# Robust Estimations for the Tail Index of Weibull-Type Distribution

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

**Lemma**

**1.**

**Definition**

**1.**

## 2. Asymptotic Results

**Theorem**

**1.**

**Remark**

**1.**

**Theorem**

**2.**

**Remark**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

## 3. Robustness

**Theorem**

**5.**

## 4. Simulations

- (i)
- The bias of the proposed M-estimations is smaller than that of Hill-type estimation and MLE estimation (see columns 2–5 for details).
- (ii)
- The sample variance ${s}^{2}$ of our estimations is very close to zero. Note by passing that even with the optimal choice of ${k}_{n}={k}_{opt}$, the ${s}^{2}$ of Hill-type estimations is still relatively larger than the other (see columns 6–9 for details).
- (iii)
- Since the ratios of MSE satisfy $\widehat{r}\le {r}^{*}\le \tilde{r}$, we see that the best rank estimation is ${\tilde{T}}_{n}$, which coincides with the analysis of the relative efficiency (see columns 10–12 and Figure 2).
- (iv)
- For $n=30,50$, the ${p}_{Hill}$ is almost zero indicating that for very small samples ${\tilde{T}}_{n}^{(0,\infty )}$ outperforms Hill-type estimators ${\widehat{\alpha}}_{Hill}^{({k}_{n})}$ for almost all ${k}_{n}$’s. For $n=80$, ${p}_{Hill}$ does not exceed $10\%$ in most cases which means that there is a set K with at most s = 8 of ${k}_{n}\in K$ such that the Hill-type estimators would outperform ${\tilde{T}}_{n}^{(0,\infty )}$. Similar argument holds for $n=100$. Hence, the M-estimations perform better even for small samples.

## 5. Empirical Study

## 6. Proofs

**Proof**

**of**

**Lemma**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**4.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Psi-functions $\tilde{\psi}$ for the huberized M-estimators ${\tilde{T}}_{n}^{(v,\infty )}$. Here the truncated densify functions are generated from the Weibull ${F}_{W}(x;\alpha )$ and contaminated Weibull ${F}_{\u03f5}(x)=(1-\u03f5){F}_{W}(x;\alpha )+\u03f5\mathsf{\Gamma}(x;\lambda ,\beta )$ with $\alpha =2,{c}_{0}=0.5,\u03f5=0.3,\lambda =1,\beta =1$.

**Figure 2.**Relative asymptotic efficiency (AEFF) of ${\tilde{T}}_{n}^{(v,\infty )}$ and ${T}_{n}^{*(v,\infty )}$ compared to the MLE ${\tilde{T}}_{n}^{(-1,\infty )}$. Here ${F}_{W}(x;{\alpha}_{0})$ is given by (1) with ${c}_{0}=1,{\alpha}_{0}=1$.

**Figure 3.**Influence functions $IF(T;F,G)$ for $T={\tilde{T}}_{n}^{(v,\infty )}$ (

**left**) and ${T}_{n}^{*(v,\infty )}$ (

**right**). Here $G(x)=\mathsf{\Gamma}(x;\lambda ,\beta ),\phantom{\rule{0.166667em}{0ex}}$$\lambda =0.5$, $\beta \in (0,5),v=\pm 1,\pm 0.5,0$ and $F(x)={F}_{W}(x;{\alpha}_{0})$ is given by (1) with ${c}_{0}=1,{\alpha}_{0}=1$.

**Figure 4.**Graph of log mean excess function of scaled log returns of daily CRIX during 31 July 2014–1 January 2018.

**Table 1.**Comparisons of ${\tilde{T}}_{n},\phantom{\rule{0.166667em}{0ex}}{T}_{n}^{*}$ with ${\widehat{\alpha}}_{mle},\phantom{\rule{0.166667em}{0ex}}{\widehat{\alpha}}_{Hill}^{({k}_{n})}$. Here we take $m=1000$ samples of size $n=30$, 50, 80, 100 from ${F}_{\u03f5}(x)=(1-\u03f5){F}_{W}(x;\alpha )+\u03f5\mathsf{\Gamma}(x;0.5,0.5)$.

