# A New Quantile Regression Model and Its Diagnostic Analytics for a Weibull Distributed Response with Applications

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## Abstract

**:**

## 1. Introduction, Motivations, and Outline

#### 1.1. Bibliographical Review

#### 1.2. Limitations of the Usual Regression Model

`R`software by the package

`survival`; see [27,28]. The characteristics of the insulating fluid defined in various standards can be broadly classified into chemical, electrical, and physical features. For example, the electrical characteristics (breakdown voltages) of the insulating fluid are affected by elements such as water content and electrostatic charges, but also possibly affected by trace components in this fluid.

#### 1.3. Objective and Outline

## 2. A New Weibull Quantile Regression Model

#### 2.1. A Reparameterized Weibull Distribution

#### 2.2. Shape Analysis

#### 2.3. The Weibull Quantile Regression Model

## 3. Estimation, Inference and Goodness of Fit

#### 3.1. Parameter Estimation

`R`software, including the BFGS approach for constrained and unconstrained maximization; see [27].

#### 3.2. Inference and Hypothesis Testing

#### 3.3. Residuals

## 4. Monte Carlo Simulation

#### 4.1. Setting

#### 4.2. Scenario 1: Maximum Likelihood Estimation

`R`software and its

`maxBFGS`function, which implements the BFGS algorithm with constraints for maximization and requires initial values for estimating $\mathit{\beta}={({\beta}_{0},{\beta}_{1})}^{\top}$ and k. We utilize the least square estimator of $\mathit{\beta}$ assuming a usual linear regression and the maximum likelihood estimate of k based on the observations ${y}_{1},\dots ,{y}_{n}$ without considering covariates. The maximum likelihood estimates are presented in Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7 wherein the empirical mean, bias, variance, root mean squared error (RMSE), CS, and CK are all reported. A look at the results in Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7 allows us to conclude that, in general, as the sample size increases, the bias, variance, and $\mathrm{RMSE}$ of the estimators decrease, as expected. Moreover, ${\widehat{\beta}}_{0}$, ${\widehat{\beta}}_{1}$, and $\widehat{k}$ seem all to be consistent and asymptotically normal distributed. Our study was conducted on a Dell Inspirion 5748 personal computer with an Intel core i7-4510U CPU, 2.00 GHz × 4, and 8 GB of RAM.

#### 4.3. Scenario 2: Empirical Distribution of the Residuals

## 5. Local Influence

#### 5.1. Perturbation Matrix and Potentially Influential Cases

#### 5.2. Perturbation Schemes

#### 5.2.1. Case-Weight Perturbation

#### 5.2.2. Perturbation on the Response

#### 5.2.3. Perturbation in the Continuous Covariate

#### 5.2.4. Perturbation of the Parameter $\mathit{k}$

## 6. Illustrative Example

#### 6.1. The Adjusted Weibull Quantile Regression

`quant.weibull.reg()`in the

`R`software, which allows us to fit Weibull quantile regression models to a data set, computing information criteria and residuals. To select the best model amongst a set of options, the AIC, BIC, and CAIC can be used. These information criteria assume the existence of an unknown “true model”. The AIC chooses the model whose divergence in relation to the “true model” is the minimum within the competing models and may be computed by

#### 6.2. Local Influence Analysis

#### 6.3. Coefficients across Quantiles

## 7. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Histogram (

**a**) and boxplots (

**b**) for the data of times to electrical breakdown with the full data set, and histogram (

**c**) and boxplots (

**d**) for the data set without cases #2 and #3.

**Figure 2.**QQ plot with envelopes of the Pearson residual for normal regression with the data of times to electrical breakdown.

**Figure 3.**Plots of the $\mathrm{Wei}(Q,k)$ probability density function for $q=0.25$ (

**left**), $q=0.5$ (

**center**) and $q=0.75$ (

**right**), with $Q=1.0$ (

**a**–

**c**), $k=1.0$ (

**d**–

**f**) and $k=2.0$ (

**g**–

**i**).

**Figure 4.**QQ plot with envelope of ${r}_{i}^{\mathrm{RQ}}$ (

**a**) and ${r}_{i}^{\mathrm{CGS}}$ (

**b**) for the Weibull median regression and of ${r}_{i}^{\mathrm{RQ}}$ for the Birnbaum–Saunders quantile regression model with logarithm link (

**c**), using the data of the time to electrical breakdown of an insulating fluid.

**Figure 5.**Index plots of ${\mathrm{C}}_{i}\left(\mathit{\theta}\right)$ under case-weight perturbation (

**a**), response perturbation (

**b**), perturbation of the parameter k (

**c**), and covariate perturbation X (

**d**) for the data of time to electrical breakdown of an insulating fluid and the Weibull quantile regression.

Median | Mean | SD | CV | CS | CK | ${\mathit{y}}_{\left(1\right)}$ | ${\mathit{y}}_{\left(\mathit{n}\right)}$ | n |
---|---|---|---|---|---|---|---|---|

