# Classical and Bayesian Inference of a Progressive-Stress Model for the Nadarajah–Haghighi Distribution with Type II Progressive Censoring and Different Loss Functions

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## Abstract

**:**

## 1. Introduction

## 2. A Cumulative Exposure Model of NH Distribution

- -
- Under usual conditions, the lifetime of a unit follows NH($\theta ,\lambda $).
- -
- The progressive-stress $\phi $(t) is directly proportional to the time t with constant rate $\beta $, i.e., $\phi \left(t\right)=\beta t,\beta 0$.
- -
- The scale parameter $\lambda $ of the CDF in (2) satisfies the inverse power law, as follows$$\lambda \left(t\right)=\frac{1}{a{\left[\phi \left(t\right)\right]}^{b}}$$
- -
- It is assumed that a and b are unknown physical positive parameters and need to be estimated.
- -
- Assume n is the total number of units tested, ${\phi}_{0}<{\phi}_{1}\left(t\right)<\dots <{\phi}_{k}\left(t\right)$ are the stress levels in the test, and ${\phi}_{0}$ is the use stress. Under each progressive-stress level, identical units ${\phi}_{i}\left(t\right)={\beta}_{i}t,i=1,2,\dots ,k,{n}_{i}$ are tested, and the progressive type II censoring is performed as follows: When the first failure ${t}_{i1:{m}_{i}:{n}_{i}}$ occurs, ${R}_{i1}$ units are picked at random from the remaining ${n}_{1}-1$ surviving units. When the second failure ${t}_{i2:{m}_{i}:{n}_{i}}$ occurs, ${R}_{i2}$ items from the remaining ${n}_{i}-2-{R}_{i1}$ units are withdrawn at random. When the ${m}_{i}-th$ failure occurs, ${t}_{i{m}_{i}:{m}_{i}:{n}_{i}}$, the test is terminated, and all remaining ${R}_{i{m}_{i}:{m}_{i}:{n}_{i}}={n}_{i}-{m}_{i}-{\displaystyle \sum}_{j=1}^{{m}_{i}-1}{R}_{i{m}_{i}}$ items are removed.
- -
- The complete samples and type II censored samples are clearly specific examples of this technique. Under the progressive-stress ${\phi}_{i}\left(t\right)$, the observed progressive-censoring data are ${t}_{i1:{m}_{i}:{n}_{i}}$ < ${t}_{i2:{m}_{i}:{n}_{i}}\xb7\xb7\xb7{t}_{i{m}_{i}:{m}_{i}:{n}_{i}}$, $i=1,2,\dots ,k$.
- -
- The linear cumulative exposure model (CEM) accounts for the effect of changing stress; for more details, see [13].
- -
- The CDF in progressive stress, ${\phi}_{i}\left(t\right)$, and the linear cumulative exposure model is given as follows$${G}_{i}\left(t\right)={F}_{i}\left(\u2206t\right),i=1,2,\dots ,k,$$$${G}_{i}\left(t\right)=1-{e}^{\left(1-{\left(1+\frac{a{\beta}_{i}^{b}{t}^{b+1}}{b+1}\right)}^{\theta}\right)},t0,a,b,\theta 0,i=1,2,\dots ,k.$$The corresponding PDF is given by$${g}_{i}\left(t\right)=a\theta {\beta}_{i}^{b}{t}^{b}{\left(1+\frac{a{\beta}_{i}^{b}{t}^{b+1}}{b+1}\right)}^{\theta -1}{e}^{\left[1-{\left(1+\frac{a{\beta}_{i}^{b}{t}^{b+1}}{b+1}\right)}^{\theta}\right]},$$

## 3. Maximum Likelihood Estimation

## 4. Bayesian Estimation

#### Bayes Estimation Using BSEL and BLINEX Loss Functions

- Step 1:
- For the parameters ($a,b$, $\theta $), set the initial guess to (${a}^{0},{b}^{0}$, ${\theta}^{0}$).
- Step 2:
- Set j = 1.
- Step 3:
- Create $a~N\left({a}^{j},{\sigma}_{11}\right),b~N\left({b}^{j},{\sigma}_{22}\right)$ and $\theta ~N\left({\theta}^{j},{\sigma}_{33}\right),$ where $\sigma $ is the variance–covariance matrix.
- Step 4:
- Compute $p=\frac{\pi \left({a}^{j},{b}^{j},{\theta}^{j}|\underset{\_}{x}\right)}{\pi \left({a}^{j-1},{b}^{j-1},{\theta}^{j-1}|\underset{\_}{x}\right)}$.
- Step 5:
- With probability $min\left(1,p\right),$ accept (${a}_{j},{b}_{j}$, ${\theta}_{j}$),
- Step 6:
- To obtain B number of samples for the parameters ($a,b$, $\theta $), repeat steps (3) to (5) B times.

## 5. Interval Estimation

#### 5.1. Asymptotic Confidence Interval

#### 5.2. HPD Interval of Credibility

#### 5.3. Bootstrap Confidence Intervals

- (1)
- Calculate the MLE values of the parameters using the original data $\theta ,a$, and $b$.
- (2)
- To make a bootstrap sample ${t}^{*},$ use the variables $\widehat{\theta}$, $\widehat{a},$ and $\widehat{b}$.
- (3)
- The bootstrap estimates ${\widehat{\theta}}^{*}$, ${\widehat{a}}^{*}$, and ${\widehat{b}}^{*}$, respectively, are obtained based on ${t}^{*}$.
- (4)
- To obtain the bootstrap samples, repeat steps 1–3 several times and organize each estimate in ascending order$$\left\{{\widehat{\theta}}^{*\left[1\right]},{\widehat{\theta}}^{*\left[2\right]},\dots ,{\widehat{\theta}}^{*\left[I\right]}\right\},\left\{{\widehat{a}}^{*\left[1\right]},{\widehat{a}}^{*\left[2\right]},\dots ,{\widehat{a}}^{*\left[I\right]}\right\}\mathrm{and}\left\{{\widehat{b}}^{*\left[1\right]},{\widehat{b}}^{*\left[2\right]},\dots ,{b}^{*\left[I\right]}\right\},$$The $100\left(1-\delta \right)\%$ percentile bootstrap CIs for $\omega $ are then calculated as follows:$$\left({\widehat{\omega}}_{iL},{\widehat{\omega}}_{iU}\right)=\left({\widehat{\omega}}_{i}{}^{*\left[\frac{\delta}{2}I\right]},{\widehat{\omega}}_{i}{}^{*\left[\left(1-\frac{\delta}{2}\right)I\right]}\right),i=1,2,3,$$

