# Skewness of Maximum Likelihood Estimators in the Weibull Censored Data

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## Abstract

**:**

## 1. Introduction

## 2. The Weibull Censored Data

## 3. Skewness Coefficient

**1**is a n-dimensional vector of ones. Finally, by (4) and the Fisher information matrix, the asymmetry coefficient of the distribution of $\widehat{\mathit{\beta}}$ to order ${n}^{-1/2}$ is given by

## 4. Simulation Study

- The terms ${\widehat{\gamma}}_{1}^{\u2606}$ and ${\widehat{\gamma}}_{1}$ are closer in all the considered combinations, suggesting that ${\widehat{\gamma}}_{1}$ approaches ${\widehat{\gamma}}_{1}^{\u2606}$ in a reasonable way, even when the sample size is small.
- In general terms, ${\widehat{\gamma}}_{1}$ approaches well $\rho $ for ${\widehat{\beta}}_{1}$ and ${\widehat{\beta}}_{2}$. However, for ${\widehat{\beta}}_{0}$ the terms seem discrepant even for $n=100$.
- Considering the 90 cases for $p=3$, $\rho $ ranges from $(-0.245,0.255)$, $(-0.429,0.340)$ and $(-0.819,1.181)$ for C 10%, 25% and 50%, respectively. For $p=5$, $\rho $ ranges from $(-0.373,0.252)$, $(-0.402,0.198)$ and $(-0.787,0.495)$ for C 10%, 25% and 50%, respectively. This suggest that a higher percentage of censorship produce a higher skewness in the MLE estimators for the components of $\mathit{\beta}$.
- Considering the 90 cases for $p=3$, $\rho $ ranges from $(-0.819,1.181)$, $(-0.363,0.867)$, $(-0.351,0.411)$, $(-0.305,0.346)$ and $(-0.273,0.255)$, for $n=20,30,40,60$ and 100, respectively. For $p=5$, $\rho $ ranges from $(-1.015,0.740)$, $(-0.529,0.426)$, $(-0.372,0.413)$, $(-0.318,0.320)$ and $(-0.225,0.243)$ for $n=20,30,40,60$ and 100, respectively. This suggest that, as expected, when n increases the skewness coefficient of the MLE estimators for the components of $\mathit{\beta}$ will be more symmetric.

## 5. Applications

#### 5.1. Smokers Dataset

`time`(in days) to first relapse (return to smoking). The study lasted 182 days (26 weeks). Therefore, the times are subject to a censoring type I (32% of times were censored). We only considered the 113 patients where such observed time was positive (non-zero). Other measures were assigned randomly treatment group with levels combination or patch only (

`grp`), age in years at time of randomization (

`age`) and

`employment`(full-time or non-full-time). We consider that

`time`${}_{i}\sim $WE$({\theta}_{i};\sigma )$, where $log{\theta}_{i}={\mathit{X}}_{i}^{\top}\mathit{\beta}$, $\mathit{\beta}={({\beta}_{\mathtt{intercept}},{\beta}_{\mathtt{grp}},{\beta}_{\mathtt{age}},{\beta}_{\mathtt{employment}})}^{\top}$ and

`grp`,

`age`and

`employment`. Note that the estimated skewness for all parameters were closer to zero, suggesting a symmetric distribution for the estimators which is corroborated by the estimated density based on the bootstrap.

#### 5.2. Insulating Fluids Dataset

`time`(in minutes) to breakdown and

`voltage`(in kilovolts). The authors assumed a regression structure based on the Weibull model and a common censoring time at $L=200$ (type I censoring), i.e.,

`time`${}_{i}\sim $WEI$({\theta}_{i},\sigma )$, where $log{\theta}_{i}={\mathit{X}}_{i}^{\top}\mathit{\beta}$, $i=1,\dots ,76$, ${\mathit{X}}_{i}^{\top}=({\beta}_{\mathtt{Intercept}},{\beta}_{\mathtt{log}}-\mathtt{voltage})$. We estimated ${\widehat{\sigma}}_{J}=1.296704$ based on the jackknife method, which was used as known. Table 3 shows the estimates, standard errors and estimated skewness coefficient for the MLE estimators and Figure 2 shows the estimated density function for ${\widehat{\beta}}_{\mathtt{Intercept}}$ and ${\widehat{\beta}}_{\mathtt{log}}-\mathtt{voltage}$. Newly, the estimated skewness for both parameters are closer to zero, suggesting a symmetric distribution for the estimators as also suggest the estimated density based on bootstrap.

## 6. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. W’s Quantities

#### Appendix A.2. Derivatives and Cumulants

## References

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**Figure 1.**Estimated density function based on 1000 bootstrap samples and the asymptotic distribution for ${\widehat{\beta}}_{\mathtt{grp}}$ (left panel), ${\widehat{\beta}}_{\mathtt{age}}$ (center panel) and ${\widehat{\beta}}_{\mathtt{employment}}$ (right panel). The red line denotes the estimated parameter.

**Figure 2.**Estimated density function based on 1000 bootstrap samples and the asymptotic distribution for ${\widehat{\beta}}_{\mathtt{Intercept}}$ (left panel) and ${\widehat{\beta}}_{\mathtt{log}-\mathtt{voltage}}$ (right panel). The red line denotes the estimated parameter.

