# Statistical Analysis and Applications of Adaptive Progressively Type-II Hybrid Poisson–Exponential Censored Data

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Estimate the parameters ($\mu ,\gamma $), RF $R(t)$, and HRF $h(t)$ functions of the PE distribution using the LF and SF methods using an adaptive progressively Type-II hybrid censoring. Approximate confidence interval (ACI) estimates of $\mu $, $\gamma $, $R(t)$, and $h(t)$, based on both frequentist approaches, were also obtained.
- Drive the Bayesian estimates via the LF- and SF-based estimation of the same objective parameters under the squared error loss (SEL) function from independent gamma priors. These estimates cannot be computed mathematically, so Monte Carlo Markov chain (MCMC) techniques were utilized.
- Determine the best progressive design, based on four different optimality criteria, that conveys a significant quantity of information about the model parameter(s) of interest.
- Evaluate the suggested approaches’ effectiveness in terms of the root-mean-squared error, relative absolute bias, average interval length, and coverage probability through several numerical comparisons.
- Two applications based on real-world chemical engineering datasets illustrate the PE distribution’s ability to fit different data types and adapt the proposed approaches to actual practical situations.

## 2. Frequentist Estimators

#### 2.1. Likelihood Inference

#### 2.2. Product of Spacing Inference

## 3. Bayesian Inference

#### 3.1. Bayes LF-Based Estimation

**Step****1.**- Put $i=1$.
**Step****2.**- Set $\left({\mu}^{(0)},{\gamma}^{(0)}\right)=(\widehat{\mu},\widehat{\gamma}).$
**Step****3.**- Generate ${\mu}^{*}$ from (21) using a normal distribution, i.e., $N\left(\widehat{\mu},{\widehat{\sigma}}_{11}=\right)$, then apply the following Steps (a)–(d):
- (a)
- Calculate ${q}_{1}=\frac{{\mathcal{H}}_{L}^{\mu}\left(\left.{\mu}^{*}\right|\mathbf{x},{\gamma}^{(i-1)}\right)}{{\mathcal{H}}_{L}^{\mu}\left(\left.{\mu}^{(i-1)}\right|\mathbf{x},{\gamma}^{(i-1)}\right)}$.
- (b)
- Obtain ${Q}_{1}=min\{1,{q}_{1}\}$.
- (c)
- Obtain ${u}_{1}$ from a uniform distribution.
- (d)
- If ${u}_{1}\u2a7d{Q}_{\mu}$, set ${\mu}^{(d)}={\mu}^{*}$; else, set ${\mu}^{(i)}={\mu}^{(i-1)}$.

**Step****4.**- Repeat Steps 2–3 for $\gamma $.
**Step****5:**- Obtain ${R}^{(i)}(t)$ and ${h}^{(i)}(t)$ for given $t>0$, respectively, as$${R}^{(i)}(t)=\frac{1-exp(-{\gamma}^{(i)}{e}^{-{\mu}^{(i)}t})}{1-{e}^{-{\gamma}^{(i)}}},\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{h}^{(i)}(t)=\frac{{\gamma}^{(i)}{\mu}^{(i)}exp(-{\mu}^{(i)}t-{\gamma}^{(i)}{e}^{-{\mu}^{(i)}t})}{1-exp(-{\gamma}^{(i)}{e}^{-{\mu}^{(i)}t})}.$$
**Step****6.**- Set $i=i+1$.
**Step****7.**- Reperform Steps 3–6, $\mathcal{G}$ times to obtain$$\left[{\mu}^{(1)},{\gamma}^{(1)},{R}^{(1)}(t),{h}^{(1)}(t)\right],\dots ,\left[{\mu}^{(\mathcal{G})},{\gamma}^{(\mathcal{G})},{R}^{(\mathcal{G})}(t),{h}^{(\mathcal{G})}(t)\right].$$
**Step****8:**- Obtain the Bayes estimates of $\mu $, $\gamma $, $R(t)$, or $h(t)$ (say ${\tilde{\vartheta}}_{L}$) under the SEL (19) after burn-in (say ${\mathcal{G}}^{\u2022}$) as$${\tilde{\vartheta}}_{L}=\frac{1}{\overline{\mathcal{G}}}{\sum}_{i={\mathcal{G}}^{\u2022}+1}^{\mathcal{G}}{\vartheta}^{(i)},$$
**Step****9:**- Obtain the $100(1-\pi )\%$ BCI of $\vartheta $ by ordering ${\vartheta}^{(i)},i={\mathcal{G}}^{\u2022}+1,{\mathcal{G}}^{\u2022}+2,\dots ,\mathcal{G}$ as ${\vartheta}_{({\mathcal{G}}^{\u2022}+1)},{\vartheta}_{({\mathcal{G}}^{\u2022}+2)},\dots ,{\vartheta}_{(\mathcal{G})}$. Thus, the $100(1-\pi )\%$ BCI of $\vartheta $ is obtained as$$\left\{{\vartheta}_{(\overline{\mathcal{G}}\frac{\pi}{2})},\phantom{\rule{0.166667em}{0ex}}{\vartheta}_{(\overline{\mathcal{G}}(1-\frac{\pi}{2}))}\right\}.$$

