Estimation of Reliability in a Multicomponent Stress–Strength System for the Exponentiated Moment-Based Exponential Distribution
Abstract
:1. Introduction
2. Estimation of by Maximum Likelihood
Algorithm 1 Estimation of |
Step 1: For given values of , , using Monte Carlo simulation, generate 10,000 samples such that X~EMED and Y~EMED . |
Step 2: Determine using given parametric values. |
Step 3: Find maximum likelihood estimate for and via simulation. |
Step 4: Estimate multicomponent stress–strength reliability estimate using maximum likelihood function for and . |
Step 5: Compute average bias, average mean squared errors, average length of the confidence interval, and average coverage probability (ACP)of . Where average bias = and average mean squared error (MSE) = , N = 10,000. |
Step 6: Draw a conclusion. |
3. Monte Carlo Simulation
4. Application
4.1. Example with Forest Data
4.2. Example with Breaking Strength Data
5. Comparison with Existing Distribution
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Glossary
EMED—Exponentiated moment-based exponential distribution |
α—Shape parameter |
—Scale parameter |
—Multicomponent stress–strength reliability |
—Beta first kind function |
MSE—Mean squared error |
ACP—Average coverage probability |
ACL—Average confidence length |
ML—Maximum likelihood |
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Components (s, k) | Sample Size (n,m) | Parameters (α1,α2) | ||||||
---|---|---|---|---|---|---|---|---|
(2.25,1.5) | (2.0,1.5) | (1.75,1.5) | (1.5,1.5) | (1.5,1.75) | (1.5,2.0) | (1.5,2.25) | ||
(1, 3) | (15,15) | 0.0223 | 0.0148 | 0.0069 | −0.0072 | −0.0218 | −0.0354 | −0.0510 |
(20,20) | 0.0217 | 0.0147 | 0.0066 | −0.0057 | −0.0199 | −0.0335 | −0.0475 | |
(25,25) | 0.0216 | 0.0147 | 0.0062 | −0.0051 | −0.0188 | −0.0308 | −0.0446 | |
(30,30) | 0.0215 | 0.0146 | 0.0057 | −0.0032 | −0.0172 | −0.0291 | −0.0436 | |
(35,35) | 0.0215 | 0.0133 | 0.0052 | −0.0037 | −0.0152 | −0.0291 | −0.0419 | |
(40,40) | 0.0214 | 0.0132 | 0.0049 | −0.0028 | −0.0152 | −0.0273 | −0.0412 | |
(45,45) | 0.0212 | 0.0125 | 0.0043 | −0.0019 | −0.0151 | −0.0272 | −0.0409 | |
(50,50) | 0.0211 | 0.0123 | 0.0041 | −0.0016 | −0.0144 | −0.0264 | −0.0396 | |
(2, 4) | (15,15) | 0.0344 | 0.0240 | 0.0125 | −0.0062 | −0.0251 | −0.0409 | −0.0581 |
(20,20) | 0.0341 | 0.0236 | 0.0121 | −0.0052 | −0.0237 | −0.0401 | −0.0559 | |
(25,25) | 0.0339 | 0.0233 | 0.0118 | −0.0051 | −0.0231 | −0.0376 | −0.0533 | |
(30,30) | 0.0338 | 0.0231 | 0.0115 | −0.0027 | −0.0214 | −0.0360 | −0.0528 | |
(35,35) | 0.0337 | 0.0227 | 0.0111 | −0.0037 | −0.0192 | −0.0364 | −0.0512 | |
(40,40) | 0.0336 | 0.0225 | 0.0107 | −0.0027 | −0.0194 | −0.0344 | −0.0508 | |
(45,45) | 0.0332 | 0.0223 | 0.0102 | −0.0016 | −0.0194 | −0.0349 | −0.0507 | |
(50,50) | 0.0331 | 0.0218 | 0.0101 | −0.0013 | −0.0187 | −0.0337 | −0.0493 |
Components (s, k) | Sample Size (n,m) | Parameters (α1,α2) | ||||||
---|---|---|---|---|---|---|---|---|
