# Tampered Random Variable Analysis in Step-Stress Testing: Modeling, Inference, and Applications

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## Abstract

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## 1. Introduction

- Step-stress partially accelerated life testing with a large amount of censored data. This approach addresses the gap in estimating non-homogeneous distribution and acceleration factor parameters under multiple censored data conditions. For more details, one can refer to Khan and Aslam [10].

- Performing an inferential analysis to obtain point and interval estimation of the unknown parameters of the distribution and the acceleration factor using both the maximum likelihood estimator and the Bayesian method.
- Applying numerical methods like Monte Carlo simulation to assess the performance of estimators obtained from Maximum Likelihood Estimation (MLE) and Bayesian methods, focusing on their bias, mean squared error, and the coverage probability (CP) for the confidence intervals.
- Evaluating real-world data sets from the medical field concerning AIDS infection, alongside another study from electrical engineering involving the causes of the failure of electronic components, serves to empirically assess the effectiveness of the newly proposed model.

## 2. Model Description

## 3. Point Estimation

#### 3.1. Maximum Likelihood Estimation

#### 3.2. Bayesian Inference

- (1)
- Set initial values $\left({\lambda}_{1}^{\left(0\right)},{\lambda}_{2}^{\left(0\right)},{\gamma}^{\left(0\right)},{\beta}^{\left(0\right)}\right).$
- (2)
- Set $j=1.$
- (3)
- Using the following M-H algorithm, from ${\pi}_{1}^{\ast}\left({\lambda}_{1}^{(j-1)}\right|{\lambda}_{2}^{(j-1)},{\gamma}^{(j-1)},{\beta}^{(j-1)},t,c){\pi}_{2}^{\ast}\left({\lambda}_{2}^{(j-1)}\right|{\lambda}_{1}^{\left(j\right)},{\gamma}^{(j-1)},{\beta}^{(j-1)},t,c)\phantom{\rule{3.33333pt}{0ex}}$, ${\pi}_{3}^{\ast}\left({\gamma}^{(j-1)}\right|{\lambda}_{1}^{\left(j\right)},{\lambda}_{2}^{\left(j\right)},{\beta}^{(j-1)},t,c)\phantom{\rule{3.33333pt}{0ex}}$, and ${\pi}_{4}^{\ast}\left({\beta}^{(j-1)}\right|{\lambda}_{1}^{\left(j\right)},{\lambda}_{2}^{\left(j\right)},{\gamma}^{\left(j\right)},t,c)$ generate ${\lambda}_{1}^{\left(j\right)},{\lambda}_{2}^{\left(j\right)},{\gamma}^{\left(j\right)}$, and ${\beta}^{\left(j\right)}\phantom{\rule{3.33333pt}{0ex}}$ with the normal proposal distributions$$N\left({\lambda}_{1}^{(j-1)},var\left({\lambda}_{1}\right)\right),\phantom{\rule{3.33333pt}{0ex}}N\left({\lambda}_{2}^{(j-1)},var\left({\lambda}_{2}\right)\right),N\left({\gamma}^{(j-1)},var\left(\gamma \right)\right),\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}N\left({\beta}^{(j-1)},var\left(\beta \right)\right),$$
- (4)
- Generate a proposal for ${\lambda}_{1}^{\ast}\phantom{\rule{3.33333pt}{0ex}}$ from $N\left({\lambda}_{1}^{(j-1)},var\left({\lambda}_{1}\right)\right)$, ${\lambda}_{2}^{\ast}$ from $N\left({\lambda}_{2}^{(j-1)},var\left({\lambda}_{2}\right)\right)\phantom{\rule{3.33333pt}{0ex}}$, ${\gamma}^{\ast}\phantom{\rule{3.33333pt}{0ex}}$ from $N\left({\gamma}^{(j-1)},var\left(\gamma \right)\right)$, and ${\beta}^{\ast}\phantom{\rule{3.33333pt}{0ex}}$ from $N\left({\beta}^{(j-1)},var\left(\beta \right)\right).$
- (i)
- The acceptance probabilities are$$\left.\begin{array}{c}{\mu}_{{\lambda}_{1}}=\mathrm{min}\left[1,\frac{{\pi}_{1}^{\ast}\left({\lambda}_{1}^{\ast}\right|{\lambda}_{2}^{(j-1)},{\gamma}^{(j-1)},{\beta}^{(j-1)},t,c)}{{\pi}_{1}^{\ast}\left({\lambda}_{1}^{(j-1)}\right|{\lambda}_{2}^{(j-1)},{\gamma}^{(j-1)},{\beta}^{(j-1)},t,c)}\right],\hfill \\ \\ {\mu}_{{\lambda}_{2}}=\mathrm{min}\left[1,\frac{{\pi}_{2}^{\ast}\left({\lambda}_{2}^{\ast}\right|{\lambda}_{1}^{\left(j\right)},{\gamma}^{(j-1)},{\beta}^{(j-1)},t,c)}{{\pi}_{2}^{\ast}\left({\lambda}_{2}^{(j-1)}\right|{\lambda}_{1}^{\left(j\right)},{\gamma}^{(j-1)},{\beta}^{(j-1)},t,c)}\right],\hfill \\ \\ {\mu}_{\gamma}=\mathrm{min}\left[1,\frac{{\pi}_{3}^{\ast}\left({\gamma}^{\ast}\right|{\lambda}_{1}^{\left(j\right)},{\lambda}_{2}^{\left(j\right)},{\beta}^{(j-1)},t,c)}{{\pi}_{3}^{\ast}\left({\gamma}^{(j-1)}\right|{\lambda}_{1}^{\left(j\right)},{\lambda}_{2}^{\left(j\right)},{\beta}^{(j-1)},t,c)}\right]\hfill \\ \\ and\hfill \\ \\ {\mu}_{\beta}=\mathrm{min}\left[1,\frac{{\pi}_{4}^{\ast}\left({\beta}^{\ast}\right|{\lambda}_{1}^{\left(j\right)},{\lambda}_{2}^{\left(j\right)},{\gamma}^{\left(j\right)},t,c)}{{\pi}_{4}^{\ast}\left({\beta}^{(j-1)}\right|{\lambda}_{1}^{\left(j\right)},{\lambda}_{2}^{\left(j\right)},{\gamma}^{\left(j\right)},t,c)}\right].\hfill \end{array}\right\}$$
- (ii)
- From a Uniform $(0,1)$ distribution ${u}_{1},{u}_{2},$${u}_{3}$, and ${u}_{4}$ are generated.
- (iii)
- If ${u}_{1}<{\mu}_{{\lambda}_{1}}$, accept the proposal and set ${\lambda}_{1}^{\left(j\right)}={\lambda}_{1}^{\ast}$, otherwise set ${\lambda}_{1}^{\left(j\right)}={\lambda}_{1}^{(j-1)}$.
- (iv)
- If ${u}_{2}<{\mu}_{{\lambda}_{2}}$, accept the proposal and set ${\lambda}_{2}^{\left(j\right)}={\lambda}_{2}^{\ast}$, otherwise set ${\lambda}_{2}^{\left(j\right)}={\lambda}_{2}^{(j-1)}$.
- (v)
- If ${u}_{3}<{\mu}_{\gamma}$, accept the proposal and set ${\gamma}^{\left(j\right)}={\gamma}^{\ast}$, otherwise set ${\gamma}^{\left(j\right)}={\gamma}^{(j-1)}$.
- (vi)
- If ${u}_{4}<{\mu}_{\beta}$, accept the proposal and set ${\beta}^{\left(j\right)}={\beta}^{\ast}$, otherwise set ${\beta}^{\left(j\right)}={\beta}^{(j-1)}.$

- (5)
- Set $j=j+1.$
- (6)
- Steps (3)–(5), are repeated N times to obtain ${\lambda}_{1}^{\left(j\right)},{\lambda}_{2}^{\left(j\right)},{\gamma}^{\left(j\right)}$, and ${\beta}^{\left(j\right)}$, j = 1, 2, …N.