#### 3.2. Bayes SF-Based Estimation

## 4. Numerical Comparisons

#### 4.1. Simulation Design

**Step****1:**- Generate a T2PC sample $({X}_{i},{R}_{i}),\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,r,$ as:
- (a)
- Simulate ${\tau}_{1},{\tau}_{2},\dots ,{\tau}_{r}$ from the uniform $U(0,1)$ distribution.
- (b)
- Put ${\eta}_{i}={\tau}_{i}^{{\left(i+{\sum}_{j=r-i+1}^{r}{R}_{j}\right)}^{-1}},$ for $i=1,2,\dots ,r.$
- (c)
- Set ${u}_{i}=1-{\eta}_{r}{\eta}_{r-1}\cdots {\eta}_{r-i+1}$ for $i=1,2,\dots ,r$.
- (d)
- Set ${X}_{i}={\mu}^{-1}\left[log(\gamma )-log(-log({u}_{i}-({u}_{i}-1){e}^{-\gamma}))\right],\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,r,$ and the T2PC from $\mathrm{PE}(\mu ,\gamma )$ is created.

**Step****2:**- Find m, and ignore ${X}_{i}$ for $i=m+2,\dots ,r$.
**Step****3:**- Use a truncated distribution $f\left(x\right){\left[1-F\left({x}_{m+1}\right)\right]}^{-1}$ to simulate the first-order statistics ${X}_{m+2},\dots ,{X}_{r}$ of size $n-m-{\sum}_{j=1}^{m}{R}_{j}-1$.

#### 4.2. Simulation Discussions

- All derived estimates of $\mu $, $\gamma $, $R(t)$, or $h(t)$ behaved satisfactorily in terms of the minimum RMSE, MRAB, and ACL values, as well as the highest CPs.
- As n increased, the acquired point (or interval) estimates were good. An identical pattern was noted when ${\sum}_{i=1}^{r}{R}_{i}$ (or $n-r$) narrowed down.
- As T increased, the RMSEs and MRABs for various estimates of $\mu $ and $\gamma $ increased, while those of $R(t)$ and $h(t)$ decreased.
- As T increased, the ACLs for various estimates of $\mu $, $\gamma $, $R(t)$, and $h(t)$ increased, while their CPs decreased.
- All Bayes results against Prior 2 were superior compared to Prior 1. This was to be expected given that Prior 2’s variance was lower than Prior 1’s.
- In most cases, the CPs of the calculated asymptotic (or Bayes) intervals of $\mu $, $\gamma $, $R(t)$, or $h(t)$ were near the specified nominal level.
- Comparing the suggested schemes, it was observed in the point inference that the unknown parameters $\mu $ and $\gamma $ behaved well using Schemes 1 “left censoring” and 3 “right censoring”, respectively, whereas $R(t)$ and $h(t)$ behaved well using Scheme 2 “middle censoring”.
- Comparing the suggested schemes, it was observed in the interval inference that the unknown parameters $\mu $ and $\gamma $ behaved well using Schemes 1 “left censoring” and 2 “middle censoring”, respectively, whereas $R(t)$ and $h(t)$ behaved well using Scheme 3 “left censoring”.
- Comparing the point estimation methodologies, in most tests, it was noted that:
- (i)
- In a classical setup, the estimates of $\mu $ and $\gamma $ obtained from the product of spacings approach, as well as those of $R(t)$ and $h(t)$ obtained from the likelihood approach performed satisfactorily compared to the others.
- (ii)
- In a Bayes setup, the estimates of $\mu $, $\gamma $, $R(t)$, and $h(t)$ obtained from the likelihood approach performed superior to others.

