# Estimations of Modified Lindley Parameters Using Progressive Type-II Censoring with Applications

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## Abstract

**:**

## 1. Introduction

## 2. Classical Estimation

#### 2.1. Maximum Lilekihood Estimation

#### 2.2. Maximum Product of Spacing Estimation

## 3. Bayesian Estimation

#### 3.1. Bayesian Estimation Using LF

**Step****1.**- Specify the initial value for $\delta $, say ${\delta}^{\left(0\right)}=\widehat{\delta}$.
**Step****2.**- Set $k=1$.
**Step****3.**- Generate ${\delta}^{\left(k\right)}$ from (19) with normal proposal distribution using the MH steps.
**Step****4.**- Taking a specific value for t, determine ${R}^{\left(k\right)}\left(t\right)$ and ${h}^{\left(k\right)}\left(t\right)$ as$${R}^{\left(k\right)}\left(t\right)=\left[1+{\delta}^{k}t{(1+{\delta}^{k})}^{-1}{e}^{-{\delta}^{k}t}\right]{e}^{-{\delta}^{k}t}$$$${h}^{\left(j\right)}\left(t\right)={\delta}^{k}+\frac{{\delta}^{k}({\delta}^{k}t-1)}{(1+{\delta}^{k}){e}^{{\delta}^{k}t}+{\delta}^{k}t}.$$
**Step****5.**- Set $k=k+1$.
**Step****6.**- Replicate Steps 3 through 5, M times to obtain$$\left[{\delta}^{\left(1\right)},\phantom{\rule{0.166667em}{0ex}}{R}^{\left(1\right)}\left(t\right),\phantom{\rule{0.166667em}{0ex}}{h}^{\left(1\right)}\left(t\right)\right],\dots ,\left[{\delta}^{\left(M\right)},\phantom{\rule{0.166667em}{0ex}}{R}^{\left(M\right)}\left(t\right),\phantom{\rule{0.166667em}{0ex}}{h}^{\left(M\right)}\left(t\right)\right],k=1,\dots ,M.$$
**Step****7.**- Compute the Bayes estimates, after removing ${M}^{*}$ samples as a burn-in period, as follows$$\begin{array}{c}\hfill {\widehat{\delta}}_{B}=\frac{{\sum}_{k={M}^{*}+1}^{M}{\delta}^{\left(k\right)}}{M-{M}^{*}},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\widehat{R}}_{B}\left(t\right)=\frac{{\sum}_{k={M}^{*}+1}^{M}{R}^{\left(k\right)}\left(t\right)}{M-{M}^{*}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{and}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\widehat{h}}_{B}\left(t\right)=\frac{{\sum}_{k={M}^{*}+1}^{M}{h}^{\left(k\right)}\left(t\right)}{M-{M}^{*}}.\end{array}$$
**Step****8.**- To compute the HPD credible intervals, order ${\delta}^{\left(k\right)},{R}^{\left(k\right)}\left(t\right)$ and ${h}^{\left(k\right)}\left(t\right),k={M}^{*}+1,\dots ,M$. Then, one can follow the method proposed by Chen and Shao [20] to obtain the required interval for the parameter $\delta $, as follows$$\begin{array}{c}\hfill \left[{\delta}^{\left({k}^{*}\right)},{\delta}^{\left({k}^{*}+\left(1-\alpha \right)\left(M-{M}^{*}\right)\right)}\right],\end{array}$$$${\delta}^{\left({k}^{*}+\left[\left(1-\alpha \right)\left(M-{M}^{*}\right)\right]\right)}-{\delta}^{\left({k}^{*}\right)}=\underset{1\u2a7dk\u2a7d\alpha \left(M-{M}^{*}\right)}{min}\left({\delta}^{\left(k+\left[\left(1-\alpha \right)\left(M-{M}^{*}\right)\right]\right)}-{\delta}^{\left(k\right)}\right).$$The largest integer less than or equal to x is denoted by $\left[x\right]$. The same process can be applied to obtain the HPD credible intervals of $R\left(t\right)$ and $h\left(t\right)$.

