# An Extension of the Truncated-Exponential Skew- Normal Distribution

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## Abstract

**:**

## 1. Introduction

## 2. Incorporating Kurtosis

#### 2.1. Representation

**Definition**

**1.**

#### 2.2. Probability Density Function

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Remark**

**1.**

#### 2.3. Reliability Analysis

#### 2.4. Moments

**Proposition**

**3.**

**Proof.**

**Corollary**

**1.**

#### 2.5. Incorporation of Parameters

**Proposition**

**4.**

#### 2.6. Log Likelihood Equations

#### 2.7. STESN or TESN Model?

## 3. Simulation Study

## 4. Application to a Data Set

## 5. Final Comments

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Johnson, N.L.; Kotz, S.; Balakrishnan, N. Continuous Univariate Distributions, 2nd ed.; Wiley Series in Probability and Statistics; Wiley: New York, NY, USA, 1995. [Google Scholar]
- Rogers, W.H.; Tukey, J.W. Understanding some long-tailed symmetrical distributions. Stat. Neerl.
**1972**, 26, 211–226. [Google Scholar] [CrossRef] - Mosteller, F.; Tukey, J.W. Data Analysis and Regression: A Second Course in Statistics; Addison-Wesley: Reading, MA, USA, 1977. [Google Scholar]
- Kadafar, K. A biweight approach to the one-sample problem. J. Am. Stat. Assoc.
**1982**, 77, 416–424. [Google Scholar] [CrossRef] - Wang, J.; Boyer, J.; Genton, M.G. A skew-symmetric representation of multivariate distributions. Stat. Sin.
**2004**, 14, 1259–1270. [Google Scholar] - Gómez, H.W.; Quintana, F.A.; Torres, F.J. New family of slash-distributions with elliptical contours. Stat. Probab. Lett.
**2007**, 77, 717–725. [Google Scholar] [CrossRef] - Gómez, H.W.; Olivares-Pacheco, J.F.; Bolfarine, H. An extension of the generalized Birnbaum-Saunders distribution. Stat. Probab. Lett.
**2009**, 79, 331–338. [Google Scholar] [CrossRef] - Nadarajah, S.; Nassiri, V.; Mohammadpour, A. Truncated-exponential skew-symmetric distributions. Statistics
**2014**, 48, 872–895. [Google Scholar] [CrossRef] - Azzalini, A. A Class of Distributions Which Includes the Normal Ones. Scand. J. Stat.
**1985**, 12, 171–178. [Google Scholar] - Ferreira, J.T.A.S.; Steel, M.F.J. A constructive representation of univariate skewed distributions. J. Am. Stat. Assoc.
**2006**, 101, 823–829. [Google Scholar] [CrossRef] [Green Version] - Barreto-Souza, W.; Simas, A.B. The exp- G family of probability distributions. Braz. J. Probab. Stat.
**2013**, 27, 84–109. [Google Scholar] [CrossRef] - Gomes, A.E.; Da-Silva, C.Q.; Cordeiro, G.M. The Exponentiated G Poisson Model. Commun. Stat.-Theory Methods
**2015**, 44, 4217–4240. [Google Scholar] [CrossRef] - Maurya, S.K.; Nadarajah, S. Poisson Generated Family of Distributions: A Review. Sankhya B
**2020**, 1–57. [Google Scholar] [CrossRef] - Lehmann, E.L. Elements of Large-Sample Theory; Springer: New York, NY, USA, 1999. [Google Scholar]
- R Development Core Team. A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2021. [Google Scholar]
- Borchers, H.W. Pracma: Practical Numerical Math Functions, R package version 2.3.3; 2021. Available online: https://CRAN.R-project.org/package=pracma (accessed on 24 June 2021).
- Akaike, H. A new look at the statistical model identification. IEEE Trans. Automat. Contr.
**1974**, 19, 716–723. [Google Scholar] [CrossRef] - Schwarz, G. Estimating the dimension of a model. Ann. Stat.
**1978**, 6, 461–464. [Google Scholar] [CrossRef] - Chernoff, H. On the distribution of the likelihood ratio. Ann. Stat.
**1954**, 54, 573–578. [Google Scholar] [CrossRef] - Stram, D.O.; Lee, J.W. Variance components testing in the longitudinal mixed effects model. Biometrics
**1994**, 50, 1171–1177. [Google Scholar] [CrossRef] [PubMed] - Gallardo, D.I.; Bolfarine, H.; Pedroso-de-Lima, A.C. A clustering cure rate model with application to a sealant study. J. Appl. Stat.
**2017**, 44, 2949–2962. [Google Scholar] [CrossRef] - Maller, R.; Zhou, S. Testing for the Presence of Immune or Cured Individuals in Censored Survival Data. Biometrics
**1995**, 51, 1197–1205. [Google Scholar] [CrossRef] [PubMed] - Barlow, R.E.; Toland, R.H.; Freeman, T. A Bayesian analysis of stress rupture life of Kevlar 49/epoxy spherical pressure vessels. In Procedings Conference on Applications of Statistics; Marcel Dekker: New York, NY, USA, 1984. [Google Scholar]

