# Statistical Inference for the Weibull Distribution Based on δ-Record Data

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

#### 2.1. Likelihood of $\delta $-Record Observations

**Proposition**

**1.**

**Proof.**

**Remark**

**1.**

#### 2.2. Density of Future Records

**Proposition**

**2.**

**Proof.**

#### 2.3. Weibull Distribution

**Definition**

**4.**

- P
_{1}: - ${f}_{\lambda ,\beta}(x)={\lambda}^{-\beta}\beta {x}^{\beta -1}{e}^{-{(x/\lambda )}^{\beta}},\phantom{\rule{1.em}{0ex}}{F}_{\lambda ,\beta}(x)=1-{e}^{-{(x/\lambda )}^{\beta}}$,
- P
_{2}: - ${f}_{\alpha ,\beta}(x)=\alpha \beta {x}^{\beta -1}{e}^{-\alpha {x}^{\beta}},\phantom{\rule{1.em}{0ex}}{F}_{\alpha ,\beta}(x)=1-{e}^{-\alpha {x}^{\beta}}$,

## 3. Maximum Likelihood Analysis

**Remark**

**2.**

#### 3.1. MLE of Parameters $\lambda ,\beta $

**Proposition**

**3.**

**Proof.**

**Remark**

**3.**

#### 3.2. Strong Consistency

**Theorem**

**1.**

**Remark**

**4.**

#### 3.3. Maximum Likelihood Prediction of Future Records

**Definition**

**5.**

**Definition**

**6.**

**Remark**

**5.**

**Proposition**

**4.**

**Remark**

**6.**

#### 3.4. Simulation Study

#### 3.5. Real Data

## 4. Bayesian Analysis

**Definition**

**7.**

#### 4.1. Posterior Distributions

#### 4.2. Estimation of Hyperparameters in Soland’s Prior

**Lemma**

**1.**

**Proof.**

#### 4.3. Bayes Estimators of Parameters $\alpha ,\beta $

**Corollary**

**1.**

**Proof.**

#### 4.4. Prediction of Future Records

#### 4.5. Simulation Study

#### 4.5.1. Known $\beta $

#### 4.5.2. Unknown $\alpha ,\beta $

#### 4.6. Real Data

## 5. Final Comments

## 6. Technical Results and Proofs

#### 6.1. Consistency of the MLE $\widehat{\lambda}(\beta )$

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

**Proof of**

**Theorem 1.**

#### 6.2. MLP of ${R}_{m}$

**Proof of**

**Proposition 4.**

- If $m=n+1$, then $\rho =0$, $\frac{\partial {\widehat{l}}_{1}}{\partial x}$ is negative on $(0,+\infty )$ and so, ${\widehat{l}}_{1}(x)$ has no maximum on $(0,+\infty )$. In this case we do, however, define $\tilde{z}(\beta )={r}_{n}$, that is, ${\tilde{z}}^{\beta}(\beta )=\rho +{r}_{n}^{\beta}$.
- If $m>n+1$, then $\rho >0$, $\frac{\partial {\widehat{l}}_{1}}{\partial x}$ is positive on $(0,\rho )$ and is negative on $(\rho ,+\infty )$. Hence ${\tilde{z}}^{\beta}(\beta )=\rho +{r}_{n}^{\beta}$.

**Lemma**

**8.**

**Proof.**

**Lemma**

**9.**

**Proof.**

**Lemma**

**10.**

**Proof.**

**Remark**

**7.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Simulated values of estimated mean square error (EMSE) of $\widehat{\lambda}$ (left panel) and $\widehat{\beta}$ (right panel), from 1000 runs, for $n\in \{10,15,20,30,40,50\}$, $\lambda =0.5$ and $\beta =1$.

**Figure 2.**Frequentist coverage and average length of Bayes HPD intervals, based of records and $\delta $-records, for several values of $\alpha $, $\beta =1.5$ and $\gamma (\alpha |4,4)$ as $\alpha $ prior.

**Table 1.**Estimated Mean Square Errors (EMSE) of maximum likelihood estimators (MLE) $\widehat{\lambda},\widehat{\beta}$.