#### 3.2. Bayesian Estimation Using PS Function

**Step****1.**- Determine the initial value of $\delta $ by setting ${\delta}^{\left(0\right)}=\tilde{\delta}$.
**Step****2.**- Put $k=1$.
**Step****3.**- Use MH steps to obtain ${\delta}^{\left(k\right)}$ from (22) with normal proposal distribution.
**Step****4.**- At a mission time t, obtain ${R}^{\left(k\right)}\left(t\right)$ and ${h}^{\left(k\right)}\left(t\right)$.
**Step****5.**- Put $k=k+1$.
**Step****6.**- Repeat Steps 3–5, M times to obtain ${\delta}^{\left(k\right)},\phantom{\rule{0.166667em}{0ex}}{R}^{\left(k\right)}\left(t\right)$ and ${h}^{\left(k\right)}\left(t\right),k=1,\dots ,M$.
**Step****7.**- Calculate the Bayes estimates as$$\begin{array}{c}\hfill {\tilde{\delta}}_{B}=\frac{{\sum}_{k={M}^{*}+1}^{M}{\delta}^{\left(k\right)}}{M-{M}^{*}},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\tilde{R}}_{B}\left(t\right)=\frac{{\sum}_{k={M}^{*}+1}^{M}{R}^{\left(k\right)}\left(t\right)}{M-{M}^{*}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{and}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\tilde{h}}_{B}\left(t\right)=\frac{{\sum}_{k={M}^{*}+1}^{M}{h}^{\left(k\right)}\left(t\right)}{M-{M}^{*}}.\end{array}$$
**Step****8.**- The same method mentioned in the previous part can be used to determine the HPD credible intervals.

## 4. Monte Carlo Simulations

- All proposed estimates of $\delta $, $R\left(t\right)$ or $h\left(t\right)$ behave well; this is the general comment.
- When n(or m) grows, all estimates of $\delta $, $R\left(t\right)$, and $h\left(t\right)$ perform satisfactorily. When ${\sum}_{i=1}^{m}{S}_{i}$ decreases, the same outcome is observed.
- Due to the gamma prior, the MCMC estimates of $\delta $, $R\left(t\right)$, and $h\left(t\right)$ (from LF or PS) outperform the other estimates as expected. When comparing the HPD credible intervals to the ACIs, the same conclusion is reached.
- It is obvious that the Bayes findings based on Prior 2 perform better than others for all unknown parameters, since the associated variance of Prior 2 is smaller than the associated variance of Prior 1.
- All proposed estimates of $\delta $, $R\left(t\right)$, and $h\left(t\right)$ behave better utilizing scheme 3 than others when comparing the proposed censoring plans 1, 2 and 3.
- Using the frequentist perspective, it can be observed that the proposed point estimates of $\delta $, $R\left(t\right)$, and $h\left(t\right)$ using the PS technique become even better than those derived from the LF approach in terms of the least RMSEs, MRABs, and ACLs. The ACI-PS interval estimates of $\delta $, $R\left(t\right)$, and $h\left(t\right)$ perform better than others in terms of ACLs criteria whereas the ACI-LF interval estimates of $\delta $, $R\left(t\right)$, and $h\left(t\right)$ perform better than others in terms of CPs criteria.
- From the Bayesian perspective, it is clear that the proposed point estimates of $\delta $, $R\left(t\right)$, and $h\left(t\right)$ produced using the BE-LF approach are superior to those obtained using the BE-PS approach. It is also noted that the HPD-LF interval estimates $\delta $, $R\left(t\right)$ and $h\left(t\right)$ perform better than others.
- As $\delta $ increases, in most cases, the RMSEs, MRABs, and ACLs of $\delta $, $R\left(t\right)$ and $h\left(t\right)$ increase while their CPs decrease.
- To sum up, in order to estimate the unknown parameters of life $\delta $, $R\left(t\right)$, and $h\left(t\right)$ of the ML model using the PT-IIC data, we recommend using the PS approach (as a frequentist technique) and BE-LF (as a Bayesian method).

## 5. Real-Life Applications

#### 5.1. Mechanical Equipment

#### 5.2. Motor Vehicle Deaths

## 6. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**The log-likelihood function and the associated first derivative of $\delta $ from mechanical equipment data.

