# A New Birnbaum–Saunders Distribution and Its Mathematical Features Applied to Bimodal Real-World Data from Environment and Medicine

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

`R`software is provided, which helps to exemplify the obtained results with two real-world data sets from environmental and medical sciences and to show potential applications.

## 2. The Birnbaum–Saunders Distribution

#### 2.1. Definition

#### 2.2. Justifying the BS Distribution in Environmental and Medical Contexts

## 3. The BS Distribution under Bimodality

#### 3.1. Probability Density Function

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

#### 3.2. Cumulative Distribution Function and BBS3 Properties

**Proposition**

**3.**

- (i)
- ${Y}^{-1}\sim \mathrm{BBS}3(\alpha ,{\beta}^{-1},\delta )$.
- (ii)
- If $\delta =0$, $Y\sim \mathrm{BS}(\alpha ,\beta )$.

**Proposition**

**4.**

**Proof.**

#### 3.3. Cumulant Generating Function

**Proposition**

**5.**

**Proof.**

#### 3.4. Moments

**Proposition**

**6.**

**Proof.**

## 4. Inference

#### 4.1. Moment Estimators

#### 4.2. Maximum Likelihood Estimators

## 5. Numerical Applications

`R`software produced by the authors of this study is utilized to illustrate the obtained results with real-world data sets from environmental and medical sciences to show potential applications. The data analyzed in this paper and the computational codes are available from the authors under request.

#### 5.1. Simulation Algorithm and Computer Characteristics

- n: The length of the $n\_\mathtt{vector}$.
- Y: A random variable with $\mathrm{BBS}3(\alpha ,\beta ,\delta )$ distribution.
- ${f}_{Y}\left(y\right)$: The BBS3 PDF with $y>0$.
- ${l}_{1}$: A lower limit for the BBS3 numbers to be generated with ${l}_{1}>0$.
- ${l}_{2}$: The maximum value of ${f}_{Y}$ with ${l}_{2}>0$.
- ${U}_{1}$: A random variable with a uniform distribution in $(0,{l}_{1})$, U$(0,{l}_{1})$ in short.
- ${U}_{2}$: A random variable with a U$(0,{l}_{2})$ distribution.

Algorithm 1: Acceptance–rejection algorithm to generate numbers from the $\mathrm{BBS}3(\alpha ,\beta ,\delta )$ distribution |

`R`programming language, using a computer with the following characteristics: (i) OS: Windows 10 Pro for 64 bits; (ii) RAM: 8 Gigabytes; and (iii) Processor: Intel(R) Core(TM) i7-8550U CPU @ 1.99 GigaHertz. Algorithm 1 was executed 100 times with $n=100,000$, and an average processing time equal to $2.9$ seconds was obtained. When executing this algorithm 5000 times, with $n=50,200$, the average processing times were 0.035 and 0.039 seconds, respectively.