Known $\mathit{\beta}$ | Unknown $\mathit{\beta}$ | ||||||
---|---|---|---|---|---|---|---|

$\mathbf{\delta}=\mathbf{0}$ | $\mathbf{\delta}=-\mathbf{0.5}$ | $\mathbf{\delta}=\mathbf{0}$ | $\mathbf{\delta}=-\mathbf{0.5}$ | ||||

$\mathbf{\lambda}$ | $\mathbf{\beta}$ | EMSE $\widehat{\mathbf{\lambda}}(\mathbf{\beta})$ | EMSE $\widehat{\mathbf{\lambda}}(\mathbf{\beta})$ | EMSE $\widehat{\mathbf{\lambda}}$ | EMSE $\widehat{\mathbf{\beta}}$ | EMSE $\widehat{\mathbf{\lambda}}$ | EMSE $\widehat{\mathbf{\beta}}$ |

0.75 | 0.024 | 0.015 | 0.370 | 1.200 | 0.230 | 0.376 | |

0.25 | 1 | 0.013 | 0.004 | 0.118 | 1.814 | 0.033 | 0.174 |

1.5 | 0.005 | 0.001 | 0.032 | 4.187 | 0.001 | 0.067 | |

0.75 | 0.102 | 0.076 | 1.488 | 1.196 | 1.252 | 0.671 | |

0.5 | 1 | 0.051 | 0.027 | 0.463 | 1.871 | 0.298 | 0.631 |

1.5 | 0.022 | 0.006 | 0.128 | 3.688 | 0.038 | 0.388 |

Known $\mathit{\beta}$ | Unknown $\mathit{\beta}$ | ||||
---|---|---|---|---|---|

$\mathbf{\lambda}$ | $\mathbf{\beta}$ | $\mathbf{\delta}=\mathbf{0}$ | $\mathbf{\delta}=-\mathbf{0.5}$ | $\mathbf{\delta}=\mathbf{0}$ | $\mathbf{\delta}=-\mathbf{0.5}$ |

0.75 | 1.462 | 1.353 | 1.739 | 1.525 | |

0.25 | 1 | 0.223 | 0.193 | 0.260 | 0.213 |

1.5 | 0.027 | 0.022 | 0.034 | 0.024 | |

0.75 | 5.893 | 5.591 | 6.963 | 6.445 | |

0.5 | 1 | 0.884 | 0.793 | 1.072 | 0.917 |

1.5 | 0.114 | 0.092 | 0.139 | 0.106 |

**Table 3.**Records and near-records ($\delta $-records) from Castellote rainfall data in millimeters. Near records are shown to the right of their corresponding records (in boldface).

$\phantom{\rule{1.em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathit{\delta}$ | $\mathit{\delta}$-Record Values |
---|---|

$\phantom{\rule{1.em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0$ | 164.6, 184.9, 224.9, 247.1, 278.8 |

$\phantom{\rule{0.277778em}{0ex}}-25$ | 164.6, 184.9, 164.7, 224.9, 247.1, 278.8 |

$\phantom{\rule{0.277778em}{0ex}}-50$ | 164.6, 138.5, 184.9, 164.7, 224.9, 179.5, 247.1, 278.8, 230.5 |

$\phantom{\rule{0.277778em}{0ex}}-75$ | 164.6, 138.5, 108.0, 184.9, 164.7, 130.1, 224.9, 179.5, 154.1, 151.1, 157.3, |

247.1, 278.8, 204.3, 230.5, 214.0, 206.0, 209.1 |

**Table 4.**MLE $\widehat{\lambda},\widehat{\beta}$ and maximum likelihood prediction (MLP) ${\tilde{R}}_{7},{\tilde{R}}_{8},{\tilde{R}}_{9}$ for Castellote rainfall data.

$\mathit{\delta}$ | $\widehat{\mathit{\lambda}}$ | $\widehat{\mathit{\beta}}$ | ${\tilde{\mathit{R}}}_{7}$ | ${\tilde{\mathit{R}}}_{8}$ | ${\tilde{\mathit{R}}}_{9}$ |
---|---|---|---|---|---|

0 | 185 | 3.93 | 289.3 | 298.5 | 306.6 |

−25 | 188 | 3.48 | 294.1 | 306.8 | 317.7 |

−50 | 182 | 3.44 | 293.4 | 305.9 | 316.7 |

−75 | 150 | 2.90 | 292.7 | 305.1 | 316.3 |

$-\infty $ | 130 | 2.04 | 305.2 | 328.6 | 349.9 |

**Table 5.**EMSE of Bayes estimator ${\widehat{\alpha}}_{{}_{B}}(\beta )$; length (LHPD) and coverage (%CHPD) of highest posterior density (HPD) interval of parameter $\alpha $, with known $\beta $.