## 6. Simulation Study

**Step****1:**- Using the algorithm presented in [10], $k\ge 2$ progressively type II censored random samples are generated from the uniform (0,1) distribution $\left({U}_{i1},{U}_{i2},\dots ,{U}_{i{m}_{i,}}\right)$, for given values of ${m}_{i,},$ $i=1,2,\dots ,k$.
**Step****2:**- To compare the performance of the estimation procedures developed in the study, we consider the following two schemes for each stress:Scheme 1: ${R}_{i{m}_{i}}={n}_{i}-{m}_{i}$, ${R}_{ij}=0;j=1,\dots ,{m}_{i}-1$.Scheme 2: ${R}_{i1}={n}_{i}-{m}_{i}$, ${R}_{ij}=0;j=1,\dots ,{m}_{i}$.
**Step****3:**- Progressively type II censored random samples $\left({t}_{i1},\dots ,{t}_{i{m}_{i}}\right)$ are produced, and from inverse CDF (3), we specify the values of parameters as follows:In Table 1 $\left(\theta =1.7,a=1.3,b=2\right),k=2,\mathrm{and}{\beta}_{1}=40,{\beta}_{2}=80$.In Table 4 ($\theta =1.7,a=1.3,b=2),k=4,$ and ${\beta}_{1}=40,{\beta}_{2}=80,{\beta}_{3}=110,{\beta}_{4}=150$.In Table 5 ($\theta =0.8,a=0.5,b=1.3$), $k=4$, and ${\beta}_{1}=40,{\beta}_{2}=80,{\beta}_{3}=110,{\beta}_{4}=150$.In Table 6 ($\theta =3,a=2.5,b=0.6)$, $k=4$, and ${\beta}_{1}=40,{\beta}_{2}=80,{\beta}_{3}=110,{\beta}_{4}=150$.
**Step****4:**- The MLEs $\left(\widehat{\theta},\widehat{a},\widehat{b}\right)$ are obtained numerically by solving the likelihood equations with respect to ($\theta ,a,b$) in (8)–(10) by using an iterative Newton–Raphson algorithm using the maxlik function of the “maxlik” package in the R program; for more information in this topic see [30].
**Step****5:**- Based on (15)–(17), and the MH algorithm, the Bayesian estimations with the BSEL and BLINEX loss functions of the parameters ($\theta ,a,b$) are computed by (13) and (14), respectively.
**Step****6:**- The above steps are repeated I times based on I different samples, and then the average of likelihood and Bayesian estimations are computed, with their MSE, bias, and length of confidence intervals (LCI) of the parameters ($\theta ,a,b$).
**Step****7:**- In length of CI (LCI) of the MLE of each parameter, we compute the ACI for likelihood estimators and bootstrap CIs with the percentile algorithm and t algorithm, which can be denoted as LBP and LBT, respectively. In the LCI of Bayesian estimation, we compute the HPD for each loss function, denoted by the LCCI.

**Table 1.**MLE and Bayesian estimation with different loss functions in the simple ramp when θ = 1.7, a = 1.3, b = 2.

k = 2 | θ = 1.7, a = 1.3, b = 2 | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

n_{1}, n_{2} | Scheme | r | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | LCI | LBP | LBP | LCCI | LCCI | LCCI | |

20, 15 | 1 | 65% | θ | −0.6751 | 1.5008 | −0.1938 | 0.0886 | −0.1172 | 0.0801 | −0.2619 | 0.1099 | 4.0094 | 0.1331 | 0.1271 | 0.8592 | 0.9821 | 0.7748 |

a | 0.3824 | 1.5335 | −0.0521 | 0.0788 | 0.0209 | 0.0994 | −0.1174 | 0.0740 | 4.6193 | 0.1432 | 0.1453 | 1.0048 | 1.1145 | 0.8922 | |||

b | 0.0673 | 0.6760 | −0.2972 | 0.1145 | −0.2616 | 0.0964 | −0.3314 | 0.1345 | 3.2139 | 0.0998 | 0.0993 | 0.6315 | 0.6551 | 0.6181 | |||

85% | θ | −0.4879 | 0.7308 | −0.0879 | 0.0657 | −0.0097 | 0.0736 | −0.1581 | 0.0724 | 2.7529 | 0.0899 | 0.0894 | 0.9552 | 1.0524 | 0.8647 | ||

a | 0.5008 | 0.9984 | 0.0504 | 0.0612 | 0.0195 | 0.0890 | 0.0426 | 0.0691 | 3.3911 | 0.1105 | 0.1101 | 0.9821 | 0.9834 | 0.9084 | |||

b | 0.0625 | 0.6469 | −0.1622 | 0.0558 | −0.1260 | 0.0471 | −0.1967 | 0.0667 | 3.1026 | 0.0978 | 0.0965 | 0.6718 | 0.6934 | 0.6551 | |||

2 | 65% | θ | 0.0872 | 0.5633 | 0.0550 | 0.0583 | 0.1391 | 0.0909 | −0.0206 | 0.0450 | 2.9235 | 0.0894 | 0.0875 | 0.9262 | 1.0470 | 0.8281 | |

a | −0.2725 | 0.5104 | 0.0517 | 0.1058 | 0.1262 | 0.1418 | −0.0158 | 0.0866 | 2.5902 | 0.0805 | 0.0818 | 1.2014 | 1.3056 | 1.0823 | |||

b | 0.5035 | 0.8594 | 0.0607 | 0.0411 | 0.1011 | 0.0500 | 0.0218 | 0.0360 | 3.0529 | 0.0908 | 0.0915 | 0.7305 | 0.7490 | 0.7048 | |||

85% | θ | 0.0823 | 0.5078 | 0.0542 | 0.0580 | 0.1346 | 0.0879 | −0.0171 | 0.0448 | 2.8345 | 0.0804 | 0.0803 | 0.9137 | 0.9103 | 0.8448 | ||

a | −0.1801 | 0.4902 | 0.0863 | 0.1041 | 0.1611 | 0.1509 | 0.0185 | 0.0778 | 2.6534 | 0.0858 | 0.0860 | 1.0977 | 1.2237 | 1.0492 | |||

b | 0.3257 | 0.5593 | 0.0494 | 0.0363 | 0.0879 | 0.0437 | 0.0126 | 0.0323 | 2.6404 | 0.0854 | 0.0845 | 0.7149 | 0.7384 | 0.6916 | |||

40, 50 | 1 | 65% | θ | −0.9678 | 1.4864 | −0.3139 | 0.1585 | −0.2489 | 0.1345 | −0.3726 | 0.1900 | 3.7765 | 0.1250 | 0.1174 | 0.9207 | 1.0006 | 0.8607 |

a | 0.3126 | 1.0050 | 0.1260 | 0.1244 | 0.1974 | 0.1754 | 0.0601 | 0.0913 | 4.4408 | 0.1433 | 0.1373 | 1.2116 | 1.3391 | 1.0992 | |||

b | −0.0262 | 0.5537 | −0.4534 | 0.2334 | −0.4262 | 0.2114 | −0.4791 | 0.2558 | 2.9167 | 0.0938 | 0.0929 | 0.6567 | 0.6774 | 0.6325 | |||