7.7400 | 122.51 | 430.24 | 3.51 | 4.36 | 20.93 | 0.09 | 2323.70 | 41 |

**Table 2.**Statistics from simulated Weibull regression data ($q=0.10,{\beta}_{0}=0.50,{\beta}_{1}=1.00$).

$\mathit{k}=0.5$ | $\mathit{k}=1.00$ | $\mathit{k}=2.00$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Statistic | $n=50$ | $n=200$ | $n=600$ | $n=50$ | $n=200$ | $n=600$ | $n=50$ | $n=200$ | $n=600$ | ||

${\widehat{\beta}}_{0}$ | ${\widehat{\beta}}_{0}$ | ${\widehat{\beta}}_{0}$ | |||||||||

True value | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 | ||

Mean | 0.6011 | 0.5491 | 0.5138 | 0.5506 | 0.5244 | 0.5069 | 0.5253 | 0.5122 | 0.5034 | ||

Bias | 0.1011 | 0.0491 | 0.0138 | 0.0506 | 0.0244 | 0.0069 | 0.0253 | 0.0122 | 0.0034 | ||

Variance | 0.6747 | 0.1746 | 0.0537 | 0.1687 | 0.0437 | 0.0134 | 0.0422 | 0.0109 | 0.0034 | ||

RMSE | 0.8276 | 0.4207 | 0.2322 | 0.4138 | 0.2104 | 0.1161 | 0.2069 | 0.1052 | 0.0581 | ||

CS | −0.1288 | −0.1332 | −0.1183 | −0.1287 | −0.1331 | −0.1179 | −0.1286 | −0.1327 | −0.1180 | ||

CK | 3.1481 | 3.0124 | 2.9364 | 3.1475 | 3.0105 | 2.9359 | 3.1476 | 3.0094 | 2.9360 | ||

${\widehat{\beta}}_{1}$ | ${\widehat{\beta}}_{1}$ | ${\widehat{\beta}}_{1}$ | |||||||||

True value | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | ||

Mean | 1.0193 | 0.9831 | 0.9954 | 1.0097 | 0.9915 | 0.9977 | 1.0048 | 0.9958 | 0.9988 | ||

Bias | 0.0193 | −0.0169 | −0.0046 | 0.0097 | −0.0085 | −0.0023 | 0.0048 | −0.0042 | −0.0012 | ||

Variance | 0.8966 | 0.2356 | 0.0745 | 0.2241 | 0.0589 | 0.0186 | 0.0560 | 0.0147 | 0.0047 | ||

RMSE | 0.9471 | 0.4856 | 0.2730 | 0.4735 | 0.2428 | 0.1365 | 0.2368 | 0.1214 | 0.0682 | ||

CS | 0.0619 | −0.0344 | 0.1067 | 0.0621 | −0.0347 | 0.1068 | 0.0621 | −0.0348 | 0.1067 | ||

CK | 2.8443 | 3.0606 | 3.0311 | 2.8454 | 3.0607 | 3.0311 | 2.8440 | 3.0633 | 3.0311 | ||

$\widehat{k}$ | $\widehat{k}$ | $\widehat{k}$ | |||||||||

True value | 0.5000 | 0.5000 | 0.5000 | 1.0000 | 1.0000 | 1.0000 | 2.0000 | 2.0000 | 2.0000 | ||

Mean | 0.5210 | 0.5061 | 0.5021 | 1.0419 | 1.0122 | 1.0043 | 2.0838 | 2.0244 | 2.0086 | ||

Bias | 0.0210 | 0.0061 | 0.0021 | 0.0419 | 0.0122 | 0.0043 | 0.0838 | 0.0244 | 0.0086 | ||

Variance | 0.0036 | 0.0008 | 0.0003 | 0.0144 | 0.0033 | 0.0010 | 0.0576 | 0.0130 | 0.0041 | ||

RMSE | 0.0636 | 0.0292 | 0.0162 | 0.1271 | 0.0584 | 0.0324 | 0.2543 | 0.1168 | 0.0648 | ||

CS | 0.5824 | 0.2446 | 0.0840 | 0.5826 | 0.2450 | 0.0840 | 0.5831 | 0.2447 | 0.0841 | ||

CK | 3.7567 | 2.9277 | 2.6255 | 3.7563 | 2.9247 | 2.6253 | 3.7577 | 2.9246 | 2.6250 |

**Table 3.**Statistics from simulated Weibull regression data ($q=0.50,{\beta}_{0}=0.50,{\beta}_{1}=1.00$).

$\mathit{k}=0.5$ | $\mathit{k}=1.00$ | $\mathit{k}=2.00$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Statistic | $n=50$ | $n=200$ | $n=600$ | $n=50$ | $n=200$ | $n=600$ | $n=50$ | $n=200$ | $n=600$ | ||

${\widehat{\beta}}_{0}$ | ${\widehat{\beta}}_{0}$ | ${\widehat{\beta}}_{0}$ | |||||||||

True value | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 | ||

Mean | 0.4963 | 0.5152 | 0.5015 | 0.4982 | 0.5076 | 0.5008 | 0.4991 | 0.5038 | 0.5004 | ||