**Figure 7.**Trace (

**top**) and histograms (

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

**Figure 10.**The log-likelihood function and the associated first derivative of $\delta $ from MVAD data.

**Figure 11.**Trace (

**top**) and histogram (

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

1.1 | 3.0 | 4.0 | 4.5 | 5.9 | 6.3 | 7.0 | 7.1 | 7.4 | 7.7 |

9.4 | 10.6 | 11.7 | 12.3 | 12.3 | 12.4 | 14.3 | 14.6 | 14.9 | 17.4 |

18.2 | 18.6 | 19.7 | 22.3 | 23.7 | 24.6 | 26.3 | 34.6 | 43.6 | 47.3 |

Model | MLE (St.E) | NL | A | CA | B | HQ | KS (p-Value) | |
---|---|---|---|---|---|---|---|---|

$\mathbf{\beta}$ | $\mathbf{\delta}$ | |||||||

ML | - | 0.0792 (0.0111) | 108.707 | 219.770 | 219.913 | 221.171 | 220.219 | 0.0630 (0.999) |

L | - | 0.1225 (0.0159) | 108.942 | 219.884 | 220.027 | 221.285 | 220.332 | 0.0688 (0.999) |

E | - | 0.0648 (0.0118) | 112.083 | 226.166 | 226.309 | 227.567 | 226.614 | 0.1844 (0.259) |

G | 1.9732 (0.4712) | 7.8105 (2.1200) | 108.711 | 221.414 | 221.859 | 224.217 | 222.311 | 0.0669 (0.999) |

W | 1.4633 (0.2029) | 17.099 (2.2539) | 108.988 | 221.976 | 222.420 | 224.778 | 222.872 | 0.0749 (0.996) |

TL | 0.0242 (1.4529) | 0.1294 (0.0206) | 108.771 | 221.422 | 221.867 | 224.225 | 222.319 | 0.0692 (0.999) |

EL | 1.1767 (0.3244) | 0.1330 (0.0243) | 108.885 | 221.542 | 221.986 | 224.344 | 222.438 | 0.0751 (0.996) |

MOL | 1.1345 (0.8926) | 0.1286 (0.0412) | 108.929 | 221.859 | 222.303 | 224.661 | 222.755 | 0.0697 (0.999) |

Sample | Scheme | Censored Data |
---|---|---|

1 | (15, 0${}^{*}$14) | 1.1, 3.0, 4.0, 4.5, 5.9, 7.0, 7.1, 7.7, 9.4, 12.3, 14.3, 17.4, 22.3, 24.6, 26.3 |

2 | $({0}^{*}6,{5}^{*}3,{0}^{*}6)$ | 1.1, 3.0, 4.0, 4.5, 5.9, 6.3, 7.0, 7.4, 9.4, 10.6, 12.4, 14.6, 18.2, 22.3, 24.6 |

3 | $({0}^{*}14,15)$ | 1.1, 3.0, 4.0, 4.5, 5.9, 6.3, 7.0, 7.1, 7.4, 7.7, 9.4, 10.6, 11.7, 12.3, 12.3 |

**Table 4.**Point and interval estimates of $\delta $, $R\left(t\right)$, and $h\left(t\right)$ from mechanical equipment data.

Sample | Par. | MLE | BE-LF | ACI-LF | HPD-LF | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Est. | St.E | Est. | St.E | Lower | Upper | Length | Lower | Upper | Length | ||

1 | $\delta $ | 0.1059 | 0.0209 | 0.1046 | 0.0091 | 0.0649 | 0.1468 | 0.0819 | 0.0872 | 0.1223 | 0.0351 |

$R\left(2\right)$ | 0.9346 | 0.0219 | 0.9357 | 0.0094 | 0.8916 | 0.9775 | 0.0859 | 0.9176 | 0.9540 | 0.0364 | |

$h\left(2\right)$ | 0.0530 | 0.0174 | 0.0521 | 0.0075 | 0.0188 | 0.0871 | 0.0683 | 0.0375 | 0.0664 | 0.0290 | |

2 | $\delta $ | 0.0808 | 0.0145 | 0.0797 | 0.0082 | 0.0524 | 0.1092 | 0.0568 | 0.0640 | 0.0959 | 0.0319 |

$R\left(2\right)$ | 0.9590 | 0.0130 | 0.9598 | 0.0073 | 0.9336 | 0.9845 | 0.0509 | 0.9456 | 0.9737 | 0.0281 | |

