# Statistical Evaluations and Applications for IER Parameters from Generalized Progressively Type-II Hybrid Censored Data

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## Abstract

**:**

## 1. Introduction

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

T1PH | Kundu and Joarder [24] | ${T}_{1}\to 0$ |

T2PH | Childs et al. [25] | ${T}_{2}\to \infty $ |

T1H | Epstein [2] | ${T}_{1}\to 0$, ${R}_{i}=0,\phantom{\rule{4pt}{0ex}}i=1,2,\cdots ,m-1,$ and ${R}_{m}=n-m$ |

T2H | Childs et al. [3] | ${T}_{2}\to \infty $, ${R}_{i}=0,\phantom{\rule{4pt}{0ex}}i=1,2,\cdots ,m-1,$ and ${R}_{m}=n-m$ |

Type-I | Bain and Engelhardt [1] | ${T}_{1}=0$, $m=n$, ${R}_{i}=0,\phantom{\rule{4pt}{0ex}}i=1,2,\cdots ,m-1,$ and ${R}_{m}=n-m$ |

Type-II | Bain and Engelhardt [1] | ${T}_{1}=0$, ${T}_{2}\to \infty $, ${R}_{i}=0,\phantom{\rule{4pt}{0ex}}i=1,2,\cdots ,m-1,$ and ${R}_{m}=n-m$ |

- Maximum likelihood estimators (MLEs) along with their approximate confidence intervals (ACIs) of $\delta $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ are obtained.
- Bayes’ estimators under independent gamma assumptions of $\delta $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ are created relative to the squared-error (SE) and general-entropy (GE) losses.
- Bayes estimators cannot be estimated in explicit form, so Markov-chain Monte-Carlo (MCMC) approximation techniques are recommended to compute the acquired Bayes MCMC estimates and the associated highest posterior density (HPD) intervals.
- Numerical solutions for the offered estimators of $\delta $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ are done by installing two useful packages, namely: ‘$\mathsf{coda}$’ (proposed by Plummer et al. [26]) and ‘$\mathsf{maxLik}$’ (proposed by Henningsen and Toomet [27]) on the $\mathsf{R}$ 4.2.2 programming platform.
- Extensive Monte Carlo comparisons, 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 behavior of the acquired estimators of $\delta $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ is discussed.
- To benefit from the practicality and flexibility of the IER model in data analysis, from the engineering and chemistry areas, we analyzed different real data sets that reflect failure times of mechanical components and cumin essential oil.

## 2. Inferences

#### 2.1. Likelihood Estimation

#### 2.2. Bayes Estimation

**Step 1:**- Set the initial guesses $({\delta}^{\left(0\right)},{\mu}^{\left(0\right)})=(\widehat{\delta},\widehat{\mu})$.
**Step 2:**- Set $\u03f5=1$.
**Step 3:**- Generate ${\delta}^{\left(\right)}$ from $G\left({D}_{\xi}+{a}_{1},\vartheta \left(\left.\omega \right|\mathbf{y}\right)\right)$.
**Step 4:**- Generate ${\mu}^{*}$ from $N(\widehat{\mu},{\widehat{\sigma}}_{22})$ via the M-H sampler as:
- (a)
- Obtain $\mathcal{Q}=\frac{{K}_{2}\left(\left.{\mu}^{*}\right|\mathbf{y},{\delta}^{\left(\u03f5\right)}\right)}{{K}_{2}\left(\left.{\mu}^{(\u03f5-1)}\right|\mathbf{y},{\delta}^{\left(\u03f5\right)}\right)}$.
- (b)
- Obtain ${\mathcal{Q}}^{*}=\mathrm{min}\{1,\mathcal{Q}\}$.
- (c)
- Obtain $u\sim U(0,1)$ from uniform distribution.
- (d)
- If $u\u2a7d{\mathcal{Q}}^{*}$, set ${\mu}^{\left(\u03f5\right)}={\mu}^{*}$ else set ${\mu}^{\left(\u03f5\right)}={\mu}^{(\u03f5-1)}$.