- Comparing the interval estimation methodologies, in most tests, it was noted that:
- (i)
- In a classical setup, the ACIs of $\mu $, $\gamma $, $R(t)$, and $h(t)$ developed from the product of spacings function performed better than the others.
- (ii)
- In a Bayes setup, the BCIs of $\mu $ developed from the likelihood function, as well as those of $\gamma $, $R(t)$, and $h(t)$ obtained from the product of spacings function performed superior to the others.

- The estimated duration of a study based on Scheme 1 (or 2) is known to be larger than that of any other, and hence, the sample acquired under the AT2PHC plan provided more extra information than those produced using any other strategy.
- As a result, from the AT2PHC plan, the Bayes technique through the MH-G algorithm is recommended to estimate the PE’s parameters of life.

## 5. Optimal Progressive Fashions

## 6. Chemical Engineering Applications

#### 6.1. Cumin Essential Oil

#### 6.2. Coating Weights of Iron Sheets

## 7. Conclusions

## Supplementary Materials

^{st}column), RMSEs (2

^{nd}column) and MRABs (3

^{rd}column) of $\mu $; Table S2: Average estimates (1

^{st}column), RMSEs (2

^{nd}column) and MRABs (3

^{rd}column) of $\gamma $; Table S3: Average estimates (1

^{st}column), RMSEs (2

^{nd}column) and MRABs (3

^{rd}column) of $R(t)$; Table S4: Average estimates (1

^{st}column), RMSEs (2

^{nd}column) and MRABs (3

^{rd}column) of $h(t)$; Table S5: The ACLs (1

^{st}column) and CPs (2

^{nd}column) of 95% Asymptotic/Bayes intervals of $\mu $; Table S6: The ACLs (1

^{st}column) and CPs (2

^{nd}column) of 95% Asymptotic/Bayes intervals of $\gamma $; Table S7: The ACLs (1

^{st}column) and CPs (2

^{nd}column) of 95% Asymptotic/Bayes intervals of $R(t)$; Table S8: The ACLs (1

^{st}column) and CPs (2

^{nd}column) of 95% Asymptotic/Bayes intervals of $h(t)$; Table S9: Summary for MCMC variates based on Samples ${\mathcal{S}}_{2}$ and ${\mathcal{S}}_{3}$ from CEO data; Table S10: Summary for MCMC variates based on Samples ${\mathcal{S}}_{2}$ and ${\mathcal{S}}_{3}$ from coating weights data; Figure S1: Marginal density (left) and Trace (right) plots of $\mu $, $\gamma $, $R(t)$ and $h(t)$ using Sample ${S}_{2}$ from CEO data; Figure S2: Marginal density (left) and Trace (right) plots of $\mu $, $\gamma $, $R(t)$ and $h(t)$ using Sample ${S}_{3}$ from CEO data; Figure S3: Marginal density (left) and Trace (right) plots of $\mu $, $\gamma $, $R(t)$ and $h(t)$ using Sample ${S}_{2}$ from coating weights data; Figure S4: Marginal density (left) and Trace (right) plots of $\mu $, $\gamma $, $R(t)$ and $h(t)$ using Sample ${S}_{3}$ from coating weights data.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 6.**Empirical/fit RF (