$h\left(2\right)$ | 0.0335 | 0.0104 | 0.0329 | 0.0058 | 0.0131 | 0.0538 | 0.0407 | 0.0217 | 0.0442 | 0.0225 | |

3 | $\delta $ | 0.0772 | 0.0135 | 0.0761 | 0.0081 | 0.0508 | 0.1036 | 0.0527 | 0.0608 | 0.0922 | 0.0314 |

$R\left(2\right)$ | 0.9622 | 0.0117 | 0.9629 | 0.0069 | 0.9393 | 0.9851 | 0.0458 | 0.9494 | 0.9763 | 0.0269 | |

$h\left(2\right)$ | 0.0309 | 0.0094 | 0.0304 | 0.0056 | 0.0126 | 0.0493 | 0.0367 | 0.0196 | 0.0411 | 0.0215 | |

MPSE | BE-PS | ACI-PS | HPD-PS | ||||||||

1 | $\delta $ | 0.1020 | 0.0199 | 0.1007 | 0.0090 | 0.0629 | 0.1410 | 0.0781 | 0.0838 | 0.1186 | 0.0348 |

$R\left(2\right)$ | 0.9386 | 0.0205 | 0.9397 | 0.0092 | 0.8985 | 0.9787 | 0.0802 | 0.9219 | 0.9573 | 0.0354 | |

$h\left(2\right)$ | 0.0498 | 0.0163 | 0.0489 | 0.0073 | 0.0178 | 0.0817 | 0.0639 | 0.0350 | 0.0632 | 0.0282 | |

2 | $\delta $ | 0.0792 | 0.0141 | 0.0781 | 0.0082 | 0.0515 | 0.1069 | 0.0554 | 0.0621 | 0.0941 | 0.0320 |

$R\left(2\right)$ | 0.9605 | 0.0125 | 0.9612 | 0.0072 | 0.9360 | 0.9849 | 0.0489 | 0.9467 | 0.9746 | 0.0279 | |

$h\left(2\right)$ | 0.0323 | 0.0100 | 0.0317 | 0.0058 | 0.0127 | 0.0519 | 0.0392 | 0.0211 | 0.0434 | 0.0223 | |

3 | $\delta $ | 0.0768 | 0.0133 | 0.0757 | 0.0080 | 0.0507 | 0.1029 | 0.0522 | 0.0603 | 0.0911 | 0.0307 |

$R\left(2\right)$ | 0.9626 | 0.0115 | 0.9632 | 0.0068 | 0.9400 | 0.9852 | 0.0452 | 0.9496 | 0.9759 | 0.0263 | |

$h\left(2\right)$ | 0.0306 | 0.0092 | 0.0301 | 0.0055 | 0.0125 | 0.0488 | 0.0362 | 0.0199 | 0.0410 | 0.0211 |

**Table 5.**Some properties of MCMC draws of $\delta $, $R\left(t\right)$ and $h\left(t\right)$ from mechanical equipment data.

Sample | Par. | Mean | Mode | ${\mathit{Q}}_{1}$ | ${\mathit{Q}}_{2}$ | ${\mathit{Q}}_{3}$ | St.D | Skewness |
---|---|---|---|---|---|---|---|---|

BE-LF | ||||||||

1 | $\delta $ | 0.10456 | 0.09494 | 0.09847 | 0.10451 | 0.11058 | 0.00897 | 0.05709 |

$R\left(2\right)$ | 0.93570 | 0.93287 | 0.92955 | 0.93597 | 0.94216 | 0.00934 | −0.19348 | |

$h\left(2\right)$ | 0.05208 | 0.04413 | 0.04694 | 0.05186 | 0.05697 | 0.00744 | 0.19173 | |

2 | $\delta $ | 0.07970 | 0.07904 | 0.07409 | 0.07965 | 0.08510 | 0.00814 | 0.06236 |

$R\left(2\right)$ | 0.95975 | 0.96057 | 0.95509 | 0.96003 | 0.96485 | 0.00722 | −0.25378 | |

$h\left(2\right)$ | 0.03289 | 0.03224 | 0.02881 | 0.03267 | 0.03662 | 0.00578 | 0.24463 | |