**Step 5:**- Obtain ${R}^{\left(\u03f5\right)}\left(t\right)$ and ${h}^{\left(\u03f5\right)}\left(t\right)$ for $t>0$, respectively, as$${R}^{\left(\u03f5\right)}\left(t\right)={\left(1-{e}^{-{\mu}^{\left(\u03f5\right)}{t}^{-2}}\right)}^{{\delta}^{\left(\u03f5\right)}},$$$${h}^{\left(\u03f5\right)}\left(t\right)=2{\delta}^{\left(\u03f5\right)}{\mu}^{\left(\u03f5\right)}{t}^{-3}{e}^{-{\mu}^{\left(\u03f5\right)}{t}^{-2}}{\left(1-{e}^{-{\mu}^{\left(\u03f5\right)}{t}^{-2}}\right)}^{-1}.$$
**Step 6:**- Set $\u03f5=\u03f5+1$.
**Step 7:**- Redo Steps 3–6 $\mathcal{S}$ times and disregard the first ${\mathcal{S}}^{\u2605}$ times (burn-in) of $\delta $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ (say $\zeta $) as$${\zeta}^{\left(\u03f5\right)}=\left({\delta}^{\left(\u03f5\right)},{\mu}^{\left(\u03f5\right)},{R}^{\left(\u03f5\right)}\left(t\right),{h}^{\left(\u03f5\right)}\left(t\right)\right),\phantom{\rule{4pt}{0ex}}\u03f5={\mathcal{S}}^{\u2605}+1,{\mathcal{S}}^{\u2605}+2,\cdots ,\mathcal{S}.$$
**Step 8:**- Draw the Bayes’ point estimates of $\zeta $ from (12) and (13), respectively, as$${\tilde{\zeta}}_{S}=\frac{1}{\overline{\mathcal{S}}}\sum _{\u03f5={\mathcal{S}}^{\u2605}+1}^{\mathcal{S}}{\zeta}^{\left(\u03f5\right)},$$$${\tilde{\zeta}}_{G}={\left[\frac{1}{\overline{\mathcal{S}}}\sum _{\u03f5={\mathcal{S}}^{\u2605}+1}^{\mathcal{S}}{\left({\zeta}^{\left(\u03f5\right)}\right)}^{-\nu}\right]}^{-\frac{1}{\nu}\phantom{\rule{0.277778em}{0ex}}},\phantom{\rule{4pt}{0ex}}\nu \ne 0,$$
**Step 9:**- Construct the $(1-q)100\%$ HPD interval of $\zeta $ via arrange its MCMC variates as ${\zeta}^{\left(\u03f5\right)}$ for $\u03f5={\mathcal{S}}^{\u2605}+1,\cdots ,\mathcal{S}$ as$$\left({\zeta}_{\left({\u03f5}^{*}\right)},{\zeta}_{({\u03f5}^{*}+(1-q)\overline{\mathcal{S}})}\right),$$$${\zeta}_{\left({\u03f5}^{*}+\left[(1-q)\overline{\mathcal{S}}\right]\right)}-{\zeta}_{\left({\u03f5}^{*}\right)}=\underset{1\u2a7d\u03f5\u2a7dq\overline{\mathcal{S}}}{\mathrm{min}}\phantom{\rule{0.166667em}{0ex}}\left({\zeta}_{\left(\u03f5+\left[(1-q)\overline{\mathcal{S}}\right]\right)}-{\zeta}_{\left(\u03f5\right)}\right),$$

- (1)
- Trace: It depicts the evolution of a parameter value across the chain’s iterations.
- (2)
- Brooks-Gelman-Rubin (BGR): It metrics the convergence of a chain by measuring the difference among the variances within and between chains.
- (3)
- Autocorrelation: It evaluates the relationship between an iteration’s current value and its past values.

## 3. Monte Carlo Simulations

**Step 1.**- Set the actual values of $\delta $ and $\mu $.
**Step 2.**- For given values of n, m, ${T}_{1}$, ${T}_{2}$ and $\mathbf{R}$, following Balakrishnan and Cramer [4], generate a traditional progressive Type-II sample with size m units.
**Step 3.**- Obtain the values of ${d}_{i},\phantom{\rule{4pt}{0ex}}i=1,2$ at ${T}_{i},\phantom{\rule{4pt}{0ex}}i=1,2$.
**Step 4.**- Determine the generalized-T2PH case as:
- a.
- Case-I: If ${Y}_{m}<{T}_{1}$, set ${R}_{i}=0,\phantom{\rule{4pt}{0ex}}for\phantom{\rule{4pt}{0ex}}i=m,m+1,\cdots ,{D}_{1}$ end the experiment at ${T}_{1}$. Then, replace ${Y}_{i},\phantom{\rule{4pt}{0ex}}i=m,\cdots ,{d}_{1}$ by those items collected from a truncated distribution $f\left(y\right){[1-F\left({y}_{m}\right)]}^{-1}$ with size $n-m-{\sum}_{i=1}^{m-1}{R}_{i}$.
- b.
- Case-II: If ${T}_{1}<{Y}_{m}<{T}_{2}$, end the experiment at ${Y}_{m}$.
- c.
- Case-III: If ${T}_{1}<{T}_{2}<{Y}_{m}$, end the experiment at ${T}_{2}$.