**Table 1.**The ${n}^{-1/2}$ and sample skewness coefficients of the distributions of the MLEs in the Weibull censored data with $p=3$ regressors and $\mathit{\beta}=(-2,0.5,1)$.

${\widehat{\mathit{\beta}}}_{0}$ | ${\widehat{\mathit{\beta}}}_{1}$ | ${\widehat{\mathit{\beta}}}_{2}$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

C | $\mathit{\sigma}$ | $\mathit{n}$ | $\mathit{\rho}$ | ${\widehat{\mathit{\gamma}}}_{\mathbf{1}}^{\u2606}$ | ${\widehat{\mathit{\gamma}}}_{\mathbf{1}}$ | $\mathit{\rho}$ | ${\widehat{\mathit{\gamma}}}_{\mathbf{1}}^{\u2606}$ | ${\widehat{\mathit{\gamma}}}_{\mathbf{1}}$ | $\mathit{\rho}$ | ${\widehat{\mathit{\gamma}}}_{\mathbf{1}}^{\u2606}$ | ${\widehat{\mathit{\gamma}}}_{\mathbf{1}}$ |

20 | −0.235 | −0.080 | −0.095 | 0.052 | 0.207 | 0.197 | 0.071 | 0.245 | 0.234 | ||

30 | −0.169 | 0.009 | 0.001 | 0.037 | 0.040 | 0.041 | 0.097 | 0.212 | 0.204 | ||

0.5 | 40 | −0.114 | −0.013 | −0.019 | 0.038 | 0.107 | 0.104 | 0.075 | 0.196 | 0.193 | |

60 | −0.139 | −0.082 | −0.083 | 0.011 | 0.061 | 0.061 | 0.033 | 0.171 | 0.170 | ||

100 | −0.059 | −0.055 | −0.055 | 0.054 | 0.074 | 0.073 | −0.005 | 0.087 | 0.087 | ||

20 | −0.198 | 0.017 | 0.004 | −0.008 | 0.225 | 0.197 | 0.068 | 0.298 | 0.284 | ||

30 | −0.219 | −0.101 | −0.107 | 0.110 | 0.200 | 0.192 | 0.216 | 0.304 | 0.299 | ||

10% | 1.0 | 40 | −0.210 | 0.004 | −0.002 | 0.083 | 0.140 | 0.137 | 0.109 | 0.185 | 0.182 |

60 | −0.147 | −0.005 | −0.010 | 0.085 | 0.143 | 0.137 | 0.057 | 0.173 | 0.171 | ||

100 | −0.092 | −0.013 | −0.015 | 0.019 | 0.048 | 0.047 | 0.116 | 0.132 | 0.131 | ||

20 | −0.232 | 0.006 | 0.005 | 0.094 | 0.143 | 0.131 | 0.022 | 0.093 | 0.087 | ||

30 | −0.178 | −0.041 | −0.040 | 0.004 | 0.054 | 0.048 | −0.007 | 0.147 | 0.136 | ||

3.0 | 40 | −0.185 | 0.005 | 0.003 | 0.002 | 0.024 | 0.023 | 0.064 | 0.156 | 0.151 | |

60 | −0.128 | −0.020 | −0.020 | 0.044 | 0.049 | 0.047 | 0.073 | 0.124 | 0.120 | ||

100 | −0.117 | −0.028 | −0.028 | 0.084 | 0.100 | 0.095 | 0.039 | 0.077 | 0.075 |

Parameter | Estimate | s.e. | ${\mathit{\gamma}}_{1}$ |
---|---|---|---|

${\beta}_{\mathtt{intercept}}$ | 3.1690 | 0.8136 | −0.0478 |

${\beta}_{\mathtt{grp}}$ | −1.0303 | 0.3694 | −0.0529 |

${\beta}_{\mathtt{age}}$ | 0.0541 | 0.0167 | 0.1251 |

${\beta}_{\mathtt{employment}}$ | −1.1460 | 0.3935 | −0.0753 |

Parameter | Estimate | s.e. | ${\mathit{\gamma}}_{1}$ |
---|---|---|---|

${\beta}_{\mathtt{intercept}}$ | 20.4342 | 1.8772 | 0.1451 |

${\beta}_{\mathtt{log}}-\mathtt{voltage}$ | −0.5311 | 0.0557 | −0.1517 |

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**MDPI and ACS Style**

Magalhães, T.M.; Gallardo, D.I.; Gómez, H.W.
Skewness of Maximum Likelihood Estimators in the Weibull Censored Data. *Symmetry* **2019**, *11*, 1351.
https://doi.org/10.3390/sym11111351

**AMA Style**

Magalhães TM, Gallardo DI, Gómez HW.
Skewness of Maximum Likelihood Estimators in the Weibull Censored Data. *Symmetry*. 2019; 11(11):1351.
https://doi.org/10.3390/sym11111351

**Chicago/Turabian Style**

Magalhães, Tiago M., Diego I. Gallardo, and Héctor W. Gómez.
2019. "Skewness of Maximum Likelihood Estimators in the Weibull Censored Data" *Symmetry* 11, no. 11: 1351.
https://doi.org/10.3390/sym11111351