# An Optimal Tail Selection in Risk Measurement

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Optimization Approaches for Threshold Selection

- The Mean Absolute Deviation Distance metric (MAD-Distance metric) method,
- The Kolmogorov–Smirnov Distance metric (KS-Distance metric) method,
- The Reiss and Thomas (RT) procedures,
- The Path Stability (PS) method,
- The automated Eyeball (Eyeball) method,
- The Guillou and Hall (GH) procedure,
- The minimization of the Asymptotic Mean Squared Error (dAMSE) method,
- The Hall and Welsh (HW) procedure,
- The single bootstrap (Hall) procedure proposed by Hall,
- The single bootstrap (Himp) procedure proposed by Caeiro and Gomes,
- The double bootstrap (Gomes) procedure proposed by Gomes, Figueiredo and Neves,
- The double bootstrap (Danielsson) procedure proposed by Danielsson, de Haan, Peng and de Vries.

#### 2.1. Mean Absolute Deviation Distance Metric (MAD-Distance Metric) and Kolmogorov–Smirnov Distance Metric (KS-Distance Metric) Methods

#### 2.2. The Reiss and Thomas (RT) Procedures

#### 2.3. The Path Stability (PS) Method

#### 2.4. The Automated Eyeball (Eyeball) Method

#### 2.5. The Guillou and Hall (GH) Procedure

#### 2.6. Minimization of the Asymptotic Mean Squared Error (dAMSE) Method

#### 2.7. The Hall and Welsh (HW) Procedure

#### 2.8. The Hall Single Bootstrap (Hall) Procedure

#### 2.9. The Single Bootstrap (Himp) Procedure Proposed by Caeiro and Gomes

#### 2.10. The Double Bootstrap (Gomes) Procedure Proposed by Gomes, Figueiredo and Neves

#### 2.11. The Double Bootstrap (Danielsson) Procedure Proposed by Danielsson, de Haan, Peng and de Vries

## 3. Data

## 4. Results of the Empirical Study

_{min}= 2, Eyeball with the tuning parameter w = 0.01, GH and Danielsson methods systematically produce high threshold estimates, since the median exceeds the 98th percentile. The findings support the results of Danielsson et al. (2016), who argued that the Eyeball and KS Dis methods tend to pick the threshold close to the maximum of distribution. For shorter time-series (in sub-periods), the RT1, RT2 for k

_{min}= 2 and Danielsson methods are the most restrictive and systematically set the threshold at too high levels to be used in financial applications. This finding is in line with the observation by Scarrott and MacDonald (2012), who pointed out that the RT1 approach is unreliable for a small k despite the weighting by ${i}^{\beta}$. In turn, Reiss and Thomas (2007) suggested using alternative distance metrics or weighting schemes when we deal with limited data. Methods based on minimizing the asymptotic MSE, especially the bootstrap-based methods, do not perform well in empirical studies (Danielsson et al. 2016). Similarly, Ferreira et al. (2003) noted that these methods do not give satisfactory results for samples of size under approximately 2000.

## 5. Conclusions

_{aux}= $\sqrt{{n}^{+}}$) methods, their maximum is below the 97.5th percentile. The PS method sets the optimal threshold much lower than the other methodologies (below the 95th percentile for the entire research period and sub-periods). The estimates of PS are relatively volatile, suggesting a difference from the fixed percentage of the total sample size, commonly presented in the literature. The PS method is based on a rather simple algorithm; thus, it can be easily implemented in a risk management process.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Index | Min | Mean | Max | Sd. Dev. | Skewness | Ex. Kurtosis |
---|---|---|---|---|---|---|

