# On the Arcsecant Hyperbolic Normal Distribution. Properties, Quantile Regression Modeling and Applications

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## Abstract

**:**

## 1. Introduction

## 2. The New Unit Distribution and Its Properties

**Proposition**

**1.**

- When x tends to 0, since $arcsechx\sim -logx\to +\infty $ and it appears in power 2 the exponential term, we have $f(x,\mu ,\sigma )\to 0$.
- When x tends to 1, since $arcsech1=0$, we have$$f(x,\mu ,\sigma )\sim \frac{1}{\sqrt{\pi}}\frac{1}{\sigma \sqrt{1-x}}{e}^{-\frac{{\mu}^{2}}{2{\sigma}^{2}}}\to +\infty .$$If $\sigma $ is large and ${\mu}^{2}\approx 2{\sigma}^{2}$, or ${\mu}^{2}/2{\sigma}^{2}$ is large, the point $x=1$ appears as a “special singularity” in the following sense: The function $f(x,\mu ,\sigma )$ can decrease to 0 in the neighborhood of $x=1$, then suddenly explodes at $x=1$. This phenomenon is only punctual; this is not a particular disadvantage for statistical modeling purposes.

## 3. Distributional Properties

#### 3.1. A Likelihood Ratio Order Result

**Proposition**

**2.**

#### 3.2. Quantile Function

#### 3.3. Moments

**Proposition**

**3.**

#### 3.4. Order Statistics

## 4. Different Methods of the Parameter Estimation

#### 4.1. Maximum Likelihood Estimation

#### 4.2. Maximum Product Spacing Estimation

#### 4.3. Least Squares Estimation

#### 4.4. Weighted Least Squares Estimation

#### 4.5. Anderson-Darling Estimation

#### 4.6. The Cramér-von Mises Estimation

`constrOptim`and

`optim`), S-Plus and Matlab to numerically optimize $\ell \left(\mathsf{\Theta}\right)$, $MPS\left(\mathsf{\Theta}\right)$, $LSE\left(\mathsf{\Theta}\right)$, $WLSE\left(\mathsf{\Theta}\right)$, $AD\left(\mathsf{\Theta}\right)$ and $CVM\left(\mathsf{\Theta}\right)$ functions.

## 5. Empirical Simulations

`constrOptim`function in the R program. Further, we calculate the empirical mean, bias and mean square error (MSE) of the estimates for comparisons between the methods. For $\u03f5=\mu $ or $\u03f5=\sigma $, the bias and MSE associated to $\u03f5$ are calculated by

## 6. A New Quantile Regression Model Based on the Special $\mathit{ASHN}$ Distribution

#### 6.1. Motivation

#### 6.2. Proposed Quantile Regression Model

#### 6.3. Parameter Estimation

`maxLik`function implemented in the R software to maximize Equation (23) (see [35]). This function also gives asymptotic SEs numerically, which are obtained by the observed information matrix.

#### 6.4. Residual Analysis

## 7. Data Analysis

#### 7.1. Univariate Real Data Modeling

- Beta distribution.The two-parameter beta pdf is given by$${f}_{Beta}(x,\mu ,\sigma )=\frac{1}{\mathrm{B}\left(\mu ,\sigma \right)}{x}^{\mu -1}{\left(1-x\right)}^{\sigma -1},\phantom{\rule{1.em}{0ex}}x\in (0,1),$$
- Kumaraswamy (Kw) distribution (see [3]).The two-parameter Kw pdf is expressed as$${f}_{Kw}(x,\mu ,\sigma )=\mu \sigma {x}^{\mu -1}{\left(1-{x}^{\mu}\right)}^{\sigma -1},\phantom{\rule{1.em}{0ex}}x\in (0,1),$$
- Johnson ${S}_{B}$ distribution (see [1]).The two-parameter Johnson ${S}_{B}$ pdf is given by$${f}_{{S}_{B}}(x,\mu ,\sigma )=\frac{\sigma}{x\left(1-x\right)}\varphi \left[\sigma log\left(\frac{x}{1-x}\right)+\mu \right],\phantom{\rule{1.em}{0ex}}x\in (0,1),$$

`maxLik`[35] and

`goftest`routines in the R software.