**a**), scaled TTT transform (

**b**), and contour diagram for the log-likelihood of $\mu $ and $\gamma $ (

**c**), from the CEO data.

**Figure 7.**The profile log-LF (

**left**) and profile log-SF (

**right**) of $\mu $ and $\gamma $ from the CEO data.

**Figure 8.**Marginal density (

**left**) and trace (

**right**) plots of $\mu $, $\gamma $, $R(t)$, and $h(t)$ using Sample ${S}_{1}$ from the CEO data.

**Figure 9.**Empirical/fit RF (

**a**), scaled TTT-transform (

**b**), and contour diagram for the log-likelihood of $\mu $ and $\gamma $ (

**c**), from the coating weights data.

**Figure 10.**The profile log-LF (

**left**) and profile log-SF (

**right**) of $\mu $ and $\gamma $ from the coating weights data.

**Figure 11.**Marginal density (

**left**) and trace (

**right**) plots of $\mu $, $\gamma $, $R(t)$, and $h(t)$ using Sample ${S}_{1}$ from the coating weights data.

Criterion | Method |
---|---|

${\mathcal{O}}_{1}$ | $\mathrm{Maximize}\phantom{\rule{4pt}{0ex}}\mathrm{trace}({\mathbf{I}}_{2\times 2}(\widehat{\mathsf{\Omega}}))$ |

${\mathcal{O}}_{2}$ | $\mathrm{Minimize}\phantom{\rule{4pt}{0ex}}\mathrm{trace}({\mathbf{I}}_{2\times 2}^{-1}(\widehat{\mathsf{\Omega}}))$ |

${\mathcal{O}}_{3}$ | $\mathrm{Minimize}\phantom{\rule{4pt}{0ex}}det({\mathbf{I}}_{2\times 2}^{-1}(\widehat{\mathsf{\Omega}}))$ |

${\mathcal{O}}_{4}$ | $\mathrm{Minimize}\phantom{\rule{4pt}{0ex}}\widehat{var}(log({\widehat{\Psi}}_{\xi})),0<\xi <1$ |

33.86 | 36.55 | 37.89 | 37.96 | 39.60 | 42.46 | 43.54 | 44.81 | 45.23 | 47.58 | 48.67 | 49.85 |

50.54 | 50.91 | 53.98 | 55.89 | 57.87 | 59.39 | 64.98 | 66.68 | 66.76 | 67.57 | 68.45 | 70.89 |

Sample | Scheme | $\mathbf{T}\text{}(\mathbf{m})$ | ${\mathit{R}}_{\mathit{r}}$ | Data |
---|---|---|---|---|

${\mathcal{S}}_{1}$ | $({3}^{4},{0}^{8})$ | 40 (2) | 6 | 33.86, 39.60, 42.46, 44.81, 45.23, 47.58, 48.67, 50.91, 53.98, 55.89, 57.87, 59.39 |

${\mathcal{S}}_{2}$ | $({0}^{4},{3}^{4},{0}^{4})$ | 46 (7) | 3 | 33.86, 36.55, 37.89, 37.96, 39.60, 43.54, 45.23, 47.58, 48.67, 57.87, 59.39, 66.76 |

${\mathcal{S}}_{3}$ | $({0}^{8},{3}^{4})$ | 51 (12) | 0 | 33.86, 36.55, 37.89, 37.96, 39.60, 42.46, 43.54, 44.81, 45.23, 48.67, 49.85, 50.54 |

Sample | Par. | MLE | MCMC-LF | ACI-LF | BCI-LF | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

MPSE | MCMC-SF | ACI-SF | BCI-SF | ||||||||

Est. | St.E | Est. | St.E | Lower | Upper | IW | Lower | Upper | IW | ||

${\mathcal{S}}_{1}$ | $\mu $ | 0.0904 | 0.0059 | 0.0904 | 0.0010 | 0.0788 | 0.1019 | 0.0231 | 0.0902 | 0.0906 | 0.0004 |

0.0770 | 0.0062 | 0.0770 | 0.0010 | 0.0647 | 0.0892 | 0.0245 | 0.0768 | 0.0772 | 0.0004 | ||

$\gamma $ | 92.367 | 14.332 | 92.366 | 0.0014 | 64.276 | 120.46 | 56.180 | 92.366 | 92.367 | 0.0004 | |

47.347 | 8.8766 | 47.342 | 0.0014 | 29.949 | 64.744 | 34.796 | 47.342 | 47.343 | 0.0004 | ||

$R(50)$ | 0.6350 | 0.0900 | 0.6351 | 0.0184 | 0.4587 | 0.8114 | 0.3527 | 0.6314 | 0.6387 | 0.0073 | |

0.6356 | 0.0887 | 0.6349 | 0.0183 | 0.4618 | 0.8094 | 0.3476 | 0.6313 | 0.6384 | 0.0071 | ||