## 4. Interval Estimation

#### 4.1. Asymptotic Confidence Interval

#### 4.2. Credible Interval

#### 4.3. Bootstrap Interval

#### 4.3.1. Parametric Boot-p

- (1)
- Based on $x={x}_{1:n},{x}_{2:n},\dots ,{x}_{m:n},$ obtain $\widehat{{\lambda}_{1}},\widehat{{\lambda}_{2}},\widehat{\gamma}$, and $\widehat{\beta}$ by maximizing Equations (10)–(13).
- (2)
- Generate ${x}^{\ast}={x}_{1:n}^{\ast},{x}_{2:n}^{\ast},\dots ,{x}_{m:n}^{\ast}$ from the PR distribution with parameters $\widehat{{\lambda}_{1}},\widehat{{\lambda}_{2}},\widehat{\gamma}$, and $\widehat{\beta}$ based on Type-II censoring under TRV, by considering the algorithm presented in [27].
- (3)
- Obtain the bootstrap parameter estimation ${\widehat{\psi}}_{i}^{\ast}=\left({\widehat{{\lambda}_{1}}}_{i}^{\ast},{\widehat{{\lambda}_{2}}}_{i}^{\ast},{\widehat{\gamma}}_{i}^{\ast},{\widehat{\beta}}_{i}^{\ast},\right)$, with $i=1,2,3,\dots ,N$ boots using the MLEs under the bootstrap sampling.
- (4)
- Repeat steps (2) and (3) $N$ boot times, and obtain ${\widehat{\psi}}_{1}^{\ast},{\widehat{\psi}}_{2}^{\ast},\dots ,{\widehat{\psi}}_{N\phantom{\rule{4.pt}{0ex}}boot}^{\ast}$
- (5)
- Obtain ${\widehat{\psi}}_{\left(1\right)}^{\ast},{\widehat{\psi}}_{\left(2\right)}^{\ast},\dots ,{\widehat{\psi}}_{\left(N\phantom{\rule{4.pt}{0ex}}boot\right)}^{\ast}$ by arrange ${\widehat{\psi}}_{i}^{\ast},i=1,2,3,\dots ,N$$boot$ in ascending orders.

#### 4.3.2. Parametric Boot-t

- (1)
- Repeat the initial three steps of the parametric Boot-p procedure.
- (2)
- Calculate the variance–covariance matrix ${I}^{-1\ast}$ utilizing the delta method.
- (3)
- Define the statistic ${T}^{\ast \psi}$ as

- (4)
- Generate ${T}_{1}^{\ast \psi},{T}_{2}^{\ast \psi},\dots ,{T}_{N\phantom{\rule{4.pt}{0ex}}boot}^{\ast \psi}$ from repeating steps $2-5$ N Boot times
- (5)
- Sort the sequence ${T}_{\left(1\right)}^{\ast \psi},{T}_{\left(2\right)}^{\ast \psi},\dots ,{T}_{\left(N\phantom{\rule{4.pt}{0ex}}boot\right)}^{\ast \psi}$ by arranging ${\widehat{\psi}}_{i}^{\ast},i=1,2,3,\dots ,N$$boot$ in ${T}_{1}^{\ast \psi},{T}_{2}^{\ast \psi},\dots ,{T}_{N\phantom{\rule{4.pt}{0ex}}boot}^{\ast \psi}$ in ascending order.

## 5. Simulation Analysis

- Our observations consistently show reduced biases, MSEs, and ALs as sample sizes increase.
- The CPs mostly align closely with their anticipated 95% level.
- In general, the informative Bayes estimation method outperforms MLE, with the disparity between the two estimators decreasing as the sample size grows. This highlights the Bayesian methods’ advantage for smaller samples.
- In particular, confidence intervals based on the Highest Posterior Density (HPD) method tend to be smaller than those based on the Approximate Confidence Interval (ACI) method, while still providing similar CPs.
- Altering the pre-determined number of failures or stress change time yields comparable performances across all cases, demonstrating the consistent efficiency and productivity of the theoretical findings.
- Increasing the sample size generally leads to improvements in bias, MSE, and the precision of confidence intervals across all methods. This is expected because larger samples provide more information about the population. The number of bootstrap samples m also influences the Bootstrap method’s accuracy and precision, with a higher m usually leading to better estimates.
- changing the stress transition time point $\tau $ affects the estimation, especially for Bayesian estimation under ELF that adjusts based on the distribution’s tail properties. Different $\tau $ values can lead to variations in bias and MSE, suggesting the importance of choosing an appropriate $\tau $ value for accurate estimation.

## 6. The Optimal Stress Change Time and Sensitivity Analysis

- Step 1: Obtain the samples ${U}_{1}$, ${U}_{2}$ and $U=\mathrm{min}\{{U}_{1},{U}_{2}\}$ using given $\tau $, n, r and parameter values.
- Step 2: The objective function $\varphi \left(\tau \right)$ is calculated.
- Step 3: For N times, repeat Step 1 to Step 2, and obtain ${\varphi}^{1}\left(\tau \right),{\varphi}^{2}\left(\tau \right),\dots ,{\varphi}^{N}\left(\tau \right)$.
- Step 4: The median of the objective functions is obtained and applied to ${\varphi}^{m}\left(\tau \right)$.
- Step 5: For all possible values of $\tau $ repeat Step 1 to Step 4.
- Step 6: The optimal $\tau $ for which ${\varphi}^{m}\left(\tau \right)$ is the minimum is obtained.

## 7. Real Data Examples

#### 7.1. HIV Infection to AIDS

#### 7.2. Electrical Appliances Data

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Likelihood profile (blue line) for parameters of PR based on SSLT under TRV model with the maximum likelihood estimation (red dot): HIV infection to AIDS data.

**Figure 2.**MCMC plots for parameters of PR based on SSLT under TRV model for HIV Infection to AIDS data.(The blue color indicates the convergence line).

**Figure 3.**Likelihood profile (blue line) for parameters of PR based on SSLT under TRV model with the maximum likelihood estimation (red dot): electrical appliances data.

**Figure 4.**MCMC plots for parameters of PR based on SSLT under TRV model: electrical appliances (The blue color indicates the convergence line).

**Table 1.**Some simulation measures from MLE, bootstrap, Bayesian based on SELF, and ELF for parameters of PR distribution based on TRV: ${\lambda}_{1}=1.5,\gamma =1.2,$ ${\lambda}_{2}=1.8$, $\beta =0.8$.

MLE | Bootstrap | SELF | ELF c = −1.25 | ELF c = 1.25 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

n | $\mathit{\tau}$ | m | Bias | MSE | LACI | CP | LBP | LBT | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | |

40 | 0.6 | 25 | ${\lambda}_{1}$ | −0.2909 | 0.2916 | 1.7842 | 96.1% | 0.0561 | 0.0569 | −0.0752 | 0.0274 | 0.6008 | −0.0508 | 0.0272 | 0.5998 | −0.0631 | 0.0298 | 0.6240 |

$\gamma $ | 0.5299 | 0.6215 | 2.2892 | 95.2% | 0.0704 | 0.0691 | 0.0470 | 0.0206 | 0.5408 | 0.0382 | 0.0207 | 0.5390 | 0.0264 | 0.0200 | 0.5441 | |||

${\lambda}_{2}$ | 0.2413 | 0.7516 | 3.2657 | 94.8% | 0.0957 | 0.0962 | 0.0232 | 0.0256 | 0.6184 | 0.0241 | 0.0256 | 0.6195 | 0.0146 | 0.0250 | 0.6224 | |||

$\beta $ | 0.3789 | 0.5924 | 2.6275 | 95.0% | 0.0799 | 0.0795 | −0.0552 | 0.0169 | 0.4607 | −0.0507 | 0.0167 | 0.4602 | −0.0656 | 0.0190 | 0.4694 | |||

35 | ${\lambda}_{1}$ | 0.2035 | 0.2412 | 1.3889 | 96.3% | 0.0726 | 0.0730 | 0.0741 | 0.0210 | 0.4968 | 0.0468 | 0.0211 | 0.4973 | 0.0607 | 0.0198 | 0.4895 | ||

$\gamma $ | 0.4141 | 0.4885 | 2.2082 | 95.4% | 0.0698 | 0.0699 | −0.0460 | 0.0151 | 0.3970 | −0.0359 | 0.0150 | 0.3951 | −0.0168 | 0.0166 | 0.4061 | |||

