# 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$z=\frac{\sigma}{{2}^{1/\alpha}}{\left(\right)}^{\lambda}$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

- Pewsey, A. Large-sample inference for the general half-normal distribution. Commun. Stat. Theory Methods
**2002**, 31, 1045–1054. [Google Scholar] [CrossRef] - Pewsey, A. Improved likelihood based inference for the general half-normal distribution. Commun. Stat. Theory Methods
**2004**, 33, 197–204. [Google Scholar] [CrossRef] - Wiper, M.P.; Girón, F.J.; Pewsey, A. Objective Bayesian inference for the half-normal and half-t distributions. Commun. Stat. Theory Methods
**2008**, 37, 3165–3185. [Google Scholar] [CrossRef] - Khan, M.A.; Islam, H.N. Bayesian analysis of system availability with half-normal life time. Math. Sci.
**2012**, 9, 203–209. [Google Scholar] [CrossRef] - Moral, R.A.; Hinde, J.; Demétrio, C.G.B. Half-normal plots and overdispersed models in R: The hnp package. J. Stat. Softw.
**2017**, 81, 1–23. [Google Scholar] [CrossRef] - Azzalini, A. A class of distributions which includes the normal ones. Scand. J. Stat.
**1985**, 12, 171–178. [Google Scholar] - Azzalini, A. Further results on a class of distributions which includes the normal ones. Statistics
**1986**, 12, 199–208. [Google Scholar] - Henze, N. A probabilistic representation of the skew-normal distribution. Scand. J. Stat.
**1986**, 13, 271–275. [Google Scholar] - Cooray, K.; Ananda, M.M.A. A generalization of the half-normal distribution with applications to lifetime data. Commun. Stat. Theory Methods
**2008**, 10, 195–224. [Google Scholar] [CrossRef] - Olmos, N.M.; Varela, H.; Gómez, H.W.; Bolfarine, H. An extension of the half-normal distribution. Stat. Pap.
**2012**, 53, 875–886. [Google Scholar] [CrossRef] - Cordeiro, G.M.; Pescim, R.R.; Ortega, E.M.M. The Kumaraswamy generalized half-normal distribution for skewed positive data. J. Data Sci.
**2012**, 10, 195–224. [Google Scholar] - Gómez, Y.M.; Bolfarine, H. Likelihood-based inference for power half-normal distribution. J. Stat. Theory Appl.
**2015**, 14, 383–398. [Google Scholar] [CrossRef] - Gómez, H.J.; Olmos, N.M.; Varela, H.; Bolfarine, H. Inference for a truncated positive normal distribution. Appl. Math. J. Chin. Univ.
**2015**, 33, 163–176. [Google Scholar] [CrossRef] - Bonferroni, C.E. Elementi di Statistica Generale; Libreria Seber: Firenze, Italy, 1930. [Google Scholar]
- Arcagnia, A.; Porrob, F. The Graphical Representation of Inequality. Rev. Colomb. Estad.
**2014**, 37, 419–436. [Google Scholar] [CrossRef] [Green Version] - Shannon, C.E. A Mathematical Theory of Communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] [Green Version] - Akaike, H. A new look at the statistical model identification. IEEE Trans. Auto. Contr.
**1974**, 19, 716–723. [Google Scholar] [CrossRef] - Dunn, P.K.; Smyth, G.K. Randomized Quantile Residuals. J. Comput. Graph. Stat.
**1996**, 5, 236–244. [Google Scholar] - Yazici, B.; Yocalan, S. A comparison of various tests of normality. J. Stat. Comput. Simul.
**2007**, 77, 175–183. [Google Scholar] [CrossRef] - Zakerzadeh, H.; Dolati, A. Generalized Lindley Distribution. J. Math. Ext.
**2009**, 3, 1–17. [Google Scholar] - Laslett, G.M. Kriging and Splines: An Empirical Comparison of Their Predictive Performance in Some Applications. J. Am. Stat. Assoc.
**1994**, 89, 391–400. [Google Scholar] [CrossRef] - Schwarz, G. Estimating the dimension of a model. Ann. Stat.
**1978**, 6, 461–464. [Google Scholar] [CrossRef] - Nierenberg, D.W.; Stukel, T.A.; Baron, J.A.; Dain, B.J.; Greenberg, E.R. Determinants of plasma levels of beta-carotene and retinol. Am. J. Epidemiol.
**1989**, 130, 511–521. [Google Scholar] [CrossRef] [PubMed]

**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