$h(50)$ | 0.0523 | 0.0106 | 0.0523 | 0.0021 | 0.0316 | 0.0731 | 0.0415 | 0.0519 | 0.0528 | 0.0008 | |

0.0445 | 0.0095 | 0.0446 | 0.0019 | 0.0260 | 0.0631 | 0.0371 | 0.0443 | 0.0450 | 0.0007 | ||

${\mathcal{S}}_{2}$ | $\mu $ | 0.0814 | 0.0075 | 0.0814 | 0.0010 | 0.0667 | 0.0960 | 0.0293 | 0.0812 | 0.0816 | 0.0004 |

0.0690 | 0.0056 | 0.0690 | 0.0010 | 0.0579 | 0.0800 | 0.0221 | 0.0688 | 0.0692 | 0.0004 | ||

$\gamma $ | 50.874 | 12.692 | 50.874 | 0.0014 | 25.999 | 75.750 | 49.751 | 50.874 | 50.875 | 0.0004 | |

28.629 | 3.6603 | 28.629 | 0.0014 | 21.455 | 35.803 | 14.348 | 28.628 | 28.629 | 0.0004 | ||

$R(50)$ | 0.5813 | 0.0909 | 0.5813 | 0.0183 | 0.4032 | 0.7594 | 0.3563 | 0.5777 | 0.5849 | 0.0072 | |

0.5972 | 0.0896 | 0.5972 | 0.0185 | 0.4217 | 0.7727 | 0.3511 | 0.5935 | 0.6007 | 0.0072 | ||

$h(50)$ | 0.0510 | 0.0103 | 0.0510 | 0.0019 | 0.0308 | 0.0713 | 0.0405 | 0.0506 | 0.0514 | 0.0008 | |

0.0423 | 0.0086 | 0.0423 | 0.0017 | 0.0254 | 0.0592 | 0.0338 | 0.0420 | 0.0427 | 0.0007 | ||

${\mathcal{S}}_{3}$ | $\mu $ | 0.1114 | 0.0053 | 0.1114 | 0.0010 | 0.1010 | 0.1217 | 0.0207 | 0.1112 | 0.1116 | 0.0004 |

0.0951 | 0.0051 | 0.0951 | 0.0010 | 0.0852 | 0.1051 | 0.0199 | 0.0950 | 0.0953 | 0.0004 | ||

$\gamma $ | 162.83 | 12.642 | 162.83 | 0.0014 | 138.05 | 187.60 | 49.557 | 162.83 | 162.83 | 0.0004 | |

78.358 | 4.0709 | 78.358 | 0.0014 | 70.379 | 86.337 | 15.958 | 78.358 | 78.358 | 0.0004 | ||

$R(50)$ | 0.4628 | 0.0827 | 0.4629 | 0.0168 | 0.3008 | 0.6249 | 0.3240 | 0.4595 | 0.4661 | 0.0066 | |

0.4899 | 0.0849 | 0.4899 | 0.0174 | 0.3234 | 0.6563 | 0.3328 | 0.4864 | 0.4932 | 0.0067 | ||

$h(50)$ | 0.0803 | 0.0105 | 0.0803 | 0.0021 | 0.0597 | 0.1009 | 0.0412 | 0.0799 | 0.0807 | 0.0008 | |

0.0667 | 0.0097 | 0.0667 | 0.0020 | 0.0477 | 0.0857 | 0.0380 | 0.0663 | 0.0671 | 0.0008 |

Sample | Par. | Mean | Mode | ${\mathit{Q}}_{1}$ | ${\mathit{Q}}_{2}$ | ${\mathit{Q}}_{3}$ | SD | Sk. |
---|---|---|---|---|---|---|---|---|

${\mathcal{S}}_{1}$ | $\mu $ | 0.0904 | 0.0902 | 0.0903 | 0.0904 | 0.0904 | 0.0001 | 0.0187 |