#### 7.1.1. Data Analysis I

#### 7.1.2. Data Analysis II

#### 7.2. The Quantile Modeling Application of the Reading Accuracy with the Dyslexia and Intelligence Quotient

`betareg`function [42] in the R software.

- y: reading score;
- ${x}_{1}$: Is the child dyslexic? (0 for no, 1 for yes);
- ${x}_{2}$: nonverbal intelligence quotient (IQ, converted to z scores).

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Proposition**

**1.**

**Proof**

**of**

**Proposition**

**2.**

**Proof**

**of**

**Proposition**

**3.**

## References

- Johnson, N.L. Systems of frequency curves generated by methods of translation. Biometrika
**1949**, 36, 149–176. [Google Scholar] [CrossRef] - Topp, C.W.; Leone, F.C. A family of J-shaped frequency functions. J. Am. Stat. Assoc.
**1955**, 50, 209–219. [Google Scholar] [CrossRef] - Kumaraswamy, P. A generalized probability density function for double-bounded random processes. J. Hydrol.
**1980**, 46, 79–88. [Google Scholar] [CrossRef] - Van Dorp, J.R.; Kotz, S. The standard two-sided power distribution and its properties: With applications in financial engineering. Am. Stat.
**2002**, 56, 90–99. [Google Scholar] [CrossRef] - Gómez-Déniz, E.; Sordo, M.A.; Calderín-Ojeda, E. The log–Lindley distribution as an alternative to the beta regression model with applications in insurance. Insur. Math. Econ.
**2014**, 54, 49–57. [Google Scholar] [CrossRef] - Altun, E.; Hamedani, G.G. The log-xgamma distribution with inference and application. J. Soc. Fr. Stat.
**2018**, 159, 40–55. [Google Scholar] - Mazucheli, J.; Menezes, A.F.; Dey, S. The unit-Birnbaum-Saunders distribution with applications. Chil. J. Stat.
**2018**, 9, 47–57. [Google Scholar] - Mazucheli, J.; Menezes, A.F.B.; Ghitany, M.E. The unit-Weibull distribution and associated inference. J. Appl. Probab. Stat.
**2018**, 13, 1–22. [Google Scholar] - Mazucheli, J.; Menezes, A.F.B.; Chakraborty, S. On the one parameter unit-Lindley distribution and its associated regression model for proportion data. J. Appl. Stat.
**2019**, 46, 700–714. [Google Scholar] [CrossRef] [Green Version] - Ghitany, M.E.; Mazucheli, J.; Menezes, A.F.B.; Alqallaf, F. The unit-inverse Gaussian distribution: A new alternative to two-parameter distributions on the unit interval. Commun. Stat. Theory Methods
**2019**, 48, 3423–3438. [Google Scholar] [CrossRef] - Mazucheli, J.; Menezes, A.F.; Dey, S. Unit-Gompertz distribution with applications. Statistica
**2019**, 79, 25–43. [Google Scholar] - Altun, E. The log-weighted exponential regression model: Alternative to the beta regression model. Commun. Stat. Theory Methods
**2021**. [Google Scholar] [CrossRef] - Altun, E.; Cordeiro, G.M. The unit-improved second-degree Lindley distribution: Inference and regression modeling. Comput. Stat.
**2020**, 35, 259–279. [Google Scholar] [CrossRef] - Korkmaz, M.Ç. A new heavy-tailed distribution defined on the bounded interval: The logit slash distribution and its application. J. Appl. Stat.
**2020**, 47, 2097–2119. [Google Scholar] [CrossRef] - Korkmaz, M.Ç. The unit generalized half normal distribution: A new bounded distribution with inference and application. Univ. Politeh. Buchar. Sci. Bull. Ser. Appl. Math. Phys.
**2020**, 82, 133–140. [Google Scholar] - Gündüz, S.; Korkmaz, M.Ç. A New Unit Distribution Based On The Unbounded Johnson Distribution Rule: The Unit Johnson SU Distribution. Pak. J. Stat. Oper. Res.
**2020**, 16, 471–490. [Google Scholar] [CrossRef] - Figueroa-Zu, J.I.; Niklitschek-Soto, S.A.; Leiva, V.; Liu, S. Modeling heavy-tailed bounded data by the trapezoidal beta distribution with applications. Revstat
**2021**. Available online: https://www.ine.pt/revstat/pdf/ModelingBoundedDataWithHeavyTails.pdf (accessed on 10 January 2021). - Bantan, R.A.R.; Chesneau, C.; Jamal, F.; Elgarhy, M.; Tahir, M.H.; Aqib, A.; Zubair, M.; Anam, S. Some new facts about the unit-Rayleigh distribution with applications. Mathematics
**2020**, 8, 1954. [Google Scholar] [CrossRef] - Koenker, R.; Bassett, G., Jr. Regression quantiles. Econom. J. Econom. Soc.
**1978**, 46, 33–50. [Google Scholar] [CrossRef] - Fischer, M.J. Generalized Hyperbolic Secant Distributions: With Applications to Finance; Springer-Verlag Berlin and Heidelberg GmbH & Co. KG: Berlin, Germany, 2013. [Google Scholar]
- Shaked, M.; Shanthikumar, J.G. Stochastic Orders; Wiley: New York, NY, USA, 2007. [Google Scholar]