3 | $\delta $ | 0.07610 | 0.07467 | 0.07068 | 0.07595 | 0.08148 | 0.00800 | 0.07658 |

$R\left(2\right)$ | 0.96290 | 0.96186 | 0.95840 | 0.96327 | 0.96769 | 0.00690 | −0.28396 | |

$h\left(2\right)$ | 0.03036 | 0.02920 | 0.02653 | 0.03008 | 0.03398 | 0.00553 | 0.27351 | |

BE-PS | ||||||||

1 | $\delta $ | 0.10065 | 0.10150 | 0.09462 | 0.10055 | 0.10659 | 0.00890 | 0.06119 |

$R\left(2\right)$ | 0.93972 | 0.93907 | 0.93378 | 0.94004 | 0.94599 | 0.00908 | −0.20777 | |

$h\left(2\right)$ | 0.04887 | 0.04939 | 0.04388 | 0.04862 | 0.05360 | 0.00723 | 0.20478 | |

2 | $\delta $ | 0.07806 | 0.07300 | 0.07245 | 0.07786 | 0.08358 | 0.00817 | 0.09572 |

$R\left(2\right)$ | 0.96119 | 0.95890 | 0.95649 | 0.96161 | 0.96622 | 0.00717 | −0.29196 | |

$h\left(2\right)$ | 0.03173 | 0.02808 | 0.02771 | 0.03141 | 0.03551 | 0.00574 | 0.28236 | |

3 | $\delta $ | 0.07571 | 0.07240 | 0.07029 | 0.07563 | 0.08102 | 0.00791 | 0.07238 |

$R\left(2\right)$ | 0.96323 | 0.96627 | 0.95881 | 0.96354 | 0.96801 | 0.00679 | −0.27788 | |

$h\left(2\right)$ | 0.03010 | 0.02767 | 0.02627 | 0.02986 | 0.03365 | 0.00544 | 0.26747 |

22 | 26 | 17 | 4 | 48 | 9 | 9 | 31 | 27 | 20 |

12 | 6 | 5 | 14 | 9 | 16 | 3 | 33 | 9 | 20 |

68 | 13 | 51 | 13 | 2 | 4 | 17 | 16 | 6 | 52 |

50 | 48 | 23 | 12 | 13 | 10 | 15 | 8 | 1 |

Model | MLE (St.E) | NL | A | CA | B | HQ | KS (p-Value) | |
---|---|---|---|---|---|---|---|---|

$\mathbf{\beta}$ | $\mathbf{\delta}$ | |||||||

ML | - | 0.0641(0.0080) | 152.968 | 308.786 | 308.894 | 310.450 | 309.383 | 0.1016(0.8157) |

L | - | 0.0978(0.0111) | 153.745 | 309.489 | 309.598 | 311.153 | 310.086 | 0.1180(0.6492) |

E | - | 0.0512(0.0082) | 154.923 | 311.846 | 311.954 | 313.510 | 312.443 | 0.1384(0.4440) |

G | 1.5081(0.3114) | 12.961(3.1672) | 153.393 | 310.297 | 310.630 | 313.624 | 311.491 | 0.0974(0.8532) |

W | 1.2502(0.1532) | 21.055(2.8507) | 153.439 | 310.878 | 311.211 | 314.205 | 312.071 | 0.1062(0.7711) |

TL | 2.8681(4.7327) | 0.0917(0.0165) | 153.600 | 311.200 | 311.533 | 314.527 | 312.393 | 0.1109(0.7232) |

EL | 0.8519(0.1945) | 0.0898(0.0155) | 153.493 | 310.985 | 311.319 | 314.313 | 312.179 | 0.1076(0.7577) |

MOL | 0.4195(0.3400) | 0.0695(0.0264) | 153.148 | 309.935 | 310.268 | 313.262 | 311.129 | 0.0867(0.9310) |

Sample | Scheme | Censored Data |
---|---|---|

1 | (19, 0${}^{*}$19) | 1, 2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 17, 20, 22, 23, 26, 31, 33, 48, 50 |

2 | $({0}^{*}8,{5}^{*}3,4,{0}^{*}8)$ | 1, 2, 3, 4, 4, 5, 6, 6, 8, 10, 12, 14, 15, 17, 20, 23, 26, 27, 31, 48 |