$(\mathit{\u03f5},{\mathit{c}}_{0},\mathit{\alpha})$ | n | ${\overline{\mathit{\alpha}}}_{\mathit{mle}}$ | ${\overline{\mathit{\alpha}}}_{\mathit{Hill}}$ | ${\tilde{\mathit{T}}}_{\mathit{n}}$ | ${\mathit{T}}_{\mathit{n}}^{*}$ | ${\mathit{s}}_{\mathit{mle}}^{2}$ | ${\mathit{s}}_{\mathit{Hill}}^{2}$ | ${\mathit{s}}_{\tilde{\mathit{T}}}^{2}$ | ${\mathit{s}}_{{\mathit{T}}^{*}}^{2}$ | $\widehat{\mathit{r}}$ | $\tilde{\mathit{r}}$ | r* | ${\mathit{p}}_{\mathit{Hill}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(0.3, 1, 1) | 30 | 0.9217 | 0.8255 | 1.0006 | 1.0005 | 0.0072 | 0.0451 | 0.0018 | 0.0015 | 68.6811 | 8.7774 | 8.3678 | 0.00 |

50 | 0.8609 | 0.8435 | 1.0022 | 1.0061 | 0.0068 | 0.0289 | 0.0014 | 0.0015 | 23.1288 | 14.3407 | 13.4344 | 0.00 | |

80 | 0.8274 | 0.8326 | 1.0083 | 1.0075 | 0.0041 | 0.0176 | 0.0013 | 0.0016 | 9.7485 | 19.5972 | 19.2393 | 0.00 | |

100 | 0.8161 | 0.8368 | 1.0147 | 1.0119 | 0.0030 | 0.0148 | 0.0018 | 0.0015 | 6.2607 | 20.8143 | 20.0842 | 0.00 | |

(0.1, 1, 1) | 30 | 0.9885 | 0.9343 | 0.9942 | 0.9940 | 0.0007 | 0.0469 | 0.0007 | 0.0006 | 5.7287 | 1.2283 | 1.0561 | 0.00 |

50 | 0.9834 | 0.9252 | 0.9949 | 0.9952 | 0.0009 | 0.0269 | 0.0007 | 0.0006 | 3.6407 | 1.8194 | 1.8000 | 0.00 | |

80 | 0.9776 | 0.9407 | 0.9962 | 0.9953 | 0.0009 | 0.0189 | 0.0005 | 0.0006 | 1.0006 | 2.5849 | 2.3829 | 0.00 | |

100 | 0.9735 | 0.9302 | 0.9964 | 0.9961 | 0.0010 | 0.0130 | 0.0005 | 0.0005 | 0.7138 | 3.2549 | 2.9623 | 0.15 | |

(0.3, 1, 2) | 30 | 1.2382 | 1.6039 | 1.9960 | 1.9919 | 0.0687 | 0.2408 | 0.0056 | 0.0050 | 74.7362 | 147.2932 | 126.8653 | 0.00 |

50 | 1.1347 | 1.6443 | 2.0015 | 1.9963 | 0.0268 | 0.1576 | 0.0045 | 0.0042 | 22.7834 | 158.1670 | 165.1127 | 0.00 | |

80 | 1.0851 | 1.6853 | 2.0050 | 2.0039 | 0.0127 | 0.1219 | 0.0039 | 0.0038 | 7.8035 | 197.7217 | 180.3746 | 0.00 | |

100 | 1.0731 | 1.6709 | 2.0081 | 2.0073 | 0.0085 | 0.0883 | 0.0042 | 0.0038 | 5.1102 | 223.2889 | 186.3996 | 0.00 | |