85% | θ | −0.9531 | 1.0410 | −0.2143 | 0.1193 | −0.1488 | 0.1099 | −0.2738 | 0.1380 | 1.4282 | 0.0451 | 0.0455 | 1.0419 | 1.1471 | 0.9624 | ||

a | 0.3033 | 0.9427 | 0.1230 | 0.1202 | 0.1374 | 0.1701 | 0.0582 | 0.0902 | 3.1609 | 0.1034 | 0.1027 | 1.2028 | 1.1908 | 1.1697 | |||

b | 0.2468 | 0.3993 | −0.2597 | 0.1017 | −0.2310 | 0.0904 | −0.2869 | 0.1145 | 2.2816 | 0.0720 | 0.0720 | 0.7346 | 0.7532 | 0.7076 | |||

2 | 65% | θ | 0.0189 | 0.5195 | 0.0350 | 0.0660 | 0.1067 | 0.0902 | −0.0307 | 0.0557 | 2.8259 | 0.0875 | 0.0878 | 0.9662 | 1.0601 | 0.9024 | |

a | −0.1043 | 0.3458 | 0.0534 | 0.1026 | 0.1154 | 0.1347 | −0.0034 | 0.0834 | 2.2699 | 0.0749 | 0.0750 | 1.1326 | 1.2150 | 1.0415 | |||

b | 0.2988 | 0.3316 | 0.0806 | 0.0416 | 0.1104 | 0.0489 | 0.0519 | 0.0365 | 1.9309 | 0.0610 | 0.0610 | 0.7083 | 0.7181 | 0.6939 | |||

85% | θ | 0.0199 | 0.2902 | 0.0345 | 0.0627 | 0.1023 | 0.0901 | −0.0155 | 0.0547 | 2.1112 | 0.0691 | 0.0689 | 0.9521 | 0.9602 | 0.9010 | ||

a | −0.0041 | 0.2644 | 0.0499 | 0.1013 | 0.1106 | 0.1325 | 0.0046 | 0.0829 | 2.0168 | 0.0606 | 0.0617 | 1.1228 | 1.1344 | 1.1043 | |||

b | 0.1356 | 0.1506 | 0.0437 | 0.0332 | 0.0711 | 0.0381 | 0.0173 | 0.0302 | 1.4262 | 0.0475 | 0.0476 | 0.6648 | 0.6811 | 0.6482 |

**Table 2.**MLE and Bayesian estimation with different loss functions in the simple ramp when θ = 0.8, a = 0.5, b = 1.3.

k = 2 | θ = 0.8, a = 0.5, b = 1.3 | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

n_{1}, n_{2} | Scheme | r | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | LCI | LBP | LBP | LCCI | LCCI | LCCI | |

20, 15 | 1 | 65% | θ | −0.6718 | 2.8664 | −0.1568 | 0.1203 | −0.1274 | 0.1278 | −0.1845 | 0.1151 | 6.0949 | 0.2060 | 0.1973 | 0.9950 | 1.0839 | 0.9307 |

a | 0.2440 | 1.6298 | 0.1466 | 0.1261 | 0.1736 | 0.1496 | 0.1202 | 0.1050 | 4.9146 | 0.1549 | 0.1554 | 1.0924 | 1.1438 | 1.0436 | |||

b | 0.0957 | 0.4839 | −0.0738 | 0.0664 | −0.0268 | 0.0708 | −0.1174 | 0.0675 | 2.7025 | 0.0858 | 0.0848 | 0.9578 | 1.0205 | 0.8990 | |||

85% | θ | −0.2515 | 0.1789 | −0.0586 | 0.1017 | −0.0262 | 0.1189 | −0.0889 | 0.0893 | 1.3336 | 0.0416 | 0.0414 | 1.0767 | 1.1327 | 0.9832 | ||

a | 0.1999 | 0.2632 | 0.1647 | 0.1249 | 0.1903 | 0.1468 | 0.1398 | 0.1054 | 1.8532 | 0.0598 | 0.0607 | 1.1127 | 1.1571 | 1.0571 | |||

b | 0.2494 | 0.3102 | 0.0044 | 0.0617 | 0.0480 | 0.0721 | −0.0363 | 0.0566 | 1.9532 | 0.0594 | 0.0579 | 0.9574 | 1.0115 | 0.8972 | |||

2 | 65% | θ | 0.1997 | 0.2701 | 0.1153 | 0.1081 | 0.1526 | 0.1338 | 0.0801 | 0.0876 | 1.8819 | 0.0604 | 0.0602 | 1.0837 | 1.1369 | 0.9657 | |

a | −0.1490 | 0.1301 | 0.0586 | 0.0855 | 0.0816 | 0.0993 | 0.0365 | 0.0737 | 1.2884 | 0.0409 | 0.0399 | 1.0231 | 1.0804 | 0.9543 | |||

b | 0.2680 | 0.2961 | 0.0933 | 0.0665 | 0.1342 | 0.0823 | 0.0551 | 0.0557 | 1.8574 | 0.0593 | 0.0599 | 0.9271 | 0.9837 | 0.8748 | |||

85% | θ | 0.1673 | 0.2162 | 0.0845 | 0.0495 | 0.1121 | 0.0621 | 0.0586 | 0.0400 | 1.7014 | 0.0522 | 0.0516 | 0.7796 | 0.8472 | 0.7347 | ||

a | −0.0699 | 0.0976 | 0.0664 | 0.0581 | 0.0850 | 0.0669 | 0.0486 | 0.0508 | 1.1941 | 0.0379 | 0.0380 | 0.8233 | 0.8664 | 0.7746 | |||

b | 0.1521 | 0.2192 | 0.0372 | 0.0230 | 0.0627 | 0.0273 | 0.0130 | 0.0206 | 1.7367 | 0.0583 | 0.0588 | 0.5765 | 0.6031 | 0.5623 | |||

40, 50 | 1 | 65% | θ | −0.6809 | 3.2570 | −0.3695 | 0.1771 | −0.3528 | 0.1723 | −0.3849 | 0.1822 | 6.5550 | 0.2254 | 0.2022 | 0.6022 | 0.6613 | 0.5605 |

a | 0.6260 | 1.1068 | 0.3023 | 0.1956 | 0.3278 | 0.2238 | 0.2769 | 0.1692 | 3.3163 | 0.1154 | 0.1066 | 1.1525 | 1.2097 | 1.0901 | |||

b | 0.2043 | 0.1752 | 0.0435 | 0.0965 | 0.0945 | 0.1144 | −0.0056 | 0.0849 | 1.4326 | 0.0473 | 0.0469 | 1.1800 | 1.2411 | 1.1403 | |||