- Cheng, R.C.H.; Amin, N.A.K. Maximum Product of Spacings Estimation with Application to the Lognormal Distribution; Math Report; University of Wales Institute of Science and Technology: Cardiff, Wales, 1979; p. 79-1. [Google Scholar]
- Ferrari, S.; Cribari-Neto, F. Beta regression for modelling rates and proportions. J. Appl. Stat.
**2004**, 31, 799–815. [Google Scholar] [CrossRef] - Bayes, C.L.; Bazán, J.L.; García, C. A new robust regression model for proportions. Bayesian Anal.
**2012**, 7, 841–866. [Google Scholar] [CrossRef] - Kieschnick, R.; McCullough, B.D. Regression analysis of variates observed on (0, 1): Percentages, proportions and fractions. Stat. Model.
**2003**, 3, 193–213. [Google Scholar] [CrossRef] [Green Version] - Migliorati, S.; Di Brisco, A.M.; Ongaro, A. A new regression model for bounded responses. Bayesian Anal.
**2018**, 13, 845–872. [Google Scholar] [CrossRef] - Galarza, C.E.; Zhang, P.; Lachos, V.H. Logistic quantile regression for bounded outcomes using a family of heavy-tailed distributions. Sankhya B
**2020**, 1–25. [Google Scholar] [CrossRef] - Bayes, C.L.; Bazán, J.L.; De Castro, M. A quantile parametric mixed regression model for bounded response variables. Stat. Its Interface
**2017**, 10, 483–493. [Google Scholar] [CrossRef] - Mitnik, P.A.; Baek, S. The Kumaraswamy distribution: Median-dispersion re-parameterizations for regression modeling and simulation-based estimation. Stat. Pap.
**2013**, 54, 177–192. [Google Scholar] [CrossRef] - Mazucheli, J.; Menezes, A.F.B.; Fernandes, L.B.; de Oliveira, R.P.; Ghitany, M.E. The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates. J. Appl. Stat.
**2020**, 47, 954–974. [Google Scholar] [CrossRef] - Gallardo, D.I.; Gómez-Déniz, E.; Gómez, H.W. Discrete generalized half-normal distribution and its applications in quantile regression. Sort-Stat. Oper. Res. Trans.
**2020**, 265–284. [Google Scholar] [CrossRef] - Jodra, P.; Jiménez-Gamero, M.D. A quantile regression model for bounded responses based on the exponential-geometric distribution. Revstat-Stat. J.
**2020**, 18, 415–436. [Google Scholar] - Korkmaz, M.Ç.; Chesneau, C.; Korkmaz, Z.S. Transmuted unit Rayleigh quantile regression model: Alternative to beta and Kumaraswamy quantile regression models. Univ. Politeh. Buchar. Sci. Bull. Ser. Appl. Math. Phys.
**2021**. to appear. [Google Scholar] - Sánchez, L.; Leiva, V.; Galea, M.; Saulo, H. Birnbaum-Saunders quantile regression models with application to spatial data. Mathematics
**2020**, 8, 1000. [Google Scholar] [CrossRef] - Henningsen, A.; Toomet, O. maxLik: A package for maximum likelihood estimation in R. Comput. Stat.
**2011**, 26, 443–458. [Google Scholar] [CrossRef] - Dunn, P.K.; Smyth, G.K. Randomized quantile residuals. J. Comput. Graph. Stat.
**1996**, 5, 236–244. [Google Scholar] - Cox, D.R.; Snell, E.J. A general definition of residuals. J. R. Stat. Soc. Ser. (Methodol.)
**1968**, 30, 248–265. [Google Scholar] [CrossRef] - Murthy, D.P.; Xie, M.; Jiang, R. Weibull Models; John Wiley & Sons: Hoboken, NJ, USA, 2004; Volume 505. [Google Scholar]
- Silva, R.B.; Bourguignon, M.; Dias, C.R.; Cordeiro, G.M. The compound class of extended Weibull power series distributions. Comput. Stat. Data Anal.
**2013**, 58, 352–367. [Google Scholar] [CrossRef] [Green Version] - Genç, A.A.; Korkmaz, M.Ç.; Kus, C. The Beta Moyal-Slash Distribution. J. Selçuk Univ. Nat. Appl. Sci.
**2014**, 3, 88–104. [Google Scholar] - Pammer, K.; Kevan, A. The Contribution of Visual Sensitivity, Phonological Processing and Non-Verbal IQ to Children’s Reading; The Australian National University: Canberra, Australia, 2004; Unpublished manuscript. [Google Scholar]
- Cribari-Neto, F.; Zeileis, A. Beta regression in R. J. Stat. Softw.
**2010**, 34, 1–24. [Google Scholar] [CrossRef] [Green Version] - Smithson, M.; Verkuilen, J. A better lemon squeezer? Maximum-likelihood regression with beta-distributed dependent variables. Psychol. Methods
**2006**, 11, 54. [Google Scholar] [CrossRef] [Green Version]