ATX (Austria) | −0.1467 | 0.0001 | 0.1202 | 0.0143 | −0.59 | 10.01 |

AEX (Netherlands) | −0.1138 | −0.0001 | 0.1003 | 0.0140 | −0.22 | 7.49 |

ATH (Greece) | −0.1771 | −0.0005 | 0.1343 | 0.0189 | −0.55 | 7.84 |

BEL20 (Belgium) | −0.1533 | 0.0000 | 0.0933 | 0.0126 | −0.44 | 10.36 |

BET (Romania) | −0.1190 | 0.0006 | 0.1458 | 0.0150 | −0.34 | 11.67 |

BUX (Hungary) | −0.1265 | 0.0003 | 0.1318 | 0.0149 | −0.24 | 7.08 |

CAC (France) | −0.1310 | −0.0001 | 0.1059 | 0.0144 | −0.22 | 6.52 |

DAX (Germany) | −0.1305 | 0.0001 | 0.1080 | 0.0148 | −0.17 | 5.99 |

FMIB (Italy) | −0.1854 | −0.0002 | 0.1087 | 0.0154 | −0.59 | 9.34 |

HEX (Finland) | −0.1740 | −0.0001 | 0.1456 | 0.0173 | −0.40 | 7.86 |

IBEX (Spain) | −0.1515 | −0.0001 | 0.1348 | 0.0147 | −0.32 | 8.13 |

ICEX (Iceland) | −1.0622 | 0.0000 | 0.0506 | 0.0186 | −39.03 | 2164.64 |

MOEX (Russia) | −0.2066 | 0.0005 | 0.2523 | 0.0200 | −0.25 | 15.99 |

OMXR (Latvia) | −0.1633 | 0.0004 | 0.1209 | 0.0140 | −0.56 | 20.56 |

OMXS (Sweden) | −0.1117 | 0.0000 | 0.0987 | 0.0148 | −0.08 | 4.37 |

OMXT (Estonia) | −0.1060 | 0.0004 | 0.1209 | 0.0105 | −0.25 | 12.85 |

OMXV (Lithuania) | −0.1194 | 0.0004 | 0.1100 | 0.0099 | −0.77 | 24.07 |

OSEAX (Norway) | −0.0983 | 0.0003 | 0.0919 | 0.0137 | −0.69 | 6.76 |

PSI20 (Portugal) | −0.1038 | −0.0002 | 0.1020 | 0.0119 | −0.42 | 7.49 |

PX (Czechia) | −0.1619 | 0.0001 | 0.1236 | 0.0134 | −0.54 | 13.73 |

RTS (Russia) | −0.2120 | 0.0003 | 0.2020 | 0.0215 | −0.54 | 9.27 |

SAX (Slovakia) | −0.1481 | 0.0003 | 0.1188 | 0.0118 | −0.69 | 14.81 |

SMI (Switzerland) | −0.1013 | 0.0000 | 0.1079 | 0.0117 | −0.29 | 7.95 |

SOFIX (Bulgaria) | −0.1136 | 0.0003 | 0.0839 | 0.0123 | −0.64 | 11.56 |

UKX (UK) | −0.1151 | 0.0000 | 0.0938 | 0.0119 | −0.34 | 8.34 |

UX (Ukraine) | −0.3437 | 0.0005 | 0.3908 | 0.0260 | 0.85 | 53.49 |

WIG20 (Poland) | −0.1425 | 0.0000 | 0.0815 | 0.0150 | −0.33 | 4.22 |

XU100 (Turkey) | −0.1998 | 0.0003 | 0.1777 | 0.0207 | −0.10 | 7.91 |

AOR (Australia) | −0.1001 | 0.0001 | 0.0635 | 0.0098 | −0.91 | 9.63 |

HSI (Hong Kong) | −0.1358 | 0.0001 | 0.1341 | 0.0146 | −0.11 | 7.90 |

JCI (Indonesia) | −0.1095 | 0.0004 | 0.0970 | 0.0134 | −0.63 | 7.18 |

KLCI (Malaysia) | −0.0998 | 0.0001 | 0.0663 | 0.0081 | −0.79 | 11.20 |

KOSPI (South Korea) | −0.1280 | 0.0001 | 0.1128 | 0.0150 | −0.57 | 7.16 |

NKX (Japan) | −0.1211 | 0.0000 | 0.1323 | 0.0150 | −0.38 | 6.42 |

NZ50 (New Zealand) | −0.0789 | 0.0004 | 0.0694 | 0.0072 | −0.72 | 10.19 |

PSEI (Philippines) | −0.1432 | 0.0002 | 0.1618 | 0.0131 | −0.15 | 16.91 |

SET (Thailand) | −0.1606 | 0.0002 | 0.1058 | 0.0132 | −0.93 | 11.77 |

SHBS (China) | −0.1029 | 0.0004 | 0.1840 | 0.0198 | −0.02 | 7.12 |

SNX (India) | −0.1410 | 0.0003 | 0.1599 | 0.0147 | −0.41 | 9.37 |

STI (Singapore) | −0.0909 | 0.0000 | 0.0753 | 0.0112 | −0.40 | 6.79 |

TWSE (Taiwan) | −0.0994 | 0.0000 | 0.0652 | 0.0134 | −0.29 | 3.89 |

BVP (Brazil) | −0.1599 | 0.0003 | 0.1368 | 0.0182 | −0.38 | 6.64 |

DJI (US) | −0.1384 | 0.0001 | 0.1076 | 0.0119 | −0.40 | 13.63 |

IPC (Mexico) | −0.0827 | 0.0003 | 0.1044 | 0.0128 | −0.05 | 5.47 |

IPSA (Chile) | −0.1522 | 0.0002 | 0.1180 | 0.0104 | −0.84 | 22.35 |

MRV (Argentina) | −0.4769 | 0.0008 | 0.1612 | 0.0232 | −2.02 | 39.79 |

SPX (US) | −0.1277 | 0.0001 | 0.1096 | 0.0125 | −0.40 | 11.37 |

TSX (Canada) | −0.1318 | 0.0001 | 0.1129 | 0.0114 | −0.96 | 17.30 |

Method | Lower Tail | Upper Tail | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Min | Mean | Median | Max | Sd. Dev. | Min | Mean | Median | Max | Sd. Dev. | |

MAD Dis ts = 0.25 | 0.8970 | 0.9192 | 0.9078 | 0.9996 | 0.0312 | 0.8832 | 0.9132 | 0.9021 | 0.9992 | 0.0342 |

MAD Dis ts = 0.20 | 0.9092 | 0.9337 | 0.9213 | 0.9986 | 0.0288 | 0.9021 | 0.9319 | 0.9163 | 0.9998 | 0.0336 |

MAD Dis ts = 0.15 | 0.9148 | 0.9499 | 0.9400 | 0.9986 | 0.0243 | 0.8998 | 0.9505 | 0.9361 | 0.9998 | 0.0290 |

MAD Dis ts = 0.10 | 0.9412 | 0.9649 | 0.9575 | 0.9992 | 0.0163 | 0.9389 | 0.9634 | 0.9539 | 0.9961 | 0.0188 |

MAD Dis ts = 0.05 | 0.9517 | 0.9792 | 0.9789 | 0.9968 | 0.0100 | 0.9561 | 0.9808 | 0.9834 | 0.9968 | 0.0113 |

KS Dis ts = 0.25 | 0.9605 | 0.9961 | 0.9982 | 0.9996 | 0.0074 | 0.9620 | 0.9933 | 0.9963 | 0.9996 | 0.0078 |

KS Dis ts = 0.20 | 0.9605 | 0.9959 | 0.9982 | 0.9996 | 0.0074 | 0.9620 | 0.9934 | 0.9968 | 0.9996 | 0.0078 |

KS Dis ts = 0.15 | 0.9605 | 0.9956 | 0.9980 | 0.9996 | 0.0075 | 0.9620 | 0.9933 | 0.9965 | 0.9996 | 0.0079 |

KS Dis ts = 0.10 | 0.9605 | 0.9958 | 0.9980 | 0.9996 | 0.0073 | 0.9620 | 0.9935 | 0.9965 | 0.9994 | 0.0077 |

KS Dis ts = 0.05 | 0.9605 | 0.9958 | 0.9980 | 0.9996 | 0.0073 | 0.9620 | 0.9936 | 0.9966 | 0.9994 | 0.0078 |

RT1 $\beta $ = 0, k_{min} = 2 | 0.9084 | 0.9762 | 0.9800 | 0.9998 | 0.0238 | 0.9311 | 0.9830 | 0.9920 | 0.9998 | 0.0205 |

RT1 $\beta $ = 0.1, k_{min} = 2 | 0.9185 | 0.9821 | 0.9917 | 0.9998 | 0.0214 | 0.9339 | 0.9863 | 0.9946 | 0.9998 | 0.0178 |

RT1 $\beta $ = 0.2, k_{min} = 2 | 0.9259 | 0.9861 | 0.9970 | 0.9998 | 0.0193 | 0.9344 | 0.9897 | 0.9978 | 0.9998 | 0.0166 |

RT1 $\beta $ = 0.3, k_{min} = 2 | 0.9478 | 0.9941 | 0.9990 | 0.9998 | 0.0108 | 0.9421 | 0.9931 | 0.9992 | 0.9998 | 0.0132 |