85% | θ | −0.4415 | 0.2093 | −0.2621 | 0.0908 | −0.2500 | 0.0879 | −0.2733 | 0.0942 | 0.4708 | 0.0153 | 0.0151 | 0.5155 | 0.5453 | 0.4948 | ||

a | 0.4369 | 0.3465 | 0.2701 | 0.1401 | 0.2903 | 0.1585 | 0.2502 | 0.1231 | 1.5470 | 0.0474 | 0.0479 | 0.9661 | 1.0058 | 0.9208 | |||

b | 0.2543 | 0.1467 | 0.0283 | 0.0340 | 0.0554 | 0.0390 | 0.0022 | 0.0311 | 1.1229 | 0.0349 | 0.0349 | 0.7075 | 0.7235 | 0.6965 | |||

2 | 65% | θ | 0.0872 | 0.1598 | 0.1037 | 0.0866 | 0.1337 | 0.1045 | 0.0753 | 0.0720 | 1.5300 | 0.0468 | 0.0472 | 0.9517 | 1.0225 | 0.9087 | |

a | −0.1012 | 0.0663 | 0.0297 | 0.0519 | 0.0452 | 0.0580 | 0.0149 | 0.0467 | 0.9282 | 0.0277 | 0.0276 | 0.8375 | 0.8677 | 0.7991 | |||

b | 0.1905 | 0.2101 | 0.0796 | 0.0639 | 0.1103 | 0.0757 | 0.0506 | 0.0552 | 1.6350 | 0.0535 | 0.0526 | 0.8925 | 0.9435 | 0.8638 | |||

85% | θ | 0.0839 | 0.1248 | 0.0757 | 0.0492 | 0.0973 | 0.0593 | 0.0550 | 0.0409 | 1.3462 | 0.0423 | 0.0426 | 0.7645 | 0.8162 | 0.7216 | ||

a | −0.0484 | 0.0473 | 0.0579 | 0.0489 | 0.0717 | 0.0550 | 0.0446 | 0.0435 | 0.8311 | 0.0265 | 0.0264 | 0.7049 | 0.7358 | 0.6820 | |||

b | 0.0961 | 0.1294 | 0.0243 | 0.0259 | 0.0433 | 0.0286 | 0.0059 | 0.0241 | 1.3594 | 0.0432 | 0.0432 | 0.6307 | 0.6517 | 0.6262 |

**Table 3.**MLE and Bayesian estimation with different loss functions in the simple ramp when θ = 3, a = 2.5, b = 0.6.

k = 2 | θ = 3, a = 0.5, b = 0.6 | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

n_{1}, n_{2} | Scheme | r | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | LCI | LBP | LBP | LCCI | LCCI | LCCI | |

20, 15 | 1 | 65% | θ | −1.0081 | 1.4542 | −0.3499 | 0.1451 | −0.2693 | 0.0983 | −0.4242 | 0.2008 | 2.5956 | 0.0851 | 0.0847 | 0.5826 | 0.6182 | 0.5601 |

a | 0.0679 | 1.2060 | −0.3724 | 0.1673 | −0.2938 | 0.1205 | −0.4438 | 0.2221 | 4.2987 | 0.1386 | 0.1410 | 0.6487 | 0.7208 | 0.6095 | |||

b | −0.1176 | 0.1318 | −0.2613 | 0.0778 | −0.2521 | 0.0737 | −0.2702 | 0.0819 | 1.3473 | 0.0437 | 0.0433 | 0.3603 | 0.3730 | 0.3459 | |||

85% | θ | −0.4454 | 0.4194 | −0.1202 | 0.0252 | −0.0733 | 0.0168 | −0.1652 | 0.0375 | 1.8440 | 0.0591 | 0.0591 | 0.3991 | 0.4098 | 0.3920 | ||

a | −0.0649 | 0.2608 | −0.1168 | 0.0311 | −0.0670 | 0.0235 | −0.1639 | 0.0433 | 1.9865 | 0.0654 | 0.0637 | 0.5281 | 0.5430 | 0.5133 | |||

b | −0.0581 | 0.0912 | −0.1632 | 0.0352 | −0.1555 | 0.0331 | −0.1707 | 0.0374 | 1.1620 | 0.0367 | 0.0374 | 0.3568 | 0.3624 | 0.3499 | |||

2 | 65% | θ | −0.1533 | 0.4265 | −0.0128 | 0.0410 | 0.0808 | 0.0511 | −0.0995 | 0.0484 | 2.4897 | 0.0734 | 0.0726 | 0.8016 | 0.8379 | 0.7820 | |

a | −0.2400 | 0.6128 | −0.0100 | 0.0597 | 0.0877 | 0.0755 | −0.0987 | 0.0639 | 2.9224 | 0.0901 | 0.0898 | 0.9570 | 1.0232 | 0.9205 | |||

b | 0.2240 | 0.1830 | 0.0290 | 0.0270 | 0.0438 | 0.0293 | 0.0145 | 0.0253 | 1.4291 | 0.0427 | 0.0426 | 0.6181 | 0.6302 | 0.6014 | |||

85% | θ | −0.0076 | 0.2889 | 0.0133 | 0.0152 | 0.0632 | 0.0198 | −0.0346 | 0.0158 | 2.1077 | 0.0662 | 0.0651 | 0.4780 | 0.4812 | 0.4698 | ||

a | −0.1365 | 0.3228 | 0.0253 | 0.0269 | 0.0803 | 0.0347 | −0.0268 | 0.0255 | 2.1629 | 0.0679 | 0.0684 | 0.6379 | 0.6699 | 0.6282 | |||

b | 0.1288 | 0.1135 | 0.0058 | 0.0144 | 0.0155 | 0.0150 | −0.0039 | 0.0140 | 1.2206 | 0.0386 | 0.0384 | 0.4462 | 0.4534 | 0.4430 | |||

40, 50 | 1 | 65% | θ | −1.1973 | 2.0957 | −0.4388 | 0.2123 | −0.3633 | 0.1551 | −0.5082 | 0.2761 | 3.1917 | 0.1021 | 0.0996 | 0.5524 | 0.5850 | 0.5390 |

a | 0.4242 | 1.8779 | −0.4240 | 0.2012 | −0.3566 | 0.1528 | −0.4852 | 0.2543 | 5.1106 | 0.1639 | 0.1643 | 0.5646 | 0.6197 | 0.5296 | |||

b | −0.3040 | 0.1460 | −0.3420 | 0.1222 | −0.3367 | 0.1188 | −0.3471 | 0.1255 | 0.9077 | 0.0310 | 0.0305 | 0.2663 | 0.2715 | 0.2631 | |||

