 0.7718 | 0.1604 | 0.5334 | 0.2528 | 0.4573 | 1.0862 | 0.6289 | 0.3851 | 0.7014 | 0.3163 | ||

Sch[3] | ${\mathcal{S}}_{1}$ | $\alpha $ | 21.738 | 9.609 | 21.689 | 0.0864 | 2.9050 | 40.572 | 37.667 | 21.554 | 21.828 | 0.2732 |

$\mu $ | 0.3115 | 0.0788 | 0.2876 | 0.0515 | 0.1571 | 0.4659 | 0.3088 | 0.2005 | 0.3794 | 0.1789 | ||

$R\left(0.5\right)$ | 0.7968 | 0.0572 | 0.7700 | 0.0517 | 0.6848 | 0.9088 | 0.2241 | 0.6825 | 0.8535 | 0.1710 | ||

$h\left(0.5\right)$ | 0.6493 | 0.1400 | 0.7105 | 0.1200 | 0.3749 | 0.9237 | 0.5488 | 0.5081 | 0.9130 | 0.4049 | ||

${\mathcal{S}}_{2}$ | $\alpha $ | 20.378 | 9.7182 | 20.330 | 0.0849 | 1.3309 | 39.425 | 38.095 | 20.196 | 20.470 | 0.2736 | |

$\mu $ | 0.3529 | 0.0909 | 0.3248 | 0.0571 | 0.1749 | 0.5310 | 0.3562 | 0.2262 | 0.4203 | 0.1940 | ||

$R\left(0.5\right)$ | 0.8230 | 0.0535 | 0.7972 | 0.0479 | 0.7182 | 0.9279 | 0.2097 | 0.7147 | 0.8689 | 0.1542 | ||

$h\left(0.5\right)$ | 0.5834 | 0.1361 | 0.6450 | 0.1154 | 0.3166 | 0.8502 | 0.5336 | 0.4638 | 0.8403 | 0.3765 | ||

${\mathcal{S}}_{3}$ | $\alpha $ | 24.284 | 9.3539 | 24.234 | 0.0863 | 5.9510 | 42.618 | 36.667 | 24.103 | 24.374 | 0.2706 | |

$\mu $ | 0.2922 | 0.0720 | 0.2699 | 0.0495 | 0.1511 | 0.4332 | 0.2821 | 0.1868 | 0.3597 | 0.1728 | ||

$R\left(0.5\right)$ | 0.7887 | 0.0584 | 0.7614 | 0.0538 | 0.6743 | 0.9031 | 0.2288 | 0.6685 | 0.8474 | 0.1788 | ||

$h\left(0.5\right)$ | 0.6698 | 0.1418 | 0.7316 | 0.1239 | 0.3919 | 0.9478 | 0.5559 | 0.5268 | 0.9450 | 0.4182 |

Criterion | Aim |
---|---|

${\mathcal{O}}_{1}$ | Maximize trace($\mathbf{I}\left(\widehat{\vartheta}\right)$) |

${\mathcal{O}}_{2}$ | Minimize trace(${\mathbf{I}}^{-1}\left(\widehat{\vartheta}\right)$) |

${\mathcal{O}}_{3}$ | Minimize det(${\mathbf{I}}^{-1}\left(\widehat{\vartheta}\right)$) |

${\mathcal{O}}_{4}$ | Minimize $\widehat{v}(\mathrm{log}({\widehat{\mathcal{T}}}_{\varrho}))$ |

Sample | Scheme | ${\mathcal{O}}_{1}$ | ${\mathcal{O}}_{2}$ | ${\mathcal{O}}_{3}$ | ${\mathcal{O}}_{4}$ | ||
---|---|---|---|---|---|---|---|

$\mathit{\varrho}\to $ | 0.3 | 0.6 | 0.9 | ||||

${\mathcal{S}}_{1}$ | Sch[1] | 57.813 | 376.19 | 6.5069 | 0.1030 | 0.6945 | 18.150 |

Sch[2] | 73.399 | 307.77 | 4.1931 | 0.1051 | 0.7019 | 18.223 | |

Sch[3] | 54.285 | 342.02 | 6.3005 | 0.1221 | 0.8200 | 21.367 | |

${\mathcal{S}}_{2}$ | Sch[1] | 46.118 | 295.74 | 6.4126 | 0.1860 | 1.2338 | 31.917 |

Sch[2] | 58.574 | 323.42 | 5.3421 | 0.1128 | 0.7568 | 19.711 | |

Sch[3] | 60.542 | 128.92 | 2.2010 | 0.0756 | 0.5160 | 13.586 | |

${\mathcal{S}}_{3}$ | Sch[1] | 52.612 | 301.00 | 5.7118 | 0.1527 | 1.0162 | 26.337 |

Sch[2] | 46.011 | 347.39 | 6.6029 | 0.1183 | 0.7969 | 20.809 | |

Sch[3] | 52.697 | 145.48 | 3.1619 | 0.1759 | 1.1714 | 30.362 |

Sample | Scheme | ${\mathcal{O}}_{1}$ | ${\mathcal{O}}_{2}$ | ${\mathcal{O}}_{3}$ | ${\mathcal{O}}_{4}$ | ||
---|---|---|---|---|---|---|---|

$\mathit{\varrho}\to $ | 0.3 | 0.6 | 0.9 | ||||

${\mathcal{S}}_{1}$ | Sch[1] | 240.82 | 72.074 | 0.2993 | 0.0371 | 0.2458 | 6.3534 |

Sch[2] | 185.93 | 318.51 | 1.7130 | 0.0439 | 0.2923 | 7.5784 | |

Sch[3] | 211.50 | 92.340 | 0.4366 | 0.0241 | 0.1642 | 4.3147 | |

${\mathcal{S}}_{2}$ | Sch[2] | 211.13 | 158.02 | 0.7485 | 0.0348 | 0.2328 | 6.0573 |

Sch[2] | 338.57 | 142.46 | 0.4208 | 0.0343 | 0.2240 | 5.7400 | |

Sch[3] | 162.58 | 94.451 | 0.5810 | 0.0303 | 0.2061 | 5.4156 | |

${\mathcal{S}}_{3}$ | Sch[3] | 566.15 | 46.993 | 0.0830 | 0.0262 | 0.1686 | 4.2879 |

Sch[2] | 1224.9 | 140.78 | 0.1149 | 0.0196 | 0.1227 | 3.0752 | |

Sch[3] | 240.05 | 87.500 | 0.3645 | 0.0227 | 0.1540 | 4.0421 |

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**MDPI and ACS Style**

Elshahhat, A.; Abo-Kasem, O.E.; Mohammed, H.S.
Statistical Analysis of Type-II Generalized Progressively Hybrid Alpha-PIE Censored Data and Applications in Electronic Tubes and Vinyl Chloride. *Axioms* **2023**, *12*, 601.
https://doi.org/10.3390/axioms12060601

**AMA Style**

Elshahhat A, Abo-Kasem OE, Mohammed HS.
Statistical Analysis of Type-II Generalized Progressively Hybrid Alpha-PIE Censored Data and Applications in Electronic Tubes and Vinyl Chloride. *Axioms*. 2023; 12(6):601.
https://doi.org/10.3390/axioms12060601

**Chicago/Turabian Style**

Elshahhat, Ahmed, Osama E. Abo-Kasem, and Heba S. Mohammed.
2023. "Statistical Analysis of Type-II Generalized Progressively Hybrid Alpha-PIE Censored Data and Applications in Electronic Tubes and Vinyl Chloride" *Axioms* 12, no. 6: 601.
https://doi.org/10.3390/axioms12060601