$\mathit{\delta}=0$ | $\mathit{\delta}=-0.5$ | ||||||
---|---|---|---|---|---|---|---|

$\mathit{\pi}(\mathit{\alpha})$ | $\mathit{\beta}$ | EMSE | LHPD | %CHPD | EMSE | LHPD | %CHPD |

0.75 | 0.128 | 1.262 | 94.9 | 0.098 | 1.140 | 95.1 | |

$\gamma (\alpha |4,4)$ | 1 | 0.131 | 1.263 | 94.7 | 0.090 | 1.077 | 95.0 |

1.5 | 0.120 | 1.250 | 94.9 | 0.066 | 0.919 | 94.7 | |

0.75 | 0.224 | 1.719 | 94.9 | 0.162 | 1.500 | 95.0 | |

$\gamma (\alpha |4,6)$ | 1 | 0.218 | 1.715 | 95.2 | 0.145 | 1.409 | 94.9 |

1.5 | 0.222 | 1.719 | 94.9 | 0.107 | 1.206 | 94.6 |

**Table 6.**EMSE of Bayes predictors ${\widehat{R}}_{6},{\widehat{R}}_{7}$; length (LHPD) and coverage (%CHPD) of HPD intervals for ${R}_{6},{R}_{7}$, with known $\beta $.

$\mathit{\delta}=0$ | $\mathit{\delta}=-0.5$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{R}}_{\mathbf{6}}$ | ${\mathit{R}}_{\mathbf{7}}$ | ${\mathit{R}}_{\mathbf{6}}$ | ${\mathit{R}}_{\mathbf{7}}$ | ||||||||||

$\mathbf{\pi}(\mathbf{\alpha})$ | $\mathbf{\beta}$ | EMSE${\widehat{\mathit{R}}}_{\mathbf{6}}$ | LHPD | %CHPD | EMSE${\widehat{\mathit{R}}}_{\mathbf{7}}$ | LHPD | %CHPD | EMSE${\widehat{\mathit{R}}}_{\mathbf{6}}$ | LHPD | %CHPD | EMSE${\widehat{\mathit{R}}}_{\mathbf{7}}$ | LHPD | %CHPD |

0.75 | 35.094 | 12.225 | 94.9 | 111.424 | 21.297 | 94.8 | 34.931 | 12.147 | 94.7 | 110.946 | 21.092 | 94.8 | |

$\gamma (\alpha |4,4)$ | 1 | 3.054 | 4.261 | 94.9 | 6.292 | 6.964 | 95.1 | 2.995 | 4.217 | 94.9 | 6.095 | 6.843 | 95.0 |

1.5 | 0.227 | 1.368 | 95.4 | 0.491 | 2.140 | 94.6 | 0.219 | 1.335 | 95.3 | 0.454 | 2.058 | 94.9 | |

0.75 | 6.302 | 5.961 | 94.6 | 14.945 | 10.211 | 94.9 | 6.213 | 5.916 | 94.7 | 14.671 | 10.074 | 94.7 | |

$\gamma (\alpha |4,6)$ | 1 | 0.902 | 2.516 | 94.7 | 1.965 | 4.081 | 94.9 | 0.883 | 2.485 | 94.8 | 1.910 | 3.985 | 94.7 |

1.5 | 0.118 | 0.978 | 95.1 | 0.222 | 1.522 | 94.8 | 0.113 | 0.957 | 94.9 | 0.207 | 1.464 | 94.7 |

**Table 7.**EMSE of Bayes estimators ${\widehat{\alpha}}_{{}_{B}}$, ${\widehat{\beta}}_{{}_{B}}$ and length (LHPD) of HPD intervals, using Kundu’s prior.

$\mathit{\delta}=0$ | $\mathit{\delta}=-0.5$ | ||||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{\pi}}_{1}(\mathit{\alpha})$ | ${\mathit{\pi}}_{1}(\mathit{\beta})$ | EMSE ${\widehat{\mathit{\alpha}}}_{{}_{\mathit{B}}}$ | LHPD $\mathit{\alpha}$ | EMSE ${\widehat{\mathit{\beta}}}_{{}_{\mathit{B}}}$ | LHPD $\mathit{\beta}$ | EMSE ${\widehat{\mathit{\alpha}}}_{{}_{\mathit{B}}}$ | LHPD $\mathit{\alpha}$ | EMSE ${\widehat{\mathit{\beta}}}_{{}_{\mathit{B}}}$ | LHPD $\mathit{\beta}$ |