(0.1, 1, 2) | 30 | 1.9245 | 1.8399 | 1.9903 | 1.9859 | 0.0169 | 0.2025 | 0.0026 | 0.0024 | 4.8833 | 9.3566 | 8.4148 | 0.00 |

50 | 1.8459 | 1.8506 | 1.9900 | 1.9888 | 0.0239 | 0.1223 | 0.0022 | 0.0021 | 3.1233 | 21.4576 | 18.4649 | 0.00 | |

80 | 1.7654 | 1.8392 | 1.9873 | 1.9895 | 0.0228 | 0.0754 | 0.0017 | 0.0017 | 3.0758 | 39.6547 | 35.4584 | 0.00 | |

100 | 1.7249 | 1.8681 | 1.9898 | 1.9906 | 0.0190 | 0.0660 | 0.0018 | 0.0018 | 1.9142 | 43.1656 | 45.1557 | 0.00 | |

(0.3, 2, 1) | 30 | 0.9466 | 0.8640 | 0.9974 | 0.9987 | 0.0047 | 0.0429 | 0.0016 | 0.0021 | 82.5558 | 4.6014 | 3.5335 | 0.00 |

50 | 0.9061 | 0.8881 | 0.9955 | 0.9965 | 0.0051 | 0.0298 | 0.0015 | 0.0013 | 22.7834 | 12.7013 | 10.1286 | 0.00 | |

80 | 0.8729 | 0.8848 | 0.9944 | 0.9955 | 0.0036 | 0.0172 | 0.0012 | 0.0010 | 3.2990 | 16.1109 | 16.0286 | 0.00 | |

100 | 0.8562 | 0.8938 | 0.9970 | 0.9978 | 0.0029 | 0.0152 | 0.0011 | 0.0011 | 1.7146 | 19.8407 | 18.1323 | 0.00 | |

(0.1, 2, 1) | 30 | 0.9880 | 0.9223 | 0.9953 | 0.9956 | 0.0005 | 0.2773 | 0.0483 | 0.0011 | 5.1904 | 0.8848 | 0.7232 | 0.00 |

50 | 0.9852 | 0.9557 | 0.9941 | 0.9952 | 0.0007 | 0.1438 | 0.0261 | 0.0006 | 3.4681 | 1.2380 | 1.1071 | 0.00 | |

80 | 0.9808 | 0.9452 | 0.9940 | 0.9942 | 0.0008 | 0.1775 | 0.0165 | 0.0005 | 0.9524 | 2.0353 | 1.6522 | 0.10 | |

100 | 0.9762 | 0.9560 | 0.9927 | 0.9939 | 0.0009 | 0.0444 | 0.0148 | 0.0006 | 0.8589 | 2.3773 | 2.2494 | 0.12 | |

(0.3, 2, 2) | 30 | 1.3019 | 1.6018 | 1.9855 | 1.9853 | 0.0583 | 0.2434 | 0.0030 | 0.0026 | 85.6893 | 153.3920 | 159.4866 | 0.00 |

50 | 1.2012 | 1.6589 | 1.9850 | 1.9850 | 0.0228 | 0.1209 | 0.0024 | 0.0021 | 14.7957 | 260.7202 | 270.0256 | 0.00 | |

80 | 1.1570 | 1.6701 | 1.9844 | 1.9838 | 0.0099 | 1.1006 | 0.0018 | 0.0017 | 2.5650 | 348.1278 | 345.6587 | 0.00 | |

100 | 1.1484 | 1.6771 | 1.9832 | 1.9830 | 0.0069 | 0.0712 | 0.0017 | 0.0017 | 1.4158 | 386.5744 | 370.8852 | 0.00 | |

(0.1, 2, 2) | 30 | 1.9238 | 1.8054 | 1.9885 | 1.9870 | 0.0161 | 0.1763 | 0.0027 | 0.0032 | 4.3161 | 5.8439 | 6.3442 | 0.00 |