${\lambda}_{2}$ | −0.0342 | 0.4384 | 2.5934 | 95.6% | 0.0805 | 0.0800 | 0.0157 | 0.0169 | 0.5044 | 0.0063 | 0.0169 | 0.5042 | −0.0065 | 0.0170 | 0.5058 | |||

$\beta $ | 0.2641 | 0.4174 | 2.4269 | 96.6% | 0.1088 | 0.1088 | 0.0491 | 0.0147 | 0.4050 | 0.0410 | 0.0152 | 0.4059 | 0.0590 | 0.0179 | 0.3967 | |||

0.9 | 25 | ${\lambda}_{1}$ | −0.1263 | 0.1698 | 1.5383 | 95.3% | 0.0493 | 0.0494 | −0.0609 | 0.0244 | 0.5561 | −0.0498 | 0.0242 | 0.5546 | −0.0597 | 0.0261 | 0.5569 | |

$\gamma $ | 0.4162 | 0.6104 | 2.1593 | 95.4% | 0.0684 | 0.0583 | 0.0461 | 0.0192 | 0.5148 | 0.0355 | 0.0200 | 0.5048 | 0.0235 | 0.0192 | 0.5524 | |||

${\lambda}_{2}$ | 0.2353 | 0.6907 | 2.5138 | 95.1% | 0.0777 | 0.0780 | 0.0176 | 0.0238 | 0.5957 | 0.0185 | 0.0239 | 0.5949 | 0.0094 | 0.0234 | 0.5919 | |||

$\beta $ | 0.1743 | 0.3717 | 2.2913 | 94.7% | 0.0734 | 0.0725 | −0.0528 | 0.0152 | 0.4527 | −0.0509 | 0.0152 | 0.4523 | −0.0609 | 0.0182 | 0.5492 | |||

35 | ${\lambda}_{1}$ | 0.1045 | 0.1523 | 1.1130 | 95.9% | 0.0671 | 0.0673 | 0.0607 | 0.0150 | 0.4670 | 0.0373 | 0.0200 | 0.4675 | 0.0565 | 0.0185 | 0.4511 | ||

$\gamma $ | 0.4030 | 0.4685 | 2.0774 | 95.8% | 0.0587 | 0.0609 | −0.0457 | 0.0146 | 0.3842 | −0.0327 | 0.0136 | 0.3419 | −0.0163 | 0.0160 | 0.4292 | |||

${\lambda}_{2}$ | 0.1219 | 0.1666 | 1.4238 | 95.3% | 0.0428 | 0.0429 | 0.0113 | 0.0147 | 0.4899 | 0.0051 | 0.0157 | 0.4901 | 0.0057 | 0.0169 | 0.4883 | |||

$\beta $ | 0.1530 | 0.3199 | 2.1296 | 95.8% | 0.0982 | 0.1079 | 0.0489 | 0.0132 | 0.3944 | 0.0393 | 0.0122 | 0.3544 | 0.0561 | 0.0159 | 0.4309 | |||

100 | 0.6 | 75 | ${\lambda}_{1}$ | −0.0995 | 0.1601 | 1.5198 | 95.2% | 0.0486 | 0.0484 | 0.0158 | 0.0182 | 0.5072 | 0.0167 | 0.0182 | 0.5077 | 0.0081 | 0.0179 | 0.5052 |

$\gamma $ | 0.4028 | 0.3386 | 1.6471 | 94.3% | 0.0541 | 0.0550 | 0.0348 | 0.0154 | 0.4495 | 0.0346 | 0.0155 | 0.4499 | 0.0236 | 0.0145 | 0.4477 | |||

${\lambda}_{2}$ | −0.0497 | 0.2607 | 1.9928 | 94.8% | 0.0630 | 0.0643 | 0.0291 | 0.0212 | 0.5632 | 0.0172 | 0.0213 | 0.5625 | 0.0081 | 0.0206 | 0.5556 | |||

$\beta $ | 0.4319 | 0.5261 | 2.2852 | 94.5% | 0.0714 | 0.0715 | 0.0508 | 0.0152 | 0.4397 | 0.0452 | 0.0145 | 0.4405 | 0.0407 | 0.0139 | 0.4305 | |||

90 | ${\lambda}_{1}$ | 0.0882 | 0.1522 | 1.1671 | 95.9% | 0.0529 | 0.0533 | 0.0148 | 0.0124 | 0.4386 | 0.0087 | 0.0172 | 0.4391 | 0.0071 | 0.0162 | 0.4289 | ||

$\gamma $ | 0.3253 | 0.2541 | 1.5100 | 94.5% | 0.0508 | 0.0497 | −0.0325 | 0.0118 | 0.3807 | −0.0315 | 0.0117 | 0.3081 | −0.0152 | 0.0124 | 0.3726 | |||

${\lambda}_{2}$ | −0.0392 | 0.2081 | 1.7523 | 95.0% | 0.0532 | 0.0533 | 0.0039 | 0.0136 | 0.4509 | 0.0045 | 0.0136 | 0.4514 | −0.0041 | 0.0137 | 0.4506 | |||

$\beta $ | 0.4057 | 0.4709 | 1.9445 | 95.0% | 0.0770 | 0.0787 | 0.0480 | 0.0132 | 0.3861 | 0.0349 | 0.0113 | 0.3287 | 0.0388 | 0.0121 | 0.3783 | |||

0.9 | 75 | ${\lambda}_{1}$ | 0.0911 | 0.0899 | 1.0850 | 94.4% | 0.0314 | 0.0318 | 0.0128 | 0.0174 | 0.4962 | 0.0153 | 0.0175 | 0.4961 | 0.0079 | 0.0167 | 0.4928 | |

$\gamma $ | 0.2835 | 0.1965 | 1.3365 | 95.2% | 0.0422 | 0.0419 | 0.0158 | 0.0129 | 0.4219 | 0.0167 | 0.0130 | 0.4221 | 0.0075 | 0.0125 | 0.4221 | |||

${\lambda}_{2}$ | 0.0419 | 0.0923 | 0.9197 | 95.2% | 0.0295 | 0.0298 | 0.0274 | 0.0171 | 0.4803 | 0.0162 | 0.0172 | 0.4819 | 0.0079 | 0.0163 | 0.4717 | |||

$\beta $ | 0.2108 | 0.1767 | 1.4266 | 95.2% | 0.0444 | 0.0447 | 0.0328 | 0.0149 | 0.3453 | 0.0341 | 0.0141 | 0.4054 | 0.0207 | 0.0134 | 0.4050 | |||

90 | ${\lambda}_{1}$ | 0.0810 | 0.0733 | 0.9337 | 94.7% | 0.0416 | 0.0413 | 0.0126 | 0.0120 | 0.4317 | 0.0081 | 0.0163 | 0.4332 | 0.0071 | 0.0154 | 0.4184 | ||

$\gamma $ | 0.2098 | 0.1822 | 0.9443 | 95.6% | 0.0471 | 0.0468 | −0.0237 | 0.0114 | 0.3597 | −0.0274 | 0.0103 | 0.2994 | −0.0108 | 0.0120 | 0.3628 | |||

${\lambda}_{2}$ | 0.0317 | 0.0827 | 0.9078 | 95.4% | 0.0279 | 0.0280 | 0.0029 | 0.0120 | 0.4102 | 0.0039 | 0.0112 | 0.4095 | 0.0039 | 0.0118 | 0.4081 | |||

$\beta $ | 0.1410 | 0.1238 | 1.0821 | 95.9% | 0.0606 | 0.0604 | 0.0413 | 0.0127 | 0.3042 | 0.0263 | 0.0102 | 0.3042 | 0.0116 | 0.0115 | 0.3401 |

**Table 2.**Some simulation measures from MLE, bootstrap, Bayesian based on SELF, and ELF for parameters of PR distribution based on TRV: ${\lambda}_{1}=1.5,\gamma =1.2$, ${\lambda}_{2}=1.8$, $\beta =0.3$.