0.0770 | 0.0771 | 0.0769 | 0.0770 | 0.0771 | 0.0001 | 0.0493 | ||

$\gamma $ | 92.366 | 92.366 | 92.366 | 92.366 | 92.367 | 0.0001 | 0.0106 | |

47.342 | 47.342 | 47.342 | 47.342 | 47.342 | 0.0001 | −0.0746 | ||

$R(50)$ | 0.6351 | 0.6374 | 0.6338 | 0.6351 | 0.6363 | 0.0018 | −0.0188 | |

0.6349 | 0.6333 | 0.6336 | 0.6349 | 0.6361 | 0.0018 | −0.0494 | ||

$h(50)$ | 0.0523 | 0.0521 | 0.0522 | 0.0523 | 0.0525 | 0.0002 | 0.0175 | |

0.0446 | 0.0448 | 0.0445 | 0.0446 | 0.0447 | 0.0002 | 0.0490 |

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

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

MLE | ||||||

${\mathcal{S}}_{1}$ | 41,866.25 | 205.406 | 0.00491 | 6.75192 | 9.83368 | 17.5193 |

${\mathcal{S}}_{2}$ | 40,860.91 | 161.084 | 0.00394 | 7.84087 | 12.5272 | 26.0732 |

${\mathcal{S}}_{3}$ | 40,886.40 | 159.826 | 0.00391 | 3.82899 | 5.30541 | 8.71828 |

MPSE | ||||||

${\mathcal{S}}_{1}$ | 43,115.80 | 78.7949 | 0.00183 | 8.91487 | 13.9230 | 27.4149 |

${\mathcal{S}}_{2}$ | 41,933.12 | 16.5722 | 0.00040 | 10.5710 | 17.3449 | 35.2131 |

${\mathcal{S}}_{3}$ | 40,931.40 | 13.3981 | 0.00032 | 5.19858 | 7.56814 | 13.1462 |

28.7 | 29.4 | 30.4 | 31.6 | 31.8 | 32.7 | 32.9 | 33.2 | 33.2 | 33.6 | 33.7 | 34.0 | 34.2 | 34.5 | 35.6 |

36.2 | 36.7 | 36.8 | 36.8 | 37.3 | 37.8 | 38.5 | 38.9 | 38.9 | 39.1 | 39.9 | 40.1 | 40.2 | 40.3 | 40.5 |

40.6 | 40.7 | 41.2 | 41.2 | 41.3 | 42.3 | 42.3 | 42.6 | 42.8 | 42.8 | 42.8 | 42.8 | 43.1 | 44.2 | 44.9 |

45.2 | 45.3 | 45.4 | 45.8 | 46.3 | 47.1 | 47.2 | 47.2 | 48.2 | 48.3 | 48.4 | 48.5 | 49.8 | 50.1 | 52.6 |

52.8 | 54.2 | 54.5 | 55.4 | 55.8 | 56.8 | 58.2 | 58.4 | 58.7 | 58.9 | 59.2 | 61.2 |

Sample | Scheme | $\mathit{T}\text{}(\mathit{m})$ | ${\mathit{R}}_{\mathit{r}}$ | Data |
---|---|---|---|---|

${\mathcal{S}}_{1}$ | $({5}^{8},{0}^{24})$ | 30 (2) | 30 | 28.7, 29.4, 31.8, 32.9, 33.2, 33.2, 33.6, 33.7, 34.0, 34.2, 34.5, |

35.6, 36.2, 36.7, 36.8, 36.8, 37.3, 37.8, 38.5, 39.1, 39.9, 40.1, | ||||

40.2, 40.3, 40.5, 40.6, 41.2, 41.3, 42.3, 42.6, 42.8, 42.8 | ||||

${\mathcal{S}}_{2}$ | $({0}^{12},{5}^{8},{0}^{12})$ | 37 (16) | 20 | 28.7, 29.4, 30.4, 31.6, 31.8, 32.7, 32.9, 33.2, 33.2, 33.6, 33.7, |

34.0, 34.2, 34.5, 36.7, 36.8, 38.5, 39.1, 39.9, 40.1, 40.3, 40.5, | ||||

40.6, 40.7, 42.3, 42.3, 42.6, 44.2, 44.9, 45.2, 45.3, 47.2 | ||||

${\mathcal{S}}_{3}$ | $({0}^{24},{5}^{8})$ | 42 (30) | 10 | 28.7, 29.4, 30.4, 31.6, 31.8, 32.7, 32.9, 33.2, 33.2, 33.6, 33.7, |