#### 5.2. Results of the Simulation Study

#### 5.3. Illustrative Example I with Real-World Data

#### 5.4. Illustrative Example II with Real-World Data

## 6. Conclusions, Limitations and Future Investigation

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Johnson, N.L.; Kotz, S.; Balakrishnan, N. Continuous Univariate Distributions; Wiley: New York, NY, USA, 1994; Volume 1–2. [Google Scholar]
- Balakrishnan, N.; Gupta, R.C.; Kundu, D.; Leiva, V.; Sanhueza, A. On some mixture models based on the Birnbaum-Saunders distribution and associated inference. J. Stat. Plan. Inference
**2011**, 141, 2175–2190. [Google Scholar] [CrossRef] - Birnbaum, Z.W.; Saunders, S.C. A new family of life distributions. J. Appl. Probab.
**1969**, 6, 319–327. [Google Scholar] [CrossRef] - Leiva, V.; Saunders, S.C. Cumulative damage models. In Wiley StatsRef: Statistics Reference Online; Wiley: Hoboken, NJ, USA, 2015; pp. 1–10. [Google Scholar]
- Leiva, V. The Birnbaum-Saunders Distribution; Academic Press: New York, NY, USA, 2016. [Google Scholar]
- Balakrishnan, N.; Kundu, D. Birnbaum-Saunders distribution: A review of models, analysis, and applications. Appl. Stoch. Model. Bus. Ind.
**2019**, 35, 4–49. [Google Scholar] [CrossRef][Green Version] - Leao, J.; Leiva, V.; Saulo, H.; Tomazella, V. Incorporation of frailties into a cure rate regression model and its diagnostics and application to melanoma data. Stat. Med.
**2018**, 37, 4421–4440. [Google Scholar] [CrossRef] - Balakrishnan, N.; Leiva, V.; Sanhueza, A.; Vilca, F. Estimation in the Birnbaum-Saunders distribution based on scale-mixture of normals and the EM-algorithm. Stat. Oper. Res. Trans.
**2009**, 33, 171–192. [Google Scholar] - Bhatti, C. The Birnbaum-Saunders autoregressive conditional duration model. Math. Comput. Simul.
**2010**, 80, 2062–2078. [Google Scholar] [CrossRef] - Kotz, S.; Leiva, V.; Sanhueza, A. Two new mixture models related to the inverse Gaussian distribution. Methodol. Comput. Appl. Probab.
**2010**, 12, 199–212. [Google Scholar] [CrossRef] - Aslam, M.; Jun, C.H.; Ahmad, M. New acceptance sampling plans based on life tests for Birnbaum-Saunders distributions. J. Stat. Comput. Simul.
**2011**, 81, 461–470. [Google Scholar] [CrossRef] - Santos-Neto, M.; Cysneiros, F.J.A.; Leiva, V.; Barros, M. On a reparameterized Birnbaum-Saunders distribution and its moments, estimation and applications. REVSTAT Stat. J.
**2014**, 12, 247–272. [Google Scholar] - Marchant, C.; Leiva, V.; Cysneiros, F.J.A. A multivariate log-linear model for Birnbaum-Saunders distributions. IEEE Trans. Reliab.
**2016**, 65, 816–827. [Google Scholar] [CrossRef] - Garcia-Papani, F.; Leiva, V.; Uribe-Opazo, M.A.; Aykroyd, R.G. Birnbaum-Saunders spatial regression models: Diagnostics and application to chemical data. Chemom. Intell. Lab. Syst.
**2018**, 177, 114–128. [Google Scholar] [CrossRef][Green Version] - Athayde, E.; Azevedo, A.; Barros, M.; Leiva, V. Failure rate of Birnbaum-Saunders distributions: Shape, change-point, estimation and robustness. Braz. J. Probab. Stat.
**2019**, 33, 301–328. [Google Scholar] [CrossRef][Green Version] - Chaves, N.L.; Azevedo, C.L.; Vilca, F.; Nobre, J.S. A new Birnbaum-Saunders type distribution based on the skew-normal model under a centered parameterization. Chil. J. Stat.