- (i)
- Root mean squared-errors (RMSE):$$\mathrm{RMSE}\left({\stackrel{\u02c7}{\zeta}}_{\varrho}\right)=\sqrt{\frac{1}{1000}\sum _{i=1}^{1000}{\left({\stackrel{\u02c7}{\zeta}}_{\varrho}^{\left(i\right)}-{\zeta}_{\varrho}\right)}^{2}}.$$
- (ii)
- Mean absolute biases (MAB):$$\mathrm{MAB}\left({\stackrel{\u02c7}{\zeta}}_{\varrho}\right)=\frac{1}{1000}\sum _{i=1}^{1000}\left|{\stackrel{\u02c7}{\zeta}}_{\varrho}^{\left(i\right)}-{\zeta}_{\varrho}\right|.$$

- (i)
- Average confidence length (ACL):$${\mathrm{ACL}}_{(1-q)\%}\left(\zeta \right)=\frac{1}{1000}\sum _{i=1}^{1000}\left({\mathcal{U}}_{{\stackrel{\u02c7}{\zeta}}_{\varrho}^{\left(i\right)}}-{\mathcal{L}}_{{\stackrel{\u02c7}{\zeta}}_{\varrho}^{\left(i\right)}}\right),$$
- (ii)
- Mean absolute biases (MAB):$${\mathrm{CP}}_{(1-q)\%}\left(\zeta \right)=\frac{1}{1000}\sum _{i=1}^{1000}{\Im}_{\left({\mathcal{L}}_{{\stackrel{\u02c7}{\zeta}}_{\varrho}^{\left(i\right)}};{\mathcal{U}}_{{\stackrel{\u02c7}{\zeta}}_{\varrho}^{\left(i\right)}}\right)}\left(\zeta \right),$$

- A general observation in this study is that the acquired estimates of $\delta $, $\mu $, $R\left(t\right)$ or $h\left(t\right)$ have good behavior.
- Due to gamma informations, the Bayes point estimates (or their HPD credible interval estimates) of $\delta $, $\mu $, $R\left(t\right)$ or $h\left(t\right)$ behave satisfactory compared to the frequentist estimates.
- Comparing the variance values associated with priors I and II, it can be seen that the variance of Prior-II is lower than the other, thus the Bayes calculations from this prior provide good estimates.
- As n(or m) increases, both point and interval estimates of all unknown quantities perform sufficiently. A similar note is also obtained at ${\sum}_{i=1}^{m}{R}_{i}$ decreases.
- As ${T}_{i},\phantom{\rule{4pt}{0ex}}i=1,2$ increase, the RMSEs, MABs and ACLs of $\delta $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ decrease except for those associated with $R\left(t\right)$ in the case of MCMC estimates. Opposite behavior of $\delta $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ is also reached in the case of estimated CP values.
- Comparing the proposed censoring mechanisms, all calculated point/interval estimates of $\delta $, $\mu $, $R\left(t\right)$ or $h\left(t\right)$ are more efficient using Scheme 3 than others.
- As a summary, to estimate the IER parameters $\delta $, $\mu $, $R\left(t\right)$ or $h\left(t\right)$ in presence of data generated from Type-II generalized progressively hybrid censored sampling, the Bayes’ paradigm via M-H algorithm is recommended.

## 4. Real-Life Applications

#### 4.1. Mechanical Components

#### 4.2. Cumin Essential Oil

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Autocorrelation plots of $\delta $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ (after burn-in) from generalized-T2PH censored data.