50 | 1.8519 | 1.8637 | 1.9886 | 1.9850 | 0.0212 | 0.1201 | 0.0031 | 0.0023 | 3.9388 | 17.6548 | 16.9291 | 0.00 | |

80 | 1.7646 | 1.8565 | 1.9869 | 1.9849 | 0.0200 | 0.0696 | 0.0017 | 0.0019 | 1.2588 | 37.7912 | 36.6744 | 0.00 | |

100 | 1.7402 | 1.8839 | 1.9849 | 1.9843 | 0.0181 | 0.0756 | 0.0017 | 0.0017 | 0.9791 | 47.8601 | 44.6477 | 0.05 | |

(0.3, 0.5, 1) | 30 | 0.9320 | 0.7989 | 1.0056 | 1.0065 | 0.0048 | 0.0565 | 0.0012 | 0.0014 | 7.2497 | 7.7231 | 6.7977 | 0.00 |

50 | 0.8912 | 0.8250 | 1.0116 | 1.0096 | 0.0051 | 0.0468 | 0.0012 | 0.0013 | 8.8465 | 12.2309 | 10.2765 | 0.00 | |

80 | 0.8565 | 0.8130 | 1.0185 | 1.0188 | 0.0034 | 0.0261 | 0.0013 | 0.0012 | 2.8355 | 15.4294 | 14.8902 | 0.00 | |

100 | 0.8463 | 0.8368 | 1.0218 | 1.0232 | 0.0024 | 0.0252 | 0.0011 | 0.0012 | 1.3175 | 17.0045 | 16.5285 | 0.00 | |

(0.1, 0.5, 1) | 30 | 0.9874 | 0.8848 | 0.9968 | 0.9943 | 0.0005 | 0.0428 | 0.0006 | 0.0006 | 5.3788 | 1.2157 | 1.0236 | 0.00 |

50 | 0.9853 | 0.9136 | 0.9972 | 0.9952 | 0.0005 | 0.0295 | 0.0005 | 0.0005 | 3.8457 | 1.5111 | 1.4974 | 0.00 | |

80 | 0.9799 | 0.9193 | 0.9977 | 0.9975 | 0.0006 | 0.0181 | 0.0004 | 0.0005 | 1.9436 | 2.3528 | 1.9708 | 0.00 | |

100 | 0.9783 | 0.9165 | 0.9991 | 0.9989 | 0.0006 | 0.0143 | 0.0005 | 0.0004 | 0.9241 | 2.2865 | 2.1840 | 0.10 | |

(0.3, 0.5, 2) | 30 | 1.3277 | 1.5964 | 2.0065 | 1.8144 | 0.0713 | 0.2504 | 0.0052 | 0.0004 | 61.5168 | 111.5918 | 15.1011 | 0.00 |

50 | 1.2141 | 1.6243 | 2.0185 | 1.8083 | 0.0373 | 0.1607 | 0.0049 | 0.0003 | 31.3754 | 129.6489 | 17.7850 | 0.00 | |

80 | 1.1618 | 1.6596 | 2.0357 | 1.8042 | 0.0147 | 0.1047 | 0.0046 | 0.0002 | 13.2211 | 128.0530 | 18.8915 | 0.00 | |

100 | 1.1486 | 1.6707 | 2.0386 | 1.8035 | 0.0125 | 0.0974 | 0.0047 | 0.0002 | 4.5564 | 118.1708 | 19.1617 | 0.00 | |

(0.1, 0.5, 2) | 30 | 1.9443 | 1.8589 | 1.9900 | 1.8040 | 0.0093 | 0.2091 | 0.0020 | 0.0005 | 8.5329 | 6.7537 | 0.3454 | 0.00 |

50 | 1.8936 | 1.8745 | 1.9935 | 1.7963 | 0.1316 | 0.6520 | 0.0020 | 0.0003 | 4.9060 | 12.6372 | 0.5832 | 0.00 | |