$\gamma (\alpha |8,6)$ | 0.179 | 1.488 | 0.048 | 0.731 | 0.152 | 1.389 | 0.036 | 0.662 | |

$\gamma (\alpha |4,4)$ | $\gamma (\alpha |6,6)$ | 0.192 | 1.485 | 0.080 | 0.970 | 0.160 | 1.363 | 0.058 | 0.850 |

$\gamma (\alpha |6,8)$ | 0.173 | 1.456 | 0.105 | 1.205 | 0.139 | 1.310 | 0.080 | 1.049 | |

$\gamma (\alpha |8,6)$ | 0.293 | 1.904 | 0.045 | 0.747 | 0.219 | 1.717 | 0.033 | 0.640 | |

$\gamma (\alpha |4,6)$ | $\gamma (\alpha |6,6)$ | 0.288 | 1.899 | 0.078 | 0.997 | 0.220 | 1.690 | 0.051 | 0.830 |

$\gamma (\alpha |6,8)$ | 0.269 | 1.888 | 0.111 | 1.247 | 0.206 | 1.620 | 0.077 | 1.020 |

**Table 8.**EMSE of Bayes estimator ${\widehat{\alpha}}_{{}_{B}}$ and of MP estimator ${\widehat{\beta}}_{{}_{\mathrm{MP}}}$ and length (LHPD) of HPD interval for $\alpha $, using Soland’s prior ${\pi}_{2}$.

$\mathit{\delta}=0$ | $\mathit{\delta}=-0.5$ | ||||||
---|---|---|---|---|---|---|---|

${\mathit{\pi}}_{2}(\mathit{\alpha}|\mathit{\beta})$ | ${\mathit{\pi}}_{2}(\mathit{\beta})$ | EMSE ${\widehat{\mathit{\alpha}}}_{{}_{\mathit{B}}}$ | EMSE ${\widehat{\mathit{\beta}}}_{{}_{\mathbf{MP}}}$ | LHPD $\mathit{\alpha}$ | EMSE ${\widehat{\mathit{\alpha}}}_{{}_{\mathit{B}}}$ | EMSE ${\widehat{\mathit{\beta}}}_{{}_{\mathbf{MP}}}$ | LHPD $\mathit{\alpha}$ |

$(\frac{4}{10},\frac{2}{10},\frac{2}{10},\frac{1}{10},\frac{1}{10})$ | 0.165 | 1.400 | 0.076 | 0.138 | 1.289 | 0.054 | |

$\gamma (\alpha |4,4)$ | $(\frac{1}{10},\frac{2}{10},\frac{4}{10},\frac{2}{10},\frac{1}{10})$ | 0.156 | 1.374 | 0.055 | 0.128 | 1.240 | 0.044 |

$(\frac{1}{10},\frac{1}{10},\frac{2}{10},\frac{2}{10},\frac{4}{10})$ | 0.163 | 1.354 | 0.079 | 0.121 | 1.166 | 0.065 | |

$(\frac{4}{10},\frac{2}{10},\frac{2}{10},\frac{1}{10},\frac{1}{10})$ | 0.258 | 1.837 | 0.085 | 0.204 | 1.634 | 0.053 | |

$\gamma (\alpha |4,6)$ | $(\frac{1}{10},\frac{2}{10},\frac{4}{10},\frac{2}{10},\frac{1}{10})$ | 0.252 | 1.820 | 0.061 | 0.189 | 1.578 | 0.046 |

$(\frac{1}{10},\frac{1}{10},\frac{2}{10},\frac{2}{10},\frac{4}{10})$ | 0.259 | 1.801 | 0.086 | 0.181 | 1.488 | 0.061 |

**Table 9.**Trimmed EMSE (TEMSE) of Bayes predictors and lengths (LHPD) of HPD intervals for ${R}_{6},{R}_{7}$, using Kundu’s prior ${\pi}_{1}$.