Bias | −0.0037 | 0.0152 | 0.0015 | −0.0018 | 0.0076 | 0.0008 | −0.0009 | 0.0038 | 0.0004 | ||

Variance | 0.3423 | 0.0885 | 0.0261 | 0.0856 | 0.0221 | 0.0065 | 0.0214 | 0.0055 | 0.0016 | ||

RMSE | 0.5851 | 0.2978 | 0.1615 | 0.2925 | 0.1489 | 0.0808 | 0.1463 | 0.0745 | 0.0404 | ||

CS | −0.2084 | −0.1717 | −0.1039 | −0.2085 | −0.1716 | −0.1039 | −0.2084 | −0.1715 | −0.1038 | ||

CK | 3.0076 | 3.1321 | 2.9602 | 3.0078 | 3.1320 | 2.9601 | 3.0075 | 3.1319 | 2.9600 | ||

${\widehat{\beta}}_{1}$ | ${\widehat{\beta}}_{1}$ | ${\widehat{\beta}}_{1}$ | |||||||||

True value | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | ||

Mean | 1.0194 | 0.9831 | 0.9954 | 1.0097 | 0.9916 | 0.9977 | 1.0049 | 0.9958 | 0.9988 | ||

Bias | 0.0194 | −0.0169 | −0.0046 | 0.0097 | −0.0084 | −0.0023 | 0.0049 | −0.0042 | −0.0012 | ||

Variance | 0.8965 | 0.2355 | 0.0745 | 0.2241 | 0.0589 | 0.0186 | 0.0560 | 0.0147 | 0.0047 | ||

RMSE | 0.9470 | 0.4856 | 0.2730 | 0.4735 | 0.2428 | 0.1365 | 0.2368 | 0.1214 | 0.0683 | ||

CS | 0.0619 | −0.0343 | 0.1067 | 0.0619 | −0.0343 | 0.1068 | 0.0620 | −0.0344 | 0.1067 | ||

CK | 2.8447 | 3.0612 | 3.0311 | 2.8448 | 3.0612 | 3.0309 | 2.8450 | 3.0609 | 3.0310 | ||

$\widehat{k}$ | $\widehat{k}$ | $\widehat{k}$ | |||||||||

True value | 0.5000 | 0.5000 | 0.5000 | 1.0000 | 1.0000 | 1.0000 | 2.0000 | 2.0000 | 2.0000 | ||

Mean | 0.5210 | 0.5061 | 0.5021 | 1.0419 | 1.0122 | 1.0043 | 2.0838 | 2.0243 | 2.0086 | ||

Bias | 0.0210 | 0.0061 | 0.0021 | 0.0419 | 0.0122 | 0.0043 | 0.0838 | 0.0243 | 0.0086 | ||

Variance | 0.0036 | 0.0008 | 0.0003 | 0.0144 | 0.0033 | 0.0010 | 0.0576 | 0.0130 | 0.0041 | ||

RMSE | 0.0636 | 0.0292 | 0.0162 | 0.1271 | 0.0584 | 0.0324 | 0.2543 | 0.1168 | 0.0648 | ||

CS | 0.5826 | 0.2448 | 0.0841 | 0.5825 | 0.2448 | 0.0840 | 0.5824 | 0.2448 | 0.0840 | ||

CK | 3.7568 | 2.9256 | 2.6256 | 3.7565 | 2.9256 | 2.6254 | 3.7559 | 2.9256 | 2.6255 |

**Table 4.**Statistics from simulated Weibull regression data ($q=0.90,{\beta}_{0}=0.50,{\beta}_{1}=1.00$).

$\mathit{k}=0.5$ | $\mathit{k}=1.00$ | $\mathit{k}=2.00$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Statistic | $n=50$ | $n=200$ | $n=600$ | $n=50$ | $n=200$ | $n=600$ | $n=50$ | $n=200$ | $n=600$ | ||

${\widehat{\beta}}_{0}$ | ${\widehat{\beta}}_{0}$ | ${\widehat{\beta}}_{0}$ | |||||||||

True value | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 | ||

Mean | 0.4295 | 0.4939 | 0.4937 | 0.4648 | 0.4969 | 0.4969 | 0.4824 | 0.4985 | 0.4984 | ||

Bias | −0.0705 | −0.0061 | −0.0063 | −0.0352 | −0.0031 | −0.0031 | −0.0176 | −0.0015 | −0.0016 | ||

Variance | 0.3075 | 0.0794 | 0.0235 | 0.0769 | 0.0198 | 0.0059 | 0.0192 | 0.0050 | 0.0015 | ||

RMSE | 0.5590 | 0.2818 | 0.1534 | 0.2795 | 0.1409 | 0.0767 | 0.1397 | 0.0704 | 0.0384 | ||

CS | −0.1501 | −0.1109 | −0.1234 | −0.1505 | −0.1108 | −0.1234 | −0.1504 | −0.1109 | −0.1234 | ||

CK | 2.9205 | 3.1507 | 2.9856 | 2.9191 | 3.1504 | 2.9857 | 2.9190 | 3.1504 | 2.9857 | ||

${\widehat{\beta}}_{1}$ | ${\widehat{\beta}}_{1}$ | ${\widehat{\beta}}_{1}$ | |||||||||

