New Regression Models Based on the Unit-Sinh-Normal Distribution: Properties, Inference, and Applications
Abstract
:1. Introduction
2. Non-Negative Sinh-Normal Distribution
2.1. Distribution Function, Survival Function, and Hazard Function of the LSHN Model
2.2. Moments of the LSHN Model
2.3. Cumulant-Generating Function and Mode
2.4. Asymptotic Distribution
3. The LSHN Regression Model
3.1. Maximum Likelihood Estimation in the LSHN Regression Model
3.2. Observed and Expected Information Matrix
4. Unit-Sinh-Normal Distribution
4.1. Distribution Function, Survival Function, and Hazard Function of the USHN Model
4.2. Moments of the USHN Model
4.3. Cumulant-Generating Function and Mode
4.4. Asymptotic Distribution
4.5. The LUSHN Regression Model
5. Simulation Study
6. Applications
6.1. Fatigue Data
6.2. Body Fat Data
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Related Theorems
References
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n | RB | RMSE | RSD | CP | RB | RMSE | RSD | CP | |
---|---|---|---|---|---|---|---|---|---|
0.50 | 10 | 6.956 | 5.910 | 0.989 | 99.98 | −0.552 | 0.695 | 0.192 | 40.88 |
25 | 2.789 | 2.052 | 0.904 | 100.0 | −0.496 | 0.614 | 0.356 | 46.26 | |
50 | 1.446 | 1.038 | 0.783 | 97.16 | −0.391 | 0.551 | 0.441 | 55.46 | |
75 | 1.063 | 0.766 | 0.756 | 94.54 | −0.334 | 0.492 | 0.447 | 60.00 | |
100 | 0.868 | 0.632 | 0.748 | 93.66 | −0.301 | 0.456 | 0.466 | 63.42 | |
200 | 0.513 | 0.400 | 0.724 | 92.76 | −0.198 | 0.402 | 0.531 | 70.58 | |
500 | 0.239 | 0.240 | 0.759 | 93.70 | −0.086 | 0.330 | 0.606 | 79.22 | |
0.75 | 10 | 7.166 | 8.224 | 0.935 | 99.96 | −0.608 | 0.716 | 0.436 | 27.78 |
25 | 2.312 | 2.546 | 0.992 | 100 0.0 | −0.444 | 0.575 | 0.487 | 43.60 | |
50 | 1.045 | 1.193 | 0.911 | 97.34 | −0.288 | 0.489 | 0.528 | 57.74 | |
75 | 0.679 | 0.814 | 0.868 | 96.06 | −0.211 | 0.443 | 0.553 | 65.68 | |
100 | 0.479 | 0.623 | 0.841 | 95.48 | −0.148 | 0.415 | 0.569 | 71.48 | |
200 | 0.237 | 0.384 | 0.841 | 95.26 | −0.058 | 0.368 | 0.671 | 79.38 | |
500 | 0.080 | 0.231 | 0.896 | 95.68 | 0.006 | 0.291 | 0.812 | 86.68 | |
1.25 | 10 | 5.736 | 10.877 | 0.900 | 100.0 | −0.585 | 0.655 | 0.684 | 24.16 |
25 | 1.493 | 2.854 | 1.009 | 100.0 | −0.326 | 0.495 | 0.662 | 50.20 | |
50 | 0.581 | 1.263 | 0.976 | 99.42 | −0.147 | 0.423 | 0.705 | 67.20 | |
75 | 0.316 | 0.833 | 0.965 | 98.06 | −0.067 | 0.375 | 0.745 | 76.40 | |
100 | 0.226 | 0.668 | 0.970 | 97.74 | −0.037 | 0.352 | 0.813 | 80.24 | |
200 | 0.090 | 0.415 | 0.989 | 97.14 | 0.003 | 0.280 | 0.957 | 86.68 | |
500 | 0.032 | 0.249 | 1.011 | 95.24 | 0.002 | 0.164 | 1.023 | 90.60 | |
1.75 | 10 | 4.809 | 13.687 | 0.967 | 100.0 | −0.526 | 0.598 | 0.781 | 26.72 |
25 | 1.160 | 3.313 | 1.064 | 100.0 | −0.255 | 0.436 | 0.784 | 56.12 | |
50 | 0.407 | 1.380 | 1.007 | 99.92 | −0.089 | 0.386 | 0.873 | 73.86 | |