85% | θ | −1.1130 | 1.6602 | −0.1645 | 0.0369 | −0.1219 | 0.0252 | −0.2053 | 0.0517 | 2.5459 | 0.0791 | 0.0792 | 0.3912 | 0.3950 | 0.3890 | ||

a | 0.3356 | 0.3522 | −0.1170 | 0.0290 | −0.0745 | 0.0218 | −0.1575 | 0.0393 | 4.9114 | 0.1509 | 0.1504 | 0.4752 | 0.4874 | 0.4623 | |||

b | −0.1376 | 0.0633 | −0.2416 | 0.0645 | −0.2372 | 0.0625 | −0.2459 | 0.0665 | 0.8264 | 0.0268 | 0.0265 | 0.2928 | 0.2964 | 0.2898 | |||

2 | 65% | θ | −0.0736 | 0.3191 | −0.0093 | 0.0421 | 0.0786 | 0.0525 | −0.0899 | 0.0477 | 2.1964 | 0.0688 | 0.0681 | 0.8118 | 0.8396 | 0.7883 | |

a | −0.0731 | 0.2996 | 0.0195 | 0.0535 | 0.1048 | 0.0712 | −0.0585 | 0.0517 | 2.1275 | 0.0674 | 0.0659 | 0.8891 | 0.9502 | 0.8511 | |||

b | 0.1074 | 0.0593 | 0.0351 | 0.0180 | 0.0438 | 0.0190 | 0.0264 | 0.0171 | 0.8576 | 0.0288 | 0.0285 | 0.4791 | 0.4815 | 0.4721 | |||

85% | θ | −0.0128 | 0.2589 | 0.0097 | 0.0167 | 0.0557 | 0.0204 | −0.0344 | 0.0173 | 1.3010 | 0.0590 | 0.0589 | 0.5014 | 0.5122 | 0.4937 | ||

a | 0.0694 | 0.2707 | 0.0260 | 0.0300 | 0.0731 | 0.0363 | −0.0188 | 0.0284 | 1.3429 | 0.0612 | 0.0612 | 0.6600 | 0.6807 | 0.6407 | |||

b | 0.0647 | 0.0409 | 0.0031 | 0.0099 | 0.0090 | 0.0101 | −0.0027 | 0.0097 | 0.7513 | 0.0240 | 0.0241 | 0.3942 | 0.3955 | 0.3921 |

**Table 4.**MLE and Bayesian estimation with different loss functions in multi ramp when θ = 1.7, a = 1.3, b = 2.

k = 4 | θ = 1.7, a = 1.3, b = 2 | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

n_{1}, n_{2},n _{3}, n_{4} | Scheme | r | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | LCI | LBP | LBP | LCCI | LCCI | LCCI | |

20, 15, 18, 10 | 1 | 65% | θ | −0.8715 | 1.0614 | −0.2242 | 0.1068 | −0.1519 | 0.0943 | −0.2893 | 0.1302 | 2.1561 | 0.0965 | 0.0965 | 0.8956 | 0.9632 | 0.8047 |

a | 0.8096 | 1.4846 | 0.0001 | 0.0882 | 0.0688 | 0.1142 | −0.0625 | 0.0765 | 3.7822 | 0.1691 | 0.1694 | 1.1078 | 1.2537 | 0.9950 | |||

b | 0.1157 | 0.3767 | −0.3089 | 0.1190 | −0.2801 | 0.1037 | −0.3365 | 0.1355 | 2.3651 | 0.1070 | 0.1066 | 0.6109 | 0.6291 | 0.5869 | |||

85% | θ | −0.5590 | 0.7077 | −0.1089 | 0.0307 | −0.0677 | 0.0259 | −0.1476 | 0.0388 | 2.4667 | 0.1144 | 0.1135 | 0.5240 | 0.5597 | 0.4972 | ||

a | 0.6354 | 0.9014 | 0.0470 | 0.0402 | 0.0905 | 0.0539 | 0.0062 | 0.0323 | 3.0655 | 0.1441 | 0.1385 | 0.6795 | 0.7253 | 0.6408 | |||

b | 0.0962 | 0.3299 | −0.1872 | 0.0476 | −0.1704 | 0.0419 | −0.2036 | 0.0538 | 2.2221 | 0.1003 | 0.1017 | 0.4437 | 0.4436 | 0.4419 | |||

2 | 65% | θ | 0.1463 | 0.4847 | 0.0373 | 0.0756 | 0.1170 | 0.1094 | −0.0345 | 0.0617 | 2.9452 | 0.1315 | 0.1295 | 0.9935 | 1.0840 | 0.8752 | |

a | −0.3137 | 0.3755 | 0.0228 | 0.1139 | 0.0899 | 0.1463 | −0.0388 | 0.0959 | 2.0658 | 0.0946 | 0.0937 | 1.2244 | 1.3813 | 1.1134 | |||

b | 0.3591 | 0.4138 | 0.0913 | 0.0437 | 0.1225 | 0.0519 | 0.0613 | 0.0379 | 2.0944 | 0.0977 | 0.0973 | 0.7276 | 0.7407 | 0.7164 | |||

85% | θ | 0.1331 | 0.3552 | 0.0306 | 0.0226 | 0.0738 | 0.0306 | −0.0100 | 0.0192 | 2.8690 | 0.1216 | 0.1226 | 0.5557 | 0.6039 | 0.5304 | ||

a | −0.1476 | 0.3143 | 0.0460 | 0.0416 | 0.0882 | 0.0533 | 0.0067 | 0.0348 | 2.1222 | 0.1006 | 0.0992 | 0.7528 | 0.8065 | 0.7120 | |||

b | 0.2191 | 0.2586 | 0.0238 | 0.0139 | 0.0419 | 0.0154 | 0.0060 | 0.0132 | 1.8008 | 0.0761 | 0.0764 | 0.4427 | 0.4514 | 0.4406 | |||

25, 20, 20, 25 | 1 | 65% | θ | −1.0371 | 1.2431 | −0.2741 | 0.1331 | −0.2060 | 0.1158 | −0.3356 | 0.1606 | 1.6059 | 0.0765 | 0.0765 | 0.9131 | 1.0396 | 0.8033 |

a | 0.9794 | 0.9451 | 0.0684 | 0.1174 | 0.1367 | 0.1581 | 0.0052 | 0.0917 | 2.7529 | 0.1255 | 0.1214 | 1.1129 | 1.2247 | 1.0286 | |||

b | 0.2007 | 0.4168 | −0.3360 | 0.1360 | −0.3096 | 0.1204 | −0.3612 | 0.1525 | 2.4078 | 0.0999 | 0.1001 | 0.5456 | 0.5477 | 0.5355 | |||