RT1 $\beta $ = 0, k_{min} = 3 | 0.9086 | 0.9738 | 0.9773 | 0.9998 | 0.0241 | 0.9313 | 0.9790 | 0.9881 | 0.9998 | 0.0212 |

RT1 $\beta $ = 0, k_{min} = 4 | 0.9088 | 0.9726 | 0.9772 | 0.9998 | 0.0244 | 0.9013 | 0.9749 | 0.9793 | 0.9998 | 0.0236 |

RT1 $\beta $ = 0, k_{min} = 5 | 0.8979 | 0.9672 | 0.9705 | 0.9998 | 0.0262 | 0.9015 | 0.9744 | 0.9795 | 0.9998 | 0.0230 |

RT1 $\beta $ = 0, k_{min} = 10 | 0.8989 | 0.9662 | 0.9694 | 0.9998 | 0.0249 | 0.9025 | 0.9732 | 0.9765 | 0.9998 | 0.0221 |

RT1 $\beta $ = 0, k_{min} = 0.003n | 0.8997 | 0.9662 | 0.9687 | 0.9998 | 0.0241 | 0.9033 | 0.9741 | 0.9774 | 0.9998 | 0.0220 |

RT1 $\beta $ = 0, k_{min} = 0.005n | 0.9017 | 0.9664 | 0.9702 | 0.9998 | 0.0230 | 0.9053 | 0.9710 | 0.9720 | 0.9998 | 0.0226 |

RT2 $\beta $ = 0, k_{min} = 2 | 0.9064 | 0.9816 | 0.9884 | 0.9998 | 0.0219 | 0.9352 | 0.9839 | 0.9923 | 0.9998 | 0.0196 |

RT2 $\beta $ = 0.1, k_{min} = 2 | 0.9118 | 0.9835 | 0.9917 | 0.9998 | 0.0206 | 0.9352 | 0.9848 | 0.9933 | 0.9998 | 0.0189 |

RT2 $\beta $ = 0.2, k_{min} = 2 | 0.9399 | 0.9855 | 0.9945 | 0.9998 | 0.0178 | 0.9414 | 0.9886 | 0.9972 | 0.9998 | 0.0160 |

RT2 $\beta $ = 0.3, k_{min} = 2 | 0.9441 | 0.9876 | 0.9967 | 0.9998 | 0.0169 | 0.9414 | 0.9895 | 0.9984 | 0.9998 | 0.0155 |

RT2 $\beta $ = 0, k_{min} = 3 | 0.9066 | 0.9805 | 0.9853 | 0.9998 | 0.0214 | 0.9354 | 0.9807 | 0.9898 | 0.9998 | 0.0201 |

RT2 $\beta $ = 0, k_{min} = 4 | 0.9068 | 0.9806 | 0.9855 | 0.9998 | 0.0214 | 0.9019 | 0.9761 | 0.9817 | 0.9998 | 0.0228 |

RT2 $\beta $ = 0, k_{min} = 5 | 0.9070 | 0.9767 | 0.9842 | 0.9998 | 0.0231 | 0.9021 | 0.9748 | 0.9793 | 0.9998 | 0.0220 |

RT2 $\beta $ = 0, k_{min} = 10 | 0.9080 | 0.9723 | 0.9759 | 0.9998 | 0.0219 | 0.9031 | 0.9745 | 0.9783 | 0.9994 | 0.0213 |

RT2 $\beta $ = 0, k_{min} = 0.003n | 0.9090 | 0.9724 | 0.9759 | 0.9996 | 0.0212 | 0.9039 | 0.9753 | 0.9793 | 0.9998 | 0.0212 |

RT2 $\beta $ = 0, k_{min} = 0.005n | 0.9110 | 0.9731 | 0.9765 | 0.9996 | 0.0206 | 0.9059 | 0.9733 | 0.9735 | 0.9986 | 0.0203 |

PS j = 1 | 0.7651 | 0.8791 | 0.8873 | 0.9511 | 0.0485 | 0.7513 | 0.8813 | 0.8869 | 0.9571 | 0.0472 |

PS j = 0 | 0.7324 | 0.8272 | 0.8262 | 0.9013 | 0.0420 | 0.6409 | 0.7994 | 0.8051 | 0.8961 | 0.0476 |

Eyeball w = 0.01, ε = 0.3, h = 0.9 | 0.9910 | 0.9939 | 0.9940 | 0.9956 | 0.0008 | 0.9912 | 0.9933 | 0.9933 | 0.9952 | 0.0008 |

Eyeball w = 0.02, ε = 0.3, h = 0.9 | 0.9860 | 0.9895 | 0.9896 | 0.9915 | 0.0009 | 0.9858 | 0.9886 | 0.9887 | 0.9909 | 0.0009 |

Eyeball w = 0.025, ε = 0.3, h = 0.9 | 0.9838 | 0.9874 | 0.9874 | 0.9899 | 0.0009 | 0.9836 | 0.9862 | 0.9862 | 0.9887 | 0.0009 |

GH | 0.9473 | 0.9871 | 0.9967 | 0.9994 | 0.0153 | 0.9488 | 0.9880 | 0.9932 | 0.9994 | 0.0143 |

dAMSE | 0.9721 | 0.9738 | 0.9738 | 0.9770 | 0.0007 | 0.9704 | 0.9723 | 0.9724 | 0.9747 | 0.0007 |

HW | 0.6482 | 0.8965 | 0.9086 | 0.9459 | 0.0500 | 0.7145 | 0.8788 | 0.8901 | 0.9327 | 0.0409 |

Hall B = 10,000, ε = 0.955, k_{aux} = 2$\sqrt{{n}^{+}}$ | 0.9187 | 0.9714 | 0.9751 | 0.9836 | 0.0129 | 0.9245 | 0.9731 | 0.9765 | 0.9848 | 0.0109 |

Hall B = 10,000, ε = 0.995, k_{aux} = 2$\sqrt{{n}^{+}}$ | 0.9085 | 0.9687 | 0.9723 | 0.9817 | 0.0143 | 0.9187 | 0.9711 | 0.9754 | 0.9828 | 0.0117 |

Hall B = 10,000, ε = 0.9, k_{aux} = 2$\sqrt{{n}^{+}}$ | 0.9289 | 0.9748 | 0.9782 | 0.9860 | 0.0112 | 0.9315 | 0.9757 | 0.9786 | 0.9863 | 0.0098 |

