# Computational Analysis of XLindley Parameters Using Adaptive Type-II Progressive Hybrid Censoring with Applications in Chemical Engineering

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## Abstract

**:**

## 1. Introduction

## 2. Classical Inference

## 3. Bayes MCMC Paradigm

**Step****1.**- Set the start value for the parameter $\lambda $, say ${\lambda}^{\left(0\right)}=\widehat{\lambda}$.
**Step****2.**- Put $j=1$.
**Step****3.**- Generate ${\lambda}^{\u2022}$ from the full conditional distribution in (18) from the normal distribution $N({\lambda}^{(j-1)},var\left({\lambda}^{(j-1)}\right))$.
**Step****4.**- Calculate the ratio:$$p\left({\lambda}^{(j-1)}\right|{\lambda}^{\u2022})=min\left[1,\frac{\psi \left({\lambda}^{\u2022}\right)}{\psi \left({\lambda}^{(j-1)}\right)}\right].$$
**Step****5.**- Generate w, where $w\sim U(0,1)$.
**Step****6.**- If $w\le p\left({\lambda}^{(j-1)}\right|{\lambda}^{\u2022})$, set ${\lambda}^{\left(j\right)}={\lambda}^{\u2022}$, otherwise, set ${\lambda}^{\left(j\right)}={\lambda}^{(j-1)}$.
**Step****7.**- Based on a distinct value t, obtain ${R}^{\left(j\right)}\left(t\right)$ and ${h}^{\left(j\right)}\left(t\right)$ as$${R}^{\left(j\right)}\left(t\right)=\left(1+\frac{{\lambda}^{\left(j\right)}t}{{\left(1+{\lambda}^{\left(j\right)}\right)}^{2}}\right){e}^{-{\lambda}^{\left(j\right)}t}$$$${h}^{\left(j\right)}\left(t\right)=\frac{{{\lambda}^{\left(j\right)}}^{2}\left({\lambda}^{\left(j\right)}+t+2\right)}{{\left(1+{\lambda}^{\left(j\right)}\right)}^{2}+{\lambda}^{\left(j\right)}t}.$$
**Step****8.**- Put $j=j+1$.
**Step****9.**- Redo Steps 3–8 H times to bring$$\left[{\lambda}^{\left(1\right)},\phantom{\rule{0.166667em}{0ex}}{R}^{\left(1\right)}\left(t\right),\phantom{\rule{0.166667em}{0ex}}{h}^{\left(1\right)}\left(t\right)\right],\dots ,\left[{\lambda}^{\left(H\right)},\phantom{\rule{0.166667em}{0ex}}{R}^{\left(H\right)}\left(t\right),\phantom{\rule{0.166667em}{0ex}}{h}^{\left(H\right)}\left(t\right)\right].$$
**Step****10.**- Calculate the Bayesian estimates of $\lambda $, $R\left(t\right)$ and $h\left(t\right)$ employing the SEL function, after a burn-in period M, as$$\begin{array}{c}\hfill \tilde{\lambda}=\frac{1}{H-M}{\sum}_{j=M+1}^{H}{\lambda}^{\left(j\right)},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\tilde{R}\left(t\right)=\frac{1}{H-M}{\sum}_{j=M+1}^{H}{R}^{\left(j\right)}\left(t\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{and}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\tilde{h}\left(t\right)=\frac{1}{H-M}{\sum}_{j=M+1}^{H}{h}^{\left(j\right)}\left(t\right).\end{array}$$
**Step****11.**- To acquire the HPD credible intervals of $\lambda $, RF, and HRF: First, sort the generated samples of ${\lambda}^{\left(j\right)}$, ${R}^{\left(j\right)}\left(t\right)$ and ${h}^{\left(j\right)}\left(t\right)$ after burn-in period as ${\lambda}^{(M+1)},{\lambda}^{(M+2)},\dots ,{\lambda}^{\left(H\right)}$, ${R}^{(M+1)}\left(t\right)$, ${R}^{(M+2)}\left(t\right)$, $\dots ,{R}^{\left(H\right)}\left(t\right)$ and ${h}^{(M+1)\left(t\right)},{h}^{(M+2)}\left(t\right),\dots ,{h}^{\left(H\right)}\left(t\right)$, respectively. Employing the procedure suggested by Chen and Shao [19], the $100(1-\alpha )\%$ two-sided HPD credible interval for the unknown parameter $\lambda $ is given by$$\begin{array}{c}\hfill \left({\lambda}^{\left({j}^{*}\right)},{\lambda}^{\left({j}^{*}+\left(1-\alpha \right)\left(H-M\right)\right)}\right),\end{array}$$$${\lambda}^{\left({j}^{*}+\left[\left(1-\alpha \right)\left(H-M\right)\right]\right)}-{\lambda}^{\left({j}^{*}\right)}=\underset{1\u2a7dj\u2a7d\alpha \left(H-M\right)}{min}\left({\lambda}^{\left(j+\left[\left(1-\alpha \right)\left(H-M\right)\right]\right)}-{\lambda}^{\left(j\right)}\right).$$The largest integer less than or equal to x is denoted by $\left[x\right]$. Then, the HPD credible interval of x with the smallest length is that interval. The HPD credible intervals of RF and HRF can be easily computed using the same approach.

