A novel extended Lindley lifetime model that exhibits unimodal or decreasing density shapes as well as increasing, bathtub or unimodal-then-bathtub failure rates, named the Marshall-Olkin-Lindley (MOL) model is studied. In this research, using a progressive Type-II censored, various inferences of the MOL model parameters of life are introduced. Utilizing the maximum likelihood method as a classical approach, the estimators of the model parameters and various reliability measures are investigated. Against both symmetric and asymmetric loss functions, the Bayesian estimates are obtained using the Markov Chain Monte Carlo (MCMC) technique with the assumption of independent gamma priors. From the Fisher information data and the simulated Markovian chains, the approximate asymptotic interval and the highest posterior density interval, respectively, of each unknown parameter are calculated. Via an extensive simulated study, the usefulness of the various suggested strategies is assessed with respect to some evaluation metrics such as mean squared errors, mean relative absolute biases, average confidence lengths, and coverage percentages. Comparing the Bayesian estimations based on the asymmetric loss function to the traditional technique or the symmetric loss function-based Bayesian estimations, the analysis demonstrates that asymmetric loss function-based Bayesian estimations are preferred. Finally, two data sets, representing vinyl chloride and repairable mechanical equipment items, have been investigated to support the approaches proposed and show the superiority of the proposed model compared to the other fourteen lifetime models.

One of the key research areas in the concept of distribution theory is the evolution of suggesting new statistical distributions. Such generalized distributions allow modelling for a range of disciplines, including reliability, engineering and medicine with even greater flexibility. The two-parameter Marshall-Olkin-Lindley (MOL) distribution suggested by Ghitany et al. [

Hence, its probability density function (PDF),

and

where

Frequently, life testing studies are stopped before all of the components fail. Due to financial or time restrictions, it occurs. The observations that emerge from this type of scenario are known as the censored sample. The literature has developed a number of filtering techniques for the evaluation of various life-testing strategies. The two most popular censorship techniques among the various techniques are Types I and II. The experimental units cannot be removed during a life-testing experiment, however, under any of these censorship techniques. This adaptability is featured in a life-testing experiment with progressive censoring. Since the publication of the book by Balakrishnan et al. [

In order to estimate the parameters of the MOL distribution, we work with the progressive Type-II censored (PT-IIC) sample in this study. The PT-IIC sample can be explained as follows: assume that a life testing experiment involving

where

Due to the MOL distribution’s flexibility and the PT-IIC scheme’s effectiveness in gathering sample data, no study investigated the estimation problems of the MOL distribution in the case of the PT-IIC sample. Also, in the original work of Ghitany et al. [

The remainder of the article is structured as follows. We present the MLEs and ACIs of the unknown parameters, RF and HRF, in

In this part, we estimate the unknown parameters, RF and HRF of the MOL distribution using the method of maximum likelihood based on the PT-IIC sample. The ACIs of the different parameters are explained in addition to the point estimators. Assume that

where

where

and

where

and

Utilizing the asymptotic normality of the MLEs is the most common approach for establishing confidence bounds for the parameters. The MLEs’ asymptotic distribution can be expressed as

with

and

where

where

and

where

Suppose that

As a result, with

For analyzing failure time data, the Bayesian estimation approach has attracted a lot of attention. It uses one’s past knowledge of the parameters and also takes into account the information that is readily available. In this section, the Bayesian estimators of

The posterior distribution of the unknown parameters

where

If one setting

Now, in order to derive the Bayesian estimators, we take into account the SE and GEnt loss functions. The Bayesian estimator for the SE loss function is the posterior mean, which considers overestimation and underestimation equally. In contrast hand, the GEnt loss function offers different influences for overestimation and underestimation. Calabria et al. [

where

given that

and

It is obvious that it is difficult to determine the Bayesian estimators using (

and

It is evident that the conditional distributions of

and

where

In this study, the first

To construct the HPD credible intervals of

where

To evaluate the behavior of the proposed estimators of

where

Once the 1,000 PT-IIC samples collected, the maximum likelihood and 95% ACI estimates of

To monitor whether the simulated Markovian sample is sufficiently close to the target posterior, beside the trace and autocorrelation plots, we purpose to consider the Brooks-Gelman-Rubin (BGR) diagnostic statistic, which evaluates the convergence by analyzing the difference between the variance-within chains and the variance-between chains for each model parameter, for details see [

The average estimates (Av.Es) from classical (or Bayesian) approach of

where

Further, the comparison between point estimates of

and

respectively.