**2019**, 10, 55–76. [Google Scholar] - Mazucheli, J.; Menezes, A.F.B.; Dey, S. The unit Birnbaum-Saunders distribution with applications. Chil. J. Stat.
**2018**, 9, 47–57. [Google Scholar] - Leiva, V.; Marchant, C.; Ruggeri, F.; Saulo, H. A criterion for environmental assessment using Birnbaum-Saunders attribute control charts. Environmetrics
**2015**, 26, 463–476. [Google Scholar] [CrossRef] - Arrué, J.; Arellano, R.; Gomez, H.W.; Leiva, V. On a new type of Birnbaum-Saunders models and its inference and application to fatigue data. J. Appl. Stat.
**2020**, 47, 2690–2710. [Google Scholar] [CrossRef] - Sanchez, L.; Leiva, V.; Galea, M.; Saulo, H. Birnbaum-Saunders quantile regression models with application to spatial data. Mathematics
**2020**, 8, 1000. [Google Scholar] [CrossRef] - Mazucheli, M.; Leiva, V.; Alves, B.; Menezes, A.F.B. A new quantile regression for modeling bounded data under a unit Birnbaum-Saunders distribution with applications in medicine and politics. Symmetry
**2021**, 13, 682. [Google Scholar] [CrossRef] - Balakrishnan, N.; Leiva, V.; López, J. Acceptance sampling plans from truncated life tests based on the generalized Birnbaum-Saunders distribution. Commun. Stat. Simul. Comput.
**2007**, 36, 643–656. [Google Scholar] [CrossRef] - Marchant, C.; Leiva, V.; Cavieres, M.F.; Sanhueza, A. Air contaminant statistical distributions with application to PM10 in Santiago, Chile. Rev. Environ. Contam. Toxicol.
**2013**, 223, 1–31. [Google Scholar] - Martinez, S.; Giraldo, R.; Leiva, V. Birnbaum-Saunders functional regression models for spatial data. Stoch. Environ. Res. Risk Assess.
**2019**, 33, 1765–1780. [Google Scholar] [CrossRef] - Kannan, G.; Jeyadurga, P.; Balamurali, S. Economic design of repetitive group sampling plan based on truncated life test under Birnbaum—Saunders distribution. Commun. Stat. Simul. Comput.
**2021**. [Google Scholar] [CrossRef] - Olmos, N.M.; Martínez-Flórez, G.; Bolfarine, H. Bimodal Birnbaum-Saunders distribution with applications to non negative measurements. Commun. Stat. Theory Methods
**2017**, 46, 6240–6257. [Google Scholar] [CrossRef] - Vila, R.; Leao, J.; Saulo, H.; Nabeed, M.; Santos-Neto, M. On a bimodal Birnbaum-Saunders distribution with applications to lifetime data. Braz. J. Probab. Stat.
**2020**, 34, 495–518. [Google Scholar] [CrossRef] - Leiva, V.; Tejo, M.; Guiraud, P.; Schmachtenberg, O.; Orio, P.; Marmolejo, F. Modeling neural activity with cumulative damage distributions. Biol. Cybern.
**2015**, 109, 421–433. [Google Scholar] [CrossRef] - Elal-Olivero, D. Alpha-skew-normal distribution. Proyecciones
**2010**, 29, 224–240. [Google Scholar] [CrossRef][Green Version] - Ng, H.K.T.; Kundu, D.; Balakrishnan, N. Modified moment estimation for the two-parameter Birnbaum–Saunders distribution. Comput. Stat. Data Anal.
**2003**, 43, 283–298. [Google Scholar] [CrossRef] - Azzalini, A.; Bowman, A.W. A look at some data on the old faithful geyser. J. R. Stat. Soc. C
**1990**, 39, 357–365. [Google Scholar] [CrossRef] - Reigner, B.G.; Welker, H.A. Factors influencing elimination and distribution of fleroxacin: Metaanalysis of individual data from 10 pharmacokinetic studies. Antimicrob. Agent Chemother.