3 | $({0}^{*}19,19)$ | 1, 2, 3, 4, 4, 5, 6, 6, 8, 9, 9, 9, 9, 10, 12, 12, 13, 13, 13, 14 |

**Table 9.**Point and interval estimates of $\delta $, $R\left(t\right)$ and $h\left(t\right)$ from MVAD data.

Sample | Par. | MLE | BE-LF | ACI-LF | HPD-LF | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Est. | St.E | Est. | St.E | Lower | Upper | Length | Lower | Upper | Length | ||

1 | $\delta $ | 0.0676 | 0.0116 | 0.0667 | 0.0076 | 0.0448 | 0.0904 | 0.0456 | 0.0522 | 0.0817 | 0.0294 |

$R\left(2\right)$ | 0.9702 | 0.0092 | 0.9707 | 0.0060 | 0.9521 | 0.9883 | 0.0362 | 0.9587 | 0.9817 | 0.0230 | |

$h\left(2\right)$ | 0.0245 | 0.0074 | 0.0241 | 0.0048 | 0.0099 | 0.0391 | 0.0292 | 0.0152 | 0.0338 | 0.0186 | |

2 | $\delta $ | 0.0602 | 0.0094 | 0.0631 | 0.0083 | 0.0417 | 0.0787 | 0.0370 | 0.0497 | 0.0771 | 0.0274 |

$R\left(2\right)$ | 0.9758 | 0.0069 | 0.9734 | 0.0063 | 0.9623 | 0.9894 | 0.0271 | 0.9631 | 0.9837 | 0.0206 | |

$h\left(2\right)$ | 0.0200 | 0.0056 | 0.0219 | 0.0051 | 0.0090 | 0.0309 | 0.0219 | 0.0136 | 0.0302 | 0.0166 | |

3 | $\delta $ | 0.0706 | 0.0107 | 0.0697 | 0.0073 | 0.0497 | 0.0916 | 0.0419 | 0.0558 | 0.0839 | 0.0281 |

$R\left(2\right)$ | 0.9677 | 0.0088 | 0.9683 | 0.0059 | 0.9506 | 0.9849 | 0.0343 | 0.9562 | 0.9790 | 0.0227 | |

$h\left(2\right)$ | 0.0265 | 0.0070 | 0.0260 | 0.0048 | 0.0127 | 0.0403 | 0.0276 | 0.0174 | 0.0357 | 0.0183 | |

MPSE | BE-PS | ACI-PS | HPD-PS | ||||||||

1 | $\delta $ | 0.0653 | 0.0111 | 0.0645 | 0.0075 | 0.0434 | 0.0871 | 0.0437 | 0.0504 | 0.0793 | 0.0289 |

$R\left(2\right)$ | 0.9720 | 0.0086 | 0.9724 | 0.0057 | 0.9551 | 0.9889 | 0.0339 | 0.9610 | 0.9831 | 0.0221 | |

$h\left(2\right)$ | 0.0231 | 0.0070 | 0.0227 | 0.0046 | 0.0094 | 0.0367 | 0.0273 | 0.0141 | 0.0319 | 0.0178 | |

2 | $\delta $ | 0.0588 | 0.0092 | 0.0580 | 0.0068 | 0.0408 | 0.0768 | 0.0360 | 0.0449 | 0.0713 | 0.0264 |

$R\left(2\right)$ | 0.9769 | 0.0066 | 0.9772 | 0.0048 | 0.9639 | 0.9898 | 0.0259 | 0.9677 | 0.9863 | 0.0186 | |

$h\left(2\right)$ | 0.0191 | 0.0054 | 0.0188 | 0.0039 | 0.0087 | 0.0296 | 0.0210 | 0.0114 | 0.0265 | 0.0151 | |

3 | $\delta $ | 0.0702 | 0.0106 | 0.0694 | 0.0073 | 0.0495 | 0.0910 | 0.0415 | 0.0553 | 0.0838 | 0.0285 |

$R\left(2\right)$ | 0.9680 | 0.0086 | 0.9685 | 0.0059 | 0.9511 | 0.9850 | 0.0339 | 0.9568 | 0.9796 | 0.0229 | |

$h\left(2\right)$ | 0.0262 | 0.0070 | 0.0259 | 0.0048 | 0.0126 | 0.0399 | 0.0273 | 0.0169 | 0.0353 | 0.0184 |