Furthermore, the comparison between interval estimates of the same unknown parameters is made using their average confidence lengths (ACLs) and coverage percentages (CPs) which can be computed as

and

respectively, where

Heatmap is a method of representing data graphically where values are depicted by color, making it easy to visualize complex data and understand it at a glance. So, via

From

Generally, the proposed point and interval estimates of

As

Bayesian estimates against the GEnt loss function perform superior than those obtained against the SE loss function, and both perform better compared to the other estimates due to the gamma prior information. Similar result is also observed in the case of HPD credible interval estimates.

To evaluate the effect of parameter loss, it can be seen that the asymmetric Bayes estimates of

Comparing the considered prior sets 1 and 2, due to the variance of prior 2 is smaller than the variance of prior 1, it is observed that the Bayesian estimates and associated HPD credible intervals under prior 2 of all unknown parameters have good perform than others.

Asymmetric Bayesian estimates of

Comparing the censoring schemes 1, 2 and 3, it is clear that the both proposed point and interval estimates of

Finally, to estimate the MOL distribution parameters or its reliability characteristics under PT-IIC mechanism, the Bayesian M-H algorithm method is recommended.

In order to demonstrate the significance of the suggested inferential methodologies and the applicability of study objectives to actual phenomena, this part presents two practical applications from the domains of engineering and chemistry.

Vinyl chloride is a known human carcinogen and a rapidly burning colorless gas. In this application, 34 data points (measured in milligrams/liter) as presented in see

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 |

To verify the flexibility of the MOL model, the MOL distribution is compared with fourteen well-known distributions, (for

Different goodness-of-fit metrics, including the negative log-likelihood (NL), Akaike information criterion (AIC), Bayesian information criterion (BIC), Hannan-Quinn (HQ), Consistent Akaike (CA), and Kolmogorov-Smirnov (KS) statistic with its

Model | MLE (St.E) | NL | AIC | BIC | HQ | CA | KS( |
||
---|---|---|---|---|---|---|---|---|---|

MOL | – | 0.3663 (0.3084) | 0.5464 (0.2305) | 55.2866 | 114.5732 | 117.6259 | 115.6143 | 114.9603 | 0.0800 (0.9816) |

MOE | – | 0.8103 (0.4731) | 0.4789 (0.1721) | 55.3962 | 114.7924 | 117.8451 | 115.8334 | 115.1795 | 0.0891 (0.9502) |

MOW | 1.3260 (0.3429) | 0.2584 (0.3974) | 0.1780 (0.2340) | 55.8635 | 115.7269 | 120.3060 | 117.2885 | 116.5269 | 0.0849 (0.9669) |

MOG | 0.0377 (0.0995) | 0.0615 (0.1568) | 0.3689 (0.2777) | 55.2922 | 116.2584 | 120.8375 | 117.8200 | 117.0584 | 0.0928 (0.9316) |

MOGE | 1.3774 (0.4001) | 0.3799 (0.4775) | 0.4097 (0.2202) | 55.8798 | 115.7597 | 120.3387 | 117.3213 | 116.5597 | 0.0840 (0.9701) |

MOLE | 1.2394 (0.2987) | 0.3653 (0.5681) | 0.3051 (0.2577) | 55.2965 | 116.1179 | 120.6970 | 117.6795 | 116.9179 | 0.0805 (0.9802) |

MONH | 0.2611 (0.3135) | 0.0026 (0.0007) | 0.0088 (0.0102) | 59.8893 | 125.7786 | 130.3577 | 127.3402 | 126.5786 | 0.1457 (0.4655) |

MOAPE | 5.1277 (20.023) | 0.3257 (0.8441) | 0.4509 (0.2307) | 55.3691 | 116.7382 | 121.3173 | 118.2998 | 117.5382 | 0.0817 (0.9770) |

