Accelerated life testing has been widely used in product life testing experiments because it can quickly provide information on the lifetime distributions by testing products or materials at higher than basic conditional levels of stress, such as pressure, temperature, vibration, voltage, or load to induce early failures. In this paper, a step stress partially accelerated life test (SS-PALT) is regarded under the progressive type-II censored data with random removals. The removals from the test are considered to have the binomial distribution. The life times of the testing items are assumed to follow length-biased weighted Lomax distribution. The maximum likelihood method is used for estimating the model parameters of length-biased weighted Lomax. The asymptotic confidence interval estimates of the model parameters are evaluated using the Fisher information matrix. The Bayesian estimators cannot be obtained in the explicit form, so the Markov chain Monte Carlo method is employed to address this problem, which ensures both obtaining the Bayesian estimates as well as constructing the credible interval of the involved parameters. The precision of the Bayesian estimates and the maximum likelihood estimates are compared by simulations. In addition, to compare the performance of the considered confidence intervals for different parameter values and sample sizes. The Bootstrap confidence intervals give more accurate results than the approximate confidence intervals since the lengths of the former are less than the lengths of latter, for different sample sizes, observed failures, and censoring schemes, in most cases. Also, the percentile Bootstrap confidence intervals give more accurate results than Bootstrap-t since the lengths of the former are less than the lengths of latter for different sample sizes, observed failures, and censoring schemes, in most cases. Further performance comparison is conducted by the experiments with real data.

Technology developments have been continuously improving product manufacturing. Therefore, it is difficult to obtain failure data for high-reliability items under normal operating conditions. This makes the lifetime testing under normal conditions costly and time-consuming. Consequently, under environmental conditions (stresses) that are normal conditions, the accelerated life tests (ALTs) or partially accelerated life tests (PALTs) have been often used. In the ALT, all test items are subjected to higher than usual stress levels, while, in the PALT, items are tested at both accelerated and normal conditions. In [

In many cases, when the life data are analyzed, all units in the sample may not fail. This type of data is called censored or incomplete data. The most common censoring schemes are the type-I censored scheme (or time censored scheme) and type-II censored scheme (or failure-censored scheme). These two censoring schemes do not allow for units to be removed from the experiments while they are still alive. In [

In this paper, a PTIIC with random removal is proposed to provide more economical and less time-consuming testing. The PTIIC scheme is designed as follows. Suppose _{1} of the remaining (

On another statistical level, the Lomax distribution is an important heavy-tail probability distribution for lifetime analysis, and it has been often used in many different fields, including business, economics, and actuarial modeling [

The weighted distributions arise in the context of unequal probability sampling. The weighted distributions have prominent importance in reliability, biomedicine, ecology, and other fields. The unified concept of the length-biased distribution can be used in the development of proper models for lifetime data. The length-biased distribution represents a special case of the more general form known as weighted distribution. In [

Statistical properties and applications of the LBWL distribution to real data have been presented in [

In view of the importance of the weighted distributions and SS-PALT in reliability studies, this paper applies the SS-PALT to items whose lifetimes under design conditions are assumed to follow the LBWL distribution under a PTIIC scheme with random removals. The removals from the test are considered to obey the binomial distribution. The maximum likelihood (ML) and approximate confidence intervals (CIs) of the estimators are presented. The Bayesian estimators, percentile bootstrap CIs, and bootstrap-t CIs are obtained. The Monte Carlo simulations, as well as experiments with real data, are performed to verify the theoretical analysis results.

The rest of the paper is organized as follows. Test procedure and the assumptions of the SS-PALT model are presented in Section 2. In Section 3, the ML and approximate CI estimators of the model parameters are provided. The Bayesian estimators of the model parameters are introduced in Section 4. The bootstrap CI estimates for ML and Bayesian estimation are given in Section 5. In Section 6, the simulation results are presented to illustrate accuracy of the estimates. The experiment with real data is introduced in Section 7. Finally, concluding remarks are given in Section 8.

The following assumptions are adopted in this work:

➣ Assume

➣ Progressive sample

➣ Each of

➣ At the _{i}

➣ Suppose that an individual unit removed from the test is independent of the other units but has the same removal probability

The lifetime of a unit under the SS-PALT denoted as

where

where _{2}(

This section introduces the ML estimators of the population parameters and acceleration factor based on the PTIIC data with binomial removal. Moreover, approximate CIs (ACIs) of the population parameters and acceleration factor are also presented.