80 | 1.8329 | 1.8271 | 1.9958 | 1.7937 | 0.0720 | 0.6408 | 0.0018 | 0.0002 | 1.3524 | 22.1326 | 0.9232 | 0.00 | |

100 | 1.8125 | 1.8538 | 2.0024 | 1.7930 | 0.0670 | 2.8163 | 0.0017 | 0.0002 | 0.9561 | 26.7188 | 1.0363 | 0.08 |

**Table 2.**Estimations of Weibull tail coefficient and its relative deviations via contamination level $\u03f5=0.05i$, $i=0,\dots ,10$. Data is the positive and scaled log returns of daily CRIX during 31 July 2014–1 January 2018.

$\mathit{\u03f5}$ | ${\tilde{\mathit{T}}}_{\mathit{n}}$ | ${\mathit{T}}_{\mathit{n}}^{*}$ | ${\widehat{\mathit{\alpha}}}_{\mathit{Hill}}^{(1)}$ | ${\widehat{\mathit{\alpha}}}_{\mathit{Hill}}^{(2)}$ | $\mathit{D}({\tilde{\mathit{T}}}_{\mathit{n}})$ | $\mathit{D}({\mathit{T}}_{\mathit{n}}^{*})$ | $\mathit{D}({\widehat{\mathit{\alpha}}}_{\mathit{Hill}}^{(1)})$ | $\mathit{D}({\widehat{\mathit{\alpha}}}_{\mathit{Hill}}^{(2)})$ |
---|---|---|---|---|---|---|---|---|

0.00 | 0.7711 | 0.7932 | 0.9202 | 0.9359 | 0.0072 | 0.0055 | 0.1277 | 0.2601 |

0.05 | 0.7783 | 0.7987 | 0.7925 | 0.6758 | 0.0056 | 0.0060 | 0.0084 | 0.0246 |

0.10 | 0.7839 | 0.8047 | 0.8009 | 0.6512 | 0.0002 | 0.0005 | 0.0038 | 0.0028 |

0.15 | 0.7841 | 0.8052 | 0.8047 | 0.6484 | 0.0026 | 0.0144 | 0.0258 | 0.0172 |

0.20 | 0.7867 | 0.8196 | 0.8305 | 0.6312 | 0.0093 | 0.0117 | 0.0560 | 0.0094 |

0.25 | 0.7960 | 0.8313 | 0.7745 | 0.6406 | 0.0046 | 0.0186 | 0.0168 | 0.0075 |

0.30 | 0.8006 | 0.8499 | 0.7577 | 0.6331 | 0.0046 | 0.0092 | 0.0038 | 0.0089 |

0.35 | 0.7960 | 0.8407 | 0.7539 | 0.6420 | 0.0120 | 0.0084 | 0.0168 | 0.0049 |

0.40 | 0.8080 | 0.8491 | 0.7707 | 0.6371 | 0.0008 | 0.0096 | 0.0370 | 0.0029 |

0.45 | 0.8072 | 0.8587 | 0.7337 | 0.6400 | 0.0069 | 0.0052 | 0.0208 | 0.0096 |

0.50 | 0.8003 | 0.8639 | 0.7545 | 0.6304 | - | - | - | - |

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**MDPI and ACS Style**

Gong, C.; Ling, C.
Robust Estimations for the Tail Index of Weibull-Type Distribution. *Risks* **2018**, *6*, 119.
https://doi.org/10.3390/risks6040119

**AMA Style**

Gong C, Ling C.
Robust Estimations for the Tail Index of Weibull-Type Distribution. *Risks*. 2018; 6(4):119.
https://doi.org/10.3390/risks6040119

**Chicago/Turabian Style**

Gong, Chengping, and Chengxiu Ling.
2018. "Robust Estimations for the Tail Index of Weibull-Type Distribution" *Risks* 6, no. 4: 119.
https://doi.org/10.3390/risks6040119