**Figure 2.**STESN for values of (

**a**) $\lambda =-2$; (

**b**) $\lambda =2$; and (

**c**) $\lambda =7$ for different values of q.

**Figure 3.**(

**a**) Survival function and (

**b**) hazard function for log-STESN model with different combinations of values for $\lambda $ and q.

**Figure 4.**Estimated pdf for the ML estimators of $\mu ,\sigma ,\lambda $, and q in the TESN distribution for: $\mu =0,\sigma =1,\lambda =-3,q=1$ (upper panels), and $\mu =10,\sigma =16,\lambda =1,q=3$ (lower panels).

**Figure 5.**Estimated pdf for STESN and TESN for kevlar data set (

**a**) and a zoom for the right tail (

**b**).

**Table 1.**Empirical bias, SE, RMSE, and 95% CP for the ML estimators of $\mu $, $\sigma $, $\lambda $, and q with different combinations of parameters and sample sizes.

True Values | $\mathit{n}=25$ | $\mathit{n}=50$ | $\mathit{n}=100$ | $\mathit{n}=200$ | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\lambda}$ | q | bias | SE | RMSE | CP | bias | SE | RMSE | CP | bias | SE | RMSE | CP | bias | SE | RMSE | CP | |

0 | 1 | −3 | 1 | $\mu $ | 0.053 | 1.140 | 1.141 | 0.903 | 0.031 | 0.882 | 0.885 | 0.909 | 0.013 | 0.662 | 0.675 | 0.912 | 0.009 | 0.449 | 0.458 | 0.939 |

$\sigma $ | 0.084 | 0.489 | 0.496 | 0.972 | 0.069 | 0.367 | 0.373 | 0.965 | 0.052 | 0.278 | 0.287 | 0.961 | 0.042 | 0.197 | 0.203 | 0.959 | ||||

$\lambda $ | −0.176 | 2.336 | 2.336 | 0.887 | −0.093 | 1.914 | 1.936 | 0.892 | −0.053 | 1.366 | 1.398 | 0.906 | −0.042 | 0.892 | 0.910 | 0.921 | ||||

q | 0.195 | 0.587 | 0.618 | 0.973 | 0.119 | 0.350 | 0.370 | 0.972 | 0.082 | 0.220 | 0.235 | 0.965 | 0.057 | 0.151 | 0.161 | 0.961 | ||||

0 | 1 | −1 | 1 | $\mu $ | −0.099 | 1.473 | 1.473 | 0.918 | −0.078 | 1.045 | 1.049 | 0.928 | −0.051 | 0.704 | 0.711 | 0.935 | −0.033 | 0.473 | 0.477 | 0.948 |