**1996**, 40, 575–580. [Google Scholar] [CrossRef][Green Version] - Aykroyd, R.G.; Leiva, V.; Marchant, C. Multivariate Birnbaum-Saunders distributions: Modelling and applications. Risks
**2018**, 6, 21. [Google Scholar] [CrossRef][Green Version] - Puentes, R.; Marchant, C.; Leiva, V.; Figueroa-Zúñiga, J.I.; Ruggeri, F. Predicting PM2.5 and PM10 levels during critical episodes management in Santiago, Chile, with a bivariate Birnbaum-Saunders log-linear model. Mathematics
**2021**, 9, 645. [Google Scholar] [CrossRef] - Marchant, C.; Leiva, V.; Cysneiros, F.J.A.; Liu, S. Robust multivariate control charts based on Birnbaum-Saunders distributions. J. Stat. Comput. Simul.
**2018**, 88, 182–202. [Google Scholar] [CrossRef] - Saulo, H.; Leao, J.; Vila, R.; Leiva, V.; Tomazella, V. On mean-based bivariate Birnbaum-Saunders distributions: Properties, inference and application. Commun. Stat. Theory Methods
**2020**, 49, 6032–6056. [Google Scholar] [CrossRef] - Dasilva, A.; Dias, R.; Leiva, V.; Marchant, C.; Saulo, H. Birnbaum-Saunders regression models: A comparative evaluation of three approaches. J. Stat. Comput. Simul.
**2020**, 90, 2552–2570. [Google Scholar] [CrossRef] - Leiva, V.; Saulo, H.; Souza, R.; Aykroyd, R.G.; Vila, R. A new BISARMA time series model for forecasting mortality using weather and particulate matter data. J. Forecast.
**2021**, 40, 346–364. [Google Scholar] [CrossRef] - Saulo, H.; Leao, J.; Leiva, V.; Aykroyd, R.G. Birnbaum-Saunders autoregressive conditional duration models applied to high-frequency financial data. Stat. Pap.
**2019**, 60, 1605–1629. [Google Scholar] [CrossRef][Green Version] - Huerta, M.; Leiva, V.; Liu, S.; Rodriguez, M.; Villegas, D. On a partial least squares regression model for asymmetric data with a chemical application in mining. Chemom. Intell. Lab. Syst.
**2019**, 190, 55–68. [Google Scholar] [CrossRef] - Rodriguez, M.; Leiva, V.; Huerta, M.; Lillo, M.; Tapia, A.; Ruggeri, F. An asymmetric area model-based approach for small area estimation applied to survey data. REVSTAT Stat. J.
**2021**, 19, 399–420. [Google Scholar] - Costa, E.; Santos-Neto, M.; Leiva, V. Optimal sample size for the Birnbaum-Saunders distribution under decision theory with symmetric and asymmetric loss functions. Symmetry
**2021**, 13, 926. [Google Scholar] [CrossRef] - Martin-Barreiro, C.; Ramirez-Figueroa, J.A.; Nieto, A.B.; Leiva, V.; Martin-Casado, A.; Galindo-Villardón, M.P. A new algorithm for computing disjoint orthogonal components in the three-way Tucker model. Mathematics
**2021**, 9, 203. [Google Scholar] [CrossRef] - Martin-Barreiro, C.; Ramirez-Figueroa, J.A.; Cabezas, X.; Leiva, V.; Galindo-Villardón, M.P. Disjoint and functional principal component analysis for infected cases and deaths due to COVID-19 in South American countries with sensor-related data. Sensors
**2021**, 21, 4094. [Google Scholar] [CrossRef] [PubMed] - Desousa, M.; Saulo, H.; Leiva, V.; Scalco, P. On a tobit-Birnbaum-Saunders model with an application to medical data. J. Appl. Stat.
**2018**, 45, 932–955. [Google Scholar] [CrossRef] - de La Fuente-Mella, H.; Rubilar, R.; Chahuan-Jimenez, K.; Leiva, V. Modeling COVID-19 cases statistically and evaluating their effect on the economy of countries. Mathematics
**2021**, 9, 1558. [Google Scholar] [CrossRef] - Azevedo, C.; Leiva, V.; Athayde, E.; Balakrishnan, N. Shape and change point analyses of the Birnbaum-Saunders-t hazard rate and associated estimation. Comput. Stat. Data Anal.
**2012**, 56, 3887–3897. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**PDF (