Let _{(i)}. The conditional likelihood function of observations

Suppose that

where

That is,

Based on

where _{u}_{a}

Since

Similarly, since

Furthermore, the ACIs of the parameters using the PTIIC data based on the asymptotic properties of the ML estimators can be obtained. The ACIs can be approximated by numerically inverting Fisher’s information matrix. Accordingly, the approximate 100(

where

In this section, the Bayesian estimation of the population parameters and the acceleration factor of the LBWL distribution based on the PTIIC data with binomial removal are presented. The Bayesian estimation is considered under the squared error loss function (SELF).

Assume that the prior of

To elicit the hyper-parameters of the informative priors, the same procedure as that presented in [

By equating the mean and variance of

Hence, the estimated hyper-parameters can be expressed as:

Based on the likelihood function

Therefore, the Bayesian estimators of the parameters

The integrals presented in

Set the initial value

Using the initial value, sample a candidate point

Given the candidate point

Draw a value of

Otherwise, reject

Repeat Steps 2–5 (

Obtain the Bayes estimate of

Repeat Steps (1–7)

According to [

Arrange

The

In this section, different bootstrap CIs of population parameters and acceleration factor based on the PTIIC data with binomial removal are proposed for the LBWL distribution.

Compute estimates of

Generate bootstrap samples of

Repeat Step (ii)

Arrange

A two-sided

Steps 1 and 2 are the same as Steps (i) and (ii) in the previous BP algorithm, respectively.

Step 3. Compute the

Step 4. Repeat Steps 1–3 ^{(1)},

Step 5. Arrange ^{(1)}, ^{[1]},

Step 6. A two-sided

The performance of the proposed methods for estimation of the LBWL distribution based on the SS-PALT under PTIIC was verified by the Monte-Carlo simulation via R-package. The Bayesian estimators were obtained using the gamma priors under the SELF. The main difficulty in the Bayesian procedure was obtaining the posterior distribution. The MH algorithm and the Gibbs sampling were used to simulate deviates from the posterior density. The simulation steps were as follows:

➣ Generate 10000 random samples of size

➣ Use the CDF of the LBWL distribution and the uni-root function in the R-package to generate the random number of the LBWL distribution in the numerical algorithm.

➣ For different parameters and

Set I:

Set II:

➣ In the PTIIC, select the sample size (failure items)

➣ Calculate the ML estimates and associated ACIs, and the Bayesian estimates and associated credible intervals at

➣ Evaluate the performance of the estimates based on the accuracy measures, including biases, mean square errors (MSEs), and lengths of CIs (L.CIs). The simulation results are presented in

ML | Bayesian | ML | Bayesian | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

50 | 35 | Bias | 0.1163 | 0.2855 | −0.0745 | 0.0635 | 0.2074 | 0.5616 | 0.0964 | 0.2694 | 0.0630 | 0.0480 | 0.2125 | 0.3562 |

MSE | 0.0790 | 0.2850 | 1.6826 | 0.0441 | 0.1113 | 1.7911 | 0.0935 | 0.2597 | 0.7398 | 0.0140 | 0.1146 | 0.6540 | ||

L.CI | 1.0033 | 1.7692 | 5.0790 | 0.7854 | 1.0250 | 4.7642 | 1.1341 | 1.7994 | 3.3602 | 0.4246 | 1.0332 | 2.8476 | ||

BP | 0.1401 | 0.2436 | 0.6924 | 0.1429 | 0.2444 | 0.7288 | 0.1617 | 0.2568 | 0.4502 | 0.1278 | 0.2442 | 0.4479 | ||

BT | 0.1642 | 0.2643 | 0.7215 | 0.1627 | 0.2712 | 0.7693 | 0.1843 | 0.2803 | 0.4773 | 0.1473 | 0.2707 | 0.4659 | ||

45 | Bias | 0.0834 | 0.1597 | −0.0896 | 0.0553 | 0.1343 | 0.4907 | 0.0993 | 0.2218 | 0.0760 | 0.0479 | 0.1708 | 0.3242 | |