## 4. Monte Carlo Simulation

**Step****1.**- Generate an ordinary Type-II progressive censored sample $({X}_{i},{R}_{i}),\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,m,$ as discussed in Balakrishnan and Cramer [20] as
- (i)
- Generate m independent observations as ${y}_{1},{y}_{2},\dots ,{y}_{m}$.
- (ii)
- Set ${\upsilon}_{i}={y}_{i}^{{\left(i+{\sum}_{j=m-i+1}^{m}{R}_{j}\right)}^{-1}},$ for $i=1,2,\dots ,m.$
- (iii)
- Set ${u}_{i}=1-{\upsilon}_{m}{\upsilon}_{m-1}\cdots {\upsilon}_{m-i+1}$ for $i=1,2,\dots ,m$. Hence, ${u}_{i},\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,m$ is a simulated sample of size m from the uniform $U(0,1)$ distribution.
- (iv)
- Set ${X}_{i}={F}^{-1}({u}_{i};\lambda ),\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,m,$ the Type-II progressive censored sample from $\mathrm{XL}\left(\lambda \right)$ is generated.

**Step****2.**- Obtain d-th failure and discard ${X}_{i}$ for $i=d+2,\dots ,m$.
**Step****3.**- Obtain order statistics ${X}_{d+2},\dots ,{X}_{m}$ from a truncated distribution $f\left(x\right)/\left[1-F\left({x}_{d+1}\right)\right]$ with sample size $n-d-{\sum}_{j=1}^{d}{R}_{j}-1$.

- Prior-I (say P1):(1.5, 3) and Prior-II (say P2):(2.5, 5) when $\mathrm{XL}\left(0.5\right)$.
- Prior-I (say P1):(4.5, 3) and Prior-II (say P2):(7.5, 5) when $\mathrm{XL}\left(1.5\right)$.

- Generally, the proposed estimates of the unknown parameters $\lambda $, $R\left(t\right)$, and $h\left(t\right)$ behave well in terms of lowest RMSE, MRAB, and ACL values, as well as the highest CP values;
- As n(or m) increases, all estimates of $\lambda $, $R\left(t\right)$, and $h\left(t\right)$ perform better. A similar result is found in the case of the total number of removal patterns, as ${R}_{i},\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,m$, decreases;
- Comparing PCSs 1, 2 and 3, we can observe that the RMSEs, MRABs, ACLs, and CPs of all unknown parameters are critically good based on PCS-1 (when the live items $n-m$ are removed at the first ${X}_{\left(1\right)}$ stage) compared to others. Since the expected duration of the life test experiment based on the first stage is greater than any other, the data collected under PCS-1 provided more information about the unknown parameters $\lambda $, $R\left(t\right)$, and $h\left(t\right)$ than those obtained based on any others;
- Comparing the gamma priors P1 and P2 on the Bayesian analysis, since the variance of P2 is less than the variance of P1, it can be seen that the Bayesian point/interval estimators of all unknown parameters from P2 perform more satisfactorily than those obtained from P1 in terms of the lowest RMSE, MRAB, and ACL values and largest CP values;
- As T increases, the RMSEs and MRABs of all estimates of all unknown parameters for $\mathrm{XL}\left(0.5\right)$ decrease, while there is an increase for $\mathrm{XL}\left(1.5\right)$;
- As T increases, the ACLs of all ACIs of all unknown parameters increase for both $\mathrm{XL}\left(0.5\right)$ and $\mathrm{XL}\left(1.5\right)$, whereas the associated CPs decrease.
- As T increases, the ACLs of all HPD credible interval estimates of all unknown parameters decrease for $\mathrm{XL}\left(0.5\right)$ and increase for $\mathrm{XL}\left(1.5\right)$. The opposite behavior is also observed in case of the CPs for all HPD credible interval estimates of $\lambda $, $R\left(t\right)$, and $h\left(t\right)$;
- As $\lambda $ increases, the RMSEs and MRABs of the MLEs of $\lambda $, $R\left(t\right)$, and $h\left(t\right)$ increase, while those based on the MCMC of $\lambda $ decrease and increase for $R\left(t\right)$ and $h\left(t\right)$ in most cases;
- As $\lambda $ increases, the associated ACLs of the ACIs of $\lambda $, $R\left(t\right)$, and $h\left(t\right)$ become wider, while those based on the HPD credible interval estimates of $\lambda $ decrease and increase for $R\left(t\right)$ and $h\left(t\right)$ in most cases. Additionally, as $\lambda $ increases, the opposite behavior is noted in the case of CPs for ACI/HPD credible interval estimates of $\lambda $, $R\left(t\right)$, and $h\left(t\right)$;
- To sum up, the Bayesian paradigm utilizing the M–H algorithm is advised to estimate the scale parameter $\lambda $ and the reliability indices RF and HRF of the XL distribution in the presence of the adaptive Type-II progressively hybrid censored scheme.