85% | θ | −0.8327 | 0.8583 | −0.1666 | 0.0526 | −0.1276 | 0.0443 | −0.2031 | 0.0638 | 1.5933 | 0.0719 | 0.0724 | 0.6202 | 0.6524 | 0.5959 | ||

a | 0.8746 | 0.8324 | 0.0859 | 0.0508 | 0.1282 | 0.0662 | 0.0464 | 0.0406 | 2.9330 | 0.1322 | 0.1275 | 0.7932 | 0.8515 | 0.7411 | |||

b | 0.2137 | 0.3082 | −0.1874 | 0.0491 | −0.1719 | 0.0440 | −0.2027 | 0.0547 | 2.0106 | 0.0873 | 0.0873 | 0.4630 | 0.4716 | 0.4563 | |||

2 | 65% | θ | 0.0105 | 0.4456 | 0.0441 | 0.0754 | 0.1172 | 0.1023 | −0.0232 | 0.0628 | 2.6189 | 0.1169 | 0.1171 | 1.0020 | 1.0986 | 0.9428 | |

a | −0.1869 | 0.3433 | −0.0015 | 0.1093 | 0.0559 | 0.1339 | −0.0545 | 0.0955 | 2.1788 | 0.0930 | 0.0930 | 1.1695 | 1.2602 | 1.0874 | |||

b | 0.3025 | 0.2693 | 0.0981 | 0.0430 | 0.1247 | 0.0504 | 0.0725 | 0.0374 | 1.6543 | 0.0717 | 0.0713 | 0.6976 | 0.7207 | 0.6846 | |||

85% | θ | 0.0224 | 0.2149 | 0.0203 | 0.0238 | 0.0630 | 0.0310 | −0.0196 | 0.0212 | 1.8168 | 0.0821 | 0.0822 | 0.6106 | 0.6641 | 0.5580 | ||

a | −0.0716 | 0.2182 | 0.0536 | 0.0487 | 0.0929 | 0.0604 | 0.0168 | 0.0415 | 1.8115 | 0.0809 | 0.0810 | 0.7661 | 0.8193 | 0.7406 | |||

b | 0.1356 | 0.1320 | 0.0284 | 0.0175 | 0.0444 | 0.0191 | 0.0127 | 0.0164 | 1.3223 | 0.0592 | 0.0601 | 0.4837 | 0.4910 | 0.4805 |

**Table 5.**MLE and Bayesian estimation with different loss functions in multi ramp when θ = 0.8, a = 0.5, b = 1.3.

k = 4 | θ = 0.8, a = 0.5, b = 1.3 | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

n_{1}, n_{2},n _{3}, n_{4} | Scheme | r | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | LCI | LBP | LBP | LCCI | LCCI | LCCI | |

20, 15, 18, 10 | 1 | 65% | θ | −0.4626 | 0.2492 | −0.2234 | 0.1145 | −0.1991 | 0.1159 | −0.2461 | 0.1151 | 0.7363 | 0.0344 | 0.0328 | 0.8227 | 0.8569 | 0.7749 |

a | 0.3913 | 0.3915 | 0.2019 | 0.1463 | 0.2270 | 0.1703 | 0.1772 | 0.1240 | 1.9158 | 0.0833 | 0.0839 | 1.1213 | 1.1835 | 1.0524 | |||

b | 0.1880 | 0.1968 | −0.0547 | 0.0723 | −0.0107 | 0.0790 | −0.0959 | 0.0703 | 1.5765 | 0.0739 | 0.0737 | 0.9501 | 1.0167 | 0.8885 | |||

85% | θ | −0.2908 | 0.1596 | −0.1193 | 0.0548 | −0.1007 | 0.0563 | −0.1370 | 0.0545 | 1.0747 | 0.0506 | 0.0498 | 0.7117 | 0.7482 | 0.6735 | ||

a | 0.2322 | 0.2134 | 0.1603 | 0.0924 | 0.1795 | 0.1058 | 0.1415 | 0.0804 | 1.5669 | 0.0688 | 0.0679 | 0.8484 | 0.8828 | 0.8237 | |||

b | 0.1623 | 0.1711 | −0.0366 | 0.0257 | −0.0133 | 0.0263 | −0.0588 | 0.0263 | 1.4928 | 0.0685 | 0.0673 | 0.5739 | 0.5880 | 0.5635 | |||

2 | 65% | θ | 0.1299 | 0.1748 | 0.1374 | 0.1253 | 0.1716 | 0.1523 | 0.1044 | 0.1022 | 1.5594 | 0.0733 | 0.0726 | 1.0883 | 1.1717 | 1.0231 | |

a | −0.1473 | 0.0858 | 0.0164 | 0.0596 | 0.0339 | 0.0670 | −0.0005 | 0.0536 | 0.9932 | 0.0429 | 0.0434 | 0.8881 | 0.9247 | 0.8439 | |||

b | 0.2240 | 0.2153 | 0.0817 | 0.0631 | 0.1126 | 0.0743 | 0.0525 | 0.0551 | 1.5945 | 0.0698 | 0.0698 | 0.9148 | 0.9542 | 0.8811 | |||

85% | θ | 0.1153 | 0.1522 | 0.0809 | 0.0505 | 0.1036 | 0.0606 | 0.0592 | 0.0421 | 1.4724 | 0.0720 | 0.0710 | 0.7057 | 0.7504 | 0.6762 | ||

a | −0.0763 | 0.0775 | 0.0679 | 0.0594 | 0.0842 | 0.0672 | 0.0520 | 0.0526 | 1.0509 | 0.0466 | 0.0465 | 0.8191 | 0.8463 | 0.7834 | |||

b | 0.1143 | 0.1520 | 0.0234 | 0.0241 | 0.0429 | 0.0264 | 0.0046 | 0.0228 | 1.4627 | 0.0616 | 0.0614 | 0.5870 | 0.5948 | 0.5762 | |||

25, 20, 20, 25 | 1 | 65% | θ | −0.5191 | 0.2829 | −0.2937 | 0.1456 | −0.2745 | 0.1430 | −0.3117 | 0.1490 | 0.4551 | 0.0212 | 0.0194 | 0.8007 | 0.8642 | 0.7501 |

a | 0.5151 | 0.4880 | 0.2251 | 0.1420 | 0.2473 | 0.1616 | 0.2032 | 0.1237 | 1.8517 | 0.0845 | 0.0843 | 1.0437 | 1.0777 | 0.9973 | |||

b | 0.1694 | 0.1096 | −0.0074 | 0.0847 | 0.0334 | 0.0947 | −0.0465 | 0.0789 | 1.1161 | 0.0488 | 0.0475 | 1.1251 | 1.1954 | 1.0986 | |||

