# Modified Power-Symmetric Distribution

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## Abstract

**:**

## 1. Introduction

## 2. Genesis and Properties of Modified Power-Normal Distribution

#### 2.1. Probability Density Function

**Definition**

**1.**

**Remark**

**1.**

**Remark**

**2.**

#### 2.2. Statistical Properties

#### 2.2.1. Shape of the Density

**Proposition**

**1.**

**Proof.**

**Remark**

**3.**

`R`. Table 1 below illustrates some approximations of the roots ${x}_{1}$, ${x}_{2}$ , and ${x}_{3}$ , and the corresponding figures of the pdf evaluated at these values.

#### 2.2.2. Moments

**Proposition**

**2.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**Remark**

**4.**

`R`. Below, in Table 2 , some approximations of the mean and variance for the $\mathcal{MPN}$ distribution for different values of α are displayed. Figure 3 illustrates the behavior of the $\mathbb{E}\left(X\right)$ and $\mathbb{V}ar\left(X\right)$ of the $\mathcal{MPN}$ distribution for different values of α. It is observable that when α grows, the mean increases and the variance decreases.

#### 2.2.3. Stochastic Ordering

**Proposition**

**3.**

**Proof.**

## 3. Inference

#### 3.1. Method of Moments

**Proposition**

**4.**

**Proof.**

#### 3.2. Maximum Likelihood Estimation

#### 3.3. Simulation Study

- Step 1: Generate $W\sim Uniform(0,1).$
- Step 2: Compute $X=\mu +\sigma {\mathsf{\Phi}}^{-1}\left(\right)open="["\; close="]">{\left(\right)}^{{2}^{\alpha}}1/\alpha .$

#### Fisher’s Information Matrix

## 4. Application

`http://lib.stat.cmu.edu/datasets/pollen.data`. This variable measures a geometric characteristic of a specific type of pollen. This dataset was previously used by Pewsey et al. [9] to compare the $\mathcal{PN}$ and $\mathcal{SN}$ distributions. A summary of some descriptive statistics are displayed in Table 4 below.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 2.**Plot of the first derivative of $\mathcal{MPN}$ distribution for selected values of the parameters.

**Figure 3.**Plot of the $\mathbb{E}\left(X\right)$ and $\mathbb{V}ar\left(X\right)$ of the $\mathcal{MPN}$ distribution.

**Figure 4.**Graphs of the skewness and kurtosis coefficients for the $\mathcal{MPN}$ and $\mathcal{PN}$ distributions.

**Figure 5.**

**Left**panel: Histogram of the empirical distribution and fitted densities by ML superimposed for pollen dataset.

**Right**panel: Plots of the tails for the four models.

**Figure 6.**QQ-plots: (

**a**) $\mathcal{MPN}$ model; (

**b**) $\mathcal{PN}$ model; (

**c**) $\mathcal{SN}$ model; (

**d**) $\mathcal{TS}$ model.

**Figure 7.**Profile log-likelihood of $\mu $, $\sigma $ and $\alpha $ for the $\mathcal{MPN}$ distribution.

$\mathit{\alpha}$ | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | $\mathit{f}({\mathit{x}}_{1};\mathit{\alpha})$ | $\mathit{f}({\mathit{x}}_{2};\mathit{\alpha})$ | $\mathit{f}({\mathit{x}}_{3};\mathit{\alpha})$ |
---|---|---|---|---|---|---|

0.5 | −0.136 | −1.135 | 0.886 | 0.397 | 0.239 | 0.241 |

1.0 | 0.000 | −1.000 | 1.000 | 0.399 | 0.242 | 0.242 |

2.0 | 0.243 | −0.691 | 1.173 | 0.412 | 0.261 | 0.251 |

3.0 | 0.435 | −0.414 | 1.299 | 0.433 | 0.282 | 0.266 |

4.0 | 0.586 | −0.203 | 1.396 | 0.457 | 0.298 | 0.284 |

5.0 | 0.706 | −0.041 | 1.475 | 0.481 | 0.316 | 0.301 |

**Table 2.**Approximations of $\mathbb{E}\left(X\right)$ and $\mathbb{V}ar\left(X\right)$ of the $\mathcal{MPN}$ distribution for different values of $\alpha $.

$\mathit{\alpha}$ | $\mathbb{E}\left(\mathit{X}\right)$ | $\mathbb{V}\mathit{ar}\left(\mathit{X}\right)$ |
---|---|---|

0.5 | −0.097 | 1.006 |

1.0 | 0.000 | 1.000 |

5.0 | 0.659 | 0.770 |

10.0 | 1.119 | 0.521 |

100.0 | 2.247 | 0.218 |

**Table 3.**Maximum likelihood (ML) estimates and standard deviation (SD) for the parameters $\mu $, $\sigma $ and $\alpha $ of the $\mathcal{MPN}$ model for different generated samples of sizes $n=50,$ 100, and 200.