APE | – | 0.6520 (0.8536) | 0.4767 (0.1886) | 55.3925 | 114.7850 | 117.8377 | 115.8260 | 115.1721 | 0.0880 (0.9549) |

GE | – | 1.0764 (0.2474) | 0.5580 (0.1242) | 55.4019 | 114.8037 | 117.8565 | 115.8448 | 115.1908 | 0.0978 (0.9012) |

NH | – | 0.9003 (0.3442) | 0.6320 (0.4160) | 55.4172 | 114.8345 | 117.8872 | 115.8755 | 115.2216 | 0.0838 (0.9707) |

W | – | 1.0102 (0.1327) | 1.8879 (0.3390) | 55.4496 | 114.8992 | 117.9520 | 115.9403 | 115.2863 | 0.0918 (0.9366) |

G | – | 1.0659 (0.2291) | 1.7640 (0.4795) | 55.4133 | 114.8265 | 117.8793 | 115.8676 | 115.2136 | 0.0979 (0.9001) |

L | – | – | 0.8238 (0.1054) | 56.3036 | 114.6073 | 118.1336 | 115.7807 | 115.7323 | 0.1326 (0.5883) |

E | – | – | 0.5321 (0.0912) | 55.4526 | 115.9052 | 117.9431 | 116.4257 | 116.0302 | 0.0889 (0.9507) |

We also provided the quantile-quantile (QQ) plot as a graphical demonstration, via

In

Now three different PT-IIC samples, from the complete vinyl chloride data, are generated with

Censored sample | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

0.1 | 0.2 | 0.4 | 0.4 | 0.4 | 0.5 | 0.5 | 0.6 | 0.6 | 0.9 | |

1.0 | 1.8 | 2.0 | 2.0 | 2.7 | 2.9 | 4.0 | 5.1 | 5.3 | 6.8 | |

0.1 | 0.1 | 0.2 | 0.2 | 0.4 | 0.4 | 0.4 | 0.5 | 0.5 | 0.5 | |

0.6 | 0.8 | 0.9 | 1.1 | 1.2 | 1.8 | 2.0 | 2.7 | 3.2 | 5.3 | |

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 |

Par. | ML | SE | GEnt | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

−3 | −0.03 | +3 | |||||||||

0.3922 | 0.4045 | 0.3604 | 0.0578 | 0.3668 | 0.0254 | 0.3572 | 0.0350 | 0.3468 | 0.0455 | ||

0.5459 | 0.2801 | 0.5182 | 0.0532 | 0.5222 | 0.0238 | 0.5163 | 0.0297 | 0.5101 | 0.0359 | ||

0.9037 | 0.0344 | 0.9026 | 0.0162 | 0.9028 | 0.0009 | 0.9024 | 0.0013 | 0.9020 | 0.0017 | ||

0.0900 | 0.1594 | 0.0768 | 0.0207 | 0.0801 | 0.0099 | 0.0752 | 0.0148 | 0.0700 | 0.0201 | ||

0.8223 | 0.7626 | 0.7077 | 0.1474 | 0.7197 | 0.1026 | 0.7017 | 0.1205 | 0.6828 | 0.1394 | ||

0.9355 | 0.4463 | 0.8395 | 0.1279 | 0.8479 | 0.0876 | 0.8353 | 0.1002 | 0.8223 | 0.1132 | ||

0.8921 | 0.0337 | 0.8928 | 0.0177 | 0.8931 | 0.0010 | 0.8926 | 0.0005 | 0.8921 | 0.0001 | ||

0.4148 | 0.6293 | 0.3097 | 0.1240 | 0.3234 | 0.0914 | 0.3029 | 0.1119 | 0.2816 | 0.1332 | ||

2.2517 | 2.6453 | 2.1375 | 0.1532 | 2.1424 | 0.1093 | 2.1351 | 0.1166 | 2.1277 | 0.1240 | ||

1.5514 | 0.8284 | 1.4534 | 0.1349 | 1.4593 | 0.0922 | 1.4505 | 0.1009 | 1.4414 | 0.1101 | ||