MLE | Bootstrap | SELF | ELF c = −1.25 | ELF c = 1.25 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

n | $\tau $ | m | Bias | MSE | LACI | CP | LBP | LBT | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | |

40 | 0.6 | 25 | ${\lambda}_{1}$ | −0.3308 | 0.2386 | 1.4870 | 94.7% | 0.0451 | 0.0459 | −0.0420 | 0.0254 | 0.6011 | −0.0409 | 0.0252 | 0.5991 | −0.0523 | 0.0274 | 0.6088 |

$\gamma $ | 0.4901 | 0.4934 | 1.9734 | 95.2% | 0.0623 | 0.0621 | 0.0215 | 0.0228 | 0.5649 | 0.0228 | 0.0229 | 0.5642 | 0.0101 | 0.0225 | 0.5605 | |||

${\lambda}_{2}$ | 0.1578 | 0.3716 | 2.3093 | 93.5% | 0.0742 | 0.0741 | 0.0267 | 0.0260 | 0.6403 | 0.0276 | 0.0261 | 0.6444 | 0.0184 | 0.0256 | 0.6265 | |||

$\beta $ | 0.1282 | 0.0660 | 0.8731 | 94.3% | 0.0284 | 0.0285 | −0.0988 | 0.0034 | 0.2074 | −0.0220 | 0.0034 | 0.2078 | −0.0320 | 0.0039 | 0.2060 | |||

35 | ${\lambda}_{1}$ | 0.0776 | 0.2268 | 1.0065 | 95.8% | 0.0629 | 0.0632 | 0.0337 | 0.0190 | 0.5233 | 0.0344 | 0.0190 | 0.5232 | 0.0271 | 0.0184 | 0.5246 | ||

$\gamma $ | 0.4732 | 0.4537 | 1.8195 | 95.5% | 0.0711 | 0.0705 | −0.0076 | 0.0151 | 0.4924 | −0.0067 | 0.0151 | 0.4915 | −0.0101 | 0.0155 | 0.4939 | |||

${\lambda}_{2}$ | −0.1569 | 0.3127 | 2.1050 | 95.6% | 0.0707 | 0.0707 | −0.0073 | 0.0177 | 0.5031 | −0.0066 | 0.0177 | 0.5032 | −0.0133 | 0.0181 | 0.5082 | |||

$\beta $ | 0.1128 | 0.0520 | 0.6350 | 95.4% | 0.0411 | 0.0411 | 0.0852 | 0.0021 | 0.2006 | 0.0209 | 0.0031 | 0.2006 | 0.0298 | 0.0036 | 0.1925 | |||

0.9 | 25 | ${\lambda}_{1}$ | 0.3061 | 0.2147 | 1.3998 | 95.4% | 0.0409 | 0.0410 | −0.0416 | 0.0220 | 0.5343 | −0.0384 | 0.0219 | 0.5342 | −0.0496 | 0.0234 | 0.5436 | |

$\gamma $ | 0.4320 | 0.4638 | 1.8635 | 95.7% | 0.0608 | 0.0606 | 0.0202 | 0.0203 | 0.5583 | 0.0213 | 0.0213 | 0.5183 | 0.0101 | 0.0226 | 0.5476 | |||

${\lambda}_{2}$ | 0.1485 | 0.3400 | 1.2760 | 95.6% | 0.0680 | 0.0678 | 0.0225 | 0.0261 | 0.6143 | 0.0234 | 0.0262 | 0.5061 | 0.0140 | 0.0256 | 0.6064 | |||

$\beta $ | 0.1196 | 0.0611 | 0.8163 | 94.6% | 0.0263 | 0.0262 | −0.0208 | 0.0032 | 0.2053 | −0.0206 | 0.0031 | 0.2001 | −0.0313 | 0.0031 | 0.2048 | |||

35 | ${\lambda}_{1}$ | −0.0674 | 0.1845 | 1.0016 | 95.9% | 0.0613 | 0.0628 | 0.0314 | 0.0171 | 0.4842 | 0.0304 | 0.0172 | 0.4832 | 0.0231 | 0.0164 | 0.4777 | ||

$\gamma $ | 0.3999 | 0.4152 | 1.7359 | 95.8% | 0.0704 | 0.0684 | −0.0070 | 0.0146 | 0.4748 | −0.0056 | 0.0146 | 0.4742 | −0.0091 | 0.0151 | 0.4795 | |||

${\lambda}_{2}$ | 0.1474 | 0.3048 | 1.1014 | 96.5% | 0.0397 | 0.0397 | −0.0061 | 0.0152 | 0.4716 | −0.0055 | 0.0152 | 0.4713 | −0.0125 | 0.0156 | 0.4710 | |||

$\beta $ | 0.0734 | 0.0486 | 0.5815 | 96.1% | 0.0370 | 0.0372 | 0.0201 | 0.0021 | 0.2005 | 0.0201 | 0.0014 | 0.1925 | 0.0283 | 0.0029 | 0.1920 | |||

100 | 0.6 | 75 | ${\lambda}_{1}$ | −0.0969 | 0.1007 | 1.1855 | 93.9% | 0.0378 | 0.0378 | −0.0105 | 0.0181 | 0.5247 | −0.0096 | 0.0181 | 0.5244 | −0.0187 | 0.0186 | 0.5230 |

$\gamma $ | 0.3355 | 0.2237 | 1.3073 | 95.2% | 0.0399 | 0.0401 | 0.0195 | 0.0203 | 0.5065 | 0.0206 | 0.0205 | 0.5063 | 0.0094 | 0.0187 | 0.4921 | |||

${\lambda}_{2}$ | −0.0389 | 0.1561 | 1.2542 | 93.5% | 0.0481 | 0.0485 | 0.0139 | 0.0224 | 0.5605 | 0.0148 | 0.0225 | 0.4956 | 0.0064 | 0.0222 | 0.5523 | |||

$\beta $ | 0.1133 | 0.0494 | 0.6976 | 94.5% | 0.0217 | 0.0216 | 0.0200 | 0.0017 | 0.1926 | 0.0200 | 0.0029 | 0.1933 | 0.0268 | 0.0031 | 0.1871 | |||

90 | ${\lambda}_{1}$ | 0.0614 | 0.0912 | 0.9262 | 95.2% | 0.0388 | 0.0388 | 0.0308 | 0.0162 | 0.4449 | 0.0277 | 0.0168 | 0.4452 | 0.0092 | 0.0159 | 0.4358 | ||

$\gamma $ | 0.3313 | 0.2126 | 1.2581 | 95.9% | 0.0407 | 0.0405 | 0.0069 | 0.0130 | 0.4336 | 0.0043 | 0.0131 | 0.4327 | 0.0089 | 0.0123 | 0.4290 | |||

${\lambda}_{2}$ | −0.0314 | 0.1277 | 1.0928 | 95.0% | 0.0385 | 0.0384 | −0.0060 | 0.0150 | 0.4520 | −0.0053 | 0.0153 | 0.4511 | −0.0114 | 0.0146 | 0.4598 | |||

$\beta $ | 0.0692 | 0.0382 | 0.4354 | 94.7% | 0.0240 | 0.0240 | 0.0201 | 0.0015 | 0.1872 | 0.0191 | 0.0011 | 0.1877 | 0.0195 | 0.0011 | 0.1126 | |||

0.9 | 75 | ${\lambda}_{1}$ | 0.0911 | 0.0791 | 1.0200 | 94.9% | 0.0334 | 0.0334 | 0.0092 | 0.0163 | 0.4957 | 0.0092 | 0.0163 | 0.4963 | 0.0084 | 0.0158 | 0.4896 | |

$\gamma $ | 0.2783 | 0.1912 | 1.1032 | 94.8% | 0.0384 | 0.0384 | 0.0184 | 0.0187 | 0.4951 | 0.0192 | 0.0188 | 0.4951 | 0.0083 | 0.0178 | 0.4507 | |||

${\lambda}_{2}$ | 0.0199 | 0.0909 | 0.9317 | 95.0% | 0.0297 | 0.0288 | 0.0124 | 0.0185 | 0.5052 | 0.0137 | 0.0186 | 0.4504 | 0.0053 | 0.0178 | 0.5045 | |||

$\beta $ | 0.0787 | 0.0246 | 0.5317 | 95.3% | 0.0171 | 0.0170 | 0.0201 | 0.0010 | 0.1823 | 0.0193 | 0.0020 | 0.1823 | 0.0189 | 0.0031 | 0.1802 | |||

90 | ${\lambda}_{1}$ | 0.0510 | 0.0691 | 0.6013 | 95.8% | 0.0351 | 0.0344 | 0.0090 | 0.0152 | 0.4206 | 0.0082 | 0.0152 | 0.4223 | 0.0021 | 0.0149 | 0.4148 | ||