$\mathit{n}=50$ | |||||
---|---|---|---|---|---|

$\mu $ | $\sigma $ | $\alpha $ | $\widehat{\mu}$ (SD) | $\widehat{\sigma}$ (SD) | $\widehat{\alpha}$ (SD) |

0 | 1 | 0.1 | −0.352478 (0.149214) | 0.994441 (0.091321) | 0.190243 (0.175202) |

0.5 | −0.19534 (0.14501) | 0.990622 (0.094550) | 0.613052 (0.272096) | ||

0.8 | −0.083183 (0.144587) | 0.990286 (0.098669) | 0.854338 (0.164924) | ||

1 | −0.009586 (0.141691) | 0.995312 (0.0997256) | 1.007328 (0.122688) | ||

5 | 0.004225 (0.100001) | 0.997408 (0.088254) | 5.030272 (0.229064) | ||

10 | 0.001108 (0.066610) | 0.999124 (0.068611) | 10.060478 (0.475019) | ||

100 | 0.002171 (0.017362) | 1.001152 (0.029604) | 100.437990 (2.668190) | ||

$n=100$ | |||||

0 | 1 | 0.1 | −0.351446 (0.104552) | 0.998513 (0.070831) | 0.180054 (0.111930) |

0.5 | −0.19268 (0.101786) | 0.997622 (0.068806) | 0.576957 (0.223378) | ||

0.8 | −0.08140 (0.099360) | 0.997674 (0.069451) | 0.830318 (0.152995) | ||

1 | 0.002786 (0.097411) | 0.996444 (0.069648) | 1.002200 (0.088749) | ||

5 | 0.002014 (0.099305) | 0.996788 (0.085987) | 5.023032 (0.221756) | ||

10 | 0.002897 (0.046109) | 1.000515 (0.050192) | 10.032857 (0.339106) | ||

100 | 0.000623 (0.012137) | 1.000185 (0.019759) | 100.168752 (1.866302) | ||

$n=200$ | |||||

0 | 1 | 0.1 | −0.348177 (0.072732) | 0.999433 (0.047548) | 0.170978 (0.076165) |

0.5 | −0.196617 (0.072015) | 0.999142 (0.047896) | 0.562935 (0.218890) | ||

0.8 | −0.076657 (0.069510) | 0.997719 (0.050718) | 0.824700 (0.127661) | ||

1 | 0.001158 (0.06877) | 0.998408 (0.050586) | 1.003651 (0.058344) | ||

5 | −0.000165 (0.053006) | 1.000623 (0.044182) | 5.005130 (0.115719) | ||

10 | −0.000239 (0.033615) | 1.000017 (0.035902) | 10.014958 (0.246652) | ||

100 | 0.000514 (0.008452) | 1.000491 (0.014599) | 100.104380 (1.295144) |

Mean | Median | Variance | Skewness | Kurtosis |
---|---|---|---|---|

0.000 | −0.030 | 9.887 | 0.109 | 3.193 |

**Table 5.**Parameter estimates; standard errors (SE); and maximum of the log-likelihood function, ${\ell}_{max}$, for the $\mathcal{TS}$, $\mathcal{SN}$, $\mathcal{PN}$, and $\mathcal{MPN}$ corresponding to the pollen density dataset.

Parameters | $\mathcal{TS}\left(\mathbf{SE}\right)$ | $\mathcal{SN}\left(\mathbf{SE}\right)$ | $\mathcal{PN}\left(\mathbf{SE}\right)$ | $\mathcal{MPN}\left(\mathbf{SE}\right)$ |
---|---|---|---|---|

$\mu $ | −0.010 (0.05) | −2.04 (0.24) | −1.74 (0.68) | −5.73 (0.43) |

$\sigma $ | 3.037 (0.05) | 3.75 (0.14) | 3.69 (0.21) | 4.62 (0.14) |

$\alpha $ | 29.995 (13.01) | 0.93 (0.14) | 1.77 (0.37) | 12.13 (1.21) |

${\ell}_{max}$ | −9864.99 | −9863.42 | −9863.37 | −9861.98 |

**Table 6.**Akaike’s information criterion (AIC), Bayesian information criterion (BIC), Kolmogorov– Smirnov (KSS) test, and the corresponding p-values for all the models considered.

Criteria | $\mathcal{TS}$ | $\mathcal{SN}$ | $\mathcal{PN}$ | $\mathcal{MPN}$ |
---|---|---|---|---|

AIC | 19,735.98 | 19,732.84 | 19,732.74 | 19,729.96 |

BIC | 19,754.74 | 19,751.61 | 19,751.50 | 19,748.72 |

KSS (p-value) | 0.014 (0.516) | 0.013 (0.559) | 0.012 (0.627) | 0.010 (0.820) |

© 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-Déniz, E.; Iriarte, Y.A.; Calderín-Ojeda, E.; Gómez, H.W.
Modified Power-Symmetric Distribution. *Symmetry* **2019**, *11*, 1410.
https://doi.org/10.3390/sym11111410

**AMA Style**

Gómez-Déniz E, Iriarte YA, Calderín-Ojeda E, Gómez HW.
Modified Power-Symmetric Distribution. *Symmetry*. 2019; 11(11):1410.
https://doi.org/10.3390/sym11111410

**Chicago/Turabian Style**

Gómez-Déniz, Emilio, Yuri A. Iriarte, Enrique Calderín-Ojeda, and Héctor W. Gómez.
2019. "Modified Power-Symmetric Distribution" *Symmetry* 11, no. 11: 1410.
https://doi.org/10.3390/sym11111410