## 5. Real-Life Applications

#### 5.1. Sodium Sulfur

#### 5.2. Vinyl Chloride

## 6. Concluding Remarks

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 10.**Trace plots (

**top-panel**) and Histograms (

**bottom-panel**) of $\lambda $, $R\left(t\right)$, and $h\left(t\right)$ from SSB data.

**Figure 14.**Trace plots (

**top panel**) and histograms (

**bottom panel**) of $\lambda $, $R\left(t\right)$, and $h\left(t\right)$ from vinyl chloride data.

Model | MLE(SE) | A | CA | B | HQ | KS | ||
---|---|---|---|---|---|---|---|---|

$\mathit{\theta}$ | $\mathit{\lambda}$ | Distance | p-Value | |||||

XL | - | 1.1897(0.2230) | 40.3272 | 40.5655 | 41.2676 | 40.4870 | 0.1009 | 0.979 |

XG | - | 1.7176(0.2910) | 40.3342 | 40.5695 | 41.2786 | 40.4940 | 0.1127 | 0.947 |

E | - | 0.9859(0.2262) | 40.5401 | 40.7754 | 41.4846 | 40.7000 | 0.1091 | 0.959 |

L | - | 1.4191(0.2466) | 41.3270 | 41.5492 | 42.3227 | 41.5213 | 0.1015 | 0.976 |

G | 1.1697(0.2119) | 1.0749(0.2228) | 41.8206 | 42.5706 | 43.7095 | 42.1403 | 0.1245 | 0.896 |

W | 1.2970(0.3788) | 1.2787(0.4537) | 41.8005 | 42.5505 | 43.6894 | 42.1202 | 0.1191 | 0.921 |

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

1 | 4(10) | $(9,{0}^{*}9)$ | 0 | 0.076, 0.775, 0.884, 1.131, 1.446, 1.824, 1.827, 2.248, 2.385, 3.077 |

2 | 0.7(5) | $({0}^{*}4,5,4,{0}^{*}4)$ | 4 | 0.076, 0.082, 0.210, 0.315, 0.385, 0.775, 0.884, 1.131, 1.446, 1.824 |

3 | 0.4(5) | $({0}^{*}9,9)$ | 9 | 0.076, 0.082, 0.210, 0.315, 0.385, 0.412, 0.491, 0.504, 0.522, 0.678 |