**Table 10.**Some properties of MCMC draws of $\delta $, $R\left(t\right)$ and $h\left(t\right)$ from MVAD data.

Sample | Par. | Mean | Mode | ${\mathit{Q}}_{1}$ | ${\mathit{Q}}_{2}$ | ${\mathit{Q}}_{3}$ | St.D | Skewness |
---|---|---|---|---|---|---|---|---|

BE-LF | ||||||||

1 | $\delta $ | 0.06668 | 0.05799 | 0.06153 | 0.06653 | 0.07168 | 0.00752 | 0.11872 |

$R\left(2\right)$ | 0.97067 | 0.96880 | 0.96687 | 0.97102 | 0.97485 | 0.00593 | −0.35553 | |

$h\left(2\right)$ | 0.02412 | 0.01868 | 0.02076 | 0.02385 | 0.02719 | 0.00478 | 0.34243 | |

2 | $\delta $ | 0.06312 | 0.06001 | 0.05826 | 0.06294 | 0.06784 | 0.00706 | 0.10234 |

$R\left(2\right)$ | 0.97344 | 0.97596 | 0.96999 | 0.97379 | 0.97723 | 0.00536 | −0.33419 | |

$h\left(2\right)$ | 0.02189 | 0.01986 | 0.01884 | 0.02162 | 0.02468 | 0.00432 | 0.32112 | |

3 | $\delta $ | 0.06965 | 0.06600 | 0.06458 | 0.06954 | 0.07459 | 0.00725 | 0.08971 |

$R\left(2\right)$ | 0.96832 | 0.96864 | 0.96442 | 0.96862 | 0.97254 | 0.00590 | −0.29753 | |

$h\left(2\right)$ | 0.02602 | 0.02351 | 0.02263 | 0.02579 | 0.02915 | 0.00474 | 0.28626 | |

BE-PS | ||||||||

1 | $\delta $ | 0.06447 | 0.06257 | 0.05934 | 0.06432 | 0.06938 | 0.00742 | 0.12590 |

$R\left(2\right)$ | 0.97239 | 0.96984 | 0.96875 | 0.97274 | 0.97645 | 0.00572 | −0.36481 | |

$h\left(2\right)$ | 0.02273 | 0.02139 | 0.01947 | 0.02247 | 0.02568 | 0.00462 | 0.35142 | |

2 | $\delta $ | 0.05799 | 0.05607 | 0.05338 | 0.05784 | 0.06241 | 0.00671 | 0.13964 |

$R\left(2\right)$ | 0.97722 | 0.97877 | 0.97419 | 0.97753 | 0.98060 | 0.00480 | −0.39356 | |

$h\left(2\right)$ | 0.01883 | 0.01759 | 0.01610 | 0.01859 | 0.02130 | 0.00388 | 0.37901 | |

3 | $\delta $ | 0.06945 | 0.06207 | 0.06445 | 0.06928 | 0.07436 | 0.00728 | 0.09881 |

$R\left(2\right)$ | 0.96848 | 0.97094 | 0.96462 | 0.96883 | 0.97264 | 0.00591 | −0.30656 | |

$h\left(2\right)$ | 0.02589 | 0.02109 | 0.02255 | 0.02561 | 0.02899 | 0.00475 | 0.29525 |

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## Share and Cite

**MDPI and ACS Style**

Alotaibi, R.; Nassar, M.; Elshahhat, A.
Estimations of Modified Lindley Parameters Using Progressive Type-II Censoring with Applications. *Axioms* **2023**, *12*, 171.
https://doi.org/10.3390/axioms12020171

**AMA Style**

Alotaibi R, Nassar M, Elshahhat A.
Estimations of Modified Lindley Parameters Using Progressive Type-II Censoring with Applications. *Axioms*. 2023; 12(2):171.
https://doi.org/10.3390/axioms12020171

**Chicago/Turabian Style**

Alotaibi, Refah, Mazen Nassar, and Ahmed Elshahhat.
2023. "Estimations of Modified Lindley Parameters Using Progressive Type-II Censoring with Applications" *Axioms* 12, no. 2: 171.
https://doi.org/10.3390/axioms12020171