$\mathit{\delta}=0$ | $\mathit{\delta}=-0.5$ | ||||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{\pi}}_{1}(\mathit{\alpha})$ | ${\mathit{\pi}}_{1}(\mathit{\beta})$ | TEMSE ${\widehat{\mathit{R}}}_{6}$ | LHPD ${\mathit{R}}_{6}$ | TEMSE ${\widehat{\mathit{R}}}_{7}$ | LHPD ${\mathit{R}}_{7}$ | TEMSE ${\widehat{\mathit{R}}}_{6}$ | LHPD ${\mathit{R}}_{6}$ | TEMSE ${\widehat{\mathit{R}}}_{7}$ | LHPD ${\mathit{R}}_{7}$ |

$\gamma (\alpha |8,6)$ | 3318.601 | 42.805 | 24,215.125 | 58,844 | 3281.805 | 42.795 | 24198.502 | 58.309 | |

$\gamma (\alpha |4,4)$ | $\gamma (\alpha |6,6)$ | 45.326 | 14.270 | 415.003 | 27.315 | 45.724 | 14.013 | 414.015 | 26.771 |

$\gamma (\alpha |6,8)$ | 1.997 | 3.735 | 7.296 | 7.446 | 1.917 | 3.602 | 6.885 | 7.288 | |

$\gamma (\alpha |8,6)$ | 134.211 | 19.175 | 1018.288 | 34.819 | 132.540 | 18.758 | 1006.229 | 34.279 | |

$\gamma (\alpha |4,6)$ | $\gamma (\alpha |6,6)$ | 10.079 | 7.689 | 67.697 | 16.994 | 9.830 | 7.636 | 64.405 | 16.586 |

$\gamma (\alpha |6,8)$ | 0.664 | 2.266 | 2.539 | 4.221 | 0.702 | 2.201 | 2.534 | 4.032 |

**Table 10.**Trimmed EMSE (TEMSE) of Bayes predictors and lengths (LHPD) of HPD intervals for ${R}_{6},{R}_{7}$, using Soland’s prior ${\pi}_{2}$.

$\mathit{\delta}=0$ | $\mathit{\delta}=-0.5$ | ||||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{\pi}}_{2}(\mathit{\alpha}|\mathit{\beta})$ | ${\mathit{\pi}}_{2}(\mathit{\beta})$ | TEMSE ${\widehat{\mathit{R}}}_{6}$ | LHPD ${\mathit{R}}_{6}$ | TEMSE ${\widehat{\mathit{R}}}_{7}$ | LHPD ${\mathit{R}}_{7}$ | TEMSE ${\widehat{\mathit{R}}}_{6}$ | LHPD ${\mathit{R}}_{6}$ | TEMSE ${\widehat{\mathit{R}}}_{7}$ | LHPD ${\mathit{R}}_{7}$ |

$(\frac{4}{10},\frac{2}{10},\frac{2}{10},\frac{1}{10},\frac{1}{10})$ | 462.880 | 25.479 | 1069.877 | 41.557 | 462.968 | 25.420 | 1054.563 | 41.486 | |

$\gamma (\alpha |4,4)$ | $(\frac{1}{10},\frac{2}{10},\frac{4}{10},\frac{2}{10},\frac{1}{10})$ | 45.904 | 7.823 | 102.881 | 12.856 | 45.608 | 7.714 | 103.401 | 12.634 |

$(\frac{1}{10},\frac{1}{10},\frac{2}{10},\frac{2}{10},\frac{4}{10})$ | 15.881 | 5.597 | 56.159 | 10.723 | 16.009 | 5.565 | 54.574 | 10.551 | |

$(\frac{4}{10},\frac{2}{10},\frac{2}{10},\frac{1}{10},\frac{1}{10})$ | 66.015 | 11.170 | 145.522 | 20.895 | 66.318 | 11.111 | 145.077 | 20.724 | |

$\gamma (\alpha |4,6)$ | $(\frac{1}{10},\frac{2}{10},\frac{4}{10},\frac{2}{10},\frac{1}{10})$ | 7.938 | 3.929 | 9.609 | 7.093 | 7.832 | 3.888 | 9.520 | 7.013 |

$(\frac{1}{10},\frac{1}{10},\frac{2}{10},\frac{2}{10},\frac{4}{10})$ | 5.827 | 3.022 | 5.961 | 5.185 | 5.769 | 2.983 | 5.943 | 5.132 |

**Table 11.**Bayes estimates ${\widehat{\alpha}}_{{}_{B}},{\widehat{\beta}}_{{}_{B}}$, Bayes predictions ${\widehat{R}}_{6},{\widehat{R}}_{7},{\widehat{R}}_{8}$ and HPD intervals, for rainfall data, using Kundu’s prior ${\pi}_{1}$.

