Generalized Truncation Positive Normal Distribution
Abstract
:1. Introduction
2. Model Properties
2.1. Stochastic Representation and Particular Cases
- .
- .
- .
2.2. Density, Cdf and Hazard Functions
2.3. Mode
- 1.
- atwheneverorand,
- 2.
- atin otherwise.
2.4. Quantiles
- 1.
- First quartile.
- 2.
- Median.
- 3.
- Third quartile.
2.5. Central Moments
2.6. Bonferroni and Lorenz Curves
2.7. Shannon Entropy
3. Inference
3.1. Maximum Likelihood Estimators
3.2. Initial Point to Obtain the Maximum Likelihood Estimators
3.2.1. A Naive Point Based on the HN Model
3.2.2. An Initial Point Based on Centiles
3.3. An Initial Point Based on the Method of Moments
3.4. Fisher Information Matrix
4. Model Discrimination
4.1. GTPN Versus Submodels
- versus (TPN versus GTPN distribution).
- versus (GHN versus GTPN distribution).
- versus (HN versus GTPN distribution).
4.1.1. Likelihood Ratio Test
4.1.2. Score Test
4.1.3. Gradient Test
4.2. Non-Nested Models
5. Simulation
- Simulate .
- Compute .
- Compute .
6. Applications
6.1. Application 1
6.2. Application 2
7. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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n = 50 | n = 150 | n = 300 | n = 1000 | |||||||||||||||||||||||
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True Valor | ||||||||||||||||||||||||||
mean | sd | mean | sd | mean | sd | mean | sd | mean | sd | mean | sd | mean | sd | mean | sd | mean | sd | mean | sd | mean | sd | mean | sd | |||
1 | 3 | 0.8 | 1.523 | 1.573 | 3.089 | 1.797 | 0.987 | 1.029 | 1.227 | 1.029 | 3.000 | 1.052 | 0.875 | 0.350 | 1.081 | 0.614 | 3.017 | 0634 | 0.830 | 0.202 | 1.011 | 0.233 | 3.014 | 0.267 | 0.803 | 0.071 |
1 | 1.286 | 1.105 | 3.118 | 1.843 | 1.227 | 0.705 | 1.111 | 0.705 | 3.027 | 1.040 | 1.083 | 0.425 | 1.044 | 0.444 | 3.020 | 0.629 | 1.031 | 0.250 | 1.006 | 0.185 | 3.012 | 0.266 | 1.004 | 0.089 | ||
2 | 1.006 | 0.510 | 3.220 | 2.036 | 2.432 | 0.309 | 1.011 | 0.309 | 3.019 | 1.055 | 2.171 | 0.850 | 1.001 | 0.196 | 3.022 | 0.614 | 2.058 | 0.481 | 0.998 | 0.093 | 3.013 | 0.266 | 2.008 | 0.177 | ||
4 | 0.8 | 1.489 | 1.799 | 4.391 | 2.010 | 0.971 | 0.860 | 1.087 | 0.860 | 4.280 | 1.199 | 0.817 | 0.282 | 1.017 | 0.513 | 4.164 | 0.786 | 0.798 | 0.148 | 1.003 | 0.301 | 4.049 | 0.413 | 0.799 | 0.082 | |
1 | 1.236 | 1.219 | 4.439 | 2.182 | 1.203 | 0.631 | 1.018 | 0.631 | 4.295 | 1.222 | 1.017 | 0.343 | 0.986 | 0.418 | 4.187 | 0.826 | 0.995 | 0.191 | 0.993 | 0.230 | 4.055 | 0.404 | 0.998 | 0.099 | ||
2 | 0.962 | 0.543 | 4.642 | 2.495 | 2.360 | 0.321 | 0.957 | 0.321 | 4.323 | 1.326 | 2.033 | 0.696 | 0.974 | 0.223 | 4.168 | 0.839 | 2.000 | 0.382 | 0.993 | 0.118 | 4.045 | 0.404 | 2.000 | 0.198 | ||