**a**) and CDF (

**b**) plots of the BBS3 and BS distributions with $\alpha =0.8$, $\beta =0.3$ and $\delta =0.1$ (solid line), $\alpha =0.8$, $\beta =0.3$ and $\delta =0$ (dashed line), and $\alpha =0.8$, $\beta =0.3$ and $\delta =0.8$ (dotted line).

**Figure 2.**Histogram with estimated $\mathrm{BBS}1$—dotted line, $\mathrm{BBS}2$—dashed line, and $\mathrm{BBS}3$—solid line, PDFs (

**a**); and plots of empirical CDF versus the $\mathrm{BBS}1$—dotted line, $\mathrm{BBS}2$—dashed line, and $\mathrm{BBS}3$—solid line, theoretical CDFs (

**b**) with eruption data.

**Figure 3.**QQ plots of the $\mathrm{BBS}1$ (

**a**), $\mathrm{BBS}2$ (

**b**) and $\mathrm{BBS}3$ (

**c**) distributions with eruption data.

**Figure 4.**Histogram with estimated $\mathrm{BBS}1$—dotted line, $\mathrm{BBS}2$—dashed line, and $\mathrm{BBS}3$—solid line, PDFs (

**a**); and plots of empirical CDF versus the $\mathrm{BBS}1$—dotted line, $\mathrm{BBS}2$—dashed line, and $\mathrm{BBS}3$—solid line, theoretical CDFs (

**b**) with Dcr data.

**Figure 5.**QQ plots of the $\mathrm{BBS}1$ (

**a**), $\mathrm{BBS}2$ (

**b**) and $\mathrm{BBS}3$ (

**c**) distributions with Dcr data.

Framework | Item | Cause | Threshold | Effect | Random Variable |
---|---|---|---|---|---|

Environmental | Chemical element | Contamination | Target | Emergence | Content |

Fatigue | Material | Damage | Rupture | Failure | Fatigue life |

Health metric | Patient | Impulse | To die | Death | Death time |

Neural spiking | Neuron | Voltage | Firing | Spike | Inter-spike time |

**Table 2.**Empirical means, SDs, ALIs, and covering probabilities of 95% confidence intervals based on simulated data from the $\mathrm{BBS}3(\alpha ,\beta ,\delta )$ distribution for its parameter estimators using the maximum likelihood method.

n | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\delta}$ | Mean$(\widehat{\mathit{\alpha}})$ | SD$(\widehat{\mathit{\alpha}})$ | ALI$(\widehat{\mathit{\alpha}})$ | C$(\widehat{\mathit{\alpha}})$ | Mean$(\widehat{\mathit{\beta}})$ | SD$(\widehat{\mathit{\beta}})$ | ALI$(\widehat{\mathit{\beta}})$ | C$(\widehat{\mathit{\beta}})$ | Mean$(\widehat{\mathit{\delta}})$ | SD$(\widehat{\mathit{\delta}})$ | ALI$(\widehat{\mathit{\delta}})$ | C$(\widehat{\mathit{\delta}})$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

50 | 0.1 | 0.2 | 0.8 | 0.0992 | 0.0067 | 0.0261 | 94.90 | 0.2003 | 0.0036 | 0.0143 | 94.80 | 1.0105 | 0.5855 | 2.2953 | 93.44 |

100 | 0.1 | 0.2 | 0.8 | 0.0995 | 0.0047 | 0.0183 | 95.22 | 0.2001 | 0.0024 | 0.0093 | 95.04 | 0.8951 | 0.3361 | 1.3174 | 94.30 |

150 | 0.1 | 0.2 | 0.8 | 0.0998 | 0.0038 | 0.0148 | 95.00 | 0.2001 | 0.0020 | 0.0078 | 95.34 | 0.8505 | 0.2367 | 0.9278 | 94.36 |

200 | 0.1 | 0.2 | 0.8 | 0.0998 | 0.0033 | 0.0128 | 95.06 | 0.2001 | 0.0017 | 0.0066 | 94.98 | 0.8489 | 0.2089 | 0.8188 | 94.64 |

50 | 0.2 | 0.3 | 0.5 | 0.1974 | 0.0165 | 0.0646 | 94.36 | 0.3019 | 0.0138 | 0.0541 | 94.70 | 0.6083 | 0.3473 | 1.3615 | 94.18 |

100 | 0.2 | 0.3 | 0.5 | 0.1983 | 0.0114 | 0.0446 | 94.58 | 0.3014 | 0.0094 | 0.0369 | 94.48 | 0.5549 | 0.1918 | 0.7518 | 93.60 |