34.0, 34.2, 34.5, 35.6, 36.2, 36.7, 36.8, 36.8, 37.3, 37.8, 38.5, | ||||

38.9, 38.9, 39.1, 39.9, 40.1, 40.6, 40.7, 41.3, 42.3, 46.3 |

Sample | Par. | MLE | MCMC-LF | ACI-LF | BCI-LF | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

MPSE | MCMC-SF | ACI-SF | BCI-SF | ||||||||

Est. | St.E | Est. | St.E | Lower | Upper | IW | Lower | Upper | IW | ||

${\mathcal{S}}_{1}$ | $\mu $ | 0.1383 | 0.0035 | 0.1383 | 0.0010 | 0.1314 | 0.1452 | 0.0138 | 0.1381 | 0.1385 | 0.0004 |

0.1233 | 0.0036 | 0.1233 | 0.0010 | 0.1162 | 0.1304 | 0.0141 | 0.1231 | 0.1235 | 0.0004 | ||

$\gamma $ | 246.00 | 6.0398 | 246.00 | 0.0014 | 234.16 | 257.84 | 23.676 | 246.00 | 246.00 | 0.0004 | |

142.99 | 4.3040 | 142.99 | 0.0014 | 134.55 | 151.43 | 16.871 | 142.99 | 142.99 | 0.0004 | ||

$R(40)$ | 0.6226 | 0.0508 | 0.6226 | 0.0147 | 0.5231 | 0.7220 | 0.1989 | 0.6197 | 0.6254 | 0.0057 | |

0.6437 | 0.0517 | 0.6437 | 0.0149 | 0.5423 | 0.7451 | 0.2028 | 0.6407 | 0.6466 | 0.0059 | ||

$h(40)$ | 0.0817 | 0.0084 | 0.0817 | 0.0024 | 0.0652 | 0.0982 | 0.0330 | 0.0812 | 0.0822 | 0.0010 | |

0.0704 | 0.0080 | 0.0704 | 0.0023 | 0.0547 | 0.0861 | 0.0314 | 0.0700 | 0.0709 | 0.0009 | ||

${\mathcal{S}}_{2}$ | $\mu $ | 0.1122 | 0.0038 | 0.1122 | 0.0010 | 0.1047 | 0.1197 | 0.0150 | 0.1120 | 0.1124 | 0.0004 |

0.1019 | 0.0037 | 0.1019 | 0.0010 | 0.0946 | 0.1091 | 0.0145 | 0.1017 | 0.1021 | 0.0004 | ||

$\gamma $ | 98.137 | 6.7694 | 98.137 | 0.0014 | 84.869 | 111.40 | 26.535 | 98.137 | 98.137 | 0.0004 | |

67.200 | 3.7342 | 67.200 | 0.0014 | 59.881 | 74.519 | 14.638 | 67.200 | 67.200 | 0.0004 | ||

$R(40)$ | 0.6682 | 0.0490 | 0.6682 | 0.0147 | 0.5721 | 0.7643 | 0.1922 | 0.6653 | 0.6711 | 0.0058 | |

0.6811 | 0.0494 | 0.6811 | 0.0145 | 0.5843 | 0.7778 | 0.1935 | 0.6782 | 0.6839 | 0.0057 | ||

$h(40)$ | 0.0615 | 0.0073 | 0.0615 | 0.0022 | 0.0472 | 0.0758 | 0.0286 | 0.0610 | 0.0619 | 0.0009 | |

0.0545 | 0.0069 | 0.0545 | 0.0020 | 0.0410 | 0.0680 | 0.0269 | 0.0541 | 0.0549 | 0.0008 | ||

${\mathcal{S}}_{3}$ | $\mu $ | 0.1445 | 0.0041 | 0.1445 | 0.0010 | 0.1365 | 0.1525 | 0.0159 | 0.1443 | 0.1447 | 0.0004 |

0.1262 | 0.0041 | 0.1262 | 0.0010 | 0.1181 | 0.1343 | 0.0163 | 0.1260 | 0.1264 | 0.0004 | ||

$\gamma $ | 237.71 | 8.6611 | 237.71 | 0.0014 | 220.74 | 254.69 | 33.951 | 237.71 | 237.71 | 0.0004 | |