**Table 3.**The point and interval estimates of $\lambda $, $R\left(t\right)$, and $h\left(t\right)$ from SSB data.

Sample | Parameter | MLE | MCMC | ACI | HPD |
---|---|---|---|---|---|

1 | $\lambda $ | 0.7976 | 0.6233 | (0.4132,1.1820) | (0.4033,0.8636) |

0.1961 | 0.0007 | 0.7688 | 0.4603 | ||

$R\left(1\right)$ | 0.5616 | 0.6654 | (0.3457,0.7775) | (0.5265,0.8049) | |

0.1101 | 0.0005 | 0.4318 | 0.2785 | ||

$h\left(1\right)$ | 0.5996 | 0.4340 | (0.3691,0.8301) | (0.2246,0.6556) | |

0.1176 | 0.0006 | 0.4610 | 0.4310 | ||

2 | $\lambda $ | 0.8261 | 0.6486 | (0.4388,1.2133) | (0.4190,0.8862) |

0.1976 | 0.0007 | 0.7746 | 0.4672 | ||

$R\left(1\right)$ | 0.5462 | 0.6502 | (0.3347,0.7578) | (0.5085,0.7878) | |

0.1079 | 0.0004 | 0.4231 | 0.2793 | ||

$h\left(1\right)$ | 0.6275 | 0.4577 | (0.3845,0.8706) | (0.2467,0.6868) | |

0.1240 | 0.0007 | 0.4861 | 0.4401 | ||

3 | $\lambda $ | 1.2491 | 0.8647 | (0.6276,1.8706) | (0.5382,1.1899) |

0.3171 | 0.0010 | 1.2430 | 0.6517 | ||

$R\left(1\right)$ | 0.3576 | 0.5318 | (0.1353,0.5798) | (0.3710,0.7054) | |

0.1134 | 0.0005 | 0.4444 | 0.3344 | ||

$h\left(1\right)$ | 1.0511 | 0.6673 | (0.3979,1.7043) | (0.3529,0.9911) | |

0.3333 | 0.0010 | 1.3065 | 0.6382 |

Model | MLE(SE) | A | CA | B | HQ | KS | ||
---|---|---|---|---|---|---|---|---|

$\mathit{\theta}$ | $\mathit{\lambda}$ | Distance | p-Value | |||||

XL | - | 0.5321(0.0913) | 112.9052 | 113.0302 | 114.4316 | 113.4257 | 0.0890 | 0.9507 |

XG | - | 1.0313(0.1235) | 114.9701 | 115.0951 | 116.4965 | 115.4906 | 0.1384 | 0.5330 |

E | - | 0.7145(0.0944) | 113.4008 | 113.5258 | 114.9271 | 113.9213 | 0.1081 | 0.8221 |

L | - | 0.8238(0.1054) | 114.6073 | 114.7323 | 116.1336 | 115.1278 | 0.1327 | 0.5878 |

G | 1.0627(0.2281) | 0.5654(0.1536) | 114.8263 | 115.2134 | 117.8790 | 115.8674 | 0.0973 | 0.9042 |

W | 1.0121(0.1328) | 1.8855(0.3377) | 114.8996 | 115.2867 | 117.9523 | 115.9407 | 0.0916 | 0.9376 |

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

1 | 0.50(1) | $(10,10,{0}^{*}12)$ | 10 | 0.1, 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 | 1.15(7) | $({0}^{*}6,10,10,{0}^{*}6)$ | 0 | 0.1, 0.1, 0.2, 0.2, 0.4, 0.4, 0.4, 1.2, 3.2, 4.0, 5.1, 5.3, 6.8, 8.0 |

3 | 2.2(13) | $({0}^{*}12,10,10)$ | 10 | 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, 2.3 |

**Table 6.**The point and interval estimates of $\lambda $, $R\left(t\right)$, and $h\left(t\right)$ from vinyl chloride data.

Sample | Parameter | MLE | MCMC | ACI | HPD |
---|---|---|---|---|---|

1 | $\lambda $ | 0.5226 | 0.4730 | (0.3274,0.7178) | (0.3464,0.6055) |

0.0996 | 0.0004 | 0.3904 | 0.2591 | ||

$R\left(1\right)$ | 0.7267 | 0.7593 | (0.5848,0.8685) | (0.6740,0.8424) | |

0.0724 | 0.0002 | 0.2837 | 0.1684 | ||

$h\left(1\right)$ | 0.3386 | 0.2951 | (0.2725,0.4047) | (0.1807,0.4094) | |

0.0337 | 0.0003 | 0.1322 | 0.2287 | ||

2 | $\lambda $ | 0.5212 | 0.4698 | (0.3192,0.7232) | (0.3399,0.6063) |

0.1301 | 0.0004 | 0.4040 | 0.2664 | ||

$R\left(1\right)$ | 0.7275 | 0.7614 | (0.5806,0.8745) | (0.6735,0.8466) | |

0.0750 | 0.0003 | 0.2939 | 0.1731 | ||

$h\left(1\right)$ | 0.3374 | 0.2923 | (0.2692,0.4055) | (0.1746,0.4093) | |

0.0348 | 0.0003 | 0.1363 | 0.2347 | ||

3 | $\lambda $ | 0.5666 | 0.5135 | (0.3577,0.7754) | (0.3764,0.6473) |

0.1066 | 0.0004 | 0.4177 | 0.2709 | ||

$R\left(1\right)$ | 0.6985 | 0.7331 | (0.5526,0.8444) | (0.6456,0.8197) | |

0.0744 | 0.0003 | 0.2917 | 0.1741 | ||

$h\left(1\right)$ | 0.3790 | 0.3313 | (0.2998,0.4581) | (0.2106,0.4546) | |

0.0404 | 0.0004 | 0.1583 | 0.2440 |

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**MDPI and ACS Style**

Alotaibi, R.; Nassar, M.; Elshahhat, A.
Computational Analysis of XLindley Parameters Using Adaptive Type-II Progressive Hybrid Censoring with Applications in Chemical Engineering. *Mathematics* **2022**, *10*, 3355.
https://doi.org/10.3390/math10183355

**AMA Style**

Alotaibi R, Nassar M, Elshahhat A.
Computational Analysis of XLindley Parameters Using Adaptive Type-II Progressive Hybrid Censoring with Applications in Chemical Engineering. *Mathematics*. 2022; 10(18):3355.
https://doi.org/10.3390/math10183355

**Chicago/Turabian Style**

Alotaibi, Refah, Mazen Nassar, and Ahmed Elshahhat.
2022. "Computational Analysis of XLindley Parameters Using Adaptive Type-II Progressive Hybrid Censoring with Applications in Chemical Engineering" *Mathematics* 10, no. 18: 3355.
https://doi.org/10.3390/math10183355