Reliability Estimation in Multicomponent StressStrength Based on Inverse Weibull Distribution
Abstract
:1. Introduction
2. Maximum Likelihood Estimation of ${\mathit{R}}_{\mathit{s},\mathit{k}}$
Asymptotic Confidence Intervals
3. Simulation Study and Data Analysis
3.1. Simulation Study
3.2. Real Data Application
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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(s, k)  (α, β)  

(3, 1.5)  (2.5, 1.5)  (2, 1.5)  (1.5, 1.5)  (1.5, 2)  (1.5, 2.5)  (1.5, 3)  
(1, 3)  0.857143  0.833333  0.800000  0.750000  0.692308  0.642857  0.600000 
(3, 5)  0.692641  0.646998  0.585812  0.500000  0.409919  0.340330  0.285714 
n  m  (α, β)  ABias  AMSE  ASE  ALCI  ACP  

$${\mathit{R}}_{1,3}$$

$${\mathit{R}}_{3,5}$$

$${\mathit{R}}_{1,3}$$

$${\mathit{R}}_{3,5}$$

$${\mathit{R}}_{1,3}$$

$${\mathit{R}}_{3,5}$$

$${\mathit{R}}_{1,3}$$

$${\mathit{R}}_{3,5}$$

$${\mathit{R}}_{1,3}$$

$${\mathit{R}}_{3,5}$$
 
10  10  (3, 1.5)  −0.0564  −0.1044  0.0045  0.0150  0.0708  0.1258  0.2774  0.4931  0.9989  0.9951 
15  15  −0.0561  −0.1032  0.0040  0.0134  0.0579  0.1030  0.2268  0.4039  0.9966  0.9863  
20  20  −0.0550  −0.1021  0.0037  0.0125  0.0500  0.0893  0.1959  0.3501  0.9905  0.9720  
25  25  −0.0548  −0.1010  0.0045  0.0120  0.0447  0.0799  0.1752  0.3131  0.9753  0.9458  
30  30  −0.0544  −0.1011  0.0034  0.0116  0.0408  0.0730  0.1598  0.2861  0.9526  0.9102  
10  10  (2.5, 1.5)  −0.0473  −0.0832  0.0036  0.0111  0.0746  0.1293  0.2924  0.5068  0.9986  0.9956 
15  15  −0.0468  −0.0826  0.0031  0.0097  0.0610  0.1060  0.2390  0.4154  0.9984  0.9957  
20  20  −0.0452  −0.0815  0.0027  0.0088  0.0526  0.0919  0.2062  0.3602  0.9977  0.9900  
25  25  −0.0450  −0.0810  0.0026  0.0083  0.0470  0.0823  0.1844  0.3225  0.9960  0.9851  
30  30  −0.0447  −0.0808  0.0025  0.0079  0.0429  0.0752  0.1682  0.2946  0.9899  0.9763  
10  10  (2, 1.5)  −0.0303  −0.0509  0.0024  0.0068  0.0786  0.1327  0.3082  0.5203  0.9992  0.9983 
15  15  −0.0293  −0.0501  0.0018  0.0053  0.0642  0.1088  0.2516  0.4266  0.9994  0.9984  
20  20  −0.0291  −0.0498  0.0016  0.0047  0.0556  0.0944  0.2180  0.3701  0.9992  0.9981  
25  25  −0.0288  −0.0497  0.0014  0.0041  0.0497  0.0846  0.1950  0.3316  0.9994  0.9981  
30  30  −0.0289  −0.0490  0.0013  0.0038  0.0454  0.0772  0.1781  0.3028  0.9989  0.9974  
10  10  (1.5, 1.5)  −0.0023  −0.0004  0.0016  0.0041  0.0837  0.1360  0.3280  0.5331  0.9989  0.9989 
15  15  −0.0095  −0.0019  0.0010  0.0028  0.0682  0.1116  0.2673  0.4375  0.9993  0.9993  
20  20  −0.0014  −0.0002  0.0008  0.0022  0.0593  0.0968  0.2324  0.3794  0.9995  0.9997  
