A New Reliability Class-Test Statistic for Life Distributions under Convolution, Mixture and Homogeneous Shock Model: Characterizations and Applications in Engineering and Medical Fields
Abstract
:1. Introduction
2. Closure Properties
- 1.
- Property of convolution: The NBRUL class is preserved under convolution, whereExample 1.The convolution of the exponential distribution with itself yields the gamma distribution of order 2: , with strictly increasing failure rate. Thus, is not NWRUL.
- 2.
- Property of mixture: The NWRUL class is preserved under mixture, whereExample 2.Let “scale parameter” and . Then the failure rate function is which is strictly decreasing; thus, is not NBRUL.
- 3.
- The shock model under a homogeneous Poisson process: Suppose the device is subjected to a series of shocks that occur at random time intervals using the Poisson process with intensity . Further suppose that the device has a probability . From surviving the first shock k, where and . Then, the survival function of the device is given by
3. NBRUL Comparative Testing Alternatives
4. The Pitman Asymptotic Efficiency (PAE) of
5. Critical Points for Monte Carlo Distribution
Estimations of Test Power
6. Censoring Data Testing
Test Power Estimates
7. Applications: Uncensored and Censored Observations
7.1. Uncensored Data
7.1.1. Data Set I: COVID-19-Italy
7.1.2. Data Set II: COVID-19-Netherlands
7.1.3. Data Set III: Aircraft Air Conditioning
7.1.4. Data Set IV: Leukemia
7.2. Censored Data
7.2.1. Data Set V: Melanoma Patients
13 | 14 | 19 | 19 | 20 | 21 | 23 | 23 | 25 | 26 | 26 | 27 |
27 | 31 | 32 | 34 | 34 | 37 | 38 | 38 | 40 | 46 | 50 | 53 |
54 | 57 | 58 | 59 | 60 | 65 | 65 | 66 | 70 | 85 | 90 | 98 |
102 | 103 | 110 | 118 | 124 | 130 | 136 | 138 | 141 | 234 |
16 | 21 | 44 | 50 | 55 | 67 | 73 | 76 | 80 | 81 | 86 | 93 |
100 | 108 | 114 | 120 | 124 | 125 | 129 | 130 | 132 | 134 | 140 | 147 |
148 | 151 | 152 | 152 | 158 | 181 | 190 | 193 | 194 | 213 | 215 |
7.2.2. Data Set VI: Blood Cancer
8. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Models | |||
---|---|---|---|
Test | Makeham | LFR | Weibull |
Mugdadi and Ahmad [23] | 0.039 | 0.408 | 0.170 |
Kango [24] | 0.144 | 0.433 | 0.132 |
Abdel-Aziz [25] | 0.184 | 0.535 | 0.223 |
Etman et al. [26] | 0.233 | 0.932 | 1.046 |
EL-Sagheer et al. [13] | 0.287 | 0.901 | 1.158 |
Proposed test | 0.280 | 0.946 | 1.116 |
Sample Size | Confidence Levels | ||
---|---|---|---|
5 | |||
10 | |||
15 | |||
20 | |||
25 | |||
29 | |||
30 | |||
35 | |||
40 | |||
43 | |||
45 | |||
50 | |||
59 |
n | Weibull | Gamma | |
---|---|---|---|
10 | 2 3 4 | ||
20 | 2 3 4 | ||
30 | 2 3 4 |
Sample Size | Confidence Intervals | ||
---|---|---|---|
10 | |||
20 | |||
30 | |||
40 | |||
50 | |||
51 | |||
60 | |||
70 | |||
80 | |||
81 |
n | Weibull | LFR | Gamma | |
---|---|---|---|---|
10 | 2 3 4 | |||
20 | 2 3 4 | |||
30 | 2 3 4 |
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Etman, W.B.; El-Morshedy, M.; Eliwa, M.S.; Almohaimeed, A.; EL-Sagheer, R.M. A New Reliability Class-Test Statistic for Life Distributions under Convolution, Mixture and Homogeneous Shock Model: Characterizations and Applications in Engineering and Medical Fields. Axioms 2023, 12, 331. https://doi.org/10.3390/axioms12040331
Etman WB, El-Morshedy M, Eliwa MS, Almohaimeed A, EL-Sagheer RM. A New Reliability Class-Test Statistic for Life Distributions under Convolution, Mixture and Homogeneous Shock Model: Characterizations and Applications in Engineering and Medical Fields. Axioms. 2023; 12(4):331. https://doi.org/10.3390/axioms12040331
Chicago/Turabian StyleEtman, Walid B., Mahmoud El-Morshedy, Mohamed S. Eliwa, Amani Almohaimeed, and Rashad M. EL-Sagheer. 2023. "A New Reliability Class-Test Statistic for Life Distributions under Convolution, Mixture and Homogeneous Shock Model: Characterizations and Applications in Engineering and Medical Fields" Axioms 12, no. 4: 331. https://doi.org/10.3390/axioms12040331
APA StyleEtman, W. B., El-Morshedy, M., Eliwa, M. S., Almohaimeed, A., & EL-Sagheer, R. M. (2023). A New Reliability Class-Test Statistic for Life Distributions under Convolution, Mixture and Homogeneous Shock Model: Characterizations and Applications in Engineering and Medical Fields. Axioms, 12(4), 331. https://doi.org/10.3390/axioms12040331