0.9125 | 0.0360 | 0.9160 | 0.0087 | 0.9160 | 0.0035 | 0.9159 | 0.0034 | 0.9158 | 0.0033 | ||

2.0483 | 3.4938 | 1.8029 | 0.2983 | 1.8188 | 0.2295 | 1.7951 | 0.2532 | 1.7700 | 0.2783 |

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

Lower | Upper | Length | Lower | Upper | Length | ||

0.0000 | 1.1850 | 1.1850 | 0.2725 | 0.4616 | 0.1891 | ||

0.0000 | 1.0950 | 1.0950 | 0.4282 | 0.6060 | 0.1778 | ||

0.8363 | 0.9711 | 0.1348 | 0.8699 | 0.9315 | 0.0616 | ||

0.0000 | 0.4025 | 0.4025 | 0.0462 | 0.1071 | 0.0609 | ||

0.0000 | 2.3169 | 2.3169 | 0.5426 | 0.8979 | 0.3553 | ||

0.0608 | 1.8102 | 1.7494 | 0.6803 | 0.9998 | 0.3195 | ||

0.8260 | 0.9582 | 0.1322 | 0.8571 | 0.9253 | 0.0682 | ||

0.0000 | 1.6482 | 1.6482 | 0.1758 | 0.4279 | 0.2521 | ||

0.0000 | 7.4364 | 7.4364 | 1.9168 | 2.3167 | 0.3999 | ||

0.0000 | 3.1750 | 3.1750 | 1.2772 | 1.6278 | 0.3506 | ||

0.8419 | 0.9831 | 0.1411 | 0.9000 | 0.9306 | 0.0307 | ||

0.0000 | 8.8960 | 8.8960 | 1.4757 | 2.1203 | 0.6447 |

From each sample in

Par. | Mean | Mode | St.D | Skewness | ||||
---|---|---|---|---|---|---|---|---|

0.36042 | 0.26397 | 0.04829 | 0.03778 | 0.32680 | 0.36034 | 0.39294 | ||

0.51820 | 0.37151 | 0.04542 | 0.04602 | 0.48720 | 0.51779 | 0.54846 | ||

0.90255 | 0.92379 | 0.01614 | −0.47811 | 0.89248 | 0.90385 | 0.91391 | ||

0.07685 | 0.03204 | 0.01596 | 0.33424 | 0.06552 | 0.07587 | 0.08718 | ||

0.70765 | 0.51841 | 0.09269 | 0.14656 | 0.64118 | 0.70368 | 0.77130 | ||

0.83947 | 0.68510 | 0.08437 | 0.13151 | 0.78220 | 0.83775 | 0.89489 | ||

0.89277 | 0.89141 | 0.01774 | −0.45987 | 0.88186 | 0.89432 | 0.90495 | ||

0.30968 | 0.17580 | 0.06571 | 0.37362 | 0.26288 | 0.30679 | 0.35120 | ||

2.13750 | 1.91678 | 0.10214 | 0.00815 | 2.06689 | 2.13791 | 2.20741 | ||

1.45340 | 1.34267 | 0.09263 | −0.02278 | 1.38835 | 1.45304 | 1.51779 | ||

0.91595 | 0.91698 | 0.00795 | −0.13721 | 0.91065 | 0.91639 | 0.92125 | ||

1.80292 | 1.47569 | 0.16970 | 0.01186 | 1.68844 | 1.80353 | 1.91608 |

In each trace plot, the sample mean and two bounds of 95% HPD credible intervals of

In this application, from the engineering field, we will explain our theoretical results based on the time between consecutive failures for repairable mechanical equipment (RME) items depicted in