MSE | 0.0468 | 0.1380 | 1.2659 | 0.0237 | 0.0588 | 1.4221 | 0.0625 | 0.2521 | 0.7194 | 0.0128 | 0.0817 | 0.5071 | ||

L.CI | 0.7829 | 1.3152 | 4.3987 | 0.5722 | 0.7923 | 4.2626 | 0.9045 | 1.6620 | 3.3172 | 0.4029 | 0.8988 | 2.4867 | ||

BP | 0.1121 | 0.1819 | 0.6282 | 0.1070 | 0.1854 | 0.6072 | 0.1282 | 0.2318 | 0.4364 | 0.1696 | 0.2644 | 0.4557 | ||

BT | 0.1264 | 0.1974 | 0.6527 | 0.1197 | 0.2015 | 0.6112 | 0.1424 | 0.2565 | 0.4674 | 0.2013 | 0.2962 | 0.4901 | ||

100 | 70 | Bias | 0.0559 | 0.1506 | −0.1022 | 0.0305 | 0.1229 | 0.3555 | 0.0579 | 0.1536 | −0.0087 | 0.0277 | 0.1181 | 0.2036 |

MSE | 0.0190 | 0.0764 | 0.7218 | 0.0052 | 0.0390 | 0.7188 | 0.0210 | 0.0802 | 0.2958 | 0.0046 | 0.0355 | 0.2417 | ||

L.CI | 0.4947 | 0.9089 | 3.3078 | 0.2551 | 0.6064 | 3.2410 | 0.5214 | 0.9331 | 2.1326 | 0.2421 | 0.5757 | 1.7552 | ||

BP | 0.0535 | 0.0966 | 0.3440 | 0.0513 | 0.0918 | 0.3253 | 0.0519 | 0.0934 | 0.2157 | 0.0519 | 0.0918 | 0.2084 | ||

BT | 0.0546 | 0.0965 | 0.3365 | 0.0532 | 0.0950 | 0.3210 | 0.0549 | 0.0984 | 0.2157 | 0.0517 | 0.0909 | 0.2107 | ||

80 | Bias | 0.0387 | 0.0954 | −0.0234 | 0.0241 | 0.0830 | 0.1366 | 0.0465 | 0.1116 | −0.0021 | 0.0237 | 0.0923 | 0.1982 | |

MSE | 0.0134 | 0.0506 | 0.4896 | 0.0037 | 0.0223 | 0.4831 | 0.0163 | 0.0563 | 0.2548 | 0.0039 | 0.0254 | 0.1899 | ||

L.CI | 0.4285 | 0.7986 | 2.7428 | 0.2182 | 0.4870 | 2.7030 | 0.4666 | 0.8211 | 1.9796 | 0.2267 | 0.5098 | 1.5222 | ||

BP | 0.0426 | 0.0800 | 0.2800 | 0.0459 | 0.0779 | 0.2862 | 0.0481 | 0.0853 | 0.1949 | 0.0460 | 0.0815 | 0.1995 | ||

BT | 0.0433 | 0.0800 | 0.2832 | 0.0463 | 0.0805 | 0.2855 | 0.0476 | 0.0879 | 0.2008 | 0.0476 | 0.0828 | 0.2008 | ||

90 | Bias | 0.0296 | 0.0632 | −0.0782 | 0.0229 | 0.0678 | 0.1274 | 0.0385 | 0.0740 | −0.0262 | 0.0217 | 0.0688 | 0.1545 | |

MSE | 0.0102 | 0.0344 | 0.4635 | 0.0034 | 0.0168 | 0.4633 | 0.0128 | 0.0355 | 0.2475 | 0.0031 | 0.0167 | 0.1650 | ||

L.CI | 0.3782 | 0.6833 | 2.6523 | 0.2108 | 0.4327 | 2.5930 | 0.4176 | 0.6799 | 1.9486 | 0.2001 | 0.4286 | 1.4733 | ||

BP | 0.0390 | 0.0692 | 0.2689 | 0.0371 | 0.0662 | 0.2660 | 0.0419 | 0.0679 | 0.1867 | 0.0405 | 0.0715 | 0.1886 | ||

BT | 0.0393 | 0.0711 | 0.2756 | 0.0390 | 0.0671 | 0.2712 | 0.0424 | 0.0703 | 0.1943 | 0.0421 | 0.0713 | 0.1940 |

Based on the obtained simulation results, the following conclusions can be drawn:

For fixed values of

For fixed values of

For fixed

In most sets of parameters, for fixed values of

For fixed

In most situations, the measures of Bayesian estimates were better than those of the ML estimates.