$\sigma $ | 0.159 | 0.532 | 0.555 | 0.970 | 0.111 | 0.372 | 0.388 | 0.965 | 0.075 | 0.252 | 0.263 | 0.958 | 0.044 | 0.179 | 0.184 | 0.953 | ||||

$\lambda $ | −0.154 | 2.398 | 2.399 | 0.887 | −0.117 | 1.810 | 1.822 | 0.889 | −0.083 | 1.183 | 1.196 | 0.907 | −0.060 | 0.765 | 0.770 | 0.915 | ||||

q | 0.214 | 0.555 | 0.594 | 0.962 | 0.163 | 0.479 | 0.506 | 0.960 | 0.080 | 0.229 | 0.243 | 0.958 | 0.041 | 0.147 | 0.152 | 0.953 | ||||

0 | 1 | 3 | 1 | $\mu $ | −0.100 | 1.166 | 1.170 | 0.901 | −0.091 | 0.907 | 0.913 | 0.928 | −0.077 | 0.615 | 0.624 | 0.936 | −0.047 | 0.448 | 0.458 | 0.945 |

$\sigma $ | 0.094 | 0.512 | 0.514 | 0.974 | 0.076 | 0.394 | 0.401 | 0.968 | 0.056 | 0.254 | 0.261 | 0.961 | 0.053 | 0.194 | 0.201 | 0.957 | ||||

$\lambda $ | 0.251 | 2.300 | 2.299 | 0.871 | 0.192 | 1.843 | 1.875 | 0.896 | 0.167 | 1.295 | 1.322 | 0.917 | 0.094 | 0.864 | 0.887 | 0.925 | ||||

q | 0.168 | 0.657 | 0.678 | 0.965 | 0.115 | 0.376 | 0.393 | 0.961 | 0.071 | 0.220 | 0.231 | 0.960 | 0.045 | 0.148 | 0.154 | 0.959 | ||||

0 | 1 | −2 | 2 | $\mu $ | −0.061 | 1.140 | 1.139 | 0.905 | −0.033 | 1.097 | 1.097 | 0.918 | −0.031 | 0.941 | 0.950 | 0.929 | −0.010 | 0.664 | 0.672 | 0.932 |

$\sigma $ | 0.053 | 0.386 | 0.389 | 0.986 | 0.042 | 0.343 | 0.357 | 0.979 | 0.034 | 0.297 | 0.318 | 0.971 | 0.025 | 0.189 | 0.200 | 0.964 | ||||

$\lambda $ | −0.192 | 2.875 | 2.880 | 0.894 | −0.151 | 2.778 | 2.783 | 0.901 | −0.122 | 2.318 | 2.355 | 0.915 | −0.097 | 1.625 | 1.652 | 0.921 | ||||

q | 0.362 | 1.301 | 1.350 | 0.971 | 0.278 | 1.254 | 1.341 | 0.969 | 0.226 | 1.008 | 1.094 | 0.963 | 0.180 | 0.510 | 0.541 | 0.959 | ||||

0 | 1 | 3 | 2 | $\mu $ | −0.205 | 1.004 | 1.024 | 0.899 | −0.079 | 0.915 | 0.918 | 0.910 | −0.061 | 0.758 | 0.762 | 0.928 | −0.044 | 0.580 | 0.581 | 0.936 |

$\sigma $ | 0.079 | 0.389 | 0.389 | 0.971 | 0.053 | 0.313 | 0.314 | 0.968 | 0.041 | 0.264 | 0.271 | 0.961 | 0.030 | 0.201 | 0.205 | 0.959 | ||||

$\lambda $ | 0.278 | 2.655 | 2.668 | 0.896 | 0.212 | 2.570 | 2.569 | 0.906 | 0.144 | 2.100 | 2.127 | 0.917 | 0.100 | 1.566 | 1.578 | 0.929 | ||||

q | 0.213 | 1.256 | 1.273 | 0.965 | 0.192 | 1.035 | 1.075 | 0.961 | 0.119 | 0.880 | 0.926 | 0.959 | 0.099 | 0.507 | 0.529 | 0.958 | ||||