(2.25,1.5) | (2.0,1.5) | (1.75,1.5) | (1.5,1.5) | (1.5,1.75) | (1.5,2.0) | (1.5,2.25) | ||
(1, 3) | (15,15) | 0.0036 | 0.0041 | 0.0048 | 0.0059 | 0.0076 | 0.0097 | 0.0122 |
(20,20) | 0.0027 | 0.0030 | 0.0034 | 0.0042 | 0.0057 | 0.0073 | 0.0092 | |
(25,25) | 0.0022 | 0.0024 | 0.0027 | 0.0033 | 0.0045 | 0.0058 | 0.0075 | |
(30,30) | 0.0019 | 0.0021 | 0.0022 | 0.0027 | 0.0036 | 0.0048 | 0.0064 | |
(35,35) | 0.0017 | 0.0018 | 0.0019 | 0.0023 | 0.0030 | 0.0043 | 0.0056 | |
(40,40) | 0.0015 | 0.0016 | 0.0016 | 0.0019 | 0.0027 | 0.0036 | 0.0050 | |
(45,45) | 0.0014 | 0.0014 | 0.0015 | 0.0017 | 0.0024 | 0.0034 | 0.0046 | |
(50,50) | 0.0013 | 0.0013 | 0.0013 | 0.0015 | 0.0021 | 0.0030 | 0.0041 | |
(2, 4) | (15,15) | 0.0085 | 0.0092 | 0.0101 | 0.0114 | 0.0132 | 0.0153 | 0.0173 |
(20,20) | 0.0065 | 0.0069 | 0.0074 | 0.0083 | 0.0101 | 0.0118 | 0.0134 | |
(25,25) | 0.0053 | 0.0056 | 0.0059 | 0.0066 | 0.0081 | 0.0096 | 0.0112 | |
(30,30) | 0.0047 | 0.0048 | 0.0048 | 0.0054 | 0.0066 | 0.0079 | 0.0097 | |
(35,35) | 0.0041 | 0.0041 | 0.0042 | 0.0047 | 0.0055 | 0.0071 | 0.0085 | |
(40,40) | 0.0037 | 0.0036 | 0.0036 | 0.0039 | 0.0050 | 0.0061 | 0.0077 | |
(45,45) | 0.0034 | 0.0033 | 0.0033 | 0.0036 | 0.0044 | 0.0058 | 0.0071 | |
(50,50) | 0.0032 | 0.0029 | 0.0030 | 0.0031 | 0.0039 | 0.0050 | 0.0065 |
Components (s, k) | Sample Size (n,m) | Parameters (α1,α2) | ||||||
---|---|---|---|---|---|---|---|---|
(2.25,1.5) | (2.0,1.5) | (1.75,1.5) | (1.5,1.5) | (1.5,1.75) | (1.5,2.0) | (1.5,2.25) | ||
(1, 3) | (15,15) | 0.1893 | 0.2122 | 0.2367 | 0.2651 | 0.2915 | 0.3105 | 0.3250 |
(20,20) | 0.1643 | 0.1839 | 0.2055 | 0.2307 | 0.2537 | 0.2710 | 0.2837 | |
(25,25) | 0.1476 | 0.1649 | 0.1838 | 0.2070 | 0.2277 | 0.2429 | 0.2545 | |
(30,30) | 0.1347 | 0.1509 | 0.1684 | 0.1887 | 0.2080 | 0.2221 | 0.2331 | |
(35,35) | 0.1250 | 0.1395 | 0.1564 | 0.1752 | 0.1924 | 0.2061 | 0.2161 | |
(40,40) | 0.1170 | 0.1309 | 0.1462 | 0.1639 | 0.1802 | 0.1927 | 0.2025 | |
(45,45) | 0.1103 | 0.1234 | 0.1378 | 0.1543 | 0.1701 | 0.1820 | 0.1911 | |
(50,50) | 0.1048 | 0.1173 | 0.1308 | 0.1464 | 0.1614 | 0.1726 | 0.1813 | |
(2, 4) | (15,15) | 0.2907 | 0.3186 | 0.3459 | 0.3732 | 0.3924 | 0.3998 | 0.3993 |
(20,20) | 0.2535 | 0.2776 | 0.3020 | 0.3266 | 0.3435 | 0.3510 | 0.3513 | |
(25,25) | 0.2281 | 0.2495 | 0.2711 | 0.2939 | 0.3094 | 0.3162 | 0.3168 | |
(30,30) | 0.2086 | 0.2287 | 0.2487 | 0.2688 | 0.2837 | 0.2902 | 0.2910 | |
(35,35) | 0.1938 | 0.2118 | 0.2312 | 0.2498 | 0.2632 | 0.2696 | 0.2705 | |
(40,40) | 0.1814 | 0.1988 | 0.2165 | 0.2341 | 0.2467 | 0.2529 | 0.2539 | |
(45,45) | 0.1712 | 0.1875 | 0.2041 | 0.2207 | 0.2331 | 0.2388 | 0.2399 | |
(50,50) | 0.1627 | 0.1783 | 0.1939 | 0.2096 | 0.2214 | 0.2270 | 0.2281 |
Components (s, k) | Sample Size (n,m) | Parameters (α1,α2) | ||||||
---|---|---|---|---|---|---|---|---|
(2.25,1.5) | (2.0,1.5) | (1.75,1.5) | (1.5,1.5) | (1.5,1.75) | (1.5,2.0) | (1.5,2.25) | ||
(1, 3) | (15,15) | 0.9375 | 0.9366 | 0.9377 | 0.9414 | 0.9389 | 0.9316 | 0.9264 |