Hall B = 10,000, ε = 0.955 k_{aux} = $\sqrt{{n}^{+}}$ | 0.9385 | 0.9773 | 0.9819 | 0.9930 | 0.0151 | 0.8947 | 0.9771 | 0.9807 | 0.9941 | 0.0168 |

Hall B = 10,000, ε = 0.955, k_{aux} = 3$\sqrt{{n}^{+}}$ | 0.9464 | 0.9681 | 0.9713 | 0.9763 | 0.0078 | 0.9436 | 0.9661 | 0.9677 | 0.9746 | 0.0076 |

Hall B = 1000, ε = 0.955, k_{aux} = 2$\sqrt{{n}^{+}}$ | 0.9183 | 0.9714 | 0.9751 | 0.9837 | 0.0128 | 0.9269 | 0.9733 | 0.9770 | 0.9838 | 0.0106 |

Hall(r) B = 10,000, ε = 0.955, k_{aux} = 2$\sqrt{{n}^{+}}$ | 0.9187 | 0.9714 | 0.9748 | 0.9836 | 0.0127 | 0.9255 | 0.9731 | 0.9762 | 0.9842 | 0.0108 |

Himp B = 10,000, ε = 0.955 | 0.9640 | 0.9848 | 0.9863 | 0.9998 | 0.0072 | 0.9618 | 0.9826 | 0.9831 | 0.9988 | 0.0088 |

Himp B = 10,000, ε = 0.995 | 0.9604 | 0.9839 | 0.9856 | 0.9994 | 0.0078 | 0.9564 | 0.9812 | 0.9831 | 0.9986 | 0.0099 |

Himp B = 10,000, ε = 0.9 | 0.9692 | 0.9862 | 0.9861 | 0.9998 | 0.0062 | 0.9662 | 0.9840 | 0.9850 | 0.9992 | 0.0077 |

Himp B = 1000, ε = 0.955 | 0.9640 | 0.9852 | 0.9863 | 0.9998 | 0.0069 | 0.9620 | 0.9827 | 0.9841 | 0.9988 | 0.0088 |

Himp (r) B = 10,000, ε = 0.955 | 0.9640 | 0.9845 | 0.9860 | 0.9998 | 0.0072 | 0.9620 | 0.9826 | 0.9833 | 0.9988 | 0.0088 |

Gomes B = 10,000, ε = 0.995 | 0.9598 | 0.9840 | 0.9851 | 0.9980 | 0.0081 | 0.9550 | 0.9808 | 0.9826 | 0.9986 | 0.0102 |

Gomes B = 10,000, ε = 0.955 | 0.9586 | 0.9827 | 0.9846 | 0.9998 | 0.0085 | 0.9548 | 0.9806 | 0.9818 | 0.9988 | 0.0105 |

Gomes B = 10,000, ε = 0.9 | 0.9596 | 0.9838 | 0.9819 | 0.9998 | 0.0098 | 0.9540 | 0.9810 | 0.9812 | 0.9998 | 0.0120 |

Gomes B = 1000, ε = 0.995 | 0.9580 | 0.9838 | 0.9851 | 0.9975 | 0.0081 | 0.9554 | 0.9819 | 0.9833 | 0.9986 | 0.0098 |

Gomes (r) B = 10,000, ε = 0.995 | 0.9600 | 0.9840 | 0.9853 | 0.9976 | 0.0080 | 0.9560 | 0.9809 | 0.9828 | 0.9986 | 0.0100 |

Danielsson B = 500, ε = 0.9 | 0.9390 | 0.9859 | 0.9899 | 0.9998 | 0.0147 | 0.9398 | 0.9886 | 0.9934 | 0.9998 | 0.0130 |

Method | Lower Tail | Upper Tail | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Min | Mean | Median | Max | Sd. Dev. | Min | Mean | Median | Max | Sd. Dev. | |

MAD Dis ts = 0.25 | 0.8611 | 0.9070 | 0.8905 | 0.9973 | 0.0414 | 0.8438 | 0.8946 | 0.8805 | 0.9973 | 0.0395 |

MAD Dis ts = 0.20 | 0.8692 | 0.9173 | 0.9039 | 0.9973 | 0.0374 | 0.8665 | 0.9056 | 0.8992 | 0.9893 | 0.0248 |

MAD Dis ts = 0.15 | 0.8732 | 0.9359 | 0.9266 | 0.9987 | 0.0302 | 0.8772 | 0.9305 | 0.9232 | 0.9973 | 0.0265 |

MAD Dis ts = 0.10 | 0.9105 | 0.9521 | 0.9486 | 0.9920 | 0.0227 | 0.9079 | 0.9530 | 0.9466 | 0.9987 | 0.0227 |

MAD Dis ts = 0.05 | 0.9573 | 0.9739 | 0.9713 | 0.9987 | 0.0100 | 0.9519 | 0.9710 | 0.9693 | 0.9920 | 0.0101 |

KS Dis ts = 0.25 | 0.9399 | 0.9830 | 0.9880 | 0.9987 | 0.0140 | 0.9519 | 0.9791 | 0.9793 | 0.9973 | 0.0118 |

KS Dis ts = 0.20 | 0.9279 | 0.9828 | 0.9840 | 0.9987 | 0.0140 | 0.9439 | 0.9790 | 0.9806 | 0.9973 | 0.0129 |

KS Dis ts = 0.15 | 0.9479 | 0.9841 | 0.9866 | 0.9987 | 0.0117 | 0.9466 | 0.9812 | 0.9813 | 0.9973 | 0.0108 |

KS Dis ts = 0.10 | 0.9479 | 0.9843 | 0.9880 | 0.9987 | 0.0118 | 0.9359 | 0.9801 | 0.9806 | 0.9973 | 0.0126 |

KS Dis ts = 0.05 | 0.9680 | 0.9877 | 0.9893 | 0.9987 | 0.0078 | 0.9519 | 0.9844 | 0.9860 | 0.9987 | 0.0091 |

RT1 $\beta $ = 0, k_{min} = 2 | 0.8024 | 0.9661 | 0.9907 | 0.9987 | 0.0440 | 0.8745 | 0.9801 | 0.9933 | 0.9987 | 0.0279 |

RT1 $\beta $ = 0.1, k_{min} = 2 | 0.8772 | 0.9728 | 0.9920 | 0.9987 | 0.0340 | 0.8852 | 0.9810 | 0.9947 | 0.9987 | 0.0267 |

RT1 $\beta $ = 0.2, k_{min} = 2 | 0.9092 | 0.9769 | 0.9947 | 0.9987 | 0.0307 | 0.9065 | 0.9854 | 0.9960 | 0.9987 | 0.0217 |