True value | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | ||

Mean | 1.0194 | 0.9831 | 0.9953 | 1.0097 | 0.9915 | 0.9977 | 1.0048 | 0.9958 | 0.9988 | ||

Bias | 0.0194 | −0.0169 | −0.0047 | 0.0097 | −0.0085 | −0.0023 | 0.0048 | −0.0042 | −0.0012 | ||

Variance | 0.8965 | 0.2355 | 0.0745 | 0.2241 | 0.0589 | 0.0186 | 0.0560 | 0.0147 | 0.0047 | ||

RMSE | 0.9470 | 0.4856 | 0.2730 | 0.4735 | 0.2428 | 0.1365 | 0.2368 | 0.1214 | 0.0682 | ||

CS | 0.0617 | −0.0343 | 0.1067 | 0.0620 | −0.0343 | 0.1067 | 0.0619 | −0.0342 | 0.1067 | ||

CK | 2.8453 | 3.0612 | 3.0310 | 2.8448 | 3.0612 | 3.0311 | 2.8447 | 3.0609 | 3.0311 | ||

$\widehat{k}$ | $\widehat{k}$ | $\widehat{k}$ | |||||||||

True value | 0.5000 | 0.5000 | 0.5000 | 1.0000 | 1.0000 | 1.0000 | 2.0000 | 2.0000 | 2.0000 | ||

Mean | 0.5210 | 0.5061 | 0.5021 | 1.0419 | 1.0122 | 1.0043 | 2.0838 | 2.0243 | 2.0086 | ||

Bias | 0.0210 | 0.0061 | 0.0021 | 0.0419 | 0.0122 | 0.0043 | 0.0838 | 0.0243 | 0.0086 | ||

Variance | 0.0036 | 0.0008 | 0.0003 | 0.0144 | 0.0033 | 0.0010 | 0.0576 | 0.0130 | 0.0041 | ||

RMSE | 0.0636 | 0.0292 | 0.0162 | 0.1271 | 0.0584 | 0.0324 | 0.2543 | 0.1168 | 0.0648 | ||

CS | 0.5825 | 0.2448 | 0.0840 | 0.5825 | 0.2449 | 0.0840 | 0.5825 | 0.2447 | 0.0840 | ||

CK | 3.7567 | 2.9257 | 2.6254 | 3.7567 | 2.9258 | 2.6255 | 3.7566 | 2.9257 | 2.6254 |

**Table 5.**Statistics from simulated Weibull regression data ($q=0.10,{\beta}_{0}=1.00,{\beta}_{1}=2.50$).

$\mathit{k}=0.5$ | $\mathit{k}=1.00$ | $\mathit{k}=2.00$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Statistic | $n=50$ | $n=200$ | $n=600$ | $n=50$ | $n=200$ | $n=600$ | $n=50$ | $n=200$ | $n=600$ | ||

${\widehat{\beta}}_{0}$ | ${\widehat{\beta}}_{0}$ | ${\widehat{\beta}}_{0}$ | |||||||||

True value | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | ||

Mean | 1.1013 | 1.0493 | 1.0138 | 1.0506 | 1.0244 | 1.0069 | 1.0253 | 1.0122 | 1.0034 | ||

Bias | 0.1013 | 0.0493 | 0.0138 | 0.0506 | 0.0244 | 0.0069 | 0.0253 | 0.0122 | 0.0034 | ||

Variance | 0.6747 | 0.1747 | 0.0537 | 0.1687 | 0.0437 | 0.0134 | 0.0422 | 0.0109 | 0.0034 | ||

RMSE | 0.8276 | 0.4208 | 0.2322 | 0.4138 | 0.2104 | 0.1161 | 0.2069 | 0.1052 | 0.0581 | ||

CS | −0.1295 | −0.1353 | −0.1183 | −0.1290 | −0.1330 | −0.1180 | −0.1292 | −0.1332 | −0.1185 | ||

CK | 3.1489 | 3.0134 | 2.9364 | 3.1480 | 3.0104 | 2.9358 | 3.1485 | 3.0116 | 2.9364 | ||

${\widehat{\beta}}_{1}$ | ${\widehat{\beta}}_{1}$ | ${\widehat{\beta}}_{1}$ | |||||||||

True value | 2.5000 | 2.5000 | 2.5000 | 2.5000 | 2.5000 | 2.5000 | 2.5000 | 2.5000 | 2.5000 | ||

Mean | 2.5193 | 2.4830 | 2.4954 | 2.5097 | 2.4915 | 2.4977 | 2.5048 | 2.4958 | 2.4989 | ||

Bias | 0.0193 | −0.0170 | −0.0046 | 0.0097 | −0.0085 | −0.0023 | 0.0048 | −0.0042 | −0.0011 | ||

Variance | 0.8962 | 0.2358 | 0.0745 | 0.2241 | 0.0589 | 0.0186 | 0.0560 | 0.0147 | 0.0047 | ||