$\mathit{\delta}$ | ${\widehat{\mathit{\alpha}}}_{{}_{\mathit{B}}}$ | HPD $\mathit{\alpha}$ | ${\widehat{\mathit{\beta}}}_{{}_{\mathit{B}}}$ | HPD $\mathit{\beta}$ | ${\widehat{\mathit{R}}}_{6}$ | HPD ${\mathit{R}}_{6}$ | ${\widehat{\mathit{R}}}_{7}$ | HPD ${\mathit{R}}_{7}$ | ${\widehat{\mathit{R}}}_{8}$ | HPD ${\mathit{R}}_{8}$ |
---|---|---|---|---|---|---|---|---|---|---|

0 | 0.41 | [0.01, 1.05] | 2.69 | [1.21, 4.32] | 3.04 | [2.79, 3.61] | 3.26 | [2.79, 4.12] | 3.47 | [2.81, 4.58] |

−0.25 | 0.41 | [0.01, 1.04] | 2.46 | [1.06, 4.02] | 3.13 | [2.79, 3.87] | 3.42 | [2.79, 4.54] | 3.24 | [2.80, 3.84] |

−0.50 | 0.39 | [0.02, 0.97] | 2.57 | [1.19, 4.09] | 3.07 | [2.79, 3.67] | 3.32 | [2.79, 4.20] | 3.55 | [2.83, 4.68] |

−0.75 | 0.48 | [0.06, 1.05] | 2.60 | [1.44, 3.82] | 2.98 | [2.79, 3.37] | 3.15 | [2.79, 3.70] | 3.32 | [2.82, 4.00] |

$-\infty $ | 0.58 | [0.42, 0.74] | 2.10 | [1.72, 2.46] | 3.05 | [2.79, 3.54] | 3.28 | [2.80, 3.94] | 3.50 | [2.87, 4.30] |

**Table 12.**Bayes estimation ${\widehat{\alpha}}_{{}_{B}}$, maximum probability estimation ${\widehat{\beta}}_{{}_{\mathrm{MP}}}$, Bayes predictions ${\widehat{R}}_{6},{\widehat{R}}_{7},{\widehat{R}}_{8}$ and HPD intervals (except $\beta $), for rainfall data, using Soland’s prior.

$\mathit{\delta}$ | ${\widehat{\mathit{\alpha}}}_{{}_{\mathit{B}}}$ | HPD $\mathit{\alpha}$ | ${\widehat{\mathit{\beta}}}_{{}_{\mathbf{MP}}}$ | ${\widehat{\mathit{R}}}_{6}$ | HPD ${\mathit{R}}_{6}$ | ${\widehat{\mathit{R}}}_{7}$ | HPD ${\mathit{R}}_{7}$ | ${\widehat{\mathit{R}}}_{8}$ | HPD ${\mathit{R}}_{8}$ |
---|---|---|---|---|---|---|---|---|---|

0 | 0.39 | [0.16, 0.62] | 2.50 | 3.01 | [2.79, 3.45] | 3.21 | [2.8, 3.79] | 3.39 | [2.85, 4.08] |

−0.25 | 0.33 | [0.15, 0.53] | 2.50 | 3.04 | [2.79, 3.53] | 3.26 | [2.8, 3.90] | 3.47 | [2.86, 4.22] |

−0.50 | 0.34 | [0.17, 0.52] | 2.50 | 3.03 | [2.79, 3.49] | 3.24 | [2.8, 3.84] | 3.44 | [2.86, 4.14] |

−0.75 | 0.43 | [0.27, 0.61] | 2.50 | 2.98 | [2.79, 3.34] | 3.15 | [2.8, 3.62] | 3.31 | [2.85, 3.86] |

$-\infty $ | 0.47 | [0.37, 0.56] | 2.50 | 2.96 | [2.79, 3.29] | 3.12 | [2.8, 3.54] | 3.27 | [2.85, 3.75] |

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Gouet, R.; López, F.J.; Maldonado, L.; Sanz, G.
Statistical Inference for the Weibull Distribution Based on *δ*-Record Data. *Symmetry* **2020**, *12*, 20.
https://doi.org/10.3390/sym12010020

**AMA Style**

Gouet R, López FJ, Maldonado L, Sanz G.
Statistical Inference for the Weibull Distribution Based on *δ*-Record Data. *Symmetry*. 2020; 12(1):20.
https://doi.org/10.3390/sym12010020

**Chicago/Turabian Style**

Gouet, Raúl, F. Javier López, Lina Maldonado, and Gerardo Sanz.
2020. "Statistical Inference for the Weibull Distribution Based on *δ*-Record Data" *Symmetry* 12, no. 1: 20.
https://doi.org/10.3390/sym12010020