# Flexible Power-Normal Models with Applications

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## Abstract

**:**

## 1. Introduction

**Lemma**

**1.**

## 2. Flexible Power-Normal Distribution

**Definition**

**1.**

#### 2.1. Some Particular Cases of the FPN Model

**Proposition**

**1.**

- $\phi (z;0,1,\alpha =1,\delta =0)=\varphi \left(z\right)$.
- $\phi (z;0,1,\alpha ,\delta =0)=\alpha \varphi \left(z\right){\left\{\mathsf{\Phi}\left(z\right)\right\}}^{\alpha -1}$.
- $\phi (z;0,1,\alpha =2,\delta =0)=2\varphi \left(z\right)\mathsf{\Phi}\left(z\right)$.
- $\phi (z;0,1,\alpha =2,\delta )={c}_{\delta}\varphi (\left|z\right|+\delta )\mathsf{\Phi}\left(z\right)$.

**Remark**

**1.**

#### 2.2. Properties of the pdf for the FPN Model

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

- If $z<0$, then it follows that the solution is given by ${z}_{1}=(\alpha -1)\varphi \left({z}_{1}\right)/\mathsf{\Phi}\left({z}_{1}\right)+\delta $.
- If $z\ge 0$, we have the solution ${z}_{2}=(\alpha -1)\varphi \left({z}_{2}\right)/\mathsf{\Phi}\left({z}_{2}\right)-\delta $.

**Corollary**

**1.**

**Proof.**

#### 2.3. Moments

**Definition**

**2.**

**Proposition**

**4.**

#### 2.4. The Location-Scale Extension

## 3. Inference for the FPN Model

#### 3.1. Standard Case

${\Delta}_{1\alpha}(\alpha ,\delta )={\left(\right)}^{\frac{\partial log{k}_{\alpha ,\delta}}{\partial \alpha}}2$ | ${\Delta}_{2\alpha}(\alpha ,\delta )={k}_{\alpha ,\delta}\frac{{\partial}^{2}{k}_{\alpha ,\delta}^{-1}}{\partial {\alpha}^{2}}$, | ${\Delta}_{1\alpha \delta}(\alpha ,\delta )=g(\alpha ,\delta )\left(\right)open="("\; close=")">\frac{\partial log\left({k}_{\alpha ,\delta}\right)}{\partial \alpha}$, |

${\Delta}_{1\delta}(\alpha ,\delta )={\left(\right)}^{\frac{\partial log{k}_{\alpha ,\delta}}{\partial \delta}}2$, | ${\Delta}_{2\delta}(\alpha ,\delta )={k}_{\alpha ,\delta}\frac{{\partial}^{2}{k}_{\alpha ,\delta}^{-1}}{\partial {\delta}^{2}}$, | ${\Delta}_{2\alpha \delta}(\alpha ,\delta )={k}_{\alpha ,\delta}\frac{{\partial}^{2}{k}_{\alpha ,\delta}^{-1}}{\partial \alpha \partial \delta}$. |

#### 3.2. The Location-Scale Model

#### 3.3. Expected Information Matrix

## 4. Simulating Values from the FPN Model

#### 4.1. Acceptance-Rejection Method: Way 1

- Simulate $U\sim U(0,1)$.
- Simulate $Y\sim PN\left(\alpha \right)$.
- Simulate ${U}_{1}\sim U(0,1)$.
- Do $Y={\mathsf{\Phi}}^{-1}\left(\right)open="("\; close=")">{U}_{1}^{1/\alpha}$.

- If $U\le exp(-\delta |Y\left|\right)$, accept Y. Otherwise, back to step 1.

#### 4.2. Acceptance-Rejection Method: Way 2

- Simulate $U\sim U(0,1)$.
- Simulate $Y\sim SFN(0,\delta )$.
- Simulate ${U}_{1},{U}_{2}\sim U(0,1)$ independently.
- Do ${Y}_{1}={\mathsf{\Phi}}^{-1}\left(\right)open="("\; close=")">\mathsf{\Phi}\left(\delta \right)+{U}_{1}[1-\mathsf{\Phi}\left(\delta \right)]$.
- If ${U}_{2}\le 1/2$, do $S=1$. Otherwise, do $S=-1$.
- Do $Y={Y}_{1}\times S$.

