# Bias Reduction for the Marshall-Olkin Extended Family of Distributions with Application to an Airplane’s Air Conditioning System and Precipitation Data

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Bias Correction of MLE

`Step 0`: choose an initial value for $\mathbf{\psi}=(\alpha ,\lambda )$, say ${\widehat{\mathbf{\psi}}}^{\left(0\right)}$. A possible value can be ${\widehat{\mathbf{\psi}}}^{\left(0\right)}=(1,{\widehat{\lambda}}^{\left(0\right)})$, where ${\widehat{\lambda}}^{\left(0\right)}$ is the MLE for $\lambda $ considering that ${X}_{1},\dots ,{X}_{n}$ are iid from $G(\xb7;\lambda )$.`Step 1`: For $k=1,2,\dots ,$ choose ${\widehat{\lambda}}^{\left(k\right)}$ as the vector that maximizes the profile log-likelihood function $\ell ({\widehat{\alpha}}^{(k-1)};\lambda )$ in relation to $\lambda $.`Step 2`: For $k=1,2,\dots ,$ do ${\widehat{\alpha}}_{M}^{\left(k\right)}$ as the solution for $\alpha $ in (6) considering $\lambda ={\widehat{\lambda}}^{\left(k\right)}$.

`Steps 1`and

`2`are repeated until a convergence rule is satisfied. For instance, $\left|\right|{\mathbf{\psi}}^{(k+1)}-{\mathbf{\psi}}^{\left(k\right)}\left|\right|$ is less than a tolerance value, where $\left|\right|x\left|\right|$ is the Euclidean norm of x.

## 3. Numerical Results

#### 3.1. Reducing the Bias in the MOEE and MOER Models

`R`software [32].

#### 3.2. A Simulated Example with Outliers

## 4. Two Applications

#### 4.1. Air Conditioning System of an Airplane Data Set

#### 4.2. Precipitation Data Set

## 5. Concluding Remarks

`R`. As mentioned by Firth [28]: “the merits of bias reduction in any particular problem will depend on several factors, including the skewness of the maximum likelihood estimator.” Since the MLE of the parameters in the Marshall-Olkin extended family is quite biased, this is an indication that its distribution is asymmetrical. For a future work, we suggest the study of the skewness of the MLE in the $MO{E}_{g}$, as, recently performed by Magalhães et al. [36] for the varying beta regression model (BRM) and Magalhães et al. [37] for Weibull censored data (WCD). In the first work, the authors showed that the MLE of the precision parameter is quite asymmetrical, while in the second, the MLE of the parameters is close to symmetry. This agrees with bias literature for the respective models, which showed that estimates of the precision parameters in the varying beta regression model are highly biased while the estimates in WCD are little biased. Since there is no closed-form for the skewness coefficient of the MLE of the $MO{E}_{g}$, it will be a greater contribution to this family of distributions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Details of the Computation of M(α)