(20,20) | 0.9372 | 0.9385 | 0.9436 | 0.9481 | 0.9398 | 0.9349 | 0.9285 | |
(25,25) | 0.9421 | 0.9418 | 0.9464 | 0.9455 | 0.9374 | 0.9365 | 0.9297 | |
(30,30) | 0.9381 | 0.9414 | 0.9537 | 0.9532 | 0.9463 | 0.9421 | 0.9350 | |
(35,35) | 0.9469 | 0.9430 | 0.9473 | 0.9476 | 0.9499 | 0.9426 | 0.9347 | |
(40,40) | 0.9450 | 0.9473 | 0.9521 | 0.9552 | 0.9474 | 0.9423 | 0.9373 | |
(45,45) | 0.9444 | 0.9449 | 0.9501 | 0.9536 | 0.9502 | 0.9391 | 0.9448 | |
(50,50) | 0.9447 | 0.9488 | 0.9513 | 0.9530 | 0.9496 | 0.9477 | 0.9418 | |
(2, 4) | (15,15) | 0.9324 | 0.9320 | 0.9348 | 0.9376 | 0.9382 | 0.9302 | 0.9248 |
(20,20) | 0.9346 | 0.9406 | 0.9406 | 0.9455 | 0.9371 | 0.9351 | 0.9311 | |
(25,25) | 0.9413 | 0.9396 | 0.9459 | 0.9448 | 0.9374 | 0.9354 | 0.9286 | |
(30,30) | 0.9359 | 0.9397 | 0.9531 | 0.9530 | 0.9451 | 0.9410 | 0.9357 | |
(35,35) | 0.9460 | 0.9406 | 0.9473 | 0.9470 | 0.9481 | 0.9412 | 0.9353 | |
(40,40) | 0.9431 | 0.9460 | 0.9514 | 0.9553 | 0.9456 | 0.9420 | 0.9383 | |
(45,45) | 0.9434 | 0.9443 | 0.9493 | 0.9538 | 0.9504 | 0.9392 | 0.9423 | |
(50,50) | 0.9440 | 0.9485 | 0.9507 | 0.9530 | 0.9480 | 0.9466 | 0.9412 |
Vanuatu Island (Group) | Area (Hectare) | Loss of Forest (Hectare) |
---|---|---|
Torres Islands | 11.52 | 45.8 |
Banks Islands | 75.359 | 56.8 |
Santo | 423.897 | 1114.4 |
Maewo | 30.39 | 217 |
Aoba | 40.566 | 210.4 |
Pentecost | 49.49 | 249 |
Malakula | 206.756 | 293.4 |
Ambrym | 73.246 | 447.4 |
Epi | 53.324 | 190.3 |
Efate | 97.004 | 302.5 |
Erromango | 88.874 | 666 |
Tanna | 56.668 | 811.6 |
Aneityum | 17.21 | 73 |
EMED | Burr XII Distribution | ||||||
---|---|---|---|---|---|---|---|
MSE | ACL | ACP | MSE | ACL | ACP | ||
(1,3) | (15,15) | 0.0059 | 0.2651 | 0.9264 | 0.0105 | 0.3722 | 0.9403 |
(20,20) | 0.0042 | 0.2307 | 0.9331 | 0.0078 | 0.3239 | 0.9430 | |
(25,25) | 0.0033 | 0.2070 | 0.9305 | 0.0059 | 0.2922 | 0.9470 | |
(30,30) | 0.0027 | 0.1887 | 0.9382 | 0.0051 | 0.2684 | 0.9380 | |
(2,4) | (15,15) | 0.0114 | 0.3732 | 0.9226 | 0.0136 | 0.4247 | 0.9347 |
(20,20) | 0.0083 | 0.3266 | 0.9305 | 0.0099 | 0.3730 | 0.9397 | |
(25,25) | 0.0066 | 0.2939 | 0.9298 | 0.0078 | 0.3367 | 0.9440 | |
(30,30) | 0.0054 | 0.2688 | 0.9380 | 0.0067 | 0.3086 | 0.9330 |
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Rao, G.S.; Bhatti, F.A.; Aslam, M.; Albassam, M. Estimation of Reliability in a Multicomponent Stress–Strength System for the Exponentiated Moment-Based Exponential Distribution. Algorithms 2019, 12, 246. https://doi.org/10.3390/a12120246
Rao GS, Bhatti FA, Aslam M, Albassam M. Estimation of Reliability in a Multicomponent Stress–Strength System for the Exponentiated Moment-Based Exponential Distribution. Algorithms. 2019; 12(12):246. https://doi.org/10.3390/a12120246
Chicago/Turabian StyleRao, G. Srinivasa, Fiaz Ahmad Bhatti, Muhammad Aslam, and Mohammed Albassam. 2019. "Estimation of Reliability in a Multicomponent Stress–Strength System for the Exponentiated Moment-Based Exponential Distribution" Algorithms 12, no. 12: 246. https://doi.org/10.3390/a12120246