- If $U\le {\left\{\mathsf{\Phi}\left(Y\right)\right\}}^{\alpha -1}$, accept Y. Otherwise, back to step 1.

#### 4.3. A Metropolis-Hastings Algorithm

- Define an initial value ${z}_{0}$.
- For $i=1,\dots ,n$, draw ${v}_{i}\sim N(0,{\sigma}^{2})$ and ${U}_{i}\sim U(0,1)$. Do ${z}_{i}^{*}={z}_{i-1}+{v}_{i}$.
- Define ${p}_{i}={\phi}^{*}({z}_{i}^{*};\alpha ,\delta )/{\phi}^{*}({z}_{i-1};\alpha ,\delta )$.
- If ${U}_{i}\le {p}_{i}$, do ${z}_{i}={z}_{i}^{*}$. Otherwise, ${z}_{i}={z}_{i-1}$.

## 5. Simulation Studies

#### 5.1. Assessing the Simulation Procedures for the FPN Model

#### 5.2. Performance of ML Estimators in Finite Samples

#### 5.3. A Model Selection Study

## 6. Numerical Illustrations

#### 6.1. Illustration 1

#### 6.2. Illustration 2

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Pdf for FPN$(0,1,\alpha ,\delta )$ with different combinations of $\alpha $ and $\delta $: (

**a**) FPN$(0,1,\alpha ,\delta =-1)$ and varying $\alpha $; (

**b**) FPN$(0,1,\alpha ,\delta =1)$ and varying $\alpha $; (

**c**) FPN$(0,1,\alpha =0.7,\delta )$ and varying $\delta $; and (

**d**) FPN$(0,1,\alpha =1.5,\delta )$ and varying $\delta $.

**Figure 3.**Constant C for the acceptance-rejection method to draw values from FPN model: (

**a**) way 1 and (

**b**) way 2.

**Figure 4.**Ten thousand values simulated from the FPN$(\xi ,\eta ,\alpha ,\delta )$ model under different scenarios: (

**a**) $\xi =2$, $\eta =3,\alpha =0.2,\delta =2.5$ and acceptance-rejection method: way 1; (

**b**) $\xi =-8,\eta =5,$$\alpha =2,$$\delta =-0.5$ and acceptance-rejection method: way 2; (

**c**) $\xi =100,\eta =50,\alpha =0.8,\delta =-2.5$ and Metropolis-Hastings method.

**Figure 5.**(

**a**) Fitted distributions: SN (dotted line), MN (dashed line) and FPN (solid line) models. (

**b**) Simulated QQ-plot for the fitted FPN model and the variable pollen.

**Figure 6.**Histogram of heights dataset and fitted models N (dotted-dashed line), SN (dotted line), PN (dashed line) and FPN (solid line).

**Table 1.**Mean of the estimates and mean of the estimated standard errors (in parentheses) based on 1000 replicates for the FPN model. In all cases $\xi =0$, $\eta =1$ are maintained and different sample sizes and values for $\alpha $ and $\delta $ are considered.

Parameter | $\mathit{\delta}=-0.5$ | $\mathit{\delta}=0$ | $\mathit{\delta}=0.5$ | ||||
---|---|---|---|---|---|---|---|

$\mathit{n}=\mathbf{200}$ | $\mathit{n}=\mathbf{500}$ | $\mathit{n}=\mathbf{200}$ | $\mathit{n}=\mathbf{500}$ | $\mathit{n}=\mathbf{200}$ | $\mathit{n}=\mathbf{500}$ | ||

$\alpha =0.1$ | $\xi $ | 0.4861 (4.4259) | −0.2412 (2.0642) | −0.1071 (1.5502) | −0.0133 (1.2664) | −0.0065 (1.0174) | −0.0029 (0.8505) |