ML | Bayesian | ML | Bayesian | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

50 | 35 | Bias | 0.1045 | 0.2351 | −0.0962 | 0.0851 | 0.1866 | 0.5419 | 0.1061 | 0.2383 | 0.0879 | 0.0560 | 0.1870 | 0.3544 |

MSE | 0.0585 | 0.2149 | 1.5883 | 0.6514 | 0.1003 | 1.7584 | 0.0687 | 0.2131 | 0.7568 | 0.0149 | 0.0912 | 0.5757 | ||

L.CI | 0.8556 | 1.5672 | 4.9284 | 3.1477 | 1.0034 | 4.7467 | 0.9399 | 1.5505 | 3.3945 | 0.4246 | 0.9303 | 2.6313 | ||

BP | 0.1248 | 0.2241 | 0.7027 | 0.1156 | 0.2154 | 0.7226 | 0.1274 | 0.2180 | 0.4610 | 0.1321 | 0.2132 | 0.4626 | ||

BT | 0.1378 | 0.2568 | 0.7506 | 0.1318 | 0.2300 | 0.7165 | 0.1373 | 0.2383 | 0.4843 | 0.1462 | 0.2392 | 0.4966 | ||

45 | Bias | 0.0724 | 0.1374 | −0.1044 | 0.0486 | 0.1295 | 0.4656 | 0.0746 | 0.1368 | 0.0337 | 0.0439 | 0.1291 | 0.3008 | |

MSE | 0.0379 | 0.1035 | 1.3028 | 0.0228 | 0.0549 | 1.4474 | 0.0409 | 0.1133 | 0.5681 | 0.0110 | 0.0593 | 0.5028 | ||

L.CI | 0.7090 | 1.1406 | 4.4577 | 0.5609 | 0.7655 | 4.3507 | 0.7372 | 1.2065 | 2.9532 | 0.3728 | 0.8094 | 2.5183 | ||

BP | 0.0985 | 0.1589 | 0.6030 | 0.0998 | 0.1571 | 0.6115 | 0.1025 | 0.1635 | 0.4318 | 0.1068 | 0.1664 | 0.4127 | ||

BT | 0.1060 | 0.1735 | 0.6538 | 0.1111 | 0.1708 | 0.6213 | 0.1107 | 0.1844 | 0.4287 | 0.1132 | 0.1778 | 0.4320 | ||

100 | 70 | Bias | 0.0554 | 0.1271 | −0.1011 | 0.0325 | 0.1068 | 0.3239 | 0.0492 | 0.1168 | 0.0153 | 0.0271 | 0.0989 | 0.2042 |

MSE | 0.0239 | 0.0798 | 0.7203 | 0.0057 | 0.0341 | 0.8516 | 0.0192 | 0.0654 | 0.3318 | 0.0048 | 0.0292 | 0.2323 | ||

L.CI | 0.5664 | 0.9897 | 3.3048 | 0.2662 | 0.5908 | 3.3891 | 0.5073 | 0.8919 | 2.2584 | 0.2487 | 0.5469 | 1.7122 | ||

BP | 0.0558 | 0.1002 | 0.3229 | 0.0543 | 0.0979 | 0.3176 | 0.0516 | 0.0908 | 0.2248 | 0.0512 | 0.0920 | 0.2229 | ||

BT | 0.0598 | 0.1066 | 0.3316 | 0.0594 | 0.1022 | 0.3242 | 0.0528 | 0.0919 | 0.2292 | 0.0543 | 0.0963 | 0.2280 | ||

80 | Bias | 0.0415 | 0.0878 | −0.0895 | 0.0280 | 0.0830 | 0.3100 | 0.0416 | 0.0816 | −0.0205 | 0.0238 | 0.0748 | 0.1725 | |

MSE | 0.0149 | 0.0481 | 0.5109 | 0.0043 | 0.0236 | 0.7193 | 0.0160 | 0.0440 | 0.2872 | 0.0042 | 0.0199 | 0.1962 | ||

L.CI | 0.4510 | 0.7879 | 2.7813 | 0.2331 | 0.5069 | 3.0960 | 0.4678 | 0.7580 | 2.1002 | 0.2350 | 0.4694 | 1.6001 | ||

BP | 0.0455 | 0.0821 | 0.2757 | 0.0456 | 0.0763 | 0.2881 | 0.0454 | 0.0761 | 0.2074 | 0.0453 | 0.0750 | 0.1983 | ||

BT | 0.0485 | 0.0810 | 0.2788 | 0.0476 | 0.0789 | 0.2874 | 0.0471 | 0.0759 | 0.2049 | 0.0474 | 0.0790 | 0.2032 | ||

90 | Bias | 0.0337 | 0.0654 | −0.0493 | 0.0230 | 0.0642 | 0.3268 | 0.0405 | 0.0725 | 0.0076 | 0.0243 | 0.0658 | 0.1825 | |