0.11 | 0.30 | 0.40 | 0.45 | 0.59 | 0.63 | 0.70 | 0.71 | 0.74 | 0.77 |

0.94 | 1.06 | 1.17 | 1.23 | 1.23 | 1.24 | 1.43 | 1.46 | 1.49 | 1.74 |

1.82 | 1.86 | 1.97 | 2.23 | 2.37 | 2.46 | 2.63 | 3.46 | 4.36 | 4.73 |

Model | MLE(SE) | NL | AIC | BIC | HQ | CA | KS( |
||
---|---|---|---|---|---|---|---|---|---|

MOL | – | 2.7991 (1.8462) | 1.4400 (0.3462) | 40.4944 | 84.9887 | 85.8852 | 87.7911 | 85.4331 | 0.0746 (0.9962) |

MOE | – | 5.1690 (3.7002) | 1.3178 (0.3523) | 40.5040 | 85.0081 | 85.9046 | 87.8105 | 85.4525 | 0.0910 (0.9648) |

MOW | 1.9211 (0.6269) | 0.2424 (0.4112) | 0.1467 (0.2446) | 40.5347 | 85.0694 | 86.4141 | 89.2730 | 85.9924 | 0.0752 (0.9958) |

MOG | 2.1276 (1.3850) | 10.314 (13.352) | 0.2181 (0.2838) | 40.5645 | 86.3289 | 87.6737 | 90.5325 | 87.2520 | 0.0790 (0.9920) |

MOGE | 2.1159 (1.5457) | 0.9994 (2.9184) | 0.9765 (0.7803) | 40.6319 | 85.2638 | 86.6086 | 89.4674 | 86.1869 | 0.0748 (0.9960) |

MOLE | 1.2345 (0.7252) | 3.5295 (12.180) | 0.9511 (1.4986) | 40.5666 | 86.1333 | 87.4780 | 90.3369 | 87.0564 | 0.0977 (0.9371) |

MONH | 0.9138 (0.2353) | 4.4984 (3.4947) | 1.4696 (1.0112) | 40.5187 | 86.8373 | 88.1821 | 91.0409 | 87.7604 | 0.0791 (0.9920) |

MOAPE | 0.0446 (0.2008) | 12.908 (16.559) | 0.9549 (0.7361) | 41.0248 | 86.0495 | 87.3943 | 90.2531 | 86.9726 | 0.0750 (0.9959) |

APE | – | 9.9305 (8.4422) | 1.0697 (0.1976) | 40.5782 | 85.0356 | 86.0253 | 87.8589 | 85.8009 | 0.0826 (0.9867) |

GE | – | 2.1234 (0.5875) | 1.0032 (0.2014) | 40.6143 | 85.2287 | 86.0125 | 87.8031 | 85.6731 | 0.0750 (0.9958) |

NH | – | 4.4288 (6.6965) | 0.0981 (0.1699) | 41.1540 | 86.3080 | 87.2045 | 89.1103 | 86.7524 | 0.1132 (0.8365) |

W | – | 1.4663 (0.2032) | 1.7103 (0.2250) | 40.9105 | 85.0821 | 86.0717 | 87.8623 | 85.6543 | 0.0751 (0.9958) |

G | – | 1.9718 (0.4717) | 0.7820 (0.2127) | 40.6296 | 85.0259 | 86.1558 | 88.0617 | 85.7037 | 0.0769 (0.9929) |

L | – | – | 0.9762 (0.1345) | 41.5473 | 85.0946 | 86.0543 | 87.9576 | 85.7416 | 0.1407 (0.5928) |

E | – | – | 0.6482 (0.1184) | 43.0054 | 88.0108 | 88.4590 | 89.4119 | 88.1536 | 0.1845 (0.2589) |

Also, using the complete RME data,

From the complete RME data, three different PT-IIC samples are generated with

Censored sample | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

0.11 | 0.59 | 0.63 | 0.70 | 0.71 | 0.77 | 1.06 | 1.23 | 1.46 | 2.46 | |

0.11 | 0.40 | 0.59 | 0.74 | 0.77 | 0.94 | 1.23 | 1.74 | 1.86 | 2.63 | |

0.11 | 0.30 | 0.40 | 0.45 | 0.59 | 0.63 | 0.70 | 0.71 | 0.74 | 0.77 |

Par. | ML | SE | GEnt | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

−5 | −0.05 | +5 | |||||||||

10.649 | 10.654 | 10.547 | 0.1423 | 10.549 | 0.0994 | 10.547 | 0.1018 | 10.544 | 0.1042 | ||

3.2107 | 1.0042 | 3.1126 | 0.1369 | 3.1185 | 0.0922 | 3.1112 | 0.0995 | 3.1039 | 0.1069 | ||