75 | 0.230 | 0.964 | 1.033 | 99.04 | −0.038 | 0.330 | 0.944 | 80.50 | |
100 | 0.169 | 0.782 | 1.039 | 98.28 | −0.030 | 0.285 | 1.005 | 83.04 | |
200 | 0.076 | 0.477 | 1.018 | 95.98 | −0.014 | 0.187 | 1.037 | 88.46 | |
500 | 0.028 | 0.276 | 1.003 | 95.80 | −0.006 | 0.109 | 1.011 | 92.36 | |
2.25 | 10 | 4.371 | 15.744 | 0.905 | 100.0 | −0.484 | 0.562 | 0.858 | 29.96 |
25 | 1.004 | 3.813 | 1.082 | 100.0 | −0.206 | 0.422 | 0.887 | 61.10 | |
50 | 0.361 | 1.663 | 1.071 | 99.94 | −0.074 | 0.342 | 0.998 | 76.34 | |
75 | 0.208 | 1.122 | 1.048 | 98.20 | −0.046 | 0.272 | 1.053 | 82.56 | |
100 | 0.130 | 0.858 | 1.025 | 96.94 | −0.025 | 0.226 | 1.049 | 86.28 | |
200 | 0.065 | 0.539 | 1.004 | 96.24 | −0.018 | 0.143 | 1.012 | 89.76 | |
500 | 0.026 | 0.330 | 1.04 | 94.88 | −0.007 | 0.090 | 1.038 | 92.28 | |
2.75 | 10 | 4.028 | 18.459 | 0.934 | 100.0 | −0.444 | 0.531 | 0.891 | 34.52 |
25 | 0.900 | 4.205 | 1.051 | 100.0 | −0.177 | 0.398 | 0.970 | 64.02 | |
50 | 0.327 | 1.882 | 1.072 | 98.60 | −0.075 | 0.277 | 1.031 | 78.30 | |
75 | 0.187 | 1.273 | 1.046 | 97.28 | −0.042 | 0.225 | 1.055 | 84.56 | |
100 | 0.137 | 1.009 | 1.025 | 96.60 | −0.035 | 0.187 | 1.052 | 87.18 | |
200 | 0.066 | 0.632 | 1.013 | 96.36 | −0.021 | 0.122 | 1.009 | 91.08 | |
500 | 0.027 | 0.368 | 0.999 | 95.98 | −0.009 | 0.076 | 1.006 | 93.40 |
n | RB | RMSE | RSD | CP | RB | RMSE | CSD | CP | |
---|---|---|---|---|---|---|---|---|---|
0.50 | 10 | −0.022 | 0.186 | 1.232 | 81.92 | 0.201 | 0.331 | 1.234 | 81.74 |
25 | −0.011 | 0.106 | 1.121 | 89.08 | 0.065 | 0.184 | 1.107 | 89.16 | |
50 | −0.006 | 0.073 | 1.069 | 91.80 | 0.029 | 0.126 | 1.063 | 92.26 | |
75 | −0.001 | 0.058 | 1.048 | 93.38 | 0.002 | 0.101 | 1.050 | 93.20 | |
100 | −0.002 | 0.050 | 1.043 | 93.34 | 0.007 | 0.086 | 1.026 | 94.10 | |
200 | −0.001 | 0.035 | 1.018 | 94.14 | 0.000 | 0.060 | 1.016 | 94.50 | |
500 | 0.000 | 0.022 | 1.008 | 94.48 | 0.001 | 0.038 | 1.016 | 94.64 | |
0.75 | 10 | −0.002 | 0.266 | 1.444 | 75.54 | 0.113 | 0.473 | 1.432 | 77.38 |
25 | −0.006 | 0.156 | 1.222 | 84.92 | 0.036 | 0.272 | 1.207 | 85.72 | |
50 | 0.000 | 0.106 | 1.109 | 90.56 | 0.000 | 0.183 | 1.106 | 90.74 | |
75 | −0.002 | 0.085 | 1.076 | 92.18 | 0.011 | 0.149 | 1.087 | 91.88 | |
100 | −0.002 | 0.071 | 1.033 | 93.44 | 0.003 | 0.124 | 1.037 | 93.70 | |
200 | 0.001 | 0.050 | 1.011 | 94.80 | −0.004 | 0.087 | 1.020 | 94.56 | |
500 | 0.000 | 0.031 | 1.004 | 94.86 | −0.001 | 0.054 | 0.999 | 94.88 | |
1.25 | 10 | −0.011 | 0.405 | 1.717 | 68.84 | 0.018 | 0.712 | 1.655 | 70.64 |
25 | 0.003 | 0.241 | 1.299 | 83.14 | −0.019 | 0.421 | 1.291 | 83.76 | |
50 | 0.002 | 0.158 | 1.131 | 89.62 | −0.011 | 0.276 | 1.135 | 89.92 | |
75 | 0.002 | 0.126 | 1.080 | 91.74 | −0.015 | 0.219 | 1.082 | 92.06 | |
100 | 0.001 | 0.107 | 1.050 | 93.22 | −0.006 | 0.185 | 1.046 | 93.30 | |
200 | 0.000 | 0.073 | 1.012 | 94.60 | −0.004 | 0.128 | 1.017 | 94.38 | |