MSE | 0.0123 | 0.0363 | 0.4597 | 0.0032 | 0.0162 | 0.7179 | 0.0131 | 0.0361 | 0.2438 | 0.0034 | 0.0152 | 0.1957 | ||

L.CI | 0.4153 | 0.7021 | 2.6521 | 0.2043 | 0.4317 | 3.0660 | 0.4194 | 0.6893 | 1.9361 | 0.2074 | 0.4091 | 1.5806 | ||

BP | 0.0401 | 0.0723 | 0.2608 | 0.0402 | 0.0686 | 0.2610 | 0.0399 | 0.0690 | 0.1979 | 0.0406 | 0.0665 | 0.1903 | ||

BT | 0.0422 | 0.0749 | 0.2710 | 0.0428 | 0.0718 | 0.2639 | 0.0418 | 0.0698 | 0.1969 | 0.0421 | 0.0684 | 0.1961 |

ML | Bayesian | ML | Bayesian | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

50 | 35 | Bias | 1.2179 | 0.4533 | 0.0846 | 0.7050 | 0.2937 | 0.2897 | 1.4356 | 0.5339 | −0.1055 | 0.7714 | 0.3238 | 0.5260 |

MSE | 5.6031 | 0.7057 | 0.6773 | 1.5621 | 0.2039 | 0.4696 | 6.3702 | 0.8539 | 1.6049 | 1.8459 | 0.2482 | 1.6412 | ||

L.CI | 7.9605 | 2.7739 | 3.2106 | 4.0475 | 1.3455 | 2.4356 | 8.1415 | 2.9579 | 4.9513 | 4.3864 | 1.4848 | 4.5813 | ||

BP | 1.0753 | 0.3777 | 0.4688 | 1.1813 | 0.4138 | 0.4666 | 1.1691 | 0.4313 | 0.6963 | 1.1846 | 0.4264 | 0.7239 | ||

BT | 1.2008 | 0.4141 | 0.4898 | 1.2680 | 0.4446 | 0.4587 | 1.2989 | 0.4809 | 0.7300 | 1.2855 | 0.4579 | 0.7698 | ||

45 | Bias | 0.7448 | 0.2492 | 0.0743 | 0.7058 | 0.2448 | 0.2233 | 0.8013 | 0.2662 | −0.0751 | 0.5727 | 0.2162 | 0.3990 | |

MSE | 2.8757 | 0.3102 | 0.5028 | 1.7947 | 0.1939 | 0.6439 | 3.3029 | 0.3520 | 1.1263 | 1.0225 | 0.1278 | 1.3917 | ||

L.CI | 5.9748 | 1.9535 | 2.7657 | 4.4656 | 1.4353 | 3.0227 | 6.3976 | 2.0793 | 4.1518 | 3.2684 | 1.1165 | 4.3541 | ||

BP | 0.8531 | 0.2843 | 0.4046 | 0.8129 | 0.2680 | 0.4024 | 0.8832 | 0.2942 | 0.5621 | 0.8853 | 0.2928 | 0.5765 | ||

BT | 0.9565 | 0.3190 | 0.4268 | 0.9027 | 0.2991 | 0.4181 | 1.0657 | 0.3558 | 0.5737 | 1.0282 | 0.3291 | 0.5940 | ||

100 | 70 | Bias | 0.6470 | 0.2326 | −0.0179 | 0.4846 | 0.1994 | 0.1737 | 0.5900 | 0.2089 | −0.0791 | 0.4906 | 0.1943 | 0.2991 |