0.8034 | 0.0849 | 0.8120 | 0.0131 | 0.8123 | 0.0088 | 0.8120 | 0.0085 | 0.8116 | 0.0082 | ||

9.7741 | 7.9925 | 9.6747 | 0.1266 | 9.6760 | 0.0982 | 9.6744 | 0.0997 | 9.6728 | 0.1013 | ||

22.117 | 8.9012 | 22.021 | 0.1350 | 22.022 | 0.0947 | 22.021 | 0.0957 | 22.020 | 0.0967 | ||

2.5831 | 0.4318 | 2.4900 | 0.1333 | 2.4973 | 0.0858 | 2.4883 | 0.0948 | 2.4790 | 0.1041 | ||

0.9296 | 0.0207 | 0.9338 | 0.0062 | 0.9338 | 0.0042 | 0.9338 | 0.0041 | 0.9337 | 0.0041 | ||

18.265 | 7.0823 | 18.076 | 0.2345 | 18.079 | 0.1864 | 18.076 | 0.1890 | 18.073 | 0.1918 | ||

58.181 | 17.183 | 58.080 | 0.1420 | 58.081 | 0.1002 | 58.080 | 0.1007 | 58.080 | 0.1011 | ||

5.0245 | 0.6102 | 4.9259 | 0.1396 | 4.9299 | 0.0946 | 4.9250 | 0.0995 | 4.9199 | 0.1045 | ||

0.8831 | 0.0294 | 0.8883 | 0.0076 | 0.8884 | 0.0053 | 0.8883 | 0.0052 | 0.8882 | 0.0052 | ||

33.580 | 7.0719 | 34.205 | 0.9234 | 34.232 | 0.6516 | 34.199 | 0.6181 | 34.164 | 0.5840 |

