Investigating the Lifetime Performance Index under Ishita Distribution Based on Progressive Type II Censored Data with Applications
Abstract
:1. Introduction
2. The Performance Index and Conforming Rate of the Ishita Distribution
3. Parameter Estimation and Testing Hypothesis
3.1. Maximum-Likelihood Estimation
3.2. Bayes Estimation
- Begin with the first suggestion .
- Specify .
- Generate from using the M-H method with the normal distribution
- (a)
- Produce a proposal from .
- (b)
- Determine the probability of acceptance
- (c)
- Produce a u from a uniform distribution.
- (d)
- Accept the proposal and set if ; otherwise set .
- Calculate the as
- Let .
- Steps (3)–(5) are repeated N times to obtain and
- Evaluate the credible intervals of and order and as and . Then, the credible intervals of be .
- Using the progressive type II censoring sample and the censoring scheme , find the MLE of the of the Ishita distribution.
- The L is predetermined; therefore, calculate the . The statistical test for lifetime performance is thus constructed as: against .
- Set the significance level.
- Determine the ’s lower bound as the lower confidence interval, .
- Lastly, the choice is made as if , and is rejected.
4. Real Data
- Step 1:
- Step 2:
- It is expected that the L is ; i.e., when the lifetime of the strength data for an aircraft window exceeds , the strength data are considered to be a conforming product. The of items must be greater than in order to address concerns raised by product buyers regarding lifetime performance. The value must exceed 1 according to Table 3. As a result, the is set at , and the following is tested: vs. .
- Step 3:
- Choose a level of significance of .
- Step 4:
- As a result, the one-sided CI for is .
- Step 5.
- As a result of the is refused.
−11 | −0.25 | 0.333439 | 0.499 | 0.640043 | |
−7 | 0.0000352065 | 0 | 0.423224 | 0.513 | 0.646582 |
−6.5 | 0.000749944 | 0.1 | 0.463129 | 0.534 | 0.656405 |
−6 | 0.000158733 | 0.2 | 0.505102 | 0.556 | 0.66671 |
−5.75 | 0.000230328 | 0.25 | 0.526791 | 0.589 | 0.682179 |
−5 | 0.000695385 | 0.3 | 0.548898 | 0.618 | 0.695765 |
−4 | 0.00293547 | 0.32 | 0.557848 | 0.634 | 0.703252 |
−3 | 0.0117921 | 0.34 | 0.566855 | 0.758 | 0.760654 |
−2.5 | 0.0230645 | 0.35 | 0.571378 | 0.897 | 0.822204 |
−2 | 0.0441834 | 0.353 | 0.572738 | 0.95 | 0.844379 |
−1 | 0.014911 | 0.412 | 0.599694 | 1 | 0.864429 |
−0.5 | 0.258413 | 0.478 | 0.630253 | 1.73215 | 1 |
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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−11 | −0.25 | 0.327713 | 0.499 | 0.611436 | |
−7 | 0.0000382405 | 0 | 0.410495 | 0.513 | 0.617739 |
−6.5 | 0.0000816901 | 0.1 | 0.447049 | 0.534 | 0.627266 |
−6 | 0.000173278 | 0.2 | 0.485503 | 0.556 | 0.63734 |
−5.75 | 0.000251629 | 0.25 | 0.505425 | 0.589 | 0.65264 |
−5 | 0.000760398 | 0.3 | 0.525805 | 0.618 | 0.666279 |
−4 | 0.0032004 | 0.32 | 0.534084 | 0.634 | 0.673885 |
−3 | 0.0127309 | 0.34 | 0.542436 | 0.758 | 0.73503 |
−2.5 | 0.0246903 | 0.35 | 0.546639 | 0.897 | 0.809544 |
−2 | 0.0467506 | 0.353 | 0.547903 | 0.95 | 0.840259 |
−1 | 0.152262 | 0.412 | 0.573104 | 1 | 0.870775 |
−0.5 | 0.257434 | 0.478 | 0.602053 | 1.1818996 | 1 |
18.83 | 20.80 | 21.657 | 23.03 | 23.23 | 24.05 | 24.321 | 25.50 | |
10 | 5 | 1 | 0 | 0 | 0 | 0 | 0 | |
25.52 | 25.80 | 26.69 | 26.77 | 26.78 | 27.05 | 27.67 | 29.90 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Parameter | MLE | SEL | LINEX | ||
---|---|---|---|---|---|
0.089449 | 4.52857 | 10.0237 | 1.47632 | 4.52819 | |
0.99997 | 1.09643 | 2.20227 | 1.01326 | 1.09641 |
Parameter | MLE | MCMC |
---|---|---|
(−136.364, 136.543) | (0.291955, 10.1095) | |
(0.99997, 1.22154) | (0.965014, 3.22197) |
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Haj Ahmad, H.; Elnagar, K.; Ramadan, D. Investigating the Lifetime Performance Index under Ishita Distribution Based on Progressive Type II Censored Data with Applications. Symmetry 2023, 15, 1779. https://doi.org/10.3390/sym15091779
Haj Ahmad H, Elnagar K, Ramadan D. Investigating the Lifetime Performance Index under Ishita Distribution Based on Progressive Type II Censored Data with Applications. Symmetry. 2023; 15(9):1779. https://doi.org/10.3390/sym15091779
Chicago/Turabian StyleHaj Ahmad, Hanan, Kariema Elnagar, and Dina Ramadan. 2023. "Investigating the Lifetime Performance Index under Ishita Distribution Based on Progressive Type II Censored Data with Applications" Symmetry 15, no. 9: 1779. https://doi.org/10.3390/sym15091779
APA StyleHaj Ahmad, H., Elnagar, K., & Ramadan, D. (2023). Investigating the Lifetime Performance Index under Ishita Distribution Based on Progressive Type II Censored Data with Applications. Symmetry, 15(9), 1779. https://doi.org/10.3390/sym15091779