MSE | 1.6544 | 0.1912 | 0.3188 | 0.7836 | 0.1077 | 0.2602 | 1.4936 | 0.1715 | 0.8438 | 0.8360 | 0.1078 | 0.7666 | ||

L.CI | 4.3598 | 1.4522 | 2.2135 | 2.9054 | 1.0221 | 1.8811 | 4.1977 | 1.4027 | 3.5892 | 3.0262 | 1.0380 | 3.2273 | ||

BP | 0.4084 | 0.1387 | 0.2146 | 0.4362 | 0.1462 | 0.2181 | 0.4045 | 0.1356 | 0.3698 | 0.4234 | 0.1425 | 0.3670 | ||

BT | 0.4297 | 0.1527 | 0.2205 | 0.4614 | 0.1551 | 0.2217 | 0.4186 | 0.1438 | 0.3664 | 0.4439 | 0.1487 | 0.3664 | ||

80 | Bias | 0.4714 | 0.1582 | −0.0047 | 0.4066 | 0.1553 | 0.1321 | 0.5382 | 0.1847 | −0.0593 | 0.4416 | 0.1705 | 0.2739 | |

MSE | 1.1551 | 0.1141 | 0.2768 | 0.5583 | 0.0689 | 0.1606 | 1.3707 | 0.1467 | 0.7824 | 0.6872 | 0.0872 | 0.6455 | ||

L.CI | 3.7880 | 1.1702 | 2.0635 | 2.4586 | 0.8299 | 1.4839 | 4.0778 | 1.3156 | 3.4613 | 2.7515 | 0.9454 | 2.9623 | ||

BP | 0.4120 | 0.1248 | 0.2115 | 0.3610 | 0.1111 | 0.2012 | 0.4050 | 0.1284 | 0.3450 | 0.3951 | 0.1221 | 0.3428 | ||

BT | 0.4214 | 0.1270 | 0.2169 | 0.3768 | 0.1161 | 0.2088 | 0.4261 | 0.1326 | 0.3546 | 0.4107 | 0.1313 | 0.3524 | ||

90 | Bias | 0.3966 | 0.1270 | −0.0065 | 0.3582 | 0.1321 | 0.1175 | 0.3510 | 0.1172 | −0.0282 | 0.3329 | 0.1232 | 0.2536 | |

MSE | 1.0418 | 0.0960 | 0.2337 | 0.5229 | 0.0600 | 0.1375 | 0.8271 | 0.0886 | 0.5825 | 0.4740 | 0.0563 | 0.5787 | ||

L.CI | 3.6885 | 1.1083 | 1.8958 | 2.4636 | 0.8092 | 1.3795 | 3.2906 | 1.0729 | 2.9913 | 2.3635 | 0.7948 | 2.8129 | ||

BP | 0.3579 | 0.1128 | 0.1867 | 0.3353 | 0.1026 | 0.1743 | 0.3382 | 0.1108 | 0.3055 | 0.3263 | 0.1077 | 0.2854 | ||

BT | 0.3806 | 0.1157 | 0.1900 | 0.3650 | 0.1058 | 0.1759 | 0.3584 | 0.1185 | 0.3040 | 0.3520 | 0.1114 | 0.2935 |

ML | Bayesian | ML | Bayesian | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

50 | 35 | Bias | 1.0128 | 0.3503 | 0.0577 | 0.6533 | 0.2534 | 0.2685 | 1.1666 | 0.4031 | −0.0919 | 0.6373 | 0.2523 | 1.0128 |

MSE | 4.3978 | 0.5125 | 0.6634 | 1.3490 | 0.1713 | 0.4901 | 5.3027 | 0.6070 | 1.6960 | 1.2438 | 0.1579 | 4.3978 | ||

L.CI | 7.2019 | 2.4488 | 3.1864 | 3.7663 | 1.2834 | 2.5357 | 7.7865 | 2.6147 | 5.0949 | 3.5897 | 1.2040 | 7.2019 | ||

BP | 0.9604 | 0.3267 | 0.4338 | 1.0201 | 0.3403 | 0.4283 | 1.0927 | 0.3657 | 0.7062 | 1.1202 | 0.3828 | 0.9604 | ||

BT | 1.0822 | 0.3700 | 0.4649 | 1.0934 | 0.3875 | 0.4630 | 1.2372 | 0.4215 | 0.7171 | 1.3084 | 0.4241 | 1.0822 | ||

45 | Bias | 0.6903 | 0.2277 | 0.0882 | 0.6393 | 0.2194 | 0.2590 | 0.8294 | 0.2734 | −0.0967 | 0.6338 | 0.2270 | 0.6903 | |