150 | 0.2 | 0.3 | 0.5 | 0.1991 | 0.0095 | 0.0371 | 94.46 | 0.3006 | 0.0076 | 0.0299 | 94.94 | 0.5307 | 0.1422 | 0.5576 | 94.36 |

200 | 0.2 | 0.3 | 0.5 | 0.1994 | 0.0083 | 0.0325 | 94.72 | 0.3005 | 0.0067 | 0.0262 | 94.76 | 0.5212 | 0.1203 | 0.4714 | 95.16 |

50 | 0.5 | 0.5 | 0.5 | 0.4890 | 0.0601 | 0.2357 | 95.08 | 0.5220 | 0.0829 | 0.3248 | 93.90 | 0.6472 | 0.4457 | 1.7472 | 94.56 |

100 | 0.5 | 0.5 | 0.5 | 0.4947 | 0.0395 | 0.1550 | 95.06 | 0.5102 | 0.0537 | 0.2105 | 94.20 | 0.5576 | 0.2155 | 0.8447 | 94.28 |

150 | 0.5 | 0.5 | 0.5 | 0.4965 | 0.0323 | 0.1267 | 94.92 | 0.5067 | 0.0431 | 0.1690 | 94.64 | 0.5322 | 0.1494 | 0.5857 | 94.02 |

200 | 0.5 | 0.5 | 0.5 | 0.4971 | 0.0273 | 0.1071 | 94.56 | 0.5050 | 0.0365 | 0.1429 | 94.56 | 0.5233 | 0.1263 | 0.4951 | 94.32 |

50 | 0.5 | 0.8 | 0.2 | 0.4939 | 0.0439 | 0.1722 | 94.96 | 0.8191 | 0.0957 | 0.3751 | 94.26 | 0.2211 | 0.0969 | 0.3797 | 94.96 |

100 | 0.5 | 0.8 | 0.2 | 0.4978 | 0.0301 | 0.1179 | 94.90 | 0.8079 | 0.0643 | 0.2520 | 94.68 | 0.2075 | 0.0547 | 0.2143 | 94.86 |

150 | 0.5 | 0.8 | 0.2 | 0.4980 | 0.0245 | 0.0960 | 94.62 | 0.8061 | 0.0516 | 0.2024 | 94.56 | 0.2051 | 0.0440 | 0.1723 | 94.98 |

200 | 0.5 | 0.8 | 0.2 | 0.4986 | 0.0216 | 0.0846 | 95.12 | 0.8053 | 0.0451 | 0.1767 | 94.96 | 0.2035 | 0.0380 | 0.1492 | 95.04 |

50 | 0.5 | 0.8 | 0.4 | 0.4898 | 0.0560 | 0.2195 | 94.94 | 0.8309 | 0.1239 | 0.4858 | 93.92 | 0.4934 | 0.3101 | 1.2156 | 95.00 |

100 | 0.5 | 0.8 | 0.4 | 0.4955 | 0.0374 | 0.1466 | 94.30 | 0.8139 | 0.0823 | 0.3226 | 94.70 | 0.4366 | 0.1479 | 0.5797 | 94.48 |

150 | 0.5 | 0.8 | 0.4 | 0.4970 | 0.0305 | 0.1196 | 94.62 | 0.8100 | 0.0652 | 0.2558 | 94.76 | 0.4199 | 0.1089 | 0.4269 | 94.42 |

200 | 0.5 | 0.8 | 0.4 | 0.4975 | 0.0258 | 0.1011 | 94.74 | 0.8073 | 0.0554 | 0.2170 | 94.68 | 0.4151 | 0.0908 | 0.3559 | 94.52 |

50 | 0.8 | 0.8 | 0.1 | 0.7923 | 0.0670 | 0.2626 | 94.82 | 0.8212 | 0.1195 | 0.4685 | 94.62 | 0.1091 | 0.0397 | 0.1557 | 93.66 |