RMSE | 0.9469 | 0.4859 | 0.2730 | 0.4735 | 0.2428 | 0.13657 | 0.2368 | 0.1214 | 0.0682 | ||

CS | 0.0622 | −0.0376 | 0.1068 | 0.0619 | −0.0346 | 0.1066 | 0.0621 | −0.0336 | 0.1070 | ||

CK | 2.8454 | 3.0685 | 3.0310 | 2.8447 | 3.0617 | 3.0290 | 2.8452 | 3.0608 | 3.0310 | ||

$\widehat{k}$ | $\widehat{k}$ | $\widehat{k}$ | |||||||||

True value | 0.5000 | 0.5000 | 0.5000 | 1.0000 | 1.0000 | 1.0000 | 2.0000 | 2.0000 | 2.0000 | ||

Mean | 0.5210 | 0.5061 | 0.5021 | 1.0419 | 1.0122 | 1.0043 | 2.0838 | 2.0243 | 2.0086 | ||

Bias | 0.0210 | 0.0061 | 0.0021 | 0.0419 | 0.0122 | 0.0043 | 0.0838 | 0.0243 | 0.0086 | ||

Variance | 0.0036 | 0.0008 | 0.0003 | 0.0144 | 0.0033 | 0.0010 | 0.0576 | 0.0130 | 0.0041 | ||

RMSE | 0.0636 | 0.0292 | 0.0162 | 0.1271 | 0.0584 | 0.0324 | 0.2543 | 0.1168 | 0.0648 | ||

CS | 0.5824 | 0.2461 | 0.0840 | 0.5826 | 0.2453 | 0.0829 | 0.5824 | 0.2456 | 0.0836 | ||

CK | 3.7574 | 2.9265 | 2.6256 | 3.7571 | 2.9272 | 2.6223 | 3.7563 | 2.9261 | 2.6256 |

**Table 6.**Statistics from simulated Weibull regression data ($q=0.50,{\beta}_{0}=1.00,{\beta}_{1}=2.50$).

$\mathit{k}=0.5$ | $\mathit{k}=1.00$ | $\mathit{k}=2.00$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Statistic | $n=50$ | $n=200$ | $n=600$ | $n=50$ | $n=200$ | $n=600$ | $n=50$ | $n=200$ | $n=600$ | ||

${\widehat{\beta}}_{0}$ | ${\widehat{\beta}}_{0}$ | ${\widehat{\beta}}_{0}$ | |||||||||

True value | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | ||

Mean | 0.9963 | 1.0152 | 1.0015 | 0.9982 | 1.0076 | 1.0008 | 0.9991 | 1.0038 | 1.0004 | ||

Bias | −0.0037 | 0.0152 | 0.0015 | −0.0018 | 0.0076 | 0.0008 | −0.0009 | 0.0038 | 0.0004 | ||

Variance | 0.3423 | 0.0885 | 0.0261 | 0.0856 | 0.0221 | 0.0065 | 0.0214 | 0.0055 | 0.0016 | ||

RMSE | 0.5851 | 0.2978 | 0.1615 | 0.2925 | 0.1489 | 0.0807 | 0.1463 | 0.0745 | 0.0404 | ||

CS | −0.2084 | −0.1718 | −0.1039 | −0.2083 | −0.1716 | −0.1038 | −0.2084 | −0.1715 | −0.1044 | ||

CK | 3.0076 | 3.1324 | 2.9603 | 3.0073 | 3.1323 | 2.9601 | 3.0069 | 3.1312 | 2.9588 | ||

${\widehat{\beta}}_{1}$ | ${\widehat{\beta}}_{1}$ | ${\widehat{\beta}}_{1}$ | |||||||||

True value | 2.5000 | 2.5000 | 2.5000 | 2.5000 | 2.5000 | 2.5000 | 2.5000 | 2.5000 | 2.5000 | ||

Mean | 2.5194 | 2.4831 | 2.4954 | 2.5097 | 2.4916 | 2.4977 | 2.5049 | 2.4958 | 2.4989 | ||

Bias | 0.0194 | −0.0169 | −0.0046 | 0.0097 | −0.0084 | −0.0023 | 0.0049 | −0.0042 | −0.0011 | ||

Variance | 0.8964 | 0.2355 | 0.0745 | 0.2241 | 0.0589 | 0.0186 | 0.0560 | 0.0147 | 0.0047 | ||

RMSE | 0.9470 | 0.4856 | 0.2730 | 0.4735 | 0.2428 | 0.1365 | 0.2368 | 0.1214 | 0.0682 | ||

CS | 0.0619 | −0.0343 | 0.1067 | 0.0618 | −0.0343 | 0.1066 | 0.0618 | −0.0342 | 0.1060 | ||

CK | 2.8447 | 3.0612 | 3.0310 | 2.8448 | 3.0615 | 3.0307 | 2.8453 | 3.0600 | 3.0309 | ||

$\widehat{k}$ | $\widehat{k}$ | $\widehat{k}$ | |||||||||

True value | 0.5000 | 0.5000 | 0.5000 | 1.0000 | 1.0000 | 1.0000 | 2.0000 | 2.0000 | 2.0000 | ||