$\gamma $ | 0.2090 | 0.1696 | 1.0311 | 95.2% | 0.0401 | 0.0394 | 0.0024 | 0.0111 | 0.4026 | 0.0032 | 0.0111 | 0.4033 | −0.0043 | 0.0109 | 0.4046 | |||

${\lambda}_{2}$ | 0.0115 | 0.0632 | 0.7802 | 96.5% | 0.0264 | 0.0259 | −0.0058 | 0.0129 | 0.4485 | −0.0051 | 0.0129 | 0.4483 | −0.0051 | 0.0091 | 0.4473 | |||

$\beta $ | 0.0615 | 0.0230 | 0.4061 | 95.7% | 0.0187 | 0.0185 | 0.0200 | 0.0010 | 0.1722 | 0.0181 | 0.0010 | 0.1722 | 0.0110 | 0.0010 | 0.1522 |

**Table 3.**Some simulation measures from MLE, bootstrap, Bayesian based on SELF, and ELF for parameters of PR distribution based on TRV: ${\lambda}_{1}=0.6,\gamma =2$, ${\lambda}_{2}=0.7$, $\beta =0.3$.

MLE | Bootstrap | SELF | ELF c = −1.25 | ELF c = 1.25 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

n | $\tau $ | m | Bias | MSE | LACI | CP | LBP | LBT | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | |

40 | 0.6 | 25 | ${\lambda}_{1}$ | −0.1998 | 0.1449 | 0.7621 | 96.0% | 0.0327 | 0.0329 | −0.0477 | 0.0182 | 0.3608 | −0.0746 | 0.0152 | 0.3615 | −0.0619 | 0.0134 | 0.3561 |

$\gamma $ | 0.2789 | 0.4008 | 1.8532 | 93.8% | 0.0814 | 0.0818 | 0.0447 | 0.0279 | 0.6292 | 0.0531 | 0.0279 | 0.6298 | 0.0297 | 0.0276 | 0.6339 | |||

${\lambda}_{2}$ | 0.1927 | 0.1262 | 1.1712 | 95.4% | 0.0514 | 0.0519 | 0.0688 | 0.0197 | 0.4464 | 0.0705 | 0.0201 | 0.4454 | 0.0532 | 0.0166 | 0.4370 | |||

$\beta $ | −0.1116 | 0.0305 | 0.3889 | 94.0% | 0.0166 | 0.0164 | −0.0692 | 0.0084 | 0.2207 | −0.0682 | 0.0084 | 0.2224 | −0.0584 | 0.0069 | 0.2084 | |||

35 | ${\lambda}_{1}$ | 0.1260 | 0.1380 | 0.7400 | 96.4% | 0.0442 | 0.0443 | 0.0465 | 0.0128 | 0.3536 | 0.0663 | 0.0130 | 0.3536 | 0.0550 | 0.0111 | 0.3459 | ||

$\gamma $ | 0.2355 | 0.3055 | 1.7076 | 95.4% | 0.0904 | 0.0910 | 0.0342 | 0.0202 | 0.5307 | 0.0348 | 0.0202 | 0.5309 | 0.0287 | 0.0199 | 0.5275 | |||

${\lambda}_{2}$ | 0.0517 | 0.0590 | 0.8553 | 95.6% | 0.0391 | 0.0388 | 0.0256 | 0.0092 | 0.3572 | 0.0267 | 0.0093 | 0.3585 | 0.0160 | 0.0086 | 0.3555 | |||

$\beta $ | 0.0581 | 0.0195 | 0.3499 | 96.4% | 0.0220 | 0.0225 | 0.0662 | 0.0075 | 0.2163 | 0.0672 | 0.0077 | 0.2162 | 0.0573 | 0.0061 | 0.2019 | |||

0.9 | 25 | ${\lambda}_{1}$ | −0.0919 | 0.0314 | 0.5944 | 96.6% | 0.0262 | 0.0264 | −0.0348 | 0.0048 | 0.2355 | −0.0340 | 0.0048 | 0.2361 | −0.0422 | 0.0052 | 0.2291 | |

$\gamma $ | 0.1929 | 0.2398 | 1.7663 | 95.2% | 0.0754 | 0.0758 | 0.0094 | 0.0205 | 0.5689 | 0.0052 | 0.0205 | 0.5686 | −0.0092 | 0.0205 | 0.5646 | |||

${\lambda}_{2}$ | 0.1410 | 0.0346 | 0.4753 | 95.8% | 0.0215 | 0.0213 | 0.0385 | 0.0095 | 0.3348 | 0.0398 | 0.0097 | 0.3357 | 0.0268 | 0.0081 | 0.3274 | |||

$\beta $ | −0.1529 | 0.0290 | 0.2949 | 94.2% | 0.0131 | 0.0132 | −0.0216 | 0.0022 | 0.2143 | −0.0593 | 0.0032 | 0.2042 | −0.0301 | 0.0032 | 0.1944 | |||

35 | ${\lambda}_{1}$ | 0.0812 | 0.0291 | 0.5708 | 96.9% | 0.0371 | 0.0366 | 0.0326 | 0.0041 | 0.2031 | 0.0326 | 0.0031 | 0.2131 | 0.0409 | 0.0041 | 0.2193 | ||

$\gamma $ | −0.1912 | 0.1355 | 1.2343 | 96.6% | 0.0557 | 0.0559 | 0.0081 | 0.0147 | 0.4481 | 0.0049 | 0.0147 | 0.4485 | 0.0239 | 0.0147 | 0.4489 | |||

${\lambda}_{2}$ | 0.0491 | 0.0325 | 0.2734 | 96.5% | 0.0126 | 0.0126 | 0.0246 | 0.0085 | 0.2672 | 0.0206 | 0.0087 | 0.2680 | 0.0145 | 0.0072 | 0.2571 | |||

$\beta $ | −0.0499 | 0.0149 | 0.2827 | 94.6% | 0.0127 | 0.0127 | 0.0204 | 0.0021 | 0.2032 | 0.0256 | 0.0031 | 0.2033 | 0.0075 | 0.0026 | 0.1831 | |||

100 | 0.6 | 75 | ${\lambda}_{1}$ | −0.0888 | 0.0181 | 0.5272 | 96.6% | 0.0238 | 0.0238 | 0.0219 | 0.0046 | 0.2130 | 0.0229 | 0.0047 | 0.2304 | 0.0130 | 0.0051 | 0.2197 |

$\gamma $ | 0.1727 | 0.1880 | 1.3465 | 94.2% | 0.0584 | 0.0584 | 0.0085 | 0.0192 | 0.5551 | 0.0051 | 0.0203 | 0.5559 | 0.0082 | 0.0202 | 0.5564 | |||

${\lambda}_{2}$ | 0.0570 | 0.0276 | 0.4613 | 96.0% | 0.0207 | 0.0207 | 0.0369 | 0.0091 | 0.2033 | 0.0370 | 0.0091 | 0.3304 | 0.0258 | 0.0071 | 0.3163 | |||

$\beta $ | 0.0301 | 0.0059 | 0.2908 | 94.8% | 0.0127 | 0.0127 | −0.0028 | 0.0014 | 0.1484 | −0.0518 | 0.0021 | 0.1482 | −0.0293 | 0.0014 | 0.1462 | |||

90 | ${\lambda}_{1}$ | 0.0712 | 0.0126 | 0.5264 | 95.4% | 0.0294 | 0.0292 | 0.0203 | 0.0040 | 0.2013 | 0.0231 | 0.0030 | 0.2031 | 0.0124 | 0.0041 | 0.2030 | ||

$\gamma $ | 0.1683 | 0.1234 | 1.2051 | 94.4% | 0.0547 | 0.0507 | 0.0075 | 0.0138 | 0.4283 | 0.0045 | 0.0138 | 0.4383 | 0.0072 | 0.0137 | 0.4380 | |||

${\lambda}_{2}$ | 0.0370 | 0.0242 | 0.2592 | 94.2% | 0.0121 | 0.0122 | 0.0203 | 0.0056 | 0.2047 | 0.0202 | 0.0057 | 0.2570 | 0.0125 | 0.0050 | 0.2470 | |||