$\eta $ | 0.9068 (0.5161) | 0.9280 (0.4248) | 0.9340 (0.3629) | 0.9258 (0.3173) | 0.9382 (0.2802) | 0.9469 (0.2468) | |

$\alpha $ | 0.1195 (0.0990) | 0.1109 (0.0774) | 0.1136 (0.0719) | 0.1101 (0.0637) | 0.1093 (0.0640) | 0.1081 (0.0601) | |

$\delta $ | −0.8799 (1.3637) | −0.6915 (0.6524) | −0.4955 (0.4797) | −0.3877 (0.3809) | −0.1621 (0.3485) | −0.0117 (0.3129) | |

$\alpha =0.8$ | $\xi $ | −0.022 (0.134) | 0.000 (0.085) | 0.054 (0.150) | 0.011 (0.124) | 0.000 (0.064) | −0.007 (0.031) |

$\eta $ | 0.971 (0.100) | 0.989 (0.065) | 0.923 (0.132) | 0.953 (0.090) | 0.977 (0.189) | 1.009 (0.118) | |

$\alpha $ | 0.835 (0.096) | 0.809 (0.060) | 0.846 (0.150) | 0.870 (0.123) | 0.895 (0.155) | 0.833 (0.079) | |

$\delta $ | −0.587 (0.225) | −0.532 (0.145) | −0.163 (0.281) | −0.109 (0.181) | 0.450 (0.439) | 0.515 (0.277) | |

$\alpha =1.0$ | $\xi $ | 0.006 (0.126) | −0.007 (0.081) | 0.018 (0.129) | 0.017 (0.105) | −0.003 (0.069) | −0.013 (0.038) |

$\eta $ | 0.965 (0.108) | 0.984 (0.070) | 0.928 (0.135) | 0.947 (0.088) | 0.991 (0.206) | 1.008 (0.126) | |

$\alpha $ | 1.011 (0.112) | 1.009 (0.072) | 1.065 (0.178) | 1.058 (0.138) | 1.184 (0.352) | 1.060 (0.104) | |

$\delta $ | −0.578 (0.243) | −0.540 (0.156) | −0.167 (0.309) | −0.122 (0.197) | 0.493 (0.516) | 0.510 (0.298) | |

$\alpha =1.5$ | $\xi $ | −0.003 (0.122) | 0.008 (0.079) | 0.050 (0.112) | −0.016 (0.096) | −0.016 (0.096) | −0.018 (0.03) |

$\eta $ | 0.970 (0.115) | 0.983 (0.074) | 0.921 (0.136) | 0.964 (0.092) | 0.963 (0.092) | 1.011 (0.126) | |

$\alpha $ | 1.515 (0.190) | 1.498 (0.115) | 1.479 (0.241) | 1.504 (0.207) | 1.604 (0.207) | 1.501 (0.154) | |

$\delta $ | −0.584 (0.273) | −0.541 (0.173) | −0.175 (0.343) | −0.104 (0.228) | 0.545 (0.527) | 0.524 (0.330) |

**Table 2.**Percentage of times where AIC and BIC choose the N, PN and FPN models based on 1000 replicates for different scenarios for the FPN model and sample size.

$\mathit{\delta}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

−1.5 | −0.5 | 0.0 | 0.5 | 2.0 | ||||||||

$\mathit{n}$ | $\mathbf{\alpha}$ | Fitted Model | AIC | BIC | AIC | BIC | AIC | BIC | AIC | BIC | AIC | BIC |