Mean | 0.5210 | 0.5061 | 0.5021 | 1.0419 | 1.0122 | 1.0043 | 2.0838 | 2.0243 | 2.0087 | ||

Bias | 0.0210 | 0.0061 | 0.0021 | 0.0419 | 0.0122 | 0.0043 | 0.0838 | 0.0243 | 0.0087 | ||

Variance | 0.0036 | 0.0008 | 0.0003 | 0.0144 | 0.0033 | 0.0010 | 0.0576 | 0.0130 | 0.0041 | ||

RMSE | 0.0636 | 0.0292 | 0.0162 | 0.1271 | 0.0584 | 0.0324 | 0.2543 | 0.1168 | 0.0648 | ||

CS | 0.5825 | 0.2447 | 0.0838 | 0.5824 | 0.2446 | 0.0838 | 0.5825 | 0.2446 | 0.0820 | ||

CK | 3.7565 | 2.9255 | 2.6254 | 3.7563 | 2.9257 | 2.6253 | 3.7571 | 2.9255 | 2.6247 |

**Table 7.**Statistics from simulated Weibull regression data ($q=0.90,{\beta}_{0}=1.00,{\beta}_{1}=2.50$).

$\mathit{k}=0.5$ | $\mathit{k}=1.00$ | $\mathit{k}=2.00$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Statistic | $n=50$ | $n=200$ | $n=600$ | $n=50$ | $n=200$ | $n=600$ | $n=50$ | $n=200$ | $n=600$ | ||

${\widehat{\beta}}_{0}$ | ${\widehat{\beta}}_{0}$ | ${\widehat{\beta}}_{0}$ | |||||||||

True value | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | ||

Mean | 0.9295 | 0.9938 | 0.9937 | 0.9648 | 0.9969 | 0.9969 | 0.9824 | 0.9985 | 0.9984 | ||

Bias | −0.0705 | −0.0062 | −0.0063 | −0.0352 | −0.0031 | −0.0031 | −0.0176 | −0.0015 | −0.0016 | ||

Variance | 0.3074 | 0.0794 | 0.0235 | 0.0769 | 0.0198 | 0.0059 | 0.0192 | 0.0050 | 0.0015 | ||

RMSE | 0.5589 | 0.2818 | 0.1534 | 0.2795 | 0.1409 | 0.0767 | 0.1397 | 0.0705 | 0.0384 | ||

CS | −0.1500 | −0.1104 | −0.1234 | −0.1505 | −0.1111 | −0.1234 | −0.1504 | −0.1107 | −0.1230 | ||

CK | 2.9199 | 3.1514 | 2.9857 | 2.9190 | 3.1513 | 2.9857 | 2.9190 | 3.1501 | 2.9853 | ||

${\widehat{\beta}}_{1}$ | ${\widehat{\beta}}_{1}$ | ${\widehat{\beta}}_{1}$ | |||||||||

True value | 2.5000 | 2.5000 | 2.5000 | 2.5000 | 2.5000 | 2.5000 | 2.5000 | 2.5000 | 2.5000 | ||

Mean | 2.5194 | 2.4832 | 2.4954 | 2.5097 | 2.4916 | 2.4977 | 2.5048 | 2.4958 | 2.4988 | ||

Bias | 0.0194 | −0.0168 | −0.0046 | 0.0097 | −0.0084 | −0.0023 | 0.0048 | −0.0042 | −0.0012 | ||

Variance | 0.8963 | 0.2355 | 0.0745 | 0.2241 | 0.0589 | 0.0186 | 0.0560 | 0.0147 | 0.0047 | ||

RMSE | 0.9469 | 0.4856 | 0.2730 | 0.4735 | 0.2428 | 0.1365 | 0.2368 | 0.1214 | 0.0683 | ||

CS | 0.0615 | −0.0349 | 0.1067 | 0.0620 | −0.0343 | 0.1068 | 0.0619 | −0.0341 | 0.1064 | ||

CK | 2.8454 | 3.0617 | 3.0310 | 2.8448 | 3.0610 | 3.0311 | 2.8448 | 3.0609 | 3.0310 | ||

$\widehat{k}$ | $\widehat{k}$ | $\widehat{k}$ | |||||||||

True value | 0.5000 | 0.5000 | 0.5000 | 1.0000 | 1.0000 | 1.0000 | 2.0000 | 2.0000 | 2.0000 | ||

Mean | 0.5210 | 0.5061 | 0.5021 | 1.0419 | 1.0122 | 1.0043 | 2.0838 | 2.0243 | 2.0086 | ||

Bias | 0.0210 | 0.0061 | 0.0021 | 0.0419 | 0.0122 | 0.0043 | 0.0838 | 0.0243 | 0.0086 | ||

Variance | 0.0036 | 0.0008 | 0.0003 | 0.0144 | 0.0033 | 0.0010 | 0.0576 | 0.0130 | 0.0041 | ||