MSE | 2.7967 | 0.3019 | 0.5431 | 1.6704 | 0.1725 | 0.7200 | 3.2640 | 0.3657 | 1.1745 | 1.4751 | 0.1595 | 2.7967 | ||

L.CI | 5.9741 | 1.9613 | 2.8696 | 4.4053 | 1.3830 | 3.1690 | 6.2948 | 2.1155 | 4.2335 | 4.0634 | 1.2884 | 5.9741 | ||

BP | 0.8230 | 0.2714 | 0.4074 | 0.8494 | 0.2699 | 0.3903 | 0.8798 | 0.2942 | 0.6169 | 0.8707 | 0.2881 | 0.8230 | ||

BT | 0.9137 | 0.3074 | 0.4189 | 0.9569 | 0.3132 | 0.4233 | 0.9962 | 0.3506 | 0.6370 | 0.9763 | 0.3226 | 0.9137 | ||

100 | 70 | Bias | 0.5706 | 0.1912 | −0.0066 | 0.4841 | 0.1830 | 0.1736 | 0.5619 | 0.1885 | −0.1175 | 0.4911 | 0.1850 | 0.5706 |

MSE | 1.5067 | 0.1583 | 0.3454 | 0.8220 | 0.0994 | 0.2654 | 1.6568 | 0.1778 | 0.7943 | 0.7772 | 0.0975 | 1.5067 | ||

L.CI | 4.2624 | 1.3687 | 2.3050 | 3.0066 | 1.0067 | 1.9025 | 4.5418 | 1.4793 | 3.4649 | 2.8716 | 0.9870 | 4.2624 | ||

BP | 0.4258 | 0.1383 | 0.2312 | 0.4354 | 0.1412 | 0.2364 | 0.4200 | 0.1436 | 0.3446 | 0.4762 | 0.1570 | 0.4258 | ||

BT | 0.4726 | 0.1513 | 0.2319 | 0.4631 | 0.1470 | 0.2440 | 0.4807 | 0.1561 | 0.3449 | 0.5112 | 0.1646 | 0.4726 | ||

80 | Bias | 0.4964 | 0.1621 | −0.0086 | 0.4005 | 0.1483 | 0.1423 | 0.4234 | 0.1434 | −0.0314 | 0.4171 | 0.1549 | 0.4964 | |

MSE | 1.2415 | 0.1291 | 0.2751 | 0.5713 | 0.0684 | 0.1589 | 1.0769 | 0.1202 | 0.6752 | 0.5825 | 0.0711 | 1.2415 | ||

L.CI | 3.9123 | 1.2575 | 2.0567 | 2.5139 | 0.8450 | 1.4602 | 3.7157 | 1.2377 | 3.2204 | 2.5066 | 0.8509 | 3.9123 | ||

BP | 0.3966 | 0.1256 | 0.2079 | 0.3765 | 0.1233 | 0.2090 | 0.3612 | 0.1181 | 0.3158 | 0.3910 | 0.1279 | 0.3966 | ||

BT | 0.4095 | 0.1299 | 0.2113 | 0.3981 | 0.1293 | 0.2092 | 0.3902 | 0.1281 | 0.3211 | 0.4113 | 0.1334 | 0.4095 | ||

90 | Bias | 0.4219 | 0.1304 | −0.0107 | 0.3645 | 0.1266 | 0.1284 | 0.3574 | 0.1176 | −0.0630 | 0.3378 | 0.1244 | 0.4219 | |

MSE | 1.0500 | 0.1046 | 0.2377 | 0.5126 | 0.0577 | 0.1604 | 0.8826 | 0.0934 | 0.6297 | 0.4146 | 0.0502 | 1.0500 | ||

L.CI | 3.6623 | 1.1610 | 1.9119 | 2.4169 | 0.8003 | 1.4880 | 3.4075 | 1.1062 | 3.1024 | 2.1500 | 0.7308 | 3.6623 | ||

BP | 0.3730 | 0.1186 | 0.1935 | 0.3590 | 0.1111 | 0.1916 | 0.3468 | 0.1142 | 0.3107 | 0.3419 | 0.1088 | 0.3730 | ||

BT | 0.3915 | 0.1283 | 0.2020 | 0.3766 | 0.1157 | 0.1954 | 0.3495 | 0.1181 | 0.3123 | 0.3649 | 0.1151 | 0.3915 |

In order to further demonstrate the performance of the proposed method, a real data set was used. The R-statistical programming language was used for computation. The dataset was an uncensored dataset consisting of the remission times (in months) of a random observation of 128 bladder cancer patients reported in [