**Figure 3.**Evaluating plots for 12,000 MCMC draws of $\delta $ and $\mu $ from generalized-T2PH censored data.

**Figure 8.**Empirical/Fitted reliability parameter (

**left-side**); Contour (

**right-side**) plots from mechanical components data.

**Figure 9.**Density (

**left**) and Trace (

**right**) plots of $\delta $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ from mechanical components data.

**Figure 10.**Empirical/Fitted reliability parameter (

**left-side**); Contour (

**right-side**) plots from cuminaldehyde data.

**Figure 11.**Density (

**left**) and Trace (

**right**) plots of $\delta $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ from cuminaldehyde data.

$\mathit{\xi}$ | ${\mathit{G}}_{\mathit{\xi}}$ | ${\mathit{D}}_{\mathit{\xi}}$ | ${\mathbf{\Psi}}_{\mathit{\xi}}({\mathit{T}}_{\mathit{\tau}};\mathit{\omega})$ | ${\mathit{R}}_{{\mathit{d}}_{\mathit{\tau}}+1}^{*}$ |
---|---|---|---|---|

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

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

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

0.67 | 0.68 | 0.76 | 0.81 | 0.84 | 0.85 | 0.85 | 0.86 | 0.89 | 0.98 |

0.98 | 1.14 | 1.14 | 1.15 | 1.21 | 1.25 | 1.31 | 1.49 | 1.60 | 4.85 |

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

1 | 4.95(11) | 5.00(11) | 0.67, 0.76, 0.84, 0.85, 0.89, 0.98, 1.14, 1.21, 1.31, 1.60, 4.85 | 0 | 4.95 |

2 | 1.25(8) | 1.65(10) | 0.67, 0.76, 0.84, 0.85, 0.89, 0.98, 1.14, 1.21, 1.31, 1.60 | 0 | 1.60 |

3 | 1.00(6) | 1.55(9) | 0.67, 0.76, 0.84, 0.85, 0.89, 0.98, 1.14, 1.21, 1.31 | 2 | 1.55 |

**Table 5.**Point estimates of $\delta $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ from mechanical components data.

Sample | Par. | MLE | SE | GE | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\nu}\to $ | −3 | −0.03 | +3 | ||||||||

Est. | St.Err | Est. | St.Err | Est. | St.Err | Est. | St.Err | Est. | St.Err | ||

1 | $\delta $ | 1.4654 | 0.6467 | 1.3402 | 0.1931 | 1.3535 | 0.1119 | 1.3312 | 0.1342 | 1.3074 | 0.1580 |

$\mu $ | 1.5668 | 0.4623 | 1.4556 | 0.1754 | 1.4661 | 0.1007 | 1.4476 | 0.1192 | 1.4284 | 0.1384 | |

$R\left(1\right)$ | 0.7096 | 0.0879 | 0.6991 | 0.0453 | 0.7015 | 0.0081 | 0.6975 | 0.0121 | 0.6931 | 0.0165 | |

$h\left(1\right)$ | 1.2112 | 0.3673 | 1.1887 | 0.1543 | 1.2074 | 0.0038 | 1.1796 | 0.0316 | 1.1512 | 0.0599 | |

2 | $\delta $ | 2.7224 | 1.6018 | 2.5983 | 0.1950 | 2.6066 | 0.1159 | 2.5939 | 0.1286 | 2.5808 | 0.1417 |

$\mu $ | 2.0465 | 0.6121 | 1.9445 | 0.1695 | 1.9479 | 0.0987 | 1.9332 | 0.1134 | 1.9176 | 0.1289 | |

$R\left(1\right)$ | 0.6862 | 0.0917 | 0.6679 | 0.0453 | 0.6686 | 0.0176 | 0.6646 | 0.0216 | 0.6604 | 0.0258 | |

$h\left(1\right)$ | 1.6530 | 0.5449 | 1.6918 | 0.1777 | 1.7167 | 0.0636 | 1.6902 | 0.0372 | 1.6630 | 0.0100 | |

3 | $\delta $ | 2.1542 | 1.3272 | 2.0273 | 0.1947 | 2.0380 | 0.1163 | 2.0220 | 0.1322 | 2.0055 | 0.1487 |

$\mu $ | 1.8663 | 0.6036 | 1.7621 | 0.1692 | 1.7721 | 0.0942 | 1.7572 | 0.1091 | 1.7417 | 0.1246 | |

$R\left(1\right)$ | 0.6963 | 0.0911 | 0.6810 | 0.0443 | 0.6835 | 0.0128 | 0.6797 | 0.0165 | 0.6757 | 0.0206 | |

$h\left(1\right)$ | 1.4715 | 0.5254 | 1.4846 | 0.1613 | 1.5020 | 0.0305 | 1.4762 | 0.0047 | 1.4501 | 0.0215 |

**Table 6.**Interval estimates of $\delta $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ from mechanical components data.