RT1 $\beta $ = 0.3, k_{min} = 2 | 0.9092 | 0.9824 | 0.9947 | 0.9987 | 0.0259 | 0.9159 | 0.9865 | 0.9960 | 0.9987 | 0.0203 |

RT1 $\beta $ = 0, k_{min} = 3 | 0.8037 | 0.9613 | 0.9766 | 0.9987 | 0.0432 | 0.8758 | 0.9782 | 0.9887 | 0.9987 | 0.0273 |

RT1 $\beta $ = 0, k_{min} = 4 | 0.8051 | 0.9620 | 0.9780 | 0.9987 | 0.0428 | 0.8772 | 0.9752 | 0.9866 | 0.9987 | 0.0287 |

RT1 $\beta $ = 0, k_{min} = 5 | 0.8064 | 0.9581 | 0.9713 | 0.9987 | 0.0438 | 0.8785 | 0.9742 | 0.9873 | 0.9987 | 0.0277 |

RT1 $\beta $ = 0, k_{min} = 10 | 0.8064 | 0.9491 | 0.9646 | 0.9987 | 0.0492 | 0.8117 | 0.9704 | 0.9813 | 0.9987 | 0.0365 |

RT1 $\beta $ = 0, k_{min} = 0.003n | 0.8024 | 0.9661 | 0.9907 | 0.9987 | 0.0440 | 0.8745 | 0.9801 | 0.9933 | 0.9987 | 0.0279 |

RT1 $\beta $ = 0, k_{min} = 0.005n | 0.8037 | 0.9613 | 0.9766 | 0.9987 | 0.0432 | 0.8758 | 0.9782 | 0.9887 | 0.9987 | 0.0273 |

RT2 $\beta $ = 0, k_{min} = 2 | 0.8825 | 0.9701 | 0.9907 | 0.9987 | 0.0353 | 0.8892 | 0.9802 | 0.9920 | 0.9987 | 0.0281 |

RT2 $\beta $ = 0.1, k_{min} = 2 | 0.8825 | 0.9721 | 0.9933 | 0.9987 | 0.0334 | 0.8892 | 0.9811 | 0.9940 | 0.9987 | 0.0270 |

RT2 $\beta $ = 0.2, k_{min} = 2 | 0.8825 | 0.9731 | 0.9933 | 0.9987 | 0.0334 | 0.8999 | 0.9826 | 0.9960 | 0.9987 | 0.0251 |

RT2 $\beta $ = 0.3, k_{min} = 2 | 0.9105 | 0.9777 | 0.9940 | 0.9987 | 0.0285 | 0.8999 | 0.9835 | 0.9960 | 0.9987 | 0.0244 |

RT2 $\beta $ = 0, k_{min} = 3 | 0.8838 | 0.9630 | 0.9760 | 0.9987 | 0.0359 | 0.8905 | 0.9794 | 0.9920 | 0.9987 | 0.0273 |

RT2 $\beta $ = 0, k_{min} = 4 | 0.8852 | 0.9641 | 0.9773 | 0.9987 | 0.0357 | 0.8919 | 0.9768 | 0.9893 | 0.9987 | 0.0275 |

RT2 $\beta $ = 0, k_{min} = 5 | 0.8865 | 0.9620 | 0.9780 | 0.9987 | 0.0356 | 0.8932 | 0.9772 | 0.9893 | 0.9987 | 0.0268 |

RT2 $\beta $ = 0, k_{min} = 10 | 0.8131 | 0.9554 | 0.9660 | 0.9987 | 0.0425 | 0.8425 | 0.9724 | 0.9866 | 0.9987 | 0.0326 |

RT2 $\beta $ = 0, k_{min} = 0.003n | 0.8825 | 0.9701 | 0.9907 | 0.9987 | 0.0353 | 0.8892 | 0.9802 | 0.9920 | 0.9987 | 0.0281 |

RT2 $\beta $ = 0, k_{min} = 0.005n | 0.8838 | 0.9630 | 0.9760 | 0.9987 | 0.0359 | 0.8905 | 0.9794 | 0.9920 | 0.9987 | 0.0273 |

PS j = 1 | 0.7370 | 0.8457 | 0.8531 | 0.9506 | 0.0542 | 0.7130 | 0.8642 | 0.8665 | 0.9840 | 0.0646 |

PS j = 0 | 0.6088 | 0.7510 | 0.7557 | 0.8852 | 0.0632 | 0.6649 | 0.7623 | 0.7644 | 0.8798 | 0.0477 |

Eyeball w = 0.01, ε = 0.3, h = 0.9 | 0.9813 | 0.9866 | 0.9866 | 0.9933 | 0.0033 | 0.9733 | 0.9846 | 0.9853 | 0.9933 | 0.0044 |

Eyeball w = 0.02, ε = 0.3, h = 0.9 | 0.9693 | 0.9801 | 0.9800 | 0.9880 | 0.0036 | 0.9680 | 0.9780 | 0.9780 | 0.9893 | 0.0043 |

GH | 0.8745 | 0.9638 | 0.9766 | 0.9960 | 0.0306 | 0.9079 | 0.9702 | 0.9780 | 0.9960 | 0.0238 |

dAMSE | 0.8451 | 0.9400 | 0.9413 | 0.9479 | 0.0142 | 0.8198 | 0.9385 | 0.9439 | 0.9479 | 0.0234 |

HW | x | x | x | x | x | x | x | x | x | x |

Hall B = 10,000, ε = 0.955, k_{aux} = 2$\sqrt{{n}^{+}}$ | 0.8331 | 0.9372 | 0.9473 | 0.9586 | 0.0271 | 0.8331 | 0.9462 | 0.9499 | 0.9653 | 0.0199 |

Hall B = 10,000, ε = 0.995, k_{aux} = 2$\sqrt{{n}^{+}}$ | 0.8198 | 0.9324 | 0.9439 | 0.9653 | 0.0294 | 0.8211 | 0.9420 | 0.9466 | 0.9546 | 0.0216 |

Hall B = 10,000, ε = 0.9, k_{aux} = 2$\sqrt{{n}^{+}}$ | 0.8505 | 0.9435 | 0.9506 | 0.9626 | 0.0239 | 0.8505 | 0.9518 | 0.9546 | 0.9693 | 0.0175 |