123.96 | 4.3089 | 123.96 | 0.0014 | 115.51 | 132.40 | 16.891 | 123.96 | 123.96 | 0.0004 | ||

$R(40)$ | 0.5202 | 0.0554 | 0.5202 | 0.0142 | 0.4115 | 0.6288 | 0.2173 | 0.5173 | 0.5229 | 0.0056 | |

0.5489 | 0.0580 | 0.5489 | 0.0145 | 0.4352 | 0.6626 | 0.2274 | 0.5461 | 0.5517 | 0.0056 | ||

$h(40)$ | 0.0979 | 0.0090 | 0.0979 | 0.0023 | 0.0802 | 0.1156 | 0.0354 | 0.0974 | 0.0983 | 0.0009 | |

0.0826 | 0.0087 | 0.0826 | 0.0022 | 0.0655 | 0.0996 | 0.0340 | 0.0821 | 0.0830 | 0.0008 |

**Table 10.**Summary for MCMC variates of $\mu $, $\gamma $, $R(t)$, and $h(t)$ from the coating weights data.

Sample | Par. | Mean | Mode | ${\mathit{Q}}_{1}$ | ${\mathit{Q}}_{2}$ | ${\mathit{Q}}_{3}$ | SD | Sk. |
---|---|---|---|---|---|---|---|---|

${\mathcal{S}}_{1}$ | $\mu $ | 0.1383 | 0.1382 | 0.1382 | 0.1383 | 0.1383 | 0.0001 | 0.0293 |

0.1233 | 0.1234 | 0.1232 | 0.1233 | 0.1234 | 0.0001 | −0.0197 | ||

$\gamma $ | 246.00 | 246.00 | 246.00 | 246.00 | 246.00 | 0.0001 | 0.0461 | |

142.99 | 142.99 | 142.99 | 142.99 | 142.99 | 0.0001 | 0.0729 | ||

$R(40)$ | 0.6226 | 0.6234 | 0.6216 | 0.6226 | 0.6236 | 0.0015 | −0.0290 | |

0.6437 | 0.6427 | 0.6427 | 0.6437 | 0.6447 | 0.0015 | 0.0193 | ||

$h(40)$ | 0.0817 | 0.0815 | 0.0815 | 0.0817 | 0.0818 | 0.0002 | 0.0274 | |

0.0704 | 0.0706 | 0.0703 | 0.0704 | 0.0706 | 0.0002 | −0.0208 |

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

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

MLE | ||||||

${\mathcal{S}}_{1}$ | 84,025.63 | 36.4797 | 0.00043 | 0.92111 | 1.24319 | 1.96602 |

${\mathcal{S}}_{2}$ | 89,273.53 | 45.8243 | 0.00051 | 1.36940 | 1.97747 | 3.45105 |

${\mathcal{S}}_{3}$ | 64,688.44 | 75.0153 | 0.00116 | 0.99059 | 1.34004 | 2.12883 |

MPSE | ||||||

${\mathcal{S}}_{1}$ | 80,825.40 | 10.5244 | 0.00013 | 1.22241 | 1.70355 | 2.80635 |

${\mathcal{S}}_{2}$ | 87,280.77 | 13.9442 | 0.00016 | 1.72264 | 2.55586 | 4.57677 |

${\mathcal{S}}_{3}$ | 61,339.85 | 18.5668 | 0.00030 | 1.38045 | 1.94120 | 3.23457 |

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**MDPI and ACS Style**

Elshahhat, A.; Mohammed, H.S.
Statistical Analysis and Applications of Adaptive Progressively Type-II Hybrid Poisson–Exponential Censored Data. *Axioms* **2023**, *12*, 533.
https://doi.org/10.3390/axioms12060533

**AMA Style**

Elshahhat A, Mohammed HS.
Statistical Analysis and Applications of Adaptive Progressively Type-II Hybrid Poisson–Exponential Censored Data. *Axioms*. 2023; 12(6):533.
https://doi.org/10.3390/axioms12060533

**Chicago/Turabian Style**

Elshahhat, Ahmed, and Heba S. Mohammed.
2023. "Statistical Analysis and Applications of Adaptive Progressively Type-II Hybrid Poisson–Exponential Censored Data" *Axioms* 12, no. 6: 533.
https://doi.org/10.3390/axioms12060533