200 | 0.5 | N | 0.0 | 0.2 | 42.0 | 85.4 | 65.2 | 93.6 | 35.0 | 77.9 | 2.6 | 19.5 |

PN | 2.2 | 6.6 | 8.9 | 3.7 | 25.0 | 6.1 | 16.0 | 9.4 | 0.9 | 1.3 | ||

FPN | 97.8 | 93.2 | 49.1 | 10.9 | 9.8 | 0.3 | 49.0 | 12.7 | 96.5 | 79.2 | ||

0.8 | N | 0.0 | 0.0 | 41.4 | 90.2 | 76.0 | 97.5 | 52.0 | 91.8 | 5.3 | 35.1 | |

PN | 0.1 | 0.1 | 9.3 | 1.8 | 12.7 | 2.1 | 9.0 | 2.1 | 1.6 | 1.3 | ||

FPN | 99.9 | 99.9 | 49.3 | 8.0 | 11.3 | 0.4 | 39.0 | 6.1 | 93.1 | 63.6 | ||

1.0 | N | 0.0 | 0.0 | 48.2 | 92.4 | 76.3 | 97.7 | 53.1 | 92.0 | 6.0 | 38.4 | |

PN | 0.0 | 0.0 | 10.9 | 3.1 | 12.9 | 1.9 | 10.2 | 2.9 | 2.3 | 2.6 | ||

FPN | 100.0 | 100.0 | 40.9 | 4.5 | 10.8 | 0.4 | 36.7 | 5.1 | 91.7 | 59.0 | ||

1.5 | N | 0.0 | 1.4 | 63.1 | 96.8 | 71.6 | 94.8 | 43.2 | 82.3 | 5.7 | 35.5 | |

PN | 4.1 | 9.9 | 11.5 | 2.1 | 16.7 | 4.8 | 25.7 | 12.3 | 6.1 | 9.3 | ||

FPN | 95.9 | 88.7 | 25.4 | 1.1 | 11.7 | 0.4 | 31.1 | 5.4 | 88.2 | 55.2 | ||

3.0 | N | 50.1 | 88.0 | 71.3 | 94.0 | 49.6 | 83.3 | 26.6 | 63.6 | 1.7 | 13.2 | |

PN | 19.3 | 11.1 | 20.5 | 6.0 | 37.7 | 15.6 | 48.2 | 33.8 | 25.7 | 48.7 | ||

FPN | 30.6 | 0.9 | 8.2 | 0.0 | 12.7 | 1.1 | 25.2 | 2.6 | 72.6 | 38.1 | ||

500 | 0.5 | N | 0.0 | 0.0 | 11.8 | 69.2 | 47.9 | 87.2 | 10.7 | 52.8 | 0.1 | 0.4 |

PN | 0.1 | 0.4 | 4.2 | 1.8 | 39.0 | 12.1 | 9.8 | 11.2 | 0.0 | 0.0 | ||

FPN | 99.9 | 99.6 | 84.0 | 29.0 | 13.1 | 0.7 | 79.5 | 36.0 | 99.9 | 99.6 | ||

0.8 | N | 0.0 | 0.0 | 13.6 | 76.3 | 72.2 | 97.6 | 25.6 | 82.6 | 0.0 | 1.4 | |

PN | 0.0 | 0.0 | 3.9 | 1.1 | 16.1 | 2.3 | 3.5 | 1.8 | 0.0 | 0.0 | ||

FPN | 100.0 | 100.0 | 82.5 | 22.6 | 11.7 | 0.1 | 70.9 | 15.6 | 100.0 | 98.6 | ||

1.0 | N | 0.0 | 0.0 | 17.4 | 80.8 | 77.4 | 98.4 | 25.4 | 84.0 | 0.0 | 2.2 | |

PN | 0.0 | 0.0 | 5.7 | 1.7 | 13.7 | 1.6 | 4.6 | 1.9 | 0.0 | 0.0 | ||

FPN | 100.0 | 100.0 | 76.9 | 17.5 | 8.9 | 0.0 | 70.0 | 14.1 | 100.0 | 97.8 | ||

1.5 | N | 0.0 | 0.0 | 41.3 | 95.2 | 60.6 | 94.1 | 15.7 | 68.5 | 0.0 | 2.1 | |

PN | 0.0 | 0.3 | 10.8 | 2.3 | 29.0 | 5.6 | 26.0 | 19.3 | 0.9 | 2.0 | ||

FPN | 100.0 | 99.7 | 47.9 | 2.5 | 10.4 | 0.3 | 58.3 | 12.2 | 99.1 | 95.9 | ||

3.0 | N | 27.6 | 91.3 | 53.1 | 93.5 | 18.3 | 61.1 | 3.1 | 25.5 | 0.0 | 0.3 | |

PN | 4.4 | 1.0 | 27.5 | 6.4 | 66.6 | 38.5 | 57.0 | 70.9 | 8.7 | 25.8 | ||

FPN | 68.0 | 7.7 | 19.4 | 0.1 | 15.1 | 0.4 | 39.9 | 3.6 | 91.3 | 73.9 |

n | Mean | Variance | $\sqrt{{\mathit{b}}_{1}}$ | ${\mathit{b}}_{2}$ |
---|---|---|---|---|