Hall B = 10,000, ε = 0.955 k_{aux} = $\sqrt{{n}^{+}}$ | 0.8892 | 0.9582 | 0.9713 | 0.9853 | 0.0233 | 0.7570 | 0.9631 | 0.9713 | 0.9813 | 0.0332 |

Hall B = 10,000, ε = 0.955, k_{aux} = 3$\sqrt{{n}^{+}}$ | 0.8438 | 0.9193 | 0.9246 | 0.9453 | 0.0219 | 0.8772 | 0.9252 | 0.9252 | 0.9413 | 0.0122 |

Hall B = 1000, ε = 0.955, k_{aux} = 2$\sqrt{{n}^{+}}$ | 0.8251 | 0.9377 | 0.9479 | 0.9613 | 0.0276 | 0.8278 | 0.9468 | 0.9493 | 0.9626 | 0.0200 |

Hall (r) B = 10,000, ε = 0.955, k_{aux} = 2$\sqrt{{n}^{+}}$ | 0.8344 | 0.9373 | 0.9459 | 0.9586 | 0.0269 | 0.8331 | 0.9461 | 0.9506 | 0.9653 | 0.0198 |

Himp B = 10,000, ε = 0.955 | 0.9226 | 0.9695 | 0.9706 | 0.9973 | 0.0177 | 0.9199 | 0.9785 | 0.9806 | 0.9987 | 0.0165 |

Himp B = 10,000, ε = 0.995 | 0.9146 | 0.9685 | 0.9693 | 0.9987 | 0.0205 | 0.9199 | 0.9774 | 0.9806 | 0.9987 | 0.0177 |

Himp B = 10,000, ε = 0.9 | 0.9332 | 0.9696 | 0.9706 | 0.9987 | 0.0156 | 0.9292 | 0.9801 | 0.9820 | 0.9987 | 0.0149 |

Himp B = 1000, ε = 0.955 | 0.9226 | 0.9696 | 0.9713 | 0.9973 | 0.0180 | 0.9212 | 0.9789 | 0.9820 | 0.9987 | 0.0167 |

Himp (r) B = 10,000, ε = 0.955 | 0.9239 | 0.9693 | 0.9700 | 0.9973 | 0.0178 | 0.9212 | 0.9785 | 0.9806 | 0.9987 | 0.0166 |

Gomes B = 10,000, ε = 0.995 | 0.9146 | 0.9684 | 0.9686 | 0.9987 | 0.0205 | 0.9226 | 0.9778 | 0.9806 | 0.9987 | 0.0177 |

Gomes B = 10,000, ε = 0.955 | 0.9132 | 0.9683 | 0.9693 | 0.9987 | 0.0209 | 0.9226 | 0.9769 | 0.9793 | 0.9987 | 0.0177 |

Gomes B = 10,000, ε = 0.9 | 0.9092 | 0.9655 | 0.9666 | 0.9987 | 0.0224 | 0.9212 | 0.9773 | 0.9786 | 0.9987 | 0.0183 |

Gomes B = 1000, ε = 0.995 | 0.9146 | 0.9684 | 0.9700 | 0.9987 | 0.0202 | 0.9065 | 0.9774 | 0.9800 | 0.9987 | 0.0186 |

Gomes (r) B = 10,000, ε = 0.995 | 0.9159 | 0.9682 | 0.9686 | 0.9987 | 0.0205 | 0.9105 | 0.9773 | 0.9806 | 0.9987 | 0.0186 |

Danielsson B = 500, ε = 0.9 | 0.9172 | 0.9842 | 0.9933 | 0.9987 | 0.0204 | 0.8665 | 0.9896 | 0.9980 | 0.9987 | 0.0218 |

Danielsson B = 500, ε = 0.955 | 0.9239 | 0.9815 | 0.9933 | 0.9987 | 0.0226 | 0.9332 | 0.9931 | 0.9987 | 0.9987 | 0.0117 |

Method | Lower Tail | Upper Tail | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Min | Mean | Median | Max | Sd. Dev. | Min | Mean | Median | Max | Sd. Dev. | |

MAD Dis ts = 0.25 | 0.8678 | 0.9095 | 0.8945 | 0.9920 | 0.0355 | 0.8411 | 0.9279 | 0.9152 | 0.9987 | 0.0480 |

MAD Dis ts = 0.20 | 0.8732 | 0.9249 | 0.9159 | 0.9933 | 0.0365 | 0.8505 | 0.9363 | 0.9399 | 0.9960 | 0.0412 |

MAD Dis ts = 0.15 | 0.8959 | 0.9458 | 0.9426 | 0.9933 | 0.0297 | 0.8665 | 0.9460 | 0.9533 | 0.9933 | 0.0336 |

MAD Dis ts = 0.10 | 0.9212 | 0.9579 | 0.9559 | 0.9933 | 0.0194 | 0.9052 | 0.9519 | 0.9519 | 0.9933 | 0.0278 |

MAD Dis ts = 0.05 | 0.9533 | 0.9690 | 0.9686 | 0.9960 | 0.0109 | 0.9519 | 0.9723 | 0.9693 | 0.9973 | 0.0136 |

KS Dis ts = 0.25 | 0.9332 | 0.9811 | 0.9826 | 0.9987 | 0.0135 | 0.8985 | 0.9753 | 0.9826 | 0.9973 | 0.0217 |

KS Dis ts = 0.20 | 0.9386 | 0.9820 | 0.9853 | 0.9987 | 0.0138 | 0.8985 | 0.9744 | 0.9833 | 0.9973 | 0.0226 |

KS Dis ts = 0.15 | 0.9386 | 0.9831 | 0.9873 | 0.9987 | 0.0138 | 0.8985 | 0.9733 | 0.9793 | 0.9973 | 0.0221 |

KS Dis ts = 0.10 | 0.9386 | 0.9826 | 0.9853 | 0.9987 | 0.0140 | 0.9079 | 0.9723 | 0.9786 | 0.9973 | 0.0213 |

KS Dis ts = 0.05 | 0.9519 | 0.9869 | 0.9893 | 0.9987 | 0.0092 | 0.9533 | 0.9787 | 0.9826 | 0.9973 | 0.0137 |

RT1 $\beta $ = 0, k_{min} = 2 | 0.7904 | 0.9759 | 0.9973 | 0.9987 | 0.0395 | 0.8238 | 0.9603 | 0.9826 | 0.9987 | 0.0475 |

RT1 $\beta $ = 0.1, k_{min} = 2 | 0.7997 | 0.9816 | 0.9973 | 0.9987 | 0.0362 | 0.8251 | 0.9685 | 0.9947 | 0.9987 | 0.0431 |