Sample | Par. | ACI | HPD | ||||
---|---|---|---|---|---|---|---|

Lower | Upper | IL | Lower | Upper | IL | ||

1 | $\delta $ | 0.1980 | 2.7328 | 2.5349 | 1.0520 | 1.6329 | 0.5809 |

$\mu $ | 0.6607 | 2.4730 | 1.8123 | 1.1967 | 1.7221 | 0.5254 | |

$R\left(1\right)$ | 0.5374 | 0.8819 | 0.3445 | 0.6131 | 0.7834 | 0.1702 | |

$h\left(1\right)$ | 0.4913 | 1.9310 | 1.4398 | 0.9049 | 1.4935 | 0.5886 | |

2 | $\delta $ | 0.4171 | 5.8619 | 5.4448 | 2.3134 | 2.8952 | 0.5818 |

$\mu $ | 0.8469 | 3.2462 | 2.3992 | 1.6652 | 2.1975 | 0.5323 | |

$R\left(1\right)$ | 0.5065 | 0.8659 | 0.3594 | 0.5858 | 0.7472 | 0.1614 | |

$h\left(1\right)$ | 0.5851 | 2.7210 | 2.1358 | 1.3687 | 2.0411 | 0.6724 | |

3 | $\delta $ | 0.4471 | 4.7555 | 4.3084 | 1.7375 | 2.3032 | 0.5657 |

$\mu $ | 0.6832 | 3.0494 | 2.3662 | 1.4869 | 2.0105 | 0.5236 | |

$R\left(1\right)$ | 0.5177 | 0.8748 | 0.3571 | 0.5986 | 0.7587 | 0.1602 | |

$h\left(1\right)$ | 0.4418 | 2.5012 | 2.0593 | 1.1731 | 1.7922 | 0.6191 |

**Table 7.**Statistics for MCMC iterations of $\delta $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ from mechanical components data.

Sample | Par. | Mean | Mode | ${\mathit{Q}}_{1}$ | Median | ${\mathit{Q}}_{3}$ | St.D | Skewness |
---|---|---|---|---|---|---|---|---|

1 | $\delta $ | 1.34023 | 0.95281 | 1.24522 | 1.34228 | 1.43862 | 0.14705 | −0.12406 |

$\mu $ | 1.45557 | 1.22232 | 1.36275 | 1.45175 | 1.54657 | 0.13566 | 0.10664 | |

$R\left(1\right)$ | 0.69911 | 0.71716 | 0.67074 | 0.70130 | 0.72989 | 0.04411 | −0.30217 | |

$h\left(1\right)$ | 1.18868 | 0.97254 | 1.08139 | 1.18309 | 1.29140 | 0.15269 | 0.20700 | |

2 | $\delta $ | 2.59829 | 2.24704 | 2.49609 | 2.59927 | 2.70160 | 0.15033 | 0.00921 |

$\mu $ | 1.94451 | 1.66521 | 1.85161 | 1.94356 | 2.03607 | 0.13535 | 0.05411 | |

$R\left(1\right)$ | 0.66789 | 0.62429 | 0.64126 | 0.66906 | 0.69711 | 0.04138 | −0.25408 | |

$h\left(1\right)$ | 1.69184 | 1.74574 | 1.57329 | 1.68643 | 1.80312 | 0.17337 | 0.23730 | |

3 | $\delta $ | 2.02727 | 1.76272 | 1.92738 | 2.02779 | 2.12651 | 0.14760 | 0.04591 |

$\mu $ | 1.76208 | 1.45968 | 1.67071 | 1.75966 | 1.85209 | 0.13325 | 0.07618 | |

$R\left(1\right)$ | 0.68098 | 0.62750 | 0.65406 | 0.68274 | 0.71047 | 0.04154 | −0.27335 | |

$h\left(1\right)$ | 1.48463 | 1.55724 | 1.37154 | 1.47701 | 1.58965 | 0.16077 | 0.26114 |

3.386 | 3.796 | 3.789 | 3.960 | 4.354 | 4.481 | 5.091 | 3.655 | 4.246 | 4.523 | 4.758 | 5.589 |

