# Generalized Truncation Positive Normal Distribution

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Model Properties

#### 2.1. Stochastic Representation and Particular Cases

- $\mathrm{GTPN}(\sigma ,\lambda ,\alpha =1)\equiv \mathrm{TPN}(\sigma ,\lambda )$.
- $\mathrm{GTPN}(\sigma ,\lambda =0,\alpha )\equiv \mathrm{GHN}(\sigma ,\alpha )$.
- $\mathrm{GTPN}(\sigma ,\lambda =0,\alpha =1)\equiv \mathrm{HN}\left(\sigma \right)$.

#### 2.2. Density, Cdf and Hazard Functions

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

#### 2.3. Mode

**Proposition**

**3.**

- 1.
- at$z=\frac{\sigma}{{2}^{1/\alpha}}{\left(\lambda +\sqrt{{\lambda}^{2}+4\left(1-\frac{1}{\alpha}\right)}\right)}^{1/\alpha}$whenever$\alpha \ge 1$or$0<\alpha <1$and$\lambda >0$,
- 2.
- at$z=0$in otherwise.

**Proof.**

**Remark**

**1.**

#### 2.4. Quantiles

**Proposition**

**4.**

**Proof.**

**Corollary**

**1.**

- 1.
- First quartile$\sigma {[{\Phi}^{-1}(1-\frac{3}{4}\Phi \left(\lambda \right))+\lambda ]}^{\frac{1}{\alpha}}$.
- 2.
- Median$\sigma {[{\Phi}^{-1}(1-\frac{\Phi \left(\lambda \right)}{2})+\lambda ]}^{\frac{1}{\alpha}}$.
- 3.
- Third quartile$\sigma {[{\Phi}^{-1}(1-\frac{1}{4}\Phi \left(\lambda \right))+\lambda ]}^{\frac{1}{\alpha}}$.

#### 2.5. Central Moments

**Proposition**

**5.**

**Proof.**

**Remark**

**2.**

#### 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

- ${H}_{0}^{\left(1\right)}:\alpha =1$ versus ${H}_{1}^{\left(1\right)}:\alpha \ne 1$ (TPN versus GTPN distribution).
- ${H}_{0}^{\left(2\right)}:\lambda =0$ versus ${H}_{1}^{\left(2\right)}:\lambda \ne 0$ (GHN versus GTPN distribution).
- ${H}_{0}^{\left(3\right)}:(\alpha ,\lambda )=(1,0)$ versus ${H}_{1}^{\left(3\right)}:(\alpha ,\lambda )\ne (1,0)$ (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 $X\sim Uniform(0,1)$.
- Compute $Y={\Phi}^{-1}(1+(X+1)\Phi \left(\lambda \right))+\lambda $.
- Compute $Z=\sigma {Y}^{\frac{1}{\alpha}}$.

## 6. Applications

#### 6.1. Application 1

#### 6.2. Application 2

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Density and hazard functions for the $\mathrm{GTPN}(\sigma =1,\lambda ,\alpha )$ model with different combinations of $\lambda $ and $\alpha $.

**Figure 3.**Plots of the (

**a**) expectation, (

**b**) variance, (

**c**) skewness and (

**d**) kurtosis for $\mathrm{GTPN}(\sigma =1,\lambda ,\alpha )$ for $\alpha \in \{0.75,1,1.5\}$ as a function of $\lambda $. In (

**d**), the dashed line represents the kurtosis of the normal distribution.

**Figure 4.**Bonferroni curve for the generalized truncation positive normal (GTPN)$(\sigma ,\lambda ,\alpha )$ model.

**Figure 7.**QR for the fitted models in the Laslett data set. The $p-values$ for the Anderson–Darling (AD), Cramer-Von-Mises (CVM) and Shapiro–Wilks (SW) normality tests are also presented to check if the RQ came from the standard normal distribution.

**Figure 9.**RQ for the fitted models in the cholesterol data set. The $p-values$ for the AD, CVM, and SW normality tests are also presented to check if the QR came from the standard normal distribution.

**Table 1.**Monte Carlo (MC) simulation study for the maximum likelihood (ML) estimators in the $(\mathrm{GTPN}\sigma ,\lambda ,\alpha )$ model in 12 combinations of $\sigma $, $\lambda $ and $\alpha $. The results summarize the mean and the standard deviation (sd) of the respective estimators obtained in the 10,000 replicates.

n = 50 | n = 150 | n = 300 | n = 1000 | |||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

True Valor | $\widehat{\mathit{\sigma}}$ | $\widehat{\mathit{\lambda}}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\sigma}}$ | $\widehat{\mathit{\lambda}}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\sigma}}$ | $\widehat{\mathit{\lambda}}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\sigma}}$ | $\widehat{\mathit{\lambda}}$ | $\widehat{\mathit{\alpha}}$ | ||||||||||||||

$\mathit{\sigma}$ | $\mathit{\lambda}$ | $\mathit{\alpha}$ | 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 | $\overline{\mathit{X}}$ | S^{2} | $\sqrt{{\mathit{b}}_{\mathbf{1}}}$ | ${\mathit{b}}_{\mathbf{2}}$ |
---|---|---|---|---|---|

Heights measured | 115 | 3.48 | 0.52 | −1.24 | 6.30 |

**Table 3.**Estimation of the parameters and their standard errors (in parentheses) for the GTPN, TPN, WEI, and GL models for the data set. The AIC and BIC criteria are also included.

Estimated | GTPN | TPN | WEI | GL |
---|---|---|---|---|

$\sigma $ | 2.354(0.379) | 0.720(0.047) | 5.855(0.435) | - |

$\lambda $ | 2.512(0.495) | 4.842(0.333) | 3.744(0.062) | 4.093(0.539) |

$\alpha $ | 2.227(0.401) | - | - | 13.467(1.902) |

$\gamma $ | - | - | - | 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 | $\overline{\mathit{X}}$ | ${\mathit{S}}^{\mathbf{2}}$ | $\sqrt{{\mathit{b}}_{\mathbf{1}}}$ | ${\mathit{b}}_{\mathbf{2}}$ |
---|---|---|---|---|---|

Heights measured | 315 | 242.46 | 17421.79 | 1.47 | 6.34 |

**Table 5.**Estimated parameters and their standard errors (in parentheses) for the GTPN, TPN, WEI, and GL models for the $cholesterol$ data set. The AIC and BIC criteria are also presented.

Estimated | GTPN | TPN | WEI | GL |
---|---|---|---|---|

$\sigma $ | 0.040(0.027) | 151.106(8.734) | 1.964(0.080) | - |

$\lambda $ | 7.888(0.459) | 1.455(0.138) | 274.717(8.353) | 0.016(0.001) |

$\alpha $ | 0.240(0.013) | - | - | 2.831(0.293) |

$\gamma $ | - | - | - | 2.067(6.720) |

AIC | 3874.92 | 3943.25 | 3905.68 | 3879.07 |

BIC | 3886.17 | 3950.75 | 3913.18 | 3890.42 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Gómez, H.J.; Gallardo, D.I.; Venegas, O.
Generalized Truncation Positive Normal Distribution. *Symmetry* **2019**, *11*, 1361.
https://doi.org/10.3390/sym11111361

**AMA Style**

Gómez HJ, Gallardo DI, Venegas O.
Generalized Truncation Positive Normal Distribution. *Symmetry*. 2019; 11(11):1361.
https://doi.org/10.3390/sym11111361

**Chicago/Turabian Style**

Gómez, 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