A New Birnbaum–Saunders Distribution and Its Mathematical Features Applied to Bimodal Real-World Data from Environment and Medicine
Abstract
:1. Introduction
2. The Birnbaum–Saunders Distribution
2.1. Definition
2.2. Justifying the BS Distribution in Environmental and Medical Contexts
3. The BS Distribution under Bimodality
3.1. Probability Density Function
3.2. Cumulative Distribution Function and BBS3 Properties
- (i)
- .
- (ii)
- If , .
3.3. Cumulant Generating Function
3.4. Moments
4. Inference
4.1. Moment Estimators
4.2. Maximum Likelihood Estimators
5. Numerical Applications
5.1. Simulation Algorithm and Computer Characteristics
- n: The length of the .
- Y: A random variable with distribution.
- : The BBS3 PDF with .
- : A lower limit for the BBS3 numbers to be generated with .
- : The maximum value of with .
- : A random variable with a uniform distribution in , U in short.
- : A random variable with a U distribution.
Algorithm 1: Acceptance–rejection algorithm to generate numbers from the distribution |
|
5.2. Results of the Simulation Study
5.3. Illustrative Example I with Real-World Data
5.4. Illustrative Example II with Real-World Data
6. Conclusions, Limitations and Future Investigation
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Framework | Item | Cause | Threshold | Effect | Random Variable |
---|---|---|---|---|---|
Environmental | Chemical element | Contamination | Target | Emergence | Content |
Fatigue | Material | Damage | Rupture | Failure | Fatigue life |
Health metric | Patient | Impulse | To die | Death | Death time |
Neural spiking | Neuron | Voltage | Firing | Spike | Inter-spike time |
n | Mean | SD | ALI | C | Mean | SD | ALI | C | Mean | SD | ALI | C | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
50 | 0.1 | 0.2 | 0.8 | 0.0992 | 0.0067 | 0.0261 | 94.90 | 0.2003 | 0.0036 | 0.0143 | 94.80 | 1.0105 | 0.5855 | 2.2953 | 93.44 |
100 | 0.1 | 0.2 | 0.8 | 0.0995 | 0.0047 | 0.0183 | 95.22 | 0.2001 | 0.0024 | 0.0093 | 95.04 | 0.8951 | 0.3361 | 1.3174 | 94.30 |
150 | 0.1 | 0.2 | 0.8 | 0.0998 | 0.0038 | 0.0148 | 95.00 | 0.2001 | 0.0020 | 0.0078 | 95.34 | 0.8505 | 0.2367 | 0.9278 | 94.36 |
200 | 0.1 | 0.2 | 0.8 | 0.0998 | 0.0033 | 0.0128 | 95.06 | 0.2001 | 0.0017 | 0.0066 | 94.98 | 0.8489 | 0.2089 | 0.8188 | 94.64 |
50 | 0.2 | 0.3 | 0.5 | 0.1974 | 0.0165 | 0.0646 | 94.36 | 0.3019 | 0.0138 | 0.0541 | 94.70 | 0.6083 | 0.3473 | 1.3615 | 94.18 |
100 | 0.2 | 0.3 | 0.5 | 0.1983 | 0.0114 | 0.0446 | 94.58 | 0.3014 | 0.0094 | 0.0369 | 94.48 | 0.5549 | 0.1918 | 0.7518 | 93.60 |
150 | 0.2 | 0.3 | 0.5 | 0.1991 | 0.0095 | 0.0371 | 94.46 | 0.3006 | 0.0076 | 0.0299 | 94.94 | 0.5307 | 0.1422 | 0.5576 | 94.36 |
200 | 0.2 | 0.3 | 0.5 | 0.1994 | 0.0083 | 0.0325 | 94.72 | 0.3005 | 0.0067 | 0.0262 | 94.76 | 0.5212 | 0.1203 | 0.4714 | 95.16 |
50 | 0.5 | 0.5 | 0.5 | 0.4890 | 0.0601 | 0.2357 | 95.08 | 0.5220 | 0.0829 | 0.3248 | 93.90 | 0.6472 | 0.4457 | 1.7472 | 94.56 |