RMSE | 0.0636 | 0.0292 | 0.0162 | 0.1271 | 0.0584 | 0.0324 | 0.2543 | 0.1168 | 0.0648 | ||

CS | 0.5825 | 0.2448 | 0.0840 | 0.5825 | 0.2449 | 0.0840 | 0.5825 | 0.2448 | 0.0838 | ||

CK | 3.7566 | 2.9255 | 2.6255 | 3.7567 | 2.9257 | 2.6254 | 3.7566 | 2.9256 | 2.6257 |

Statistic | $\mathit{k}=0.50$ | $\mathit{k}=1.00$ | $\mathit{k}=2.00$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$n=50$ | $n=200$ | $n=600$ | $n=50$ | $n=200$ | $n=600$ | $n=50$ | $n=200$ | $n=600$ | |||

$q=0.10$ | |||||||||||

Mean | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | ||

SD | 0.9882 | 0.9963 | 0.9986 | 0.9882 | 0.9963 | 0.9986 | 0.9882 | 0.9963 | 0.9986 | ||

CS | 1.5711 | 1.8525 | 1.9394 | 1.5710 | 1.8524 | 1.9394 | 1.5711 | 1.8524 | 1.9394 | ||

CK | 5.7186 | 7.6584 | 8.3894 | 5.7185 | 7.6578 | 8.3894 | 5.7187 | 7.6580 | 8.3895 | ||

$q=0.50$ | |||||||||||

Mean | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | ||

SD | 0.9882 | 0.9963 | 0.9986 | 0.9882 | 0.9963 | 0.9986 | 0.9882 | 0.9963 | 0.9986 | ||

CS | 1.5711 | 1.8524 | 1.9394 | 1.5711 | 1.8524 | 1.9394 | 1.5711 | 1.8524 | 1.9394 | ||

CK | 5.7189 | 7.6577 | 8.3894 | 5.7188 | 7.6577 | 8.3895 | 5.7188 | 7.6577 | 8.3895 | ||

$q=0.90$ | |||||||||||

Mean | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | ||

SD | 0.9882 | 0.9963 | 0.9986 | 0.9882 | 0.9963 | 0.9986 | 0.9882 | 0.9963 | 0.9986 | ||

CS | 1.5711 | 1.8524 | 1.9394 | 1.5711 | 1.8524 | 1.9394 | 1.5711 | 1.8524 | 1.9394 | ||

CK | 5.7189 | 7.6577 | 8.3895 | 5.7188 | 7.6577 | 8.3895 | 5.7189 | 7.6577 | 8.3895 |

Statistic | $\mathit{k}=0.50$ | $\mathit{k}=1.00$ | $\mathit{k}=2.00$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$n=50$ | $n=200$ | $n=600$ | $n=50$ | $n=200$ | $n=600$ | $n=50$ | $n=200$ | $n=600$ | |||

$q=0.10$ | |||||||||||

Mean | 0.0012 | 0.0004 | 0.0001 | 0.0012 | 0.0004 | 0.0001 | 0.0012 | 0.0004 | 0.0001 | ||

SD | 1.0134 | 1.0033 | 1.0011 | 1.0134 | 1.0033 | 1.0011 | 1.0133 | 1.0033 | 1.0011 | ||

CS | 0.0142 | 0.0026 | 0.0009 | 0.0142 | 0.0027 | 0.0009 | 0.0142 | 0.0027 | 0.0009 | ||

CK | 2.7487 | 2.9258 | 2.9774 | 2.7486 | 2.9258 | 2.9774 | 2.7487 | 2.9258 | 2.9774 | ||

$q=0.50$ | |||||||||||

Mean | 0.0012 | 0.0003 | 0.0001 | 0.0012 | 0.0004 | 0.0001 | 0.0012 | 0.0003 | 0.0001 | ||

SD | 1.0134 | 1.0033 | 1.0011 | 1.0134 | 1.0033 | 1.0011 | 1.0134 | 1.0033 | 1.0011 | ||

CS | 0.0142 | 0.0027 | 0.0009 | 0.0142 | 0.0027 | 0.0009 | 0.0142 | 0.0027 | 0.0009 | ||

CK | 2.7487 | 2.9258 | 2.9774 | 2.7487 | 2.9258 | 2.9774 | 2.7487 | 2.9258 | 2.9774 | ||

$q=0.90$ | |||||||||||

Mean | 0.0012 | 0.0004 | 0.0001 | 0.0012 | 0.0003 | 0.0001 | 0.0012 | 0.0003 | 0.0001 | ||

SD | 1.0134 | 1.0033 | 1.0011 | 1.0134 | 1.0033 | 1.0011 | 1.0134 | 1.0033 | 1.0011 | ||

CS | 0.0142 | 0.0027 | 0.0009 | 0.0142 | 0.0027 | 0.0009 | 0.0142 | 0.0027 | 0.0009 | ||

CK | 2.7487 | 2.9258 | 2.9774 | 2.7487 | 2.9258 | 2.9774 | 2.7487 | 2.9258 | 2.9774 |

**Table 10.**Values of AIC, BIC, CAIC, and log-likelihood function for Weibull median-regression models with the data of time to electrical breakdown of an insulating fluid.