#### Details of the Asymptotic CI Used in Application 2

## References

- Marshall, A.W.; Olkin, I. A new method for adding a parameter to a family of distributions with application to the exponential and weibull families. Biometrika
**1997**, 84, 641–652. [Google Scholar] [CrossRef] - Marshall, A.W.; Olkin, I. Life Distributions. Structure of Nonparametric, Semipara- Metric and Parametric Families; Springer: New York, NY, USA, 2007. [Google Scholar]
- Castellares, F.; Lemonte, A.J. On the Marshall-Olkin extended distributions. Commun. Stat. Theory Methods
**2016**, 45, 4537–4555. [Google Scholar] [CrossRef] - Ghitany, M.E. Marshall-Olkin extended pareto distribution and its application. Int. J. Appl. Math.
**2005**, 18, 17–32. [Google Scholar] - Ghitany, M.E.; Al-Hussaini, E.K.; Jarallah, R.A. Marshall-Olkin extended Weibull distribution and its application to censored data. J. Appl. Stat.
**2005**, 32, 1025–1034. [Google Scholar] [CrossRef] - Ristic, M.M.; Kanichukattu, J.; Joseph, A. A marshall-olkin gamma distribution and minification process. STARS Int. J. Sci.
**2007**, 1, 107–117. [Google Scholar] - Ghitany, M.E.; Al-Awadhi, F.A.; Alkhalfan, L.A. Marshall-olkin extended lomax distribution and its application to censored data. Commun. Stat. Theory Methods
**2007**, 36, 1855–1866. [Google Scholar] [CrossRef] - Ghitany, M.E.; Kotz, S. Reliability properties of extended linear failure-rate distributions. Probab. Eng. Inf. Sci.
**2007**, 21, 441–450. [Google Scholar] [CrossRef] - Jayakumar, K.; Mathew, T. On a generalization to Marshall–Olkin scheme and its application to Burr type XII distribution. Stat. Pap.
**2008**, 49, 421–439. [Google Scholar] [CrossRef] - García, V.; Gómez, E.; Vásquez-Polo, F. A new skew generalization of the normal distribution: Properties and applications. Comput. Stat. Data Anal.
**2010**, 54, 2021–2034. [Google Scholar] [CrossRef] - Gómez-Déniz, E. Another generalization of the geometric distribution. Test
**2010**, 19, 399–415. [Google Scholar] [CrossRef] - Lemonte, A. A new extension of the Birnbaum-Saunders distribution. Braz. J. Probab. Stat.
**2013**, 27, 133–149. [Google Scholar] [CrossRef] - Cordeiro, G.; Lemonte, A. On the Marshall-Olkin extended Weibull distribution. Stat. Pap.
**2013**, 54, 333–353. [Google Scholar] [CrossRef] - Cordeiro, G.; Lemonte, A.; Ortega, E. The Marshall-Olkin family of distributions: Mathematical properties and new models. J. Stat. Theory Pract.
**2013**, 8, 343–366. [Google Scholar] [CrossRef] - Alshangiti, A.M.; Kayid, M.; Alarfaj, B. A new family of Marshall-Olkin extended distributions. J. Comput. Appl. Math.
**2014**, 271, 369–379. [Google Scholar] [CrossRef] - Alizadeh, M.; Cordeiro, G.M.; De Brito, E.; Demétrio, C.G.B. The beta Marshall-Olkin family of distributions. J. Stat. Distrib. Appl.
**2015**, 2, 1–18. [Google Scholar] [CrossRef][Green Version] - Ristić, M.; Kundu, D. Marshall-Olkin generalized exponential distribution. Metron
**2015**, 73, 317–333. [Google Scholar] [CrossRef] - Benkhelifa, L. The Marshall-Olkin extended generalized Lindley distribution: Properties and applications. Commun. Stat. Simul. Computation.
**2017**, 46, 8306–8330. [Google Scholar] [CrossRef] - Afify, A.Z.; Cordeiro, G.M.; Yousof, H.M.; Saboor, A.; Ortega, E.M.M. The Marshall-Olkin additive Weibull distribution with variable shapes for the hazard rate. Hacet. J. Math. Stat.
**2018**, 47, 365–381. [Google Scholar] [CrossRef] - Javed, M.; Nawaz, T.; Irfan, M. The Marshall-Olkin Kappa distribution: Properties and applications. J. King Saud Univ. Sci.
**2019**, 31, 684–691. [Google Scholar] [CrossRef] - Mansoor, M.; Tahir, M.H.; Cordeiro, G.M.; Provost, S.B.; Alzaatreh, A. The Marshall-Olkin logistic-exponential distribution. Commun. Stat. Theory Methods
**2019**, 48, 220–234. [Google Scholar] [CrossRef] - Yousof, H.M.; Afify, A.Z.; Alizadeh, M.; Nadarajah, S.; Aryal, G.; Hamedani, G. The Marshall-Olkin generalized-G family of distributions with applications. Statistica
**2018**, 78, 273–295. [Google Scholar] - Durrans, S.R. Distributions of fractional order statistics in hydrology. Water Resour. Res.
**1992**, 28, 1649–1655. [Google Scholar] [CrossRef] - Guney, Y.; Tuac, Y.; Arslan, O. Marshall-Olkin distribution: Parameter estimation and application to cancer data. J. Appl. Stat.
**2017**, 44, 2238–2250. [Google Scholar] [CrossRef] - Korkmaz, M.Ç.; Cordeiro, G.M.; Yousof, H.M.; Pescim, R.R.; Afify, A.Z.; Nadarajah, S. The Weibull Marshall-Olkin family: Regression model and application to censored data. Commun. Stat. Theory Methods
**2019**, 48, 4171–4194. [Google Scholar] [CrossRef] - Nassar, M.; Kumar, D.; Dey, S.; Cordeiro, G.M.; Afify, A.Z. The Marshall-Olkin alpha power family of distributions with applications. J. Comput. Appl. Math.
**2019**, 1, 41–53. [Google Scholar] [CrossRef] - Cox, D.R.; Snell, E.J. A general definition of residuals. J. R. Society. Ser. B Methodol.
**1968**, 30, 248–275. [Google Scholar] [CrossRef] - Firth, D. Bias reduction of maximum likelihood estimates. Biometrika
**1993**, 80, 27–38. [Google Scholar] [CrossRef] - Sartori, N. Bias prevention of maximum likelihood estimates for scalar skew normal and skew t distributions. J. Stat. Plan. Inference
**2006**, 136, 4259–4275. [Google Scholar] [CrossRef] - Cordeiro, G.M.; Cribari-Neto, F. An introduction to Bartlett Corrections and Bias Reduction; Springer: Berlin/Heidelberg, Geramny, 2014. [Google Scholar]
- Arrué, J.; Arellano-Valle, R.B.; Gómez, H.W. Bias reduction of maximum likelihood estimates for a modified skew normal distribution. J. Stat. Comput. Simul.
**2016**, 86, 2967–2984. [Google Scholar] [CrossRef] - R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2017. [Google Scholar]
- Linhart, H.; Zucchini, W. Model Selection; John Wiley and Sons: New York, NY, USA, 1986. [Google Scholar]
- Dunn, P.; Smyth, G. Randomized quantile residuals. J. Comput. Graph. Stat.
**1996**, 5, 236–244. [Google Scholar] - Hinkley, D. On quick choice of power transformation. Am. Stat.
**1977**, 26, 67–69. [Google Scholar] [CrossRef] - Magalhães, T.M.; Botter, D.A.; Sandoval, M.C.; Pereira, G.H.A.; Cordeiro, G.M. Skewness of maximum likelihood estimators in the varying dispersion beta regression model. Commun. Stat. Theory Methods
**2019**, 48, 4250–4260. [Google Scholar] [CrossRef] - Magalhães, T.M.; Gallardo, D.I.; Gómez, H.W. Skewness of maximum likelihood estimators in the weibull censored data. Symmetry
**2019**, 11, 1351. [Google Scholar] [CrossRef][Green Version]