6.676 | 6.845 | 6.498 | 5.398 | 6.668 | 6.757 | 5.939 | 5.787 | 7.089 | 5.054 | 4.867 | 4.985 |

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

1 | 7.2(13) | 7.5(13) | 3.386, 3.789, 3.960, 4.354, 4.523, 4.867, 5.054, 5.398, 5.787, 6.498, 6.676, 6.845, 7.089 | 7.2 | 0 |

2 | 6.5(11) | 7.2(12) | 3.386, 3.789, 3.960, 4.354, 4.523, 4.867, 5.054, 5.398, 5.787, 6.498, 6.676, 6.845 | 6.845 | 0 |

3 | 6.2(9) | 6.8(11) | 3.386, 3.789, 3.960, 4.354, 4.523, 4.867, 5.054, 5.398, 5.787, 6.498, 6.676 | 6.8 | 2 |

**Table 10.**Point estimates of $\delta $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ from cuminaldehyde data.

Sample | Par. | MLE | SE | GE | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\nu}\to $ | −3 | −0.03 | +3 | ||||||||

Est. | St.Err | Est. | St.Err | Est. | St.Err | Est. | St.Err | Est. | St.Err | ||

1 | $\delta $ | 5.0778 | 2.5010 | 4.9248 | 0.2297 | 4.9307 | 0.1471 | 4.9219 | 0.1559 | 4.9128 | 0.1650 |

$\mu $ | 66.290 | 12.124 | 66.138 | 0.2321 | 66.139 | 0.1507 | 66.138 | 0.1514 | 66.137 | 0.1521 | |

$R\left(4\right)$ | 0.9220 | 0.0339 | 0.9235 | 0.0031 | 0.9235 | 0.0016 | 0.9235 | 0.0016 | 0.9235 | 0.0015 | |

$h\left(4\right)$ | 0.1697 | 0.0563 | 0.1658 | 0.0071 | 0.1660 | 0.0037 | 0.1657 | 0.0040 | 0.1653 | 0.0043 | |

2 | $\delta $ | 4.0875 | 2.0227 | 3.9287 | 0.2343 | 3.9363 | 0.1512 | 3.9251 | 0.1625 | 3.9135 | 0.1740 |

$\mu $ | 61.594 | 12.920 | 61.436 | 0.2359 | 61.436 | 0.1584 | 61.435 | 0.1592 | 61.435 | 0.1599 | |

$R\left(4\right)$ | 0.9158 | 0.0406 | 0.9182 | 0.0042 | 0.9182 | 0.0024 | 0.9182 | 0.0023 | 0.9181 | 0.0023 | |

$h\left(4\right)$ | 0.1711 | 0.0612 | 0.1657 | 0.0091 | 0.1661 | 0.0051 | 0.1656 | 0.0056 | 0.1651 | 0.0061 | |

3 | $\delta $ | 3.1790 | 1.4369 | 3.0201 | 0.2339 | 3.0298 | 0.1492 | 3.0153 | 0.1637 | 3.0003 | 0.1787 |

$\mu $ | 56.471 | 10.278 | 56.312 | 0.2361 | 56.312 | 0.1585 | 56.311 | 0.1593 | 56.311 | 0.1601 | |

$R\left(4\right)$ | 0.9097 | 0.0375 | 0.9132 | 0.0059 | 0.9132 | 0.0035 | 0.9132 | 0.0035 | 0.9132 | 0.0034 | |

$h\left(4\right)$ | 0.1695 | 0.0565 | 0.1622 | 0.0118 | 0.1627 | 0.0067 | 0.1619 | 0.0075 | 0.1611 | 0.0083 |

**Table 11.**Interval estimates of $\delta $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ from cuminaldehyde data.