500 | 0.001 | 0.047 | 1.014 | 94.58 | −0.011 | 0.081 | 1.011 | 94.46 | |
1.75 | 10 | 0.012 | 0.519 | 1.805 | 66.92 | −0.008 | 0.925 | 1.752 | 67.98 |
25 | 0.005 | 0.298 | 1.321 | 83.64 | −0.063 | 0.518 | 1.304 | 83.42 | |
50 | 0.001 | 0.190 | 1.124 | 90.26 | 0.000 | 0.335 | 1.133 | 90.14 | |
75 | 0.005 | 0.153 | 1.091 | 91.64 | −0.02 | 0.264 | 1.077 | 92.06 | |
100 | 0.000 | 0.130 | 1.062 | 92.66 | 0.000 | 0.225 | 1.057 | 92.76 | |
200 | 0.001 | 0.091 | 1.040 | 93.62 | −0.003 | 0.156 | 1.030 | 94.10 | |
500 | 0.000 | 0.056 | 1.006 | 94.90 | 0.001 | 0.096 | 1.006 | 95.00 | |
2.25 | 10 | 0.005 | 0.597 | 1.867 | 66.88 | −0.019 | 1.044 | 1.791 | 68.74 |
25 | 0.000 | 0.327 | 1.313 | 82.96 | 0.004 | 0.568 | 1.284 | 84.32 | |
50 | −0.007 | 0.210 | 1.124 | 90.60 | 0.038 | 0.366 | 1.118 | 90.50 | |
75 | −0.004 | 0.168 | 1.089 | 91.80 | 0.031 | 0.293 | 1.089 | 92.00 | |
100 | 0.000 | 0.144 | 1.070 | 92.52 | −0.006 | 0.249 | 1.063 | 93.06 | |
200 | −0.002 | 0.100 | 1.047 | 93.78 | 0.012 | 0.173 | 1.044 | 93.28 | |
500 | 0.000 | 0.062 | 1.017 | 94.94 | 0.000 | 0.106 | 1.008 | 94.58 | |
2.75 | 10 | 0.002 | 0.654 | 1.879 | 67.68 | −0.017 | 1.152 | 1.800 | 69.18 |
25 | 0.000 | 0.338 | 1.257 | 85.32 | −0.031 | 0.601 | 1.265 | 85.52 | |
50 | −0.005 | 0.225 | 1.133 | 90.16 | 0.037 | 0.392 | 1.125 | 90.48 | |
75 | −0.003 | 0.179 | 1.092 | 92.34 | 0.025 | 0.314 | 1.100 | 92.24 | |
100 | 0.003 | 0.150 | 1.059 | 93.16 | −0.023 | 0.258 | 1.049 | 93.20 | |
200 | 0.000 | 0.101 | 1.010 | 94.44 | −0.009 | 0.176 | 1.010 | 94.86 | |
500 | 0.000 | 0.064 | 1.013 | 94.78 | 0.004 | 0.112 | 1.014 | 94.60 |
Parameters | LBS | LSBS | LSHN |
---|---|---|---|
1.279(0.143) | 2.011(0.313) | 0.228(0.076) | |
0.097(0.170) | −0.961(0.166) | 0.296(0.159) | |
−14.116(1.571) | −13.870(1.602) | −12.618(1.371) | |
−0.932(0.174) | 8.675(2.933) | ||
AIC | 129.235 | 125.360 | 120.099 |
BIC | 134.296 | 132.115 | 126.854 |
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Martínez-Flórez, G.; Tovar-Falón, R. New Regression Models Based on the Unit-Sinh-Normal Distribution: Properties, Inference, and Applications. Mathematics 2021, 9, 1231. https://doi.org/10.3390/math9111231
Martínez-Flórez G, Tovar-Falón R. New Regression Models Based on the Unit-Sinh-Normal Distribution: Properties, Inference, and Applications. Mathematics. 2021; 9(11):1231. https://doi.org/10.3390/math9111231
Chicago/Turabian StyleMartínez-Flórez, Guillermo, and Roger Tovar-Falón. 2021. "New Regression Models Based on the Unit-Sinh-Normal Distribution: Properties, Inference, and Applications" Mathematics 9, no. 11: 1231. https://doi.org/10.3390/math9111231
APA StyleMartínez-Flórez, G., & Tovar-Falón, R. (2021). New Regression Models Based on the Unit-Sinh-Normal Distribution: Properties, Inference, and Applications. Mathematics, 9(11), 1231. https://doi.org/10.3390/math9111231