0 | 1 | 2 | 2 | $\mu $ | 0.164 | 1.171 | 1.191 | 0.904 | 0.115 | 1.114 | 1.115 | 0.912 | 0.079 | 0.901 | 0.904 | 0.939 | 0.052 | 0.689 | 0.693 | 0.945 |

$\sigma $ | 0.099 | 0.369 | 0.369 | 0.986 | 0.082 | 0.335 | 0.350 | 0.982 | 0.074 | 0.280 | 0.299 | 0.971 | 0.040 | 0.204 | 0.215 | 0.960 | ||||

$\lambda $ | 0.381 | 3.081 | 3.103 | 0.866 | 0.242 | 2.772 | 2.782 | 0.899 | 0.171 | 2.252 | 2.267 | 0.919 | 0.127 | 1.668 | 1.683 | 0.935 | ||||

q | 0.280 | 1.318 | 1.347 | 0.976 | 0.223 | 1.137 | 1.213 | 0.973 | 0.112 | 1.023 | 1.103 | 0.967 | 0.087 | 0.583 | 0.629 | 0.959 | ||||

0 | 1 | −1 | 3 | $\mu $ | 0.023 | 1.102 | 1.102 | 0.919 | 0.022 | 1.059 | 1.059 | 0.932 | 0.005 | 1.013 | 1.013 | 0.938 | 0.004 | 0.913 | 0.914 | 0.941 |

$\sigma $ | 0.102 | 0.321 | 0.321 | 0.979 | 0.071 | 0.246 | 0.256 | 0.975 | 0.068 | 0.222 | 0.251 | 0.971 | 0.057 | 0.191 | 0.224 | 0.961 | ||||

$\lambda $ | −0.134 | 3.189 | 3.188 | 0.972 | −0.116 | 3.001 | 3.002 | 0.965 | −0.072 | 2.726 | 2.725 | 0.961 | −0.049 | 2.414 | 2.417 | 0.955 | ||||

q | 0.386 | 1.835 | 1.837 | 0.974 | 0.320 | 1.526 | 1.582 | 0.970 | 0.202 | 1.420 | 1.484 | 0.965 | 0.170 | 1.258 | 1.244 | 0.961 | ||||

0 | 1 | 2 | 3 | $\mu $ | 0.128 | 1.014 | 1.022 | 0.899 | 0.069 | 0.930 | 0.930 | 0.912 | 0.061 | 0.830 | 0.832 | 0.943 | 0.046 | 0.813 | 0.815 | 0.944 |

$\sigma $ | 0.134 | 0.324 | 0.325 | 0.984 | 0.079 | 0.274 | 0.278 | 0.977 | 0.052 | 0.246 | 0.266 | 0.962 | 0.049 | 0.209 | 0.227 | 0.959 | ||||

$\lambda $ | 0.159 | 2.939 | 2.942 | 0.888 | 0.128 | 2.739 | 2.752 | 0.911 | 0.116 | 2.643 | 2.655 | 0.927 | 0.082 | 2.253 | 2.264 | 0.935 | ||||

q | 0.322 | 1.556 | 1.556 | 0.981 | 0.262 | 1.411 | 1.475 | 0.972 | 0.174 | 1.271 | 1.237 | 0.961 | 0.054 | 1.027 | 1.039 | 0.958 | ||||

−5 | 4 | −2 | 2 | $\mu $ | −0.354 | 4.484 | 4.495 | 0.900 | −0.225 | 4.405 | 4.405 | 0.917 | −0.137 | 3.562 | 3.600 | 0.927 | −0.040 | 2.641 | 2.670 | 0.935 |

$\sigma $ | 0.317 | 1.550 | 1.553 | 0.970 | 0.281 | 1.311 | 1.365 | 0.964 | 0.237 | 1.070 | 1.132 | 0.958 | 0.170 | 0.793 | 0.837 | 0.957 | ||||