$\beta $ | 0.0300 | 0.0051 | 0.2338 | 94.8% | 0.0126 | 0.0116 | 0.0019 | 0.0012 | 0.1317 | 0.0236 | 0.0020 | 0.1349 | 0.0054 | 0.0012 | 0.1410 | |||

0.9 | 75 | ${\lambda}_{1}$ | 0.0756 | 0.0152 | 0.3813 | 96.0% | 0.0171 | 0.0170 | 0.0193 | 0.0043 | 0.2013 | 0.0203 | 0.0044 | 0.2015 | 0.0129 | 0.0038 | 0.1981 | |

$\gamma $ | −0.0977 | 0.0595 | 0.8773 | 96.0% | 0.0383 | 0.0387 | 0.0081 | 0.0164 | 0.4916 | 0.0050 | 0.0164 | 0.4919 | 0.0079 | 0.0164 | 0.4920 | |||

${\lambda}_{2}$ | 0.0416 | 0.0274 | 0.2007 | 95.2% | 0.0092 | 0.0092 | 0.0275 | 0.0080 | 0.1807 | 0.0356 | 0.0081 | 0.1815 | 0.0247 | 0.0067 | 0.1732 | |||

$\beta $ | −0.0080 | 0.0042 | 0.1561 | 95.2% | 0.0070 | 0.0067 | −0.0018 | 0.0014 | 0.1381 | −0.0509 | 0.0018 | 0.1381 | −0.0290 | 0.0010 | 0.1372 | |||

90 | ${\lambda}_{1}$ | 0.0632 | 0.0120 | 0.3506 | 96.2% | 0.0249 | 0.0250 | 0.0173 | 0.0040 | 0.1928 | 0.0203 | 0.0028 | 0.1985 | 0.0092 | 0.0032 | 0.1827 | ||

$\gamma $ | −0.0817 | 0.0493 | 0.7108 | 96.4% | 0.0313 | 0.0314 | 0.0072 | 0.0129 | 0.4136 | 0.0042 | 0.0129 | 0.4371 | 0.0061 | 0.0127 | 0.4319 | |||

${\lambda}_{2}$ | 0.0230 | 0.0206 | 0.1993 | 96.0% | 0.0089 | 0.0088 | 0.0188 | 0.0049 | 0.1587 | 0.0189 | 0.0053 | 0.1587 | 0.0118 | 0.0046 | 0.1564 | |||

$\beta $ | −0.0074 | 0.0031 | 0.1471 | 96.2% | 0.0076 | 0.0074 | 0.0010 | 0.0012 | 0.1248 | 0.0113 | 0.0013 | 0.1285 | 0.0052 | 0.0009 | 0.1284 |

**Table 4.**Some simulation measures from MLE, bootstrap, Bayesian based on SELF, and ELF for parameters of PR distribution based on TRV: ${\lambda}_{1}=0.6,\gamma =2$, ${\lambda}_{2}=0.7$, $\beta =0.8$.

MLE | Bootstrap | SELF | ELF c = −1.25 | ELF c = 1.25 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

n | $\tau $ | m | Bias | MSE | LACI | CP | LBP | LBT | Bias | MSE | LCCI | Bias | MSE | LCCI | Bias | MSE | LCCI | |

40 | 0.6 | 25 | ${\lambda}_{1}$ | −0.0942 | 0.0975 | 1.8407 | 94.4% | 0.0379 | 0.0382 | −0.0613 | 0.0136 | 0.3633 | −0.0599 | 0.0134 | 0.3638 | −0.0732 | 0.0149 | 0.3561 |

$\gamma $ | 0.3890 | 0.5133 | 2.3610 | 96.2% | 0.1002 | 0.1019 | 0.0160 | 0.0285 | 0.6367 | 0.0168 | 0.0285 | 0.6383 | 0.0085 | 0.0284 | 0.6491 | |||

${\lambda}_{2}$ | 0.2001 | 0.1825 | 1.4812 | 95.0% | 0.0679 | 0.0678 | 0.0559 | 0.0153 | 0.4141 | 0.0576 | 0.0156 | 0.4147 | 0.0404 | 0.0127 | 0.3986 | |||

$\beta $ | −0.1482 | 0.2855 | 1.1319 | 96.4% | 0.0486 | 0.0491 | −0.0486 | 0.0159 | 0.4415 | −0.0471 | 0.0157 | 0.4391 | −0.0625 | 0.0177 | 0.4456 | |||

35 | ${\lambda}_{1}$ | 0.0820 | 0.0967 | 1.1768 | 95.0% | 0.0523 | 0.0526 | 0.0593 | 0.0128 | 0.3374 | 0.0426 | 0.0128 | 0.3276 | 0.0682 | 0.0135 | 0.3458 | ||

$\gamma $ | 0.3482 | 0.5068 | 2.2636 | 97.2% | 0.1198 | 0.1191 | 0.0148 | 0.0188 | 0.5247 | 0.0162 | 0.0188 | 0.5251 | 0.0081 | 0.0185 | 0.5277 | |||

${\lambda}_{2}$ | 0.0160 | 0.0805 | 1.1115 | 95.6% | 0.0485 | 0.0488 | 0.0539 | 0.0106 | 0.3311 | 0.0550 | 0.0108 | 0.3326 | 0.0394 | 0.0092 | 0.3227 | |||

$\beta $ | 0.1352 | 0.2146 | 1.1057 | 96.8% | 0.0723 | 0.0729 | 0.0410 | 0.0150 | 0.3672 | 0.0375 | 0.0152 | 0.3695 | 0.0615 | 0.0132 | 0.3601 | |||

0.9 | 25 | ${\lambda}_{1}$ | −0.0822 | 0.0417 | 0.7350 | 88.2% | 0.0358 | 0.0359 | −0.0486 | 0.0056 | 0.2287 | −0.0478 | 0.0056 | 0.2278 | −0.0557 | 0.0063 | 0.2278 | |

$\gamma $ | 0.1871 | 0.2469 | 1.8064 | 95.4% | 0.0837 | 0.0843 | 0.0053 | 0.0219 | 0.5513 | 0.0061 | 0.0219 | 0.5507 | −0.0072 | 0.0220 | 0.5530 | |||

${\lambda}_{2}$ | 0.1577 | 0.0430 | 0.5280 | 94.8% | 0.0249 | 0.0249 | 0.0311 | 0.0088 | 0.3443 | 0.0325 | 0.0089 | 0.3446 | 0.0182 | 0.0077 | 0.3279 | |||

$\beta $ | −0.0414 | 0.0722 | 0.8377 | 95.7% | 0.0383 | 0.0383 | −0.0459 | 0.0133 | 0.4165 | −0.0456 | 0.0132 | 0.4165 | −0.0609 | 0.0164 | 0.3715 | |||

35 | ${\lambda}_{1}$ | 0.0793 | 0.0371 | 0.6949 | 95.8% | 0.0428 | 0.0423 | 0.0381 | 0.0052 | 0.2133 | 0.0391 | 0.0052 | 0.2033 | 0.0481 | 0.0051 | 0.2070 | ||

$\gamma $ | −0.1791 | 0.1439 | 1.2860 | 95.6% | 0.0567 | 0.0565 | 0.0041 | 0.0169 | 0.5039 | 0.0046 | 0.0169 | 0.5032 | −0.0018 | 0.0170 | 0.5084 | |||

${\lambda}_{2}$ | 0.0104 | 0.0405 | 0.2861 | 95.4% | 0.0128 | 0.0126 | 0.0305 | 0.0087 | 0.2515 | 0.0305 | 0.0088 | 0.2516 | 0.0141 | 0.0073 | 0.2444 | |||

$\beta $ | −0.0392 | 0.0681 | 0.7567 | 95.8% | 0.0336 | 0.0339 | 0.0092 | 0.0106 | 0.3479 | 0.0106 | 0.0130 | 0.3048 | −0.0533 | 0.0122 | 0.3482 | |||

100 | 0.6 | 75 | ${\lambda}_{1}$ | −0.0155 | 0.0225 | 0.5852 | 95.4% | 0.0263 | 0.0265 | 0.0223 | 0.0051 | 0.2183 | 0.0382 | 0.0054 | 0.2185 | 0.0144 | 0.0056 | 0.2177 |