**Figure 3.**The results related to $\mu $ (

**top**) and $\sigma $ (

**bottom**) for the first simulation study.

**Figure 4.**The results related to $\mu $ (

**top**) and $\sigma $ (

**bottom**) for the second simulation study.

**Table 1.**MLEs, SEs of the estimates (in parentheses), $\widehat{\ell}$ and goodness-of-fits statistics for the first dataset (p-value is given in $[\xb7]$).

Model | $\widehat{\mathit{\mu}}$ | $\widehat{\mathit{\sigma}}$ | $\widehat{\mathit{\ell}}$ | $\mathit{AIC}$ | $\mathit{BIC}$ | ${\mathit{A}}^{*}$ | ${\mathit{W}}^{*}$ | $\mathit{KS}$ |
---|---|---|---|---|---|---|---|---|

ASHN | 2.9179 | 0.4322 | 33.2443 | −62.4885 | −60.4970 | 1.1850 | 0.1664 | 0.1746 |

(0.0966) | (0.0684) | [0.5754] | ||||||

Beta | 3.1126 | 21.8245 | 27.8813 | −51.7626 | −49.7711 | 2.2611 | 0.3726 | 0.2537 |

(1.0287) | (7.7997) | [0.1521] | ||||||

Kw | 1.5877 | 21.8673 | 25.6484 | −47.2968 | −45.3054 | 2.6889 | 0.4681 | 0.2626 |

(0.3966) | 17.9755 | [0.1265] | ||||||

Johnson ${S}_{B}$ | 3.8952 | 1.8605 | 31.3599 | −58.7198 | −56.7283 | 1.5531 | 0.2307 | 0.2039 |

(0.6554) | (0.2942) | [0.3765] |

Minimum | Mean | Median | Maximum | Variance | Skewness | Kurtosis | n |
---|---|---|---|---|---|---|---|

0.0240 | 0.0567 | 0.0515 | 0.1780 | 0.0007 | 2.7117 | 12.0173 | 36 |

**Table 3.**MLEs, SEs of the estimates (in parentheses), $\widehat{\ell}$ and goodness-of-fits statistics for the second dataset (p-value is given in $[\xb7]$).