100 | 0.5 | 0.5 | 0.5 | 0.4947 | 0.0395 | 0.1550 | 95.06 | 0.5102 | 0.0537 | 0.2105 | 94.20 | 0.5576 | 0.2155 | 0.8447 | 94.28 |
150 | 0.5 | 0.5 | 0.5 | 0.4965 | 0.0323 | 0.1267 | 94.92 | 0.5067 | 0.0431 | 0.1690 | 94.64 | 0.5322 | 0.1494 | 0.5857 | 94.02 |
200 | 0.5 | 0.5 | 0.5 | 0.4971 | 0.0273 | 0.1071 | 94.56 | 0.5050 | 0.0365 | 0.1429 | 94.56 | 0.5233 | 0.1263 | 0.4951 | 94.32 |
50 | 0.5 | 0.8 | 0.2 | 0.4939 | 0.0439 | 0.1722 | 94.96 | 0.8191 | 0.0957 | 0.3751 | 94.26 | 0.2211 | 0.0969 | 0.3797 | 94.96 |
100 | 0.5 | 0.8 | 0.2 | 0.4978 | 0.0301 | 0.1179 | 94.90 | 0.8079 | 0.0643 | 0.2520 | 94.68 | 0.2075 | 0.0547 | 0.2143 | 94.86 |
150 | 0.5 | 0.8 | 0.2 | 0.4980 | 0.0245 | 0.0960 | 94.62 | 0.8061 | 0.0516 | 0.2024 | 94.56 | 0.2051 | 0.0440 | 0.1723 | 94.98 |
200 | 0.5 | 0.8 | 0.2 | 0.4986 | 0.0216 | 0.0846 | 95.12 | 0.8053 | 0.0451 | 0.1767 | 94.96 | 0.2035 | 0.0380 | 0.1492 | 95.04 |
50 | 0.5 | 0.8 | 0.4 | 0.4898 | 0.0560 | 0.2195 | 94.94 | 0.8309 | 0.1239 | 0.4858 | 93.92 | 0.4934 | 0.3101 | 1.2156 | 95.00 |
100 | 0.5 | 0.8 | 0.4 | 0.4955 | 0.0374 | 0.1466 | 94.30 | 0.8139 | 0.0823 | 0.3226 | 94.70 | 0.4366 | 0.1479 | 0.5797 | 94.48 |
150 | 0.5 | 0.8 | 0.4 | 0.4970 | 0.0305 | 0.1196 | 94.62 | 0.8100 | 0.0652 | 0.2558 | 94.76 | 0.4199 | 0.1089 | 0.4269 | 94.42 |
200 | 0.5 | 0.8 | 0.4 | 0.4975 | 0.0258 | 0.1011 | 94.74 | 0.8073 | 0.0554 | 0.2170 | 94.68 | 0.4151 | 0.0908 | 0.3559 | 94.52 |
50 | 0.8 | 0.8 | 0.1 | 0.7923 | 0.0670 | 0.2626 | 94.82 | 0.8212 | 0.1195 | 0.4685 | 94.62 | 0.1091 | 0.0397 | 0.1557 | 93.66 |
100 | 0.8 | 0.8 | 0.1 | 0.7961 | 0.0478 | 0.1872 | 94.68 | 0.8116 | 0.0830 | 0.3254 | 94.88 | 0.1037 | 0.0251 | 0.0983 | 94.82 |
150 | 0.8 | 0.8 | 0.1 | 0.7971 | 0.0386 | 0.1513 | 95.06 | 0.8079 | 0.0663 | 0.2600 | 94.62 | 0.1026 | 0.0200 | 0.0783 | 95.00 |
200 | 0.8 | 0.8 | 0.1 | 0.7971 | 0.0333 | 0.1303 | 94.80 | 0.8065 | 0.0564 | 0.2210 | 94.82 | 0.1023 | 0.0170 | 0.0667 | 94.68 |
n | Mean | SD | ALI | C | Mean | SD | ALI | C | Mean | SD | ALI | C | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
10 | 0.1 | 0.2 | 0.8 | 0.0928 | 0.0233 | 0.0912 | 97.44 | 0.2014 | 0.0143 | 0.0559 | 94.94 | 0.8100 | 0.0722 | 0.2829 | 95.06 |
25 | 0.1 | 0.2 | 0.8 | 0.0973 | 0.0123 | 0.0482 | 96.16 | 0.2007 | 0.0091 | 0.0357 | 95.02 | 0.8073 | 0.0624 | 0.2447 | 95.70 |
40 | 0.1 | 0.2 | 0.8 | 0.0984 | 0.0091 | 0.0357 | 94.88 | 0.2005 | 0.0074 | 0.0289 | 95.30 | 0.8044 | 0.0528 | 0.2070 | 96.56 |
10 | 0.2 | 0.3 | 0.5 | 0.1873 | 0.0491 | 0.1927 | 94.76 | 0.3094 | 0.0495 | 0.1939 | 94.88 | 0.5334 | 0.1666 | 0.6532 | 95.62 |
25 | 0.2 | 0.3 | 0.5 | 0.1953 | 0.0313 | 0.1228 | 95.10 | 0.3035 | 0.0309 | 0.1210 | 94.82 | 0.5232 | 0.1346 | 0.5276 | 95.08 |
40 | 0.2 | 0.3 | 0.5 | 0.1970 | 0.0249 | 0.0978 | 95.24 | 0.3025 | 0.0244 | 0.0956 | 94.88 | 0.5193 | 0.1175 | 0.4604 | 95.30 |