Sample | Par. | ACI | HPD | ||||
---|---|---|---|---|---|---|---|

Lower | Upper | IL | Lower | Upper | IL | ||

1 | $\delta $ | 0.1759 | 9.9797 | 9.8037 | 4.5950 | 5.2730 | 0.6780 |

$\mu $ | 42.527 | 90.052 | 47.525 | 65.797 | 66.484 | 0.6864 | |

$R\left(4\right)$ | 0.8555 | 0.9885 | 0.1330 | 0.9183 | 0.9289 | 0.0106 | |

$h\left(4\right)$ | 0.0593 | 0.2800 | 0.2206 | 0.1543 | 0.1778 | 0.0235 | |

2 | $\delta $ | 0.1231 | 8.0519 | 7.9287 | 3.5875 | 4.2610 | 0.6735 |

$\mu $ | 36.272 | 86.917 | 50.645 | 61.096 | 61.774 | 0.6780 | |

$R\left(4\right)$ | 0.8363 | 0.9953 | 0.1590 | 0.9112 | 0.9251 | 0.0139 | |

$h\left(4\right)$ | 0.0513 | 0.2910 | 0.2397 | 0.1512 | 0.1802 | 0.0290 | |

3 | $\delta $ | 0.3628 | 5.9952 | 5.6324 | 2.6943 | 3.3658 | 0.6716 |

$\mu $ | 36.326 | 76.615 | 40.289 | 55.972 | 56.649 | 0.6776 | |

$R\left(4\right)$ | 0.8362 | 0.9833 | 0.1470 | 0.9039 | 0.9227 | 0.0188 | |

$h\left(4\right)$ | 0.0588 | 0.2802 | 0.2214 | 0.1438 | 0.1802 | 0.0364 |

**Table 12.**Statistics for MCMC iterations of $\delta $, $\mu $, $R\left(t\right)$ and $h\left(t\right)$ from cuminaldehyde data.

Sample | Par. | Mean | Mode | ${\mathit{Q}}_{1}$ | Median | ${\mathit{Q}}_{3}$ | St.D | Skewness |
---|---|---|---|---|---|---|---|---|

1 | $\delta $ | 4.92477 | 4.78637 | 4.81165 | 4.92498 | 5.03747 | 0.17136 | 0.00749 |

$\mu $ | 66.1384 | 65.8608 | 66.0193 | 66.1387 | 66.2583 | 0.17612 | −0.00900 | |

$R\left(4\right)$ | 0.92353 | 0.92433 | 0.92172 | 0.92350 | 0.92532 | 0.00269 | 0.00983 | |

$h\left(4\right)$ | 0.16577 | 0.16328 | 0.16192 | 0.16579 | 0.16973 | 0.00596 | −0.00406 | |

2 | $\delta $ | 3.92874 | 3.65246 | 3.81278 | 3.93027 | 4.04089 | 0.17234 | −0.01738 |

$\mu $ | 61.4356 | 61.1458 | 61.3167 | 61.4362 | 61.5532 | 0.17435 | 0.00399 | |

$R\left(4\right)$ | 0.91816 | 0.92233 | 0.91581 | 0.91818 | 0.92052 | 0.00353 | 0.03267 | |

$h\left(4\right)$ | 0.16573 | 0.15621 | 0.16081 | 0.16576 | 0.17056 | 0.00737 | −0.02234 | |

3 | $\delta $ | 3.02009 | 2.74396 | 2.90488 | 3.02150 | 3.13214 | 0.17159 | −0.02325 |

$\mu $ | 56.3117 | 56.0220 | 56.1925 | 56.3125 | 56.4294 | 0.17444 | 0.00396 | |

$R\left(4\right)$ | 0.91322 | 0.91941 | 0.91005 | 0.91320 | 0.91641 | 0.00478 | 0.04269 | |

$h\left(4\right)$ | 0.16220 | 0.14937 | 0.15605 | 0.16227 | 0.16826 | 0.00928 | −0.02747 |

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## Share and Cite

**MDPI and ACS Style**

Elshahhat, A.; Mohammed, H.S.; Abo-Kasem, O.E.
Statistical Evaluations and Applications for IER Parameters from Generalized Progressively Type-II Hybrid Censored Data. *Axioms* **2023**, *12*, 565.
https://doi.org/10.3390/axioms12060565

**AMA Style**

Elshahhat A, Mohammed HS, Abo-Kasem OE.
Statistical Evaluations and Applications for IER Parameters from Generalized Progressively Type-II Hybrid Censored Data. *Axioms*. 2023; 12(6):565.
https://doi.org/10.3390/axioms12060565

**Chicago/Turabian Style**

Elshahhat, Ahmed, Heba S. Mohammed, and Osama E. Abo-Kasem.
2023. "Statistical Evaluations and Applications for IER Parameters from Generalized Progressively Type-II Hybrid Censored Data" *Axioms* 12, no. 6: 565.
https://doi.org/10.3390/axioms12060565