Model | AIC | CAIC | BIC | ${\mathbf{R}}_{\mathbf{M}}^{2}$ | Log-Likelihood |
---|---|---|---|---|---|

L1 | 327.07 | 327.71 | 332.21 | 0.71 | −160.53 |

L2 | 351.63 | 352.28 | 356.77 | 0.47 | −172.81 |

**Table 11.**Estimate, SE, and p-value of the indicated parameter for the data of time to electrical breakdown of an insulating fluid.

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

Estimate | 20.97 | −0.56 | 0.82 |

SE | 1.86 | 0.06 | 0.10 |

p-value | <0.01 | <0.01 | <0.01 |

**Table 12.**RCs of maximum likelihood estimates and of the associated estimated SEs for the indicated cases, and respective p-values for the data of time to electrical breakdown of an insulating fluid and the Weibull quantile regression.

Parameter | ||||
---|---|---|---|---|

Removed Case(s) | ${\mathit{\beta}}_{\mathbf{0}}$ | ${\mathit{\beta}}_{\mathbf{1}}$ | $\mathit{k}$ | |

None | RC($\widehat{\theta}$) | N/A | N/A | N/A |

RC($\widehat{\mathrm{SE}}$) | N/A | N/A | N/A | |

p-value | <0.01 | <0.01 | <0.01 | |

$\{\#1\}$ | RC($\widehat{\theta}$) | 3.41 | 3.41 | 5.81 |

RC($\widehat{\mathrm{SE}}$) | 2.52 | 2.78 | 5.38 | |

p-value | <0.01 | <0.01 | <0.01 | |

$\{\#3\}$ | RC($\widehat{\theta}$) | 4.87 | 5.23 | 0.77 |

RC($\widehat{\mathrm{SE}}$) | 18.86 | 18.31 | 0.14 | |

p-value | <0.01 | <0.01 | <0.01 | |

$\{\#33\}$ | RC($\widehat{\theta}$) | 1.46 | 1.98 | 8.16 |

RC($\widehat{\mathrm{SE}}$) | 13.23 | 13.26 | 12.43 | |

p-value | <0.01 | <0.01 | <0.01 | |

$\{\#1,\#3\}$ | RC($\widehat{\theta}$) | 0.45 | 0.71 | 4.74 |

RC($\widehat{\mathrm{SE}}$) | 12.25 | 11.37 | 5.15 | |

p-value | <0.01 | <0.01 | <0.01 | |

$\{\#1,\#33\}$ | RC($\widehat{\theta}$) | 4.46 | 4.92 | 15.72 |

RC($\widehat{\mathrm{SE}}$) | 15.21 | 15.50 | 20.27 | |

p-value | <0.01 | <0.01 | <0.01 | |

$\{\#3,\#33\}$ | RC($\widehat{\theta}$) | 3.28 | 3.11 | 7.54 |

RC($\widehat{\mathrm{SE}}$) | 3.65 | 4.11 | 12.51 | |

p-value | <0.01 | <0.01 | <0.01 | |

$\{\#1,\#3,\#33\}$ | RC($\widehat{\theta}$) | 0.56 | 0.76 | 14.77 |

RC($\widehat{\mathrm{SE}}$) | 5.40 | 6.19 | 20.12 | |

p-value | <0.01 | <0.01 | <0.01 |

**Table 13.**Estimates of the parameters of the Weibull quantile regression model considering different quantiles, with insulating fluid data.

Estimate | $\mathit{q}=0.10$ | $\mathit{q}=0.25$ | $\mathit{q}=0.50$ | $\mathit{q}=0.75$ | $\mathit{q}=0.90$ | ${\mathit{q}}_{\mathbf{opt}}=0.32$ |
---|---|---|---|---|---|---|

$\widehat{{\beta}_{0}}$ | 18.97 | 21.80 | 20.97 | 20.19 | 21.02 | 20.50 |

$\widehat{{\beta}_{1}}$ | −0.57 | −0.62 | −0.56 | −0.52 | −0.52 | −0.57 |

$\widehat{k}$ | 0.84 | 0.81 | 0.82 | 0.84 | 0.84 | 0.85 |

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## Share and Cite

**MDPI and ACS Style**

Sánchez, L.; Leiva, V.; Saulo, H.; Marchant, C.; Sarabia, J.M. A New Quantile Regression Model and Its Diagnostic Analytics for a Weibull Distributed Response with Applications. *Mathematics* **2021**, *9*, 2768.
https://doi.org/10.3390/math9212768

**AMA Style**

Sánchez L, Leiva V, Saulo H, Marchant C, Sarabia JM. A New Quantile Regression Model and Its Diagnostic Analytics for a Weibull Distributed Response with Applications. *Mathematics*. 2021; 9(21):2768.
https://doi.org/10.3390/math9212768

**Chicago/Turabian Style**

Sánchez, Luis, Víctor Leiva, Helton Saulo, Carolina Marchant, and José M. Sarabia. 2021. "A New Quantile Regression Model and Its Diagnostic Analytics for a Weibull Distributed Response with Applications" *Mathematics* 9, no. 21: 2768.
https://doi.org/10.3390/math9212768