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

Lower | Upper | Length | Lower | Upper | Length | ||

0.0000 | 31.530 | 31.530 | 10.346 | 10.741 | 0.3948 | ||

1.2424 | 5.1790 | 3.9366 | 2.9245 | 3.2943 | 0.3697 | ||

0.6371 | 0.9697 | 0.3326 | 0.7923 | 0.8306 | 0.0383 | ||

0.0000 | 25.439 | 25.439 | 9.5034 | 9.8159 | 0.3125 | ||

4.6708 | 39.563 | 34.892 | 21.841 | 22.215 | 0.3738 | ||

1.7368 | 3.4293 | 1.6925 | 2.3026 | 2.6754 | 0.3728 | ||

0.8890 | 0.9703 | 0.0813 | 0.9247 | 0.9425 | 0.0178 | ||

4.3840 | 32.146 | 27.762 | 17.789 | 18.324 | 0.5344 | ||

24.503 | 91.859 | 67.356 | 57.898 | 58.280 | 0.3819 | ||

3.8285 | 6.2204 | 2.3919 | 4.7400 | 5.1153 | 0.3752 | ||

0.8255 | 0.9407 | 0.1151 | 0.8775 | 0.8982 | 0.0207 | ||

19.720 | 47.441 | 27.721 | 32.862 | 35.460 | 2.5979 |

Par. | Mean | Mode | St.D | Skewness | ||||
---|---|---|---|---|---|---|---|---|

10.5473 | 10.2897 | 0.09996 | 0.01633 | 10.4822 | 10.5465 | 10.6134 | ||

3.11263 | 3.09570 | 0.09553 | 0.09140 | 3.04572 | 3.11020 | 3.17706 | ||

0.81202 | 0.81011 | 0.00995 | −0.14604 | 0.80544 | 0.81218 | 0.81902 | ||

9.67470 | 9.48238 | 0.07831 | −0.00812 | 9.62314 | 9.67474 | 9.72670 | ||

22.0214 | 21.8156 | 0.09544 | 0.13513 | 21.9559 | 22.0168 | 22.0847 | ||

2.49003 | 2.39730 | 0.09541 | −0.02557 | 2.42499 | 2.49036 | 2.55517 | ||

0.93376 | 0.93764 | 0.00456 | −0.08578 | 0.93070 | 0.93382 | 0.93693 | ||

18.0765 | 17.8060 | 0.13949 | −0.43657 | 17.9895 | 18.0877 | 18.1746 | ||

58.0804 | 57.9260 | 0.10016 | 0.00309 | 58.0092 | 58.0809 | 58.1483 | ||

4.92592 | 4.75752 | 0.09890 | −0.03977 | 4.85957 | 4.92677 | 4.99215 | ||

0.88834 | 0.89721 | 0.00546 | −0.07519 | 0.88471 | 0.88844 | 0.89206 | ||

34.2049 | 35.2789 | 0.68016 | 0.06226 | 33.7409 | 34.1986 | 34.6577 |

From the PT-IIC sample generated by

In this study, we looked into the statistical inference of the Marshall-Olkin Lindley distribution’s unknown parameters, reliability, and hazard rate functions under progressively Type-II censored data. The various parameters of interest are inferred using both classical and Bayesian methods. The normal approximation of the maximum likelihood estimators is also used to create the approximate confidence intervals. The Bayesian estimations are addressed by employing independent gamma priors and symmetric and asymmetric loss functions. We have indicated that the explicit expressions of the proposed Bayesian estimators are not available. The Markov Chain Monte Carlo technique is employed as a result. For each parameter, the highest posterior density credible intervals are also attained. We conducted a thorough simulation analysis and examined two applications to real-world data sets to evaluate the effectiveness of the delivered estimations. The findings of the numerical study showed that when progressively Type-II censored data were given, the suggested point and interval estimations of the Marshall-Olkin Lindley distribution acted reasonably. More specifically, the highest posterior density credible intervals were advised and the Bayesian estimates utilizing the general entropy loss function outperformed all other estimates. In addition, the real data analysis showed that the Marshall-Olkin Lindley distribution could be used as a good model to fit vinyl chloride and repairable mechanical equipment data sets rather than some other Marshall-Olkin models, including Marshall-Olkin Weibull, Marshall-Olkin Gompertz, Marshall-Olkin generalized exponential and Marshall-Olkin logistic-exponential distributions. In future work, it is of interest to investigate the estimation problems of the considered distribution based on other censoring schemes like an adaptive progressive Type-II censoring scheme. Another significant future work to be addressed is exploring the performance of dependability metrics of the utilized model in the case of accelerated life tests.

Approximative confidence interval

Average confidence length

Akaike information criterion

Alpha power exponential

Average estimates

Bayesian information criterion

Brooks-Gelman-Rubin

Consistent Akaike

Coverage percentage

Exponential

Failure percentage

Gamma

Generalized-exponential

General entropy

Hannan-Quinn

Highest posterior density

Hazard rate function

Kolmogorov-Smirnov

Lindley

Metropolis-Hastings

Markov Chain Monte Carlo

Maximum likelihood estimator

Marshall-Olkin alpha power exponential

Marshall-Olkin exponential

Marshall-Olkin Gompertz

Marshall-Olkin generalized exponential

Marshall-Olkin-Lindley

Marshall-Olkin logistic-exponential

Marshall-Olkin Nadarajah-Haghighi

Marshall-Olkin Weibull

Mean relative absolute bias

Nadarajah-Haghighi

Negative log-likelihood

Probability density function

Progressive Type-II censored

Quantile-quantile

Reliability function

Repairable mechanical equipment

Root mean squared-error

Squared error

Standard deviation

Standard-error

Weibull

The authors would desire to express their thanks to the editor and the three anonymous referees for useful suggestions and valuable comments. Princess Nourah bint Abdulrahman University Researchers Supporting Project and Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

This research was funded by Princess Nourah bint Abdulrahman University Researchers Supporting Project Number (PNURSP2023R50), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

The authors confirm contribution to the paper as follows: study conception and design: R. Alotaibi, M. Nassar, and A. Elshahhat; data collection: A. Elshahhat; analysis and interpretation of results: A. Elshahhat; draft manuscript preparation: M. Nassar, R. Alotaibi. All authors reviewed the results and approved the final version of the manuscript.

The data that support the findings of this study are available in the text.

The authors declare that they have no conflicts of interest to report regarding the present study.

The supplementary material is available online at