100 | 0.8 | 0.8 | 0.1 | 0.7961 | 0.0478 | 0.1872 | 94.68 | 0.8116 | 0.0830 | 0.3254 | 94.88 | 0.1037 | 0.0251 | 0.0983 | 94.82 |

150 | 0.8 | 0.8 | 0.1 | 0.7971 | 0.0386 | 0.1513 | 95.06 | 0.8079 | 0.0663 | 0.2600 | 94.62 | 0.1026 | 0.0200 | 0.0783 | 95.00 |

200 | 0.8 | 0.8 | 0.1 | 0.7971 | 0.0333 | 0.1303 | 94.80 | 0.8065 | 0.0564 | 0.2210 | 94.82 | 0.1023 | 0.0170 | 0.0667 | 94.68 |

**Table 3.**Empirical means, SDs, ALIs, and covering probabilities of 95% confidence intervals based on simulated data from the $\mathrm{BBS}3(\alpha ,\beta ,\delta )$ distribution for its parameter estimators using the moment method.

n | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\delta}$ | Mean$(\widehat{\mathit{\alpha}})$ | SD$(\widehat{\mathit{\alpha}})$ | ALI$(\widehat{\mathit{\alpha}})$ | C$(\widehat{\mathit{\alpha}})$ | Mean$(\widehat{\mathit{\beta}})$ | SD$(\widehat{\mathit{\beta}})$ | ALI$(\widehat{\mathit{\beta}})$ | C$(\widehat{\mathit{\beta}})$ | Mean$(\widehat{\mathit{\delta}})$ | SD$(\widehat{\mathit{\delta}})$ | ALI$(\widehat{\mathit{\delta}})$ | C$(\widehat{\mathit{\delta}})$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

10 | 0.1 | 0.2 | 0.8 | 0.0928 | 0.0233 | 0.0912 | 97.44 | 0.2014 | 0.0143 | 0.0559 | 94.94 | 0.8100 | 0.0722 | 0.2829 | 95.06 |

25 | 0.1 | 0.2 | 0.8 | 0.0973 | 0.0123 | 0.0482 | 96.16 | 0.2007 | 0.0091 | 0.0357 | 95.02 | 0.8073 | 0.0624 | 0.2447 | 95.70 |

40 | 0.1 | 0.2 | 0.8 | 0.0984 | 0.0091 | 0.0357 | 94.88 | 0.2005 | 0.0074 | 0.0289 | 95.30 | 0.8044 | 0.0528 | 0.2070 | 96.56 |

10 | 0.2 | 0.3 | 0.5 | 0.1873 | 0.0491 | 0.1927 | 94.76 | 0.3094 | 0.0495 | 0.1939 | 94.88 | 0.5334 | 0.1666 | 0.6532 | 95.62 |

25 | 0.2 | 0.3 | 0.5 | 0.1953 | 0.0313 | 0.1228 | 95.10 | 0.3035 | 0.0309 | 0.1210 | 94.82 | 0.5232 | 0.1346 | 0.5276 | 95.08 |

40 | 0.2 | 0.3 | 0.5 | 0.1970 | 0.0249 | 0.0978 | 95.24 | 0.3025 | 0.0244 | 0.0956 | 94.88 | 0.5193 | 0.1175 | 0.4604 | 95.30 |

10 | 0.5 | 0.8 | 0.4 | 0.4925 | 0.0509 | 0.1996 | 98.22 | 0.7449 | 0.1350 | 0.5291 | 96.18 | 0.1278 | 1.3174 | 5.1644 | 98.66 |

25 | 0.5 | 0.8 | 0.4 | 0.4963 | 0.0482 | 0.1890 | 99.04 | 0.7608 | 0.1079 | 0.4231 | 93.90 | 0.1715 | 1.1857 | 4.6480 | 98.82 |

40 | 0.5 | 0.8 | 0.4 | 0.4973 | 0.0461 | 0.1806 | 98.98 | 0.7675 | 0.0948 | 0.3717 | 90.64 | 0.2090 | 0.8960 | 3.5124 | 97.96 |