RT1 $\beta $ = 0.2, k_{min} = 2 | 0.8131 | 0.9855 | 0.9973 | 0.9987 | 0.0318 | 0.8825 | 0.9837 | 0.9960 | 0.9987 | 0.0254 |

RT1 $\beta $ = 0.3, k_{min} = 2 | 0.9212 | 0.9910 | 0.9973 | 0.9987 | 0.0170 | 0.8825 | 0.9843 | 0.9960 | 0.9987 | 0.0253 |

RT1 $\beta $ = 0, k_{min} = 3 | 0.7917 | 0.9702 | 0.9880 | 0.9987 | 0.0390 | 0.8251 | 0.9547 | 0.9733 | 0.9987 | 0.0477 |

RT1 $\beta $ = 0, k_{min} = 4 | 0.7931 | 0.9620 | 0.9766 | 0.9987 | 0.0428 | 0.8264 | 0.9498 | 0.9539 | 0.9987 | 0.0475 |

RT1 $\beta $ = 0, k_{min} = 5 | 0.7944 | 0.9630 | 0.9780 | 0.9987 | 0.0425 | 0.8278 | 0.9438 | 0.9426 | 0.9987 | 0.0483 |

RT1 $\beta $ = 0, k_{min} = 10 | 0.8011 | 0.9628 | 0.9713 | 0.9987 | 0.0404 | 0.8344 | 0.9305 | 0.9319 | 0.9987 | 0.0479 |

RT1 $\beta $ = 0, k_{min} = 0.003n | 0.7904 | 0.9759 | 0.9973 | 0.9987 | 0.0395 | 0.8238 | 0.9603 | 0.9826 | 0.9987 | 0.0475 |

RT1 $\beta $ = 0, k_{min} = 0.005n | 0.7917 | 0.9702 | 0.9880 | 0.9987 | 0.0390 | 0.8251 | 0.9547 | 0.9733 | 0.9987 | 0.0477 |

RT2 $\beta $ = 0, k_{min} = 2 | 0.7664 | 0.9819 | 0.9973 | 0.9987 | 0.0394 | 0.8425 | 0.9696 | 0.9947 | 0.9987 | 0.0408 |

RT2 $\beta $ = 0.1, k_{min} = 2 | 0.7690 | 0.9829 | 0.9973 | 0.9987 | 0.0381 | 0.8425 | 0.9696 | 0.9947 | 0.9987 | 0.0408 |

RT2 $\beta $ = 0.2, k_{min} = 2 | 0.7690 | 0.9829 | 0.9973 | 0.9987 | 0.0381 | 0.8838 | 0.9799 | 0.9953 | 0.9987 | 0.0275 |

RT2 $\beta $ = 0.3, k_{min} = 2 | 0.8024 | 0.9841 | 0.9973 | 0.9987 | 0.0332 | 0.8838 | 0.9818 | 0.9960 | 0.9987 | 0.0265 |

RT2 $\beta $ = 0, k_{min} = 3 | 0.7677 | 0.9756 | 0.9933 | 0.9987 | 0.0392 | 0.8438 | 0.9640 | 0.9820 | 0.9987 | 0.0422 |

RT2 $\beta $ = 0, k_{min} = 4 | 0.7690 | 0.9689 | 0.9880 | 0.9987 | 0.0424 | 0.8451 | 0.9595 | 0.9800 | 0.9987 | 0.0427 |

RT2 $\beta $ = 0, k_{min} = 5 | 0.7704 | 0.9687 | 0.9880 | 0.9987 | 0.0420 | 0.8465 | 0.9516 | 0.9633 | 0.9987 | 0.0451 |

RT2 $\beta $ = 0, k_{min} = 10 | 0.7770 | 0.9649 | 0.9700 | 0.9987 | 0.0390 | 0.8531 | 0.9428 | 0.9453 | 0.9973 | 0.0420 |

RT2 $\beta $ = 0, k_{min} = 0.003n | 0.7664 | 0.9819 | 0.9973 | 0.9987 | 0.0394 | 0.8425 | 0.9696 | 0.9947 | 0.9987 | 0.0408 |

RT2 $\beta $ = 0, k_{min} = 0.005n | 0.7677 | 0.9756 | 0.9933 | 0.9987 | 0.0392 | 0.8438 | 0.9640 | 0.9820 | 0.9987 | 0.0422 |

PS j = 1 | 0.7370 | 0.8600 | 0.8611 | 0.9479 | 0.0552 | 0.7343 | 0.8629 | 0.8645 | 0.9666 | 0.0537 |

PS j = 0 | 0.6435 | 0.7971 | 0.8071 | 0.8985 | 0.0565 | 0.6128 | 0.7728 | 0.7697 | 0.8959 | 0.0489 |

Eyeball w = 0.01, ε = 0.3, h = 0.9 | 0.9773 | 0.9868 | 0.9873 | 0.9947 | 0.0034 | 0.9826 | 0.9878 | 0.9880 | 0.9933 | 0.0027 |

Eyeball w = 0.02, ε = 0.3, h = 0.9 | 0.9706 | 0.9802 | 0.9800 | 0.9893 | 0.0040 | 0.9760 | 0.9813 | 0.9813 | 0.9866 | 0.0027 |

GH | 0.8932 | 0.9682 | 0.9820 | 0.9960 | 0.0270 | 0.8798 | 0.9527 | 0.9706 | 0.9973 | 0.0402 |

dAMSE | 0.8264 | 0.9386 | 0.9426 | 0.9573 | 0.0216 | 0.8158 | 0.9386 | 0.9413 | 0.9546 | 0.0184 |

HW | x | x | x | x | x | x | x | x | x | x |

Hall B = 10,000, ε = 0.955, k_{aux} = 2$\sqrt{{n}^{+}}$ | 0.8505 | 0.9443 | 0.9493 | 0.9640 | 0.0189 | 0.8665 | 0.9289 | 0.9332 | 0.9626 | 0.0248 |

Hall B = 10,000, ε = 0.995, k_{aux} = 2$\sqrt{{n}^{+}}$ | 0.8425 | 0.9406 | 0.9466 | 0.9626 | 0.0202 | 0.8585 | 0.9240 | 0.9312 | 0.9586 | 0.0263 |

Hall B = 10,000, ε = 0.9, k_{aux} = 2$\sqrt{{n}^{+}}$ | 0.8678 | 0.9497 | 0.9539 | 0.9693 | 0.0170 | 0.8838 | 0.9350 | 0.9393 | 0.9653 | 0.0220 |