**Figure 2.**Estimated bias for the MLE of $\alpha $ and $\theta $ in the MOEE($\alpha $, $\theta $) under different scenarios based on 5000 replicates.

**Figure 3.**Estimated bias for the MLE of $\alpha $ and $\theta $ in the MOER($\alpha $, $\theta $) under different scenarios based on 5000 replicates.

**Figure 7.**Empirical cdf and cdf estimated by MLE and BCE in the air conditioning system of an airplane data set considering the MOEE model.

**Figure 8.**Randomized quantile residuals for MOEE in the air conditioning system of an airplane. Left panel: MLE. Right panel: BCE. The p-values for the following normality tests are also presented: Kolmogorov-Smirnov (KS), Shapiro-Wilks (SW), Anderson-Darling (AD) and Cramer-von Mises (CVM)

**Figure 9.**Histogram and estimated pdf for MLE and BCE considering the MOER distribution in the precipitation data set.

**Figure 10.**Width of 95% CI for quantiles of MOER model considering the MLE (continuous line) and BCE (dashed line).

$\mathit{\alpha}=1.5$, $\mathit{\theta}=1.5$ | $\mathit{\alpha}=2.5$, $\mathit{\theta}=1.5$ | |||||||
---|---|---|---|---|---|---|---|---|

n | Estimator of $\mathit{\alpha}$ | Estimator of $\mathit{\theta}$ | Estimator of $\mathit{\alpha}$ | Estimator of $\mathit{\theta}$ | ||||

MLE | BCE | MLE | BCE | MLE | BCE | MLE | BCE | |

10 | 2.329 | 1.374 | 0.385 | 0.062 | 2.481 | 1.588 | 0.297 | 0.116 |

(0.855) | (0.656) | (1.283) | (1.104) | (0.851) | (0.642) | (1.036) | (0.924) | |