$\gamma $ | 0.1731 | 0.2335 | 1.7403 | 95.4% | 0.0771 | 0.0773 | 0.0047 | 0.0214 | 0.5357 | 0.0059 | 0.0204 | 0.5371 | 0.0062 | 0.0203 | 0.5419 | |||

${\lambda}_{2}$ | 0.0443 | 0.0370 | 0.4735 | 94.6% | 0.0232 | 0.0239 | 0.0297 | 0.0081 | 0.2934 | 0.0317 | 0.0081 | 0.2942 | 0.0176 | 0.0071 | 0.2850 | |||

$\beta $ | 0.0413 | 0.0556 | 0.6791 | 95.8% | 0.0340 | 0.0377 | −0.0133 | 0.0086 | 0.3675 | −0.0412 | 0.0086 | 0.3666 | −0.0523 | 0.0090 | 0.3623 | |||

90 | ${\lambda}_{1}$ | 0.0147 | 0.0213 | 0.5674 | 95.9% | 0.0323 | 0.0320 | 0.0213 | 0.0042 | 0.2086 | 0.0315 | 0.0052 | 0.2019 | 0.0141 | 0.0050 | 0.1928 | ||

$\gamma $ | 0.1328 | 0.1320 | 1.1807 | 96.4% | 0.0379 | 0.0478 | 0.0039 | 0.0147 | 0.4574 | 0.0043 | 0.0148 | 0.4569 | 0.0017 | 0.0144 | 0.4606 | |||

${\lambda}_{2}$ | 0.0102 | 0.0345 | 0.2722 | 96.0% | 0.0099 | 0.0113 | 0.0283 | 0.0079 | 0.2453 | 0.0300 | 0.0081 | 0.2503 | 0.0136 | 0.0071 | 0.2408 | |||

$\beta $ | 0.0318 | 0.0510 | 0.6704 | 96.4% | 0.0305 | 0.0305 | 0.0090 | 0.0079 | 0.3334 | 0.0102 | 0.0082 | 0.3004 | 0.0483 | 0.0081 | 0.3347 | |||

0.9 | 75 | ${\lambda}_{1}$ | 0.0149 | 0.0184 | 0.4049 | 94.0% | 0.0188 | 0.0188 | 0.0217 | 0.0042 | 0.2022 | 0.0280 | 0.0043 | 0.2022 | 0.0123 | 0.0038 | 0.2130 | |

$\gamma $ | −0.1066 | 0.0595 | 0.8613 | 95.8% | 0.0386 | 0.0386 | 0.0046 | 0.0177 | 0.4874 | 0.0054 | 0.0178 | 0.4883 | 0.0053 | 0.0171 | 0.4817 | |||

${\lambda}_{2}$ | 0.0416 | 0.0274 | 0.1928 | 95.6% | 0.0093 | 0.0092 | 0.0260 | 0.0060 | 0.1798 | 0.0306 | 0.0061 | 0.1802 | 0.0153 | 0.0050 | 0.1755 | |||

$\beta $ | −0.0396 | 0.0414 | 0.4319 | 95.2% | 0.0189 | 0.0190 | −0.0116 | 0.0083 | 0.3528 | −0.0401 | 0.0083 | 0.3525 | −0.0514 | 0.0074 | 0.3550 | |||

90 | ${\lambda}_{1}$ | 0.0132 | 0.0172 | 0.3954 | 95.8% | 0.0237 | 0.0232 | 0.0202 | 0.0043 | 0.1926 | 0.0250 | 0.0042 | 0.1826 | 0.0114 | 0.0032 | 0.1821 | ||

$\gamma $ | −0.0912 | 0.0495 | 0.7302 | 96.5% | 0.0332 | 0.0323 | 0.0031 | 0.0124 | 0.4160 | 0.0040 | 0.0124 | 0.4165 | 0.0017 | 0.0123 | 0.4177 | |||

${\lambda}_{2}$ | 0.0092 | 0.0256 | 0.1896 | 96.5% | 0.0090 | 0.0089 | 0.0228 | 0.0048 | 0.1562 | 0.0249 | 0.0058 | 0.1565 | 0.0128 | 0.0048 | 0.1494 | |||

$\beta $ | −0.0265 | 0.0408 | 0.4249 | 95.6% | 0.0208 | 0.0207 | 0.0066 | 0.0070 | 0.3079 | 0.0041 | 0.0071 | 0.3179 | −0.0305 | 0.0069 | 0.3287 |

**Table 5.**Optimal stress change time $\tau $ for different sample sizes and parameter values by SVC $\varphi \left(\tau \right)$.

n | $\mathit{\tau}$ | m | Table 1 | Table 2 | Table 3 | Table 4 |
---|---|---|---|---|---|---|

40 | 0.6 | 25 | 0.4225 | 0.3565 | 0.5263 | 0.5119 |

35 | 0.3850 | 0.3477 | 0.3439 | 0.3934 | ||

0.9 | 25 | 0.4215 | 0.3301 | 0.2896 | 0.2695 | |

35 | 0.3384 | 0.3605 | 0.2680 | 0.2661 | ||

100 | 0.6 | 75 | 0.3600 | 0.2797 | 0.1999 | 0.2476 |

90 | 0.3199 | 0.2737 | 0.1683 | 0.2312 | ||

0.9 | 75 | 0.2850 | 0.2638 | 0.1686 | 0.2050 | |

90 | 0.2349 | 0.2352 | 0.1567 | 0.2008 |

${t}_{i}$ | c | ${t}_{i}$ | c | ${t}_{i}$ | c | ${t}_{i}$ | c | ${t}_{i}$ | c | ${t}_{i}$ | c | ${t}_{i}$ | c | ${t}_{i}$ | c | ${t}_{i}$ | c | ${t}_{i}$ | c |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.112 | 1 | 2.048 | 1 | 2.798 | 1 | 3.373 | 0 | 3.8 | 0 | 4.389 | 1 | 5.018 | 0 | 5.566 | 0 | 5.982 | 1 | 6.461 | 0 |

0.137 | 1 | 2.053 | 1 | 2.814 | 1 | 3.439 | 1 | 3.817 | 1 | 4.394 | 1 | 5.021 | 1 | 5.574 | 0 | 6.018 | 1 | 6.511 | 1 |

0.474 | 1 | 2.155 | 0 | 2.866 | 1 | 3.477 | 0 | 3.819 | 0 | 4.4 | 1 | 5.082 | 1 | 5.582 | 0 | 6.042 | 0 | 6.516 | 1 |

0.824 | 1 | 2.177 | 0 | 2.875 | 0 | 3.477 | 1 | 3.88 | 1 | 4.52 | 1 | 5.106 | 1 | 5.618 | 0 | 6.042 | 1 | 6.579 | 0 |

0.884 | 1 | 2.234 | 0 | 2.891 | 0 | 3.486 | 0 | 3.94 | 1 | 4.523 | 1 | 5.12 | 1 | 5.667 | 0 | 6.045 | 0 | 6.733 | 0 |

0.969 | 1 | 2.283 | 0 | 2.982 | 1 | 3.513 | 0 | 3.953 | 0 | 4.583 | 0 | 5.224 | 1 | 5.678 | 0 | 6.054 | 0 | 6.801 | 0 |

1.013 | 1 | 2.322 | 1 | 3.039 | 1 | 3.535 | 0 | 3.975 | 0 | 4.608 | 0 | 5.251 | 0 | 5.7 | 1 | 6.177 | 0 | 6.82 | 1 |

1.101 | 1 | 2.513 | 1 | 3.064 | 0 | 3.584 | 1 | 4.033 | 1 | 4.69 | 0 | 5.314 | 1 | 5.703 | 1 | 6.195 | 0 | 6.85 | 1 |

1.205 | 1 | 2.533 | 0 | 3.064 | 1 | 3.592 | 0 | 4.079 | 1 | 4.734 | 1 | 5.336 | 1 | 5.723 | 0 | 6.199 | 0 | 6.866 | 0 |

1.44 | 0 | 2.565 | 1 | 3.195 | 0 | 3.639 | 0 | 4.099 | 0 | 4.811 | 0 | 5.374 | 1 | 5.73 | 1 | 6.218 | 1 | 6.943 | 1 |