Model | $\widehat{\mathit{\mu}}$ | $\widehat{\mathit{\sigma}}$ | $\widehat{\mathit{\ell}}$ | $\mathit{AIC}$ | $\mathit{BIC}$ | ${\mathit{A}}^{*}$ | ${\mathit{W}}^{*}$ | $\mathit{KS}$ |
---|---|---|---|---|---|---|---|---|

ASHN | 3.6422 | 0.3791 | 90.1076 | −176.2152 | −173.0481 | 0.5963 | 0.0895 | 0.1261 |

(0.0632) | (0.0447) | [0.6162] | ||||||

Beta | 5.8569 | 97.1458 | 86.9760 | −169.9519 | −166.7848 | 1.1152 | 0.1768 | 0.1636 |

(0.5166) | (6.2564) | [0.2903] | ||||||

Kw | 2.1577 | 373.3878 | 82.0487 | −160.0975 | −156.9305 | 2.2041 | 0.3651 | 0.1916 |

(0.0648) | 8.4525 | [0.1422] | ||||||

Johnson ${S}_{B}$ | 7.1149 | 2.4608 | 89.6573 | −175.3146 | −172.1476 | 0.6666 | 0.1008 | 0.1322 |

(0.8440) | (0.2864) | [0.5554] |

**Table 4.**The results of the $EASHN$ and unit Weibull regression models with model selection criteria.

Parameters | EASHN | Unit-Weibull | ||||
---|---|---|---|---|---|---|

Estimate | SE | p-Value | Estimate | SE | p-Value | |

${\beta}_{0}$ | 2.2810 | 0.0025 | <0.001 | 2.4045 | 0.2589 | <0.001 |

${\beta}_{1}$ | −1.0490 | 0.0028 | <0.001 | −1.3362 | 0.3751 | 0.0003 |

${\beta}_{2}$ | 0.5918 | 0.00001 | <0.001 | 0.4837 | 0.2453 | 0.0486 |

$\alpha $ | 0.1260 | 0.00001 | <0.001 | 0.9795 | 0.1193 | <0.001 |

ℓ | 37.9466 | 37.3185 | ||||

AIC | −67.8934 | −66.6369 | ||||

BIC | −60.7566 | −59.5001 |

**Table 5.**The goodness-of-fit results of the randomized quantile residuals for the regression models.

Models | KS | p-Value | ${\mathit{A}}^{*}$ | p-Value | ${\mathit{W}}^{*}$ | p-Value |
---|---|---|---|---|---|---|

EASHN | 0.0849 | 0.9093 | 0.4211 | 0.8267 | 0.0502 | 0.8775 |

Unit-Weibull | 0.1159 | 0.5955 | 0.4989 | 0.7470 | 0.0720 | 0.7419 |

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**MDPI and ACS Style**

Korkmaz, M.Ç.; Chesneau, C.; Korkmaz, Z.S.
On the Arcsecant Hyperbolic Normal Distribution. Properties, Quantile Regression Modeling and Applications. *Symmetry* **2021**, *13*, 117.
https://doi.org/10.3390/sym13010117

**AMA Style**

Korkmaz MÇ, Chesneau C, Korkmaz ZS.
On the Arcsecant Hyperbolic Normal Distribution. Properties, Quantile Regression Modeling and Applications. *Symmetry*. 2021; 13(1):117.
https://doi.org/10.3390/sym13010117

**Chicago/Turabian Style**

Korkmaz, Mustafa Ç., Christophe Chesneau, and Zehra Sedef Korkmaz.
2021. "On the Arcsecant Hyperbolic Normal Distribution. Properties, Quantile Regression Modeling and Applications" *Symmetry* 13, no. 1: 117.
https://doi.org/10.3390/sym13010117