85% | θ | −0.3831 | 0.1711 | −0.2176 | 0.0699 | −0.2051 | 0.0671 | −0.2295 | 0.0729 | 0.6114 | 0.0285 | 0.0277 | 0.5299 | 0.5513 | 0.5081 | ||

a | 0.3178 | 0.2274 | 0.2112 | 0.1011 | 0.2299 | 0.1151 | 0.1928 | 0.0883 | 1.3947 | 0.0643 | 0.0637 | 0.8576 | 0.8927 | 0.8125 | |||

b | 0.1490 | 0.1018 | 0.0134 | 0.0287 | 0.0367 | 0.0318 | −0.0089 | 0.0269 | 1.0124 | 0.0450 | 0.0420 | 0.6588 | 0.6696 | 0.6415 | |||

2 | 65% | θ | 0.1017 | 0.1693 | 0.1241 | 0.0918 | 0.1547 | 0.1115 | 0.0951 | 0.0755 | 1.5643 | 0.0706 | 0.0690 | 1.0020 | 1.0791 | 0.9475 | |

a | −0.1549 | 0.0652 | −0.0193 | 0.0454 | −0.0055 | 0.0490 | −0.0326 | 0.0426 | 0.7969 | 0.0354 | 0.0353 | 0.7545 | 0.7783 | 0.7351 | |||

b | 0.2107 | 0.1870 | 0.0883 | 0.0566 | 0.1150 | 0.0655 | 0.0629 | 0.0499 | 1.4821 | 0.0700 | 0.0698 | 0.8434 | 0.8594 | 0.8319 | |||

85% | θ | 0.0992 | 0.1270 | 0.0802 | 0.0429 | 0.0992 | 0.0500 | 0.0619 | 0.0369 | 1.3428 | 0.0595 | 0.0595 | 0.7124 | 0.7547 | 0.6790 | ||

a | −0.0549 | 0.0520 | 0.0444 | 0.0417 | 0.0577 | 0.0465 | 0.0314 | 0.0374 | 0.8686 | 0.0394 | 0.0392 | 0.6876 | 0.7238 | 0.6680 | |||

b | 0.0887 | 0.1077 | 0.0285 | 0.0241 | 0.0455 | 0.0266 | 0.0120 | 0.0225 | 1.2399 | 0.0551 | 0.0552 | 0.5933 | 0.6070 | 0.5709 |

**Table 6.**MLE and Bayesian estimation with different loss functions in multi ramp when θ = 3, a = 2.5, b = 0.6.

k = 4 | θ = 3, a = 2.5, b = 0.6 | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | MLE | BSEL | BLINEX (c = −0.5) | BLINEX (c = 0.5) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

n_{1}, n_{2},n _{3}, n_{4} | Scheme | r | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | LCI | LBP | LBP | LCCI | LCCI | LCCI | |

20, 15, 18, 10 | 1 | 65% | θ | −1.3150 | 2.3042 | −0.3331 | 0.1341 | −0.2518 | 0.0897 | −0.4078 | 0.1876 | 2.9740 | 0.0931 | 0.0934 | 0.5858 | 0.6235 | 0.5653 |

a | 0.8989 | 3.0227 | −0.3491 | 0.1514 | −0.2737 | 0.1096 | −0.4177 | 0.2009 | 5.8365 | 0.1831 | 0.1819 | 0.6546 | 0.7089 | 0.6228 | |||

b | −0.0913 | 0.0606 | −0.2622 | 0.0764 | −0.2552 | 0.0731 | −0.2691 | 0.0798 | 0.8971 | 0.0275 | 0.0271 | 0.3234 | 0.3309 | 0.3182 | |||

85% | θ | −0.4708 | 0.4836 | −0.1116 | 0.0243 | −0.0654 | 0.0166 | −0.1557 | 0.0358 | 2.0072 | 0.0672 | 0.0661 | 0.4192 | 0.4311 | 0.4174 | ||

a | 0.1134 | 0.3691 | −0.0982 | 0.0274 | −0.0510 | 0.0217 | −0.1430 | 0.0372 | 2.3408 | 0.0769 | 0.0765 | 0.5218 | 0.5481 | 0.5058 | |||

b | −0.1124 | 0.0513 | −0.1800 | 0.0387 | −0.1749 | 0.0370 | −0.1852 | 0.0405 | 0.7709 | 0.0247 | 0.0247 | 0.2934 | 0.2941 | 0.2908 | |||

2 | 65% | θ | −0.1024 | 0.4661 | −0.0198 | 0.0413 | 0.0717 | 0.0510 | −0.1043 | 0.0486 | 2.6473 | 0.0861 | 0.0860 | 0.8006 | 0.8313 | 0.7613 | |

a | −0.1680 | 0.7442 | −0.0116 | 0.0668 | 0.0803 | 0.0824 | −0.0956 | 0.0693 | 3.3186 | 0.1090 | 0.1096 | 0.9976 | 1.0673 | 0.9505 | |||

b | 0.1489 | 0.0759 | 0.0331 | 0.0167 | 0.0426 | 0.0178 | 0.0237 | 0.0159 | 0.9086 | 0.0284 | 0.0285 | 0.4847 | 0.4906 | 0.4771 | |||

85% | θ | 0.0948 | 0.4053 | 0.0163 | 0.0164 | 0.0650 | 0.0211 | −0.0303 | 0.0166 | 2.4820 | 0.0808 | 0.0806 | 0.4926 | 0.5031 | 0.4903 | ||

a | −0.1387 | 0.3914 | 0.0311 | 0.0291 | 0.0823 | 0.0368 | −0.0174 | 0.0271 | 2.3925 | 0.0765 | 0.0772 | 0.6587 | 0.6836 | 0.6349 | |||

b | 0.0754 | 0.0501 | 0.0002 | 0.0083 | 0.0061 | 0.0085 | −0.0057 | 0.0082 | 0.8269 | 0.0263 | 0.0273 | 0.3591 | 0.3616 | 0.3564 | |||

25, 20, 20, 25 | 1 | 65% | θ | −1.0913 | 1.4984 | −0.3755 | 0.1641 | −0.2985 | 0.1155 | −0.4466 | 0.2206 | 2.1751 | 0.0671 | 0.0668 | 0.5890 | 0.6191 | 0.5681 |

a | 0.1753 | 0.6860 | −0.3636 | 0.1586 | −0.2929 | 0.1171 | −0.4282 | 0.2069 | 3.1748 | 0.1003 | 0.1006 | 0.6294 | 0.6672 | 0.6001 | |||

b | −0.1634 | 0.0712 | −0.2865 | 0.0879 | −0.2809 | 0.0849 | −0.2920 | 0.0909 | 0.8272 | 0.0254 | 0.0253 | 0.2929 | 0.2975 | 0.2878 | |||