481 | −0.0483 | 26.9980 | 0.2329 | 2.5944 |

**Table 4.**Parameter estimates (standard errors in parentheses) for N, SN, PN, MN and FPN models in the pollen dataset.

Parameter | N | SN | PN | FPN | Parameter | MN |
---|---|---|---|---|---|---|

$\xi $ | −0.0489 (0.2367) | −4.9482 (0.8164) | −11.8216 (7.2910) | −2.0300 (0.2797) | ${\xi}_{1}$ | −3.5847 (0.8948) |

$\eta $ | 5.1902 (0.1673) | 7.1379 (0.6058) | 8.4627 (1.8070) | 3.5192 (0.1982) | ${\eta}_{1}$ | 3.3009 (0.3662) |

$\alpha $ | - | 1.6568 (0.4856) | 7.5132 (7.9838) | 0.7316 (0.0464) | ${\xi}_{2}$ | 3.8531 (1.5545) |

$\delta $ | - | - | - | −0.6946 (0.1316) | ${\eta}_{2}$ | 3.9516 (0.6768) |

- | - | - | - | - | p | 0.5245 (0.1593) |

Log-likelihood | −1474.64 | −1472.08 | −1472.24 | −1466.33 | −1466.30 | |

AIC | 2953.28 | 2950.16 | 2950.47 | 2940.66 | 2942.60 | |

BIC | 2961.63 | 2962.68 | 2963.00 | 2957.36 | 2963.48 |

n | Mean | Variance | $\sqrt{{\mathit{b}}_{1}}$ | ${\mathit{b}}_{2}$ |
---|---|---|---|---|

1150 | 3.535 | 0.422 | −0.986 | 4.855 |

**Table 6.**Parameter estimates (standard errors in parentheses) for N, SN, PN and FPN models in roller dataset.

Parameter | N | SN | PN | FPN |
---|---|---|---|---|

$\xi $ | 3.5347 (0.0192) | 4.2475 (0.0284) | 4.5494 (0.0570) | 3.8529 (0.0106) |

$\eta $ | 0.6497 (0.0135) | 0.9644 (0.0304) | 0.1983 (0.0279) | 0.7589 (0.0634) |

$\alpha $ | - | −2.7578 (0.2529) | 0.0479 (0.0155) | 0.2082 (0.0568) |

$\delta $ | - | - | - | 1.3045 (0.2064) |

Log-likelihood | −1135.87 | −1071.35 | −1085.24 | −1065.92 |

AIC | 2275.73 | 2148.69 | 2176.84 | 2139.84 |

BIC | 2285.83 | 2163.84 | 2191.98 | 2160.03 |

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**MDPI and ACS Style**

Martínez-Flórez, G.; Gallardo, D.I.; Venegas, O.; Bolfarine, H.; Gómez, H.W.
Flexible Power-Normal Models with Applications. *Mathematics* **2021**, *9*, 3183.
https://doi.org/10.3390/math9243183

**AMA Style**

Martínez-Flórez G, Gallardo DI, Venegas O, Bolfarine H, Gómez HW.
Flexible Power-Normal Models with Applications. *Mathematics*. 2021; 9(24):3183.
https://doi.org/10.3390/math9243183

**Chicago/Turabian Style**

Martínez-Flórez, Guillermo, Diego I. Gallardo, Osvaldo Venegas, Heleno Bolfarine, and Héctor W. Gómez.
2021. "Flexible Power-Normal Models with Applications" *Mathematics* 9, no. 24: 3183.
https://doi.org/10.3390/math9243183