$\lambda $ | −0.247 | 2.862 | 2.861 | 0.885 | −0.223 | 2.808 | 2.817 | 0.906 | −0.191 | 2.240 | 2.273 | 0.919 | −0.130 | 1.631 | 1.658 | 0.929 | ||||

q | 0.309 | 1.292 | 1.328 | 0.974 | 0.245 | 1.155 | 1.237 | 0.966 | 0.222 | 0.851 | 0.909 | 0.961 | 0.157 | 0.581 | 0.617 | 0.955 | ||||

10 | 16 | 1 | 3 | $\mu $ | 0.677 | 17.507 | 17.571 | 0.912 | 0.594 | 13.643 | 13.944 | 0.914 | 0.430 | 11.891 | 11.889 | 0.926 | 0.184 | 9.222 | 9.240 | 0.945 |

$\sigma $ | 1.237 | 4.948 | 4.951 | 0.904 | 0.957 | 4.159 | 4.205 | 0.927 | 0.651 | 3.454 | 3.827 | 0.937 | 0.480 | 2.030 | 2.113 | 0.949 | ||||

$\lambda $ | 0.128 | 3.206 | 3.204 | 0.898 | 0.118 | 3.084 | 3.088 | 0.915 | 0.102 | 2.726 | 2.727 | 0.922 | 0.075 | 2.339 | 2.344 | 0.938 | ||||

q | 0.470 | 1.401 | 1.401 | 0.981 | 0.453 | 1.273 | 1.240 | 0.979 | 0.333 | 1.087 | 1.052 | 0.961 | 0.142 | 0.927 | 0.938 | 0.959 |

n | $\overline{\mathit{X}}$ | S | $\sqrt{{\mathit{b}}_{1}}$ | ${\mathit{b}}_{2}$ | $min\left(\mathit{X}\right)$ | $max\left(\mathit{X}\right)$ |
---|---|---|---|---|---|---|

76 | $1.959$ | $1.573$ | $1.979$ | $8.161$ | $0.025$ | $9.096$ |

**Table 3.**Estimated parameters and standard errors (in parentheses), log-likelihood, AIC and BIC values, and KSS with p-values for TESN and STESN models in kevlar dataset.

Estimations | TESN | STESN |
---|---|---|

$\mu $ | $3.374\left(0.384\right)$ | $-0.870\left(0.790\right)$ |

$\sigma $ | $1.668\left(0.166\right)$ | $1.077\left(0.247\right)$ |

$\lambda $ | $3.939\left(1.052\right)$ | $-15.562\left(11.223\right)$ |

q | – | $2.873\left(0.896\right)$ |

log-likelihood | $-133.846$ | $-122.052$ |

AIC | $273.693$ | $250.103$ |

BIC | $280.685$ | $261.426$ |

KSS | $0.122$ | $0.077$ |

p-value | $0.191$ | $0.728$ |

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**MDPI and ACS Style**

Rivera, P.A.; Gallardo, D.I.; Venegas, O.; Bourguignon, M.; Gómez, H.W.
An Extension of the Truncated-Exponential Skew- Normal Distribution. *Mathematics* **2021**, *9*, 1894.
https://doi.org/10.3390/math9161894

**AMA Style**

Rivera PA, Gallardo DI, Venegas O, Bourguignon M, Gómez HW.
An Extension of the Truncated-Exponential Skew- Normal Distribution. *Mathematics*. 2021; 9(16):1894.
https://doi.org/10.3390/math9161894

**Chicago/Turabian Style**

Rivera, Pilar A., Diego I. Gallardo, Osvaldo Venegas, Marcelo Bourguignon, and Héctor W. Gómez.
2021. "An Extension of the Truncated-Exponential Skew- Normal Distribution" *Mathematics* 9, no. 16: 1894.
https://doi.org/10.3390/math9161894