Hall B = 10,000, ε = 0.955 k_{aux} = $\sqrt{{n}^{+}}$ | 0.7837 | 0.9556 | 0.9633 | 0.9826 | 0.0336 | 0.8558 | 0.9384 | 0.9453 | 0.9826 | 0.0362 |

Hall B = 10,000, ε = 0.955, k_{aux} = 3$\sqrt{{n}^{+}}$ | 0.8665 | 0.9227 | 0.9239 | 0.9559 | 0.0163 | 0.8665 | 0.9173 | 0.9212 | 0.9519 | 0.0204 |

Hall B = 1000, ε = 0.955, k_{aux} = 2$\sqrt{{n}^{+}}$ | 0.8505 | 0.9439 | 0.9479 | 0.9653 | 0.0190 | 0.8665 | 0.9283 | 0.9332 | 0.9626 | 0.0249 |

Hall (r) B = 10,000, ε = 0.955, k_{aux} = 2$\sqrt{{n}^{+}}$ | 0.8491 | 0.9436 | 0.9486 | 0.9653 | 0.0192 | 0.8678 | 0.9289 | 0.9332 | 0.9613 | 0.0248 |

Himp B = 10,000, ε = 0.955 | 0.9186 | 0.9682 | 0.9720 | 0.9987 | 0.0187 | 0.9292 | 0.9640 | 0.9633 | 0.9987 | 0.0162 |

Himp B = 10,000, ε = 0.995 | 0.9306 | 0.9691 | 0.9706 | 0.9987 | 0.0187 | 0.9226 | 0.9611 | 0.9613 | 0.9973 | 0.0178 |

Himp B = 10,000, ε = 0.9 | 0.9266 | 0.9697 | 0.9733 | 0.9987 | 0.0163 | 0.9372 | 0.9640 | 0.9640 | 0.9987 | 0.0131 |

Himp B = 1000, ε = 0.955 | 0.9172 | 0.9694 | 0.9713 | 0.9987 | 0.0189 | 0.9266 | 0.9631 | 0.9633 | 0.9987 | 0.0172 |

Himp(r) B = 10,000, ε = 0.955 | 0.9186 | 0.9678 | 0.9706 | 0.9987 | 0.0188 | 0.9292 | 0.9632 | 0.9633 | 0.9987 | 0.0158 |

Gomes B = 10,000, ε = 0.995 | 0.9279 | 0.9687 | 0.9706 | 0.9987 | 0.0190 | 0.9212 | 0.9606 | 0.9599 | 0.9960 | 0.0178 |

Gomes B = 10,000, ε = 0.955 | 0.9105 | 0.9651 | 0.9693 | 0.9987 | 0.0209 | 0.9199 | 0.9614 | 0.9619 | 0.9987 | 0.0192 |

Gomes B = 10,000, ε = 0.9 | 0.9052 | 0.9645 | 0.9680 | 0.9987 | 0.0219 | 0.9079 | 0.9585 | 0.9593 | 0.9987 | 0.0186 |

Gomes B = 1000, ε = 0.995 | 0.9252 | 0.9690 | 0.9720 | 0.9987 | 0.0192 | 0.9226 | 0.9616 | 0.9613 | 0.9987 | 0.0183 |

Gomes (r) B = 10,000, ε = 0.995 | 0.9279 | 0.9687 | 0.9713 | 0.9987 | 0.0191 | 0.9226 | 0.9607 | 0.9613 | 0.9987 | 0.0180 |

Danielsson B = 500, ε = 0.9 | 0.8465 | 0.9839 | 0.9953 | 0.9987 | 0.0261 | 0.8892 | 0.9720 | 0.9773 | 0.9987 | 0.0275 |

Danielsson B = 500, ε = 0.955 | 0.8344 | 0.9849 | 0.9987 | 0.9987 | 0.0274 | 0.9012 | 0.9727 | 0.9780 | 0.9987 | 0.0266 |

**Table 5.**The mean absolute differences (MAE) of the threshold estimates in the left tail (below diagonal) and right tail (above diagonal) for five methods.

Method | PS | Eyeball | dAMSE | HW | Hall 10,000 | Hall 1000 |
---|---|---|---|---|---|---|

PS | 0 | 0.1148 | 0.1023 | 0.0478 | 0.1043 | 0.1044 |

Eyeball | 0.1184 | 0 | 0.0139 | 0.1147 | 0.0171 | 0.0168 |

dAMSE | 0.1060 | 0.0136 | 0 | 0.1020 | 0.0107 | 0.0105 |

HW | 0.0718 | 0.1032 | 0.0918 | 0 | 0.1044 | 0.1046 |

Hall 10,000 | 0.1069 | 0.0206 | 0.0130 | 0.0930 | 0 | 0.0018 |

Hall 1000 | 0.1070 | 0.0206 | 0.0129 | 0.0930 | 0.0013 | 0 |

**Table 6.**The root mean squared differences (RMSE) of the threshold estimates in the left tail (below diagonal) and right tail (above diagonal) for five methods.

Method | PS | Eyeball | dAMSE | HW | Hall 10,000 | Hall 1000 |
---|---|---|---|---|---|---|

PS | 0 | 0.1148 | 0.1023 | 0.0478 | 0.1043 | 0.1044 |

Eyeball | 0.1184 | 0 | 0.0139 | 0.1147 | 0.0171 | 0.0168 |

dAMSE | 0.1060 | 0.0136 | 0 | 0.1020 | 0.0107 | 0.0105 |

HW | 0.0718 | 0.1032 | 0.0918 | 0 | 0.1044 | 0.1046 |

Hall 10,000 | 0.1069 | 0.0206 | 0.0130 | 0.0930 | 0 | 0.0018 |

Hall 1000 | 0.1070 | 0.0206 | 0.0129 | 0.0930 | 0.0013 | 0 |

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**MDPI and ACS Style**

Just, M.; Echaust, K.
An Optimal Tail Selection in Risk Measurement. *Risks* **2021**, *9*, 70.
https://doi.org/10.3390/risks9040070

**AMA Style**

Just M, Echaust K.
An Optimal Tail Selection in Risk Measurement. *Risks*. 2021; 9(4):70.
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**Chicago/Turabian Style**

Just, Małgorzata, and Krzysztof Echaust.
2021. "An Optimal Tail Selection in Risk Measurement" *Risks* 9, no. 4: 70.
https://doi.org/10.3390/risks9040070