10 | 0.5 | 0.8 | 0.4 | 0.4925 | 0.0509 | 0.1996 | 98.22 | 0.7449 | 0.1350 | 0.5291 | 96.18 | 0.1278 | 1.3174 | 5.1644 | 98.66 |
25 | 0.5 | 0.8 | 0.4 | 0.4963 | 0.0482 | 0.1890 | 99.04 | 0.7608 | 0.1079 | 0.4231 | 93.90 | 0.1715 | 1.1857 | 4.6480 | 98.82 |
40 | 0.5 | 0.8 | 0.4 | 0.4973 | 0.0461 | 0.1806 | 98.98 | 0.7675 | 0.0948 | 0.3717 | 90.64 | 0.2090 | 0.8960 | 3.5124 | 97.96 |
10 | 0.5 | 0.5 | 0.5 | 0.5017 | 0.0509 | 0.1995 | 98.06 | 0.4818 | 0.0842 | 0.3302 | 90.06 | 0.4354 | 0.4059 | 1.5909 | 88.36 |
25 | 0.5 | 0.5 | 0.5 | 0.4994 | 0.0327 | 0.1282 | 98.38 | 0.4961 | 0.0466 | 0.1825 | 96.06 | 0.4963 | 0.2398 | 0.9402 | 95.68 |
40 | 0.5 | 0.5 | 0.5 | 0.4996 | 0.0171 | 0.0670 | 95.84 | 0.4986 | 0.0336 | 0.1316 | 96.50 | 0.5030 | 0.1781 | 0.6980 | 96.34 |
Estimate/Indicator | BBS1 (SE) | BBS2 (SE) | BBS3 (SE) |
---|---|---|---|
0.0898 (0.0045) | 0.1257 (0.0039) | 0.1042 (0.0027) | |
65.0852 (0.3740) | 66.9851 (0.4785) | 63.8997 (0.4378) | |
−2.1242 (0.1392) | −3.8569 (0.0225) | 1.7480 (0.4438) | |
Log-likelihood | −1054.180 | −1050.573 | −1036.824 |
AIC | 2114.360 | 2107.146 | 2079.648 |
BIC | 2129.325 | 2122.111 | 2090.465 |
KS statistic | 0.1764 | 0.0808 | 0.0735 |
KS p-value | 0.0004 | 0.3359 | 0.454 |
n | Mean | SD | CS | CK |
---|---|---|---|---|
172 | 69.623 | 38.0311 | −0.2112 | 2.1637 |
Estimate/Indicator | BBS1 (SE) | BBS1 (SE) | BBS3 (SE) |
---|---|---|---|
0.4308 (0.0252) | 0.6976 (0.0283) | 0.6784 (0.0442) | |
27.9125 (0.9800) | 31.7897 (1.4594) | 15.6625 (1.5963) | |
−2.5689 (0.1831) | −2.6272 (0.3830) | 0.4213 (0.1031) | |
Log-likelihood | −870.4175 | −851.217 | −838.790 |
AIC | 1746.835 | 1708.434 | 1683.580 |
BIC | 1756.277 | 1717.876 | 1692.602 |
KS statistic | 0.0698 | 0.0640 | 0.0407 |
KS p-value | 0.4329 | 0.4949 | 0.7521 |
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Reyes, J.; Arrué, J.; Leiva, V.; Martin-Barreiro, C. A New Birnbaum–Saunders Distribution and Its Mathematical Features Applied to Bimodal Real-World Data from Environment and Medicine. Mathematics 2021, 9, 1891. https://doi.org/10.3390/math9161891
Reyes J, Arrué J, Leiva V, Martin-Barreiro C. A New Birnbaum–Saunders Distribution and Its Mathematical Features Applied to Bimodal Real-World Data from Environment and Medicine. Mathematics. 2021; 9(16):1891. https://doi.org/10.3390/math9161891
Chicago/Turabian StyleReyes, Jimmy, Jaime Arrué, Víctor Leiva, and Carlos Martin-Barreiro. 2021. "A New Birnbaum–Saunders Distribution and Its Mathematical Features Applied to Bimodal Real-World Data from Environment and Medicine" Mathematics 9, no. 16: 1891. https://doi.org/10.3390/math9161891
APA StyleReyes, J., Arrué, J., Leiva, V., & Martin-Barreiro, C. (2021). A New Birnbaum–Saunders Distribution and Its Mathematical Features Applied to Bimodal Real-World Data from Environment and Medicine. Mathematics, 9(16), 1891. https://doi.org/10.3390/math9161891