25  25  −0.0010  −0.0003  0.0006  0.0017  0.0530  0.0867  0.2077  0.3399  0.9995  0.9997  
30  30  −0.0004  −0.0007  0.0005  0.0014  0.0483  0.0793  0.1895  0.3107  0.9995  0.9998  
10  10  (1.5, 2)  0.0321  0.0525  0.0030  0.0073  0.0884  0.1378  0.3465  0.5402  0.9928  0.9959 
15  15  0.0332  0.0517  0.0024  0.0058  0.0723  0.1132  0.2833  0.4436  0.9913  0.9958  
20  20  0.0328  0.0522  0.0020  0.0050  0.0627  0.0983  0.2459  0.3852  0.9932  0.9963  
25  25  0.0336  0.0521  0.0019  0.0045  0.0561  0.0880  0.2198  0.3451  0.9888  0.9954  
30  30  0.0332  0.0524  0.0017  0.0043  0.0513  0.0805  0.2009  0.3154  0.9887  0.9928  
10  10  (1.5, 2.5)  0.0623  0.0907  0.0061  0.0131  0.0920  0.1383  0.3607  0.5422  0.9769  0.9877 
15  15  0.0622  0.0906  0.0053  0.0115  0.0754  0.1136  0.2956  0.4454  0.9645  0.9855  
20  20  0.0615  0.0902  0.0049  0.0106  0.0655  0.0987  0.2568  0.3870  0.9536  0.9777  
25  25  0.0622  0.0902  0.0048  0.0101  0.0586  0.0885  0.2296  0.3468  0.9324  0.9677  
30  30  0.0621  0.0901  0.0046  0.0098  0.0535  0.0808  0.2098  0.3169  0.9121  0.9491  
10  10  (1.5, 3)  0.0872  0.1194  0.0101  0.0194  0.0950  0.1380  0.3725  0.5409  0.9443  0.9787 
15  15  0.0867  0.1186  0.0091  0.0175  0.0780  0.1134  0.3056  0.4444  0.9136  0.9605  
20  20  0.0868  0.1179  0.0088  0.0165  0.0676  0.0985  0.2651  0.3860  0.8618  0.9325  
25  25  0.0871  0.1176  0.0086  0.0159  0.0605  0.0883  0.2373  0.3461  0.8008  0.8984  
30  30  0.0867  0.1174  0.0083  0.0155  0.0553  0.0807  0.2169  0.3163  0.7391  0.8356 
Strength (X)  Stress (Y)  

Mean  $0.42688$  $0.33987$ 
Median  $0.40400$  $0.33400$ 
Standard Deviation  $0.09936$  $0.06703$ 
Standard Error  0.01196  0.00844 
Skewness  $1.38091$  $0.31303$ 
Kurtoses  5.38230  2.67273 
$\widehat{\alpha},\widehat{\beta}$  0.00469  0.00235 
$\widehat{\lambda}$  5.50125  5.04997 
LogLikelihood  71.12399  76.47376 
KS Test Statistic  0.05329  0.08753 
KS Test pvalue  0.98371  0.68698 
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Shawky, A.I.; Khan, K. Reliability Estimation in Multicomponent StressStrength Based on Inverse Weibull Distribution. Processes 2022, 10, 226. https://doi.org/10.3390/pr10020226
Shawky AI, Khan K. Reliability Estimation in Multicomponent StressStrength Based on Inverse Weibull Distribution. Processes. 2022; 10(2):226. https://doi.org/10.3390/pr10020226
Chicago/Turabian StyleShawky, Ahmed Ibrahim, and Khushnoor Khan. 2022. "Reliability Estimation in Multicomponent StressStrength Based on Inverse Weibull Distribution" Processes 10, no. 2: 226. https://doi.org/10.3390/pr10020226