2 | 3 | 0.8 | 3.006 | 3.131 | 3.157 | 1.876 | 0.974 | 1.996 | 2.423 | 1.996 | 3.018 | 1.045 | 0.869 | 0.340 | 2.152 | 1.185 | 3.023 | 0.624 | 0.823 | 0.194 | 2.034 | 0.472 | 3.007 | 0.269 | 0.805 | 0.072 |
1 | 2.512 | 2.198 | 3.215 | 1.942 | 1.207 | 1.437 | 2.237 | 1.437 | 3.028 | 1.077 | 1.086 | 0.431 | 2.094 | 0.897 | 3.017 | 0.642 | 1.033 | 0.250 | 2.009 | 0.373 | 3.015 | 0.269 | 1.004 | 0.089 | ||
2 | 1.999 | 1.039 | 3.318 | 2.223 | 2.417 | 0.631 | 2.017 | 0.631 | 3.035 | 1.096 | 2.171 | 0.862 | 2.000 | 0.405 | 3.027 | 0.638 | 2.061 | 0.504 | 1.995 | 0.186 | 3.016 | 0.267 | 2.006 | 0.177 | ||
4 | 0.8 | 2.880 | 3.544 | 4.562 | 2.283 | 0.970 | 1.714 | 2.161 | 1.714 | 4.326 | 1.288 | 0.813 | 0.282 | 2.024 | 1.078 | 4.194 | 0.849 | 0.796 | 0.160 | 2.003 | 0.573 | 4.053 | 0.404 | 0.799 | 0.079 | |
1 | 2.402 | 2.434 | 4.659 | 2.419 | 1.176 | 1.283 | 2.033 | 1.283 | 4.344 | 1.346 | 1.015 | 0.355 | 1.974 | 0.835 | 4.190 | 0.844 | 0.995 | 0.191 | 1.990 | 0.465 | 4.053 | 0.408 | 0.998 | 0.100 | ||
2 | 1.913 | 1.132 | 4.802 | 2.848 | 2.378 | 0.668 | 1.898 | 0.668 | 4.393 | 1.469 | 2.024 | 0.708 | 1.938 | 0.460 | 4.198 | 0.901 | 1.993 | 0.391 | 1.984 | 0.236 | 4.047 | 0.401 | 2.000 | 0.198 |
Dataset | n | S2 | |||
---|---|---|---|---|---|
Heights measured | 115 | 3.48 | 0.52 | −1.24 | 6.30 |
Estimated | GTPN | TPN | WEI | GL |
---|---|---|---|---|
2.354(0.379) | 0.720(0.047) | 5.855(0.435) | - | |
2.512(0.495) | 4.842(0.333) | 3.744(0.062) | 4.093(0.539) | |
2.227(0.401) | - | - | 13.467(1.902) | |
- | - | - | 15.528(29.638) | |
AIC | 238.43 | 254.63 | 251.74 | 308.67 |
BIC | 246.67 | 260.12 | 257.23 | 316.90 |
Data Set | n | ||||
---|---|---|---|---|---|
Heights measured | 315 | 242.46 | 17421.79 | 1.47 | 6.34 |
Estimated | GTPN | TPN | WEI | GL |
---|---|---|---|---|
0.040(0.027) | 151.106(8.734) | 1.964(0.080) | - | |
7.888(0.459) | 1.455(0.138) | 274.717(8.353) | 0.016(0.001) | |
0.240(0.013) | - | - | 2.831(0.293) | |
- | - | - | 2.067(6.720) | |
AIC | 3874.92 | 3943.25 | 3905.68 | 3879.07 |
BIC | 3886.17 | 3950.75 | 3913.18 | 3890.42 |
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Gómez, H.J.; Gallardo, D.I.; Venegas, O. Generalized Truncation Positive Normal Distribution. Symmetry 2019, 11, 1361. https://doi.org/10.3390/sym11111361
Gómez HJ, Gallardo DI, Venegas O. Generalized Truncation Positive Normal Distribution. Symmetry. 2019; 11(11):1361. https://doi.org/10.3390/sym11111361
Chicago/Turabian StyleGómez, Héctor J., Diego I. Gallardo, and Osvaldo Venegas. 2019. "Generalized Truncation Positive Normal Distribution" Symmetry 11, no. 11: 1361. https://doi.org/10.3390/sym11111361