Based on the real data, the SS-PALT of the LBWL distribution under the PTIIC with binomial removals was considered. The ML and Bayesian estimates of parameters and accelerated factor were calculated. In addition, the ACIs for parameters and accelerated factor of the LBWL distribution at a different significant level for

Methods | |||||||||
---|---|---|---|---|---|---|---|---|---|

70 | 0.25 | ML | Estimates | 7.6616 | 24.0890 | 7.9564 | 3.4136 | 7.1470 | 15.6050 |

SE | 6.3438 | 24.3121 | 3.2046 | 1.1863 | 4.1016 | 7.2071 | |||

Bayesian | Estimates | 11.6077 | 39.4794 | 8.0091 | 3.9091 | 9.1546 | 17.1497 | ||

SE | 4.7009 | 17.9081 | 3.1449 | 1.2401 | 4.5333 | 8.2739 | |||

0.5 | ML | Estimates | 3.0285 | 6.3043 | 12.2879 | 3.0321 | 6.3173 | 22.9839 | |

SE | 0.8693 | 3.2489 | 5.0929 | 0.8708 | 3.2558 | 9.5087 | |||

Bayesian | Estimates | 3.0573 | 6.5321 | 13.7553 | 3.2284 | 7.2684 | 25.0118 | ||

SE | 0.5837 | 2.1964 | 5.0300 | 0.6934 | 2.7585 | 8.9943 | |||

0.75 | ML | Estimates | 3.5528 | 7.7159 | 8.5280 | 3.5571 | 7.7294 | 15.9533 | |

SE | 1.2798 | 4.4997 | 4.0122 | 1.2818 | 4.5065 | 7.4926 | |||

Bayesian | Estimates | 3.8300 | 9.0410 | 10.3091 | 3.6750 | 8.4149 | 19.1855 | ||

SE | 0.9736 | 3.6979 | 4.0303 | 0.9067 | 3.4011 | 7.4354 | |||

100 | 0.25 | ML | Estimates | 3.7854 | 9.0819 | 9.9502 | 3.7825 | 9.0720 | 18.6744 |

SE | 1.2215 | 4.5667 | 3.5475 | 1.2194 | 4.5592 | 6.6596 | |||

Bayesian | Estimates | 4.1864 | 10.9759 | 10.8710 | 4.2122 | 10.8904 | 18.7347 | ||

SE | 1.0984 | 4.1604 | 3.5064 | 0.9592 | 3.7099 | 5.4908 | |||

0.5 | ML | Estimates | 4.0116 | 9.6960 | 8.5389 | 4.0172 | 9.7184 | 15.9979 | |

SE | 1.3900 | 5.1062 | 3.0495 | 1.3972 | 5.1340 | 5.7210 | |||

Bayesian | Estimates | 4.8383 | 13.1819 | 8.5757 | 4.4484 | 11.5132 | 16.4234 | ||

SE | 1.5162 | 5.0121 | 2.4670 | 1.2147 | 4.8149 | 4.5385 | |||

0.75 | ML | Estimates | 4.1199 | 10.0591 | 7.0695 | 4.1182 | 10.0534 | 13.2684 | |

SE | 1.5367 | 5.6271 | 2.6016 | 1.5377 | 5.6304 | 4.8911 | |||

Bayesian | Estimates | 4.3458 | 11.1512 | 7.9142 | 4.9837 | 13.5235 | 13.1067 | ||

SE | 1.2083 | 4.7402 | 2.4686 | 1.3139 | 5.0768 | 4.2847 |

The history plots, approximate marginal posterior density, and MCMC convergence of

The ACIs for parameters of the LBWL distribution based on the SS-PALT under the PTIIC data with binomial removals at a different level of significance for

In this paper, the Bayesian and maximum likelihood estimation methods for the LBWL distribution are discussed based on SS-PALT using the PTIIC data with binomial removals. The approximate confidence intervals of the ML estimators of the model parameters are assessed based on the Fisher information matrix. In addition, the percentile bootstrap and bootstrap-t confidence intervals are determined. Moreover, the effects of sample size

The simulation results show that, for fixed values of

As a future work, this study can be extended to explore the situation under type-I progressive censoring. Evaluation of the coverage probabilities can also be computed rather than the lengths of the CIs.

This project was funded by the Deanship of Scientific Research (DSR), at King Abdulaziz University, Jeddah, under Grant No. FP-190-42. The authors, therefore, thank DSR’s technical and financial support.