85% | θ | −0.6934 | 0.7946 | −0.1339 | 0.0307 | −0.0891 | 0.0214 | −0.1768 | 0.0435 | 2.1972 | 0.0689 | 0.0680 | 0.4372 | 0.4488 | 0.4279 | ||

a | 0.1442 | 0.6040 | −0.0950 | 0.0278 | −0.0492 | 0.0224 | −0.1385 | 0.0372 | 3.0604 | 0.1013 | 0.0911 | 0.5472 | 0.5677 | 0.5313 | |||

b | −0.1170 | 0.0424 | −0.1894 | 0.0414 | −0.1851 | 0.0398 | −0.1936 | 0.0429 | 0.6642 | 0.0197 | 0.0199 | 0.2797 | 0.2791 | 0.2786 | |||

2 | 65% | θ | −0.1862 | 0.6036 | −0.0300 | 0.0419 | 0.0594 | 0.0488 | −0.1123 | 0.0510 | 2.9582 | 0.0956 | 0.0961 | 0.7994 | 0.8209 | 0.7782 | |

a | −0.0428 | 0.6685 | −0.0307 | 0.0599 | 0.0549 | 0.0696 | −0.1088 | 0.0656 | 3.2022 | 0.1008 | 0.1016 | 0.9397 | 1.0007 | 0.9127 | |||

b | 0.1438 | 0.0634 | 0.0441 | 0.0147 | 0.0515 | 0.0157 | 0.0367 | 0.0139 | 0.8105 | 0.0261 | 0.0261 | 0.4317 | 0.4365 | 0.4286 | |||

85% | θ | −0.0443 | 0.1186 | 0.0114 | 0.0177 | 0.0587 | 0.0219 | −0.0340 | 0.0182 | 1.3394 | 0.0427 | 0.0432 | 0.5242 | 0.5270 | 0.5207 | ||

a | −0.0603 | 0.1282 | 0.0288 | 0.0309 | 0.0779 | 0.0382 | −0.0179 | 0.0290 | 1.3843 | 0.0435 | 0.0431 | 0.6507 | 0.6920 | 0.6365 | |||

b | 0.0485 | 0.0237 | 0.0034 | 0.0071 | 0.0082 | 0.0072 | −0.0014 | 0.0070 | 0.5732 | 0.0183 | 0.0181 | 0.3187 | 0.3206 | 0.3177 |

#### Simulation Results

- (1)
- For fixed values of the sample sizes ${n}_{i}$, by increasing the censored sample sizes, ${m}_{i}$, the bias, MSE, and LCI of the estimates decrease for the two different censored schemes.
- (2)
- For fixed values of ${m}_{i}$, by increasing the sample sizes ${n}_{i}$, the bias, MSE, and LCI decrease for different censored schemes.
- (3)
- For fixed values of ${m}_{i}$ or ${n}_{i}$ or scheme, by increasing the level of stress k, the bias, MSE, and LCI decrease.
- (4)
- For fixed values of ${m}_{i}$ or ${n}_{i}$ or scheme, we note that Scheme 2 is better than Scheme 1 for some or all parameters.
- (5)
- The bias and MSE reduce significantly, and the symmetric and asymmetric Bayesian estimations are better than the MLE in the considered scenarios.
- (6)
- The LCI reduces significantly, the symmetric and asymmetric Bayesian estimations of the HPD are better than the ACI of MLE.
- (7)
- We observe that the shortest lengths of the CI are the bootstrap CI.

## 7. An Illustrative Example

## 8. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Profile–likelihood for the three parameters for the real data set. Where ge-03 is $g\times {10}^{-3}$ and $g$ is value of parameter $g$ without virgule.

**Figure 4.**Iterations of MCMC results for progressive-stress model. Where ge-06 is $g\times {10}^{-6}$ and $g$ is value of parameter $g$ without virgule.

**Figure 6.**Bayes point and interval estimates of the sub-survivor functions and the overall survivor function.

First Level | Second Level | |||
---|---|---|---|---|

Estimates | SE | Estimates | SE | |

$\theta $ | 3.3823 | 0.8429 | 2.6078 | 0.5590 |

a | $1.10\times {10}^{-5}$ | $3.96\times {10}^{-6}$ | $8.33\times {10}^{-6}$ | $3.08\times {10}^{-6}$ |

b | 1.3756 | 0.0320 | 1.6994 | 0.0507 |

AIC | 554.7177 | 493.5818 | ||

KSD | 0.1124 | 0.1113 | ||

p-value | 0.3852 | 0.4362 |

MLE | Bayesian | |||||||
---|---|---|---|---|---|---|---|---|

Estimates | SE | Lower | Upper | Estimates | SE | Lower | Upper | |

θ | 11.9739 | 0.5590 | 0.0000 | 66.3388 | 12.3834 | 0.4691 | 4.0420 | 21.6353 |

a | $1.84\times {10}^{-5}$ | $3.08\times {10}^{-6}$ | $1.84\times {10}^{-5}$ | $1.84\times {10}^{-5}$ | $1.92\times {10}^{-5}$ | $2.75\times {10}^{-6}$ | $1.08\times {10}^{-6}$ | $1.71\times {10}^{-5}$ |

b | 1.0260 | 0.0507 | 0.9979 | 1.0541 | 1.0322 | 0.0409 | 0.9997 | 1.0539 |

AIC | 1074.528 |

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**MDPI and ACS Style**

Alotaibi, R.; Alamri, F.S.; Almetwally, E.M.; Wang, M.; Rezk, H.
Classical and Bayesian Inference of a Progressive-Stress Model for the Nadarajah–Haghighi Distribution with Type II Progressive Censoring and Different Loss Functions. *Mathematics* **2022**, *10*, 1602.
https://doi.org/10.3390/math10091602

**AMA Style**

Alotaibi R, Alamri FS, Almetwally EM, Wang M, Rezk H.
Classical and Bayesian Inference of a Progressive-Stress Model for the Nadarajah–Haghighi Distribution with Type II Progressive Censoring and Different Loss Functions. *Mathematics*. 2022; 10(9):1602.
https://doi.org/10.3390/math10091602

**Chicago/Turabian Style**

Alotaibi, Refah, Faten S. Alamri, Ehab M. Almetwally, Min Wang, and Hoda Rezk.
2022. "Classical and Bayesian Inference of a Progressive-Stress Model for the Nadarajah–Haghighi Distribution with Type II Progressive Censoring and Different Loss Functions" *Mathematics* 10, no. 9: 1602.
https://doi.org/10.3390/math10091602