20 | 0.711 | 0.311 | 0.184 | 0.033 | 0.715 | 0.385 | 0.144 | 0.011 |

(0.780) | (0.693) | (0.743) | (0.670) | (0.780) | (0.664) | (0.619) | (0.578) | |

30 | 0.414 | 0.170 | 0.122 | 0.030 | 0.408 | 0.193 | 0.096 | 0.017 |

(0.749) | (0.699) | (0.564) | (0.520) | (0.749) | (0.700) | (0.474) | (0.447) | |

40 | 0.295 | 0.120 | 0.089 | 0.021 | 0.290 | 0.130 | 0.070 | 0.014 |

(0.732) | (0.692) | (0.465) | (0.435) | (0.736) | (0.700) | (0.394) | (0.376) | |

50 | 0.233 | 0.098 | 0.072 | 0.019 | 0.228 | 0.099 | 0.057 | 0.012 |

(0.713) | (0.686) | (0.417) | (0.395) | (0.717) | (0.690) | (0.354) | (0.339) |

Parameter | True | Original | with Outlier | ||
---|---|---|---|---|---|

Value | MLE | BCE | MLE | BCE | |

$\alpha $ | 2.5 | 3.873 | 2.174 | 5.223 | 2.866 |

$\theta $ | 1.5 | 1.839 | 1.464 | 1.817 | 1.467 |

**Table 3.**Estimates for MOEE distribution in the air conditioning system of an airplane. Standard errors are presented in parenthesis.

Parameter | MLE | BCE |
---|---|---|

$\alpha $ | 0.380 (0.272) | 0.285 (0.225) |

$\theta $ | 0.010 (0.005) | 0.008 (0.005) |

**Table 4.**Estimates, standard errors (SE) and bias (estimated via bootstrap) for MOER distribution in precipitation data set.

Parameter | MLE | BCE | ||||
---|---|---|---|---|---|---|

Estimate | SE | Bias | Estimate | SE | Bias | |

$\alpha $ | 0.514 | 0.351 | 0.255 | 0.394 | 0.270 | 0.213 |

$\theta $ | 0.185 | 0.087 | 0.041 | 0.157 | 0.080 | 0.039 |

**Table 5.**Approximated 95% CI for the q-th quantile of the MOER model in precipitation data set based on MLE and BCE.

q | MLE | BCE | ||
---|---|---|---|---|

95% IC | Width | 95% IC | Width | |

0.10 | (0.5313–0.5629) | 0.0316 | (0.5106–0.5361) | 0.0256 |

0.25 | (0.8845–0.9619) | 0.0773 | (0.8553–0.9210) | 0.0657 |

0.50 | (1.4225–1.5678) | 0.1452 | (1.3882–1.5261) | 0.1379 |

0.75 | (2.1286–2.3566) | 0.2281 | (2.1067–2.3593) | 0.2526 |

0.90 | (2.8078–3.2958) | 0.4880 | (2.8108–3.4102) | 0.5994 |

0.99 | (3.5118–5.7177) | 2.2059 | (3.3545–6.3541) | 2.9996 |

© 2020 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/).

## Share and Cite

**MDPI and ACS Style**

Magalhães, T.M.; Gómez, Y.M.; Gallardo, D.I.; Venegas, O.
Bias Reduction for the Marshall-Olkin Extended Family of Distributions with Application to an Airplane’s Air Conditioning System and Precipitation Data. *Symmetry* **2020**, *12*, 851.
https://doi.org/10.3390/sym12050851

**AMA Style**

Magalhães TM, Gómez YM, Gallardo DI, Venegas O.
Bias Reduction for the Marshall-Olkin Extended Family of Distributions with Application to an Airplane’s Air Conditioning System and Precipitation Data. *Symmetry*. 2020; 12(5):851.
https://doi.org/10.3390/sym12050851

**Chicago/Turabian Style**

Magalhães, Tiago M., Yolanda M. Gómez, Diego I. Gallardo, and Osvaldo Venegas.
2020. "Bias Reduction for the Marshall-Olkin Extended Family of Distributions with Application to an Airplane’s Air Conditioning System and Precipitation Data" *Symmetry* 12, no. 5: 851.
https://doi.org/10.3390/sym12050851