1.462 | 1 | 2.571 | 1 | 3.214 | 0 | 3.647 | 0 | 4.219 | 0 | 4.854 | 1 | 5.454 | 1 | 5.736 | 1 | 6.224 | 0 | 6.955 | 0 |

1.503 | 1 | 2.631 | 1 | 3.22 | 1 | 3.663 | 0 | 4.219 | 0 | 4.909 | 1 | 5.478 | 1 | 5.886 | 1 | 6.267 | 0 | 6.979 | 1 |

1.593 | 1 | 2.672 | 0 | 3.242 | 0 | 3.707 | 0 | 4.23 | 1 | 4.966 | 1 | 5.525 | 1 | 5.889 | 1 | 6.311 | 1 | 7.006 | 0 |

1.837 | 0 | 2.683 | 0 | 3.258 | 1 | 3.724 | 0 | 4.334 | 1 | 4.981 | 0 | 5.555 | 0 | 5.908 | 1 | 6.412 | 0 | 7.17 | 1 |

1.889 | 1 | 2.705 | 0 | 3.315 | 0 | 3.797 | 1 | 4.375 | 1 | 5.013 | 0 | 5.563 | 1 | 5.938 | 0 | 6.439 | 1 | 7.302 | 0 |

${\mathit{t}}_{\mathit{i}}<\mathit{\tau}$ | Size | KSD | PVKS | AIC | BIC | CAIC | HQIC | |||
---|---|---|---|---|---|---|---|---|---|---|

${t}_{i}<max\left(t\right)$ | 220 | $\lambda $ | 5.6859 | 0.9111 | 0.0890 | 0.1853 | 602.8644 | 608.8856 | 602.9460 | 605.3106 |

$\gamma $ | 1.3300 | 0.0913 | ||||||||

${t}_{i}<\tau $ | 121 | $\lambda $ | 3.9018 | 0.7312 | 0.1217 | 0.1813 | 260.7185 | 265.5074 | 260.8724 | 262.6399 |

$\gamma $ | 1.4365 | 0.1381 | ||||||||

$\tau <{t}_{i}<max\left(t\right)$ | 101 | $\lambda $ | 44.2060 | 20.9781 | 0.0612 | 0.9585 | 145.3549 | 152.0572 | 145.7241 | 148.0139 |

$\gamma $ | 1.1368 | 0.1974 | ||||||||

$\beta $ | 0.0432 | 0.0353 |

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

Estimates | StEr | Lower | Upper | Estimates | StEr | Lower | Upper | |

${\lambda}_{1}$ | 6.0925 | 1.0225 | 4.0884 | 8.0966 | 6.1924 | 0.7669 | 4.6570 | 7.6641 |

$\gamma $ | 1.1113 | 0.1052 | 0.9050 | 1.3175 | 1.1176 | 0.0774 | 0.9679 | 1.2601 |

${\lambda}_{2}$ | 6.6004 | 1.1181 | 4.4089 | 8.7919 | 6.7538 | 0.8359 | 5.3139 | 8.4690 |

$\beta $ | 0.5539 | 0.0991 | 0.3596 | 0.7482 | 0.5801 | 0.1257 | 0.3312 | 0.8086 |

$1-{F}_{1}(\overline{t};\gamma ,{\lambda}_{1})$ | 0.70965 | 0.71305 | ||||||

$1-{F}_{2}(\overline{t};\gamma ,{\lambda}_{2})$ | 0.74660 | 0.75253 | ||||||

$1-{F}_{1}(\overline{t};\gamma ,{\lambda}_{1},\beta )$ | 0.74056 | 0.74077 | ||||||

$1-{F}_{2}(\overline{t};\gamma ,{\lambda}_{2},\beta )$ | 0.77422 | 0.77705 | ||||||

$1-\left(1-{F}_{1}\left({t}_{r}\right)\right)\left(1-{F}_{2}\left({t}_{r}\right)\right)$ | 0.97519 | 0.97895 |

${t}_{i}$ | c | ${t}_{i}$ | c | ${t}_{i}$ | c | ${t}_{i}$ | c | ${t}_{i}$ | c |

11 | 1 | 381 | 1 | 1594 | 1 | 2400 | 0 | 2694 | 0 |

35 | 1 | 708 | 1 | 1925 | 0 | 2451 | 1 | 2702 | 1 |

49 | 1 | 958 | 1 | 1990 | 0 | 2471 | 0 | 2761 | 1 |

170 | 1 | 1062 | 1 | 2223 | 0 | 2551 | 0 | 2831 | 1 |

329 | 1 | 1167 | 0 | 2327 | 1 | 2568 | 0 | 3034 | 0 |

${\mathit{x}}_{\mathit{i}}<\mathit{\tau}$ | Size | Estimate | StEr | KSD | PVKS | AIC | BIC | CAIC | HQIC | |
---|---|---|---|---|---|---|---|---|---|---|

${x}_{i}<max\left({x}_{i}\right)$ | 25 | $\lambda $ | 53.3420 | 43.2098 | 0.2403 | 0.0938 | 423.9574 | 426.3951 | 424.5028 | 424.6335 |

$\gamma $ | 0.5803 | 0.1044 | ||||||||

${x}_{i}<\tau $ | 18 | $\lambda $ | 22.3780 | 16.9145 | 0.1772 | 0.5644 | 296.2920 | 298.0727 | 297.0920 | 296.5375 |

$\gamma $ | 0.4859 | 0.1002 | ||||||||

$\tau <{x}_{i}<max\left({x}_{i}\right)$ | 7 | $\lambda $ | 2814.8644 | 5262.1741 | 0.2066 | 0.8722 | 94.8192 | 94.6569 | 102.8192 | 92.8135 |

$\gamma $ | 0.7401 | 0.1678 | ||||||||

$\beta $ | 0.0036 | 0.0012 |

**Table 11.**MLE and Bayesian estimation for the parameters of PR based on SSLT under TRV: electrical appliances data.

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

Estimates | StEr | Lower | Upper | Estimates | StEr | Lower | Upper | |

${\lambda}_{1}$ | 30.436 | 11.750 | 7.407 | 73.065 | 30.475 | 2.933 | 24.635 | 36.016 |

$\gamma $ | 0.456 | 0.091 | 0.277 | 0.635 | 0.453 | 0.022 | 0.412 | 0.495 |

${\lambda}_{2}$ | 37.277 | 16.856 | 4.240 | 89.915 | 37.313 | 3.475 | 31.022 | 44.277 |

$\beta $ | 0.086 | 0.043 | 0.0015 | 0.179 | 0.096 | 0.042 | 0.020 | 0.171 |

$1-{F}_{1}({t}_{r};\gamma ,{\lambda}_{1})$ | 0.44465 | 0.46349 | ||||||

$1-{F}_{2}({t}_{r};\gamma ,{\lambda}_{2})$ | 0.58258 | 0.59872 | ||||||

$1-{F}_{1}({t}_{r};\gamma ,{\lambda}_{1},\beta )$ | 0.12041 | 0.15590 | ||||||

$1-{F}_{2}({t}_{r};\gamma ,{\lambda}_{2},\beta )$ | 0.24386 | 0.28945 | ||||||

$1-\left(1-{F}_{1}\left({t}_{r}\right)\right)\left(1-{F}_{2}\left({t}_{r}\right)\right)$ | 0.97064 | 0.97549 |

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**MDPI and ACS Style**

Ahmad, H.H.; Ramadan, D.A.; Almetwally, E.M.
Tampered Random Variable Analysis in Step-Stress Testing: Modeling, Inference, and Applications. *Mathematics* **2024**, *12*, 1248.
https://doi.org/10.3390/math12081248

**AMA Style**

Ahmad HH, Ramadan DA, Almetwally EM.
Tampered Random Variable Analysis in Step-Stress Testing: Modeling, Inference, and Applications. *Mathematics*. 2024; 12(8):1248.
https://doi.org/10.3390/math12081248

**Chicago/Turabian Style**

Ahmad, Hanan Haj, Dina A. Ramadan, and Ehab M. Almetwally.
2024. "Tampered Random Variable Analysis in Step-Stress Testing: Modeling, Inference, and Applications" *Mathematics* 12, no. 8: 1248.
https://doi.org/10.3390/math12081248