10 | 0.5 | 0.5 | 0.5 | 0.5017 | 0.0509 | 0.1995 | 98.06 | 0.4818 | 0.0842 | 0.3302 | 90.06 | 0.4354 | 0.4059 | 1.5909 | 88.36 |

25 | 0.5 | 0.5 | 0.5 | 0.4994 | 0.0327 | 0.1282 | 98.38 | 0.4961 | 0.0466 | 0.1825 | 96.06 | 0.4963 | 0.2398 | 0.9402 | 95.68 |

40 | 0.5 | 0.5 | 0.5 | 0.4996 | 0.0171 | 0.0670 | 95.84 | 0.4986 | 0.0336 | 0.1316 | 96.50 | 0.5030 | 0.1781 | 0.6980 | 96.34 |

**Table 4.**Maximum likelihood estimates (and SEs in parenthesis) of the indicated distribution parameters with eruption data and model selection indicators.

Estimate/Indicator | BBS1 (SE) | BBS2 (SE) | BBS3 (SE) |
---|---|---|---|

$\widehat{\alpha}$ | 0.0898 (0.0045) | 0.1257 (0.0039) | 0.1042 (0.0027) |

$\widehat{\beta}$ | 65.0852 (0.3740) | 66.9851 (0.4785) | 63.8997 (0.4378) |

$\widehat{\delta}$ | −2.1242 (0.1392) | −3.8569 (0.0225) | 1.7480 (0.4438) |

Log-likelihood | −1054.180 | −1050.573 | −1036.824 |

AIC | 2114.360 | 2107.146 | 2079.648 |

BIC | 2129.325 | 2122.111 | 2090.465 |

KS statistic | 0.1764 | 0.0808 | 0.0735 |

KS p-value | 0.0004 | 0.3359 | 0.454 |

n | Mean | SD | CS | CK |
---|---|---|---|---|

172 | 69.623 | 38.0311 | −0.2112 | 2.1637 |

**Table 6.**Maximum likelihood estimates (and SEs in parenthesis) of the indicated distribution parameters with Dcr data and model selection indicators.

Estimate/Indicator | BBS1 (SE) | BBS1 (SE) | BBS3 (SE) |
---|---|---|---|

$\widehat{\alpha}$ | 0.4308 (0.0252) | 0.6976 (0.0283) | 0.6784 (0.0442) |

$\widehat{\beta}$ | 27.9125 (0.9800) | 31.7897 (1.4594) | 15.6625 (1.5963) |

$\widehat{\delta}$ | −2.5689 (0.1831) | −2.6272 (0.3830) | 0.4213 (0.1031) |

Log-likelihood | −870.4175 | −851.217 | −838.790 |

AIC | 1746.835 | 1708.434 | 1683.580 |

BIC | 1756.277 | 1717.876 | 1692.602 |

KS statistic | 0.0698 | 0.0640 | 0.0407 |

KS p-value | 0.4329 | 0.4949 | 0.7521 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Reyes, J.; Arrué, J.; Leiva, V.; Martin-Barreiro, C.
A New Birnbaum–Saunders Distribution and Its Mathematical Features Applied to Bimodal Real-World Data from Environment and Medicine. *Mathematics* **2021**, *9*, 1891.
https://doi.org/10.3390/math9161891

**AMA Style**

Reyes J, Arrué J, Leiva V, Martin-Barreiro C.
A New Birnbaum–Saunders Distribution and Its Mathematical Features Applied to Bimodal Real-World Data from Environment and Medicine. *Mathematics*. 2021; 9(16):1891.
https://doi.org/10.3390/math9161891

**Chicago/Turabian Style**

Reyes, Jimmy, Jaime Arrué, Víctor Leiva, and Carlos Martin-Barreiro.
2021. "A New Birnbaum–Saunders Distribution and Its Mathematical Features Applied to Bimodal Real-World Data from Environment and Medicine" *Mathematics* 9, no. 16: 1891.
https://doi.org/10.3390/math9161891