Tail Dependence - ETH - Entrepreneurial Risks - ETH Zürich

4.17 Establishing bias and error bars of ˆ β calculated by all data and effect on
ˆλ +,− estimated by applying the non-parametric approach according to
Sornette & Malevergne to 1000 synthetic samples consisting of N = 2507
data points. The bias is calculated by comparing mean values ˆ β all data
and ˆ λ +,−
( ˆ βalldata) to original values denoted by βorig and corresponding
tail dependence estimator ˆ λ +,− (βorig) and for the estimation of error bars
standard deviation ’std’ and 95% quantiles of the estimates are provided.
Relative bias (’rel.bias’) is calculated as percentage of ˆ λ +,−
(βorig). Tail
indexes α were set equal to three to avoid a second source of error and
’inf’ means ·/0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.18 Establishing bias and error bars of ˆ βSI 2 calculated by second β-smile
condition: � Y ≥ Y (k) � and effect on ˆ λ +,− estimated by applying the
non-parametric approach according to Sornette & Malevergne to 1000
synthetic samples consisting of N = 2507 data points. The bias is cal-
culated by comparing mean values ˆ β alldata and ˆ λ +,−
( ˆ βalldata) to original
values denoted by βorig and corresponding tail dependence estimator
ˆλ +,− (βorig) and for the estimation of error bars standard deviation ’std’
and 95% quantiles of the estimates are provided. Tail indexes α were set
equal to three to avoid a second source of error and ∗ means that relative
bias (’rel.bias’) calculated as percentage of ˆ λ +,−
(βorig) ≥ 100%. . . . . . 151
4.19 Establishing bias and error bars of ˆν and ˆb γ n , estimated for 1000 synthetic
samples consisting of N = 2507 data points. The bias is calculated by
comparing mean values ˆν and ˆb γ
n to original values denoted by α and for
the estimation of error bars standard deviation ’std’ and 95% quantiles
of the estimates are provided. Relative bias (’rel.bias’) is calculated as
percentage of α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4.20 Establishing bias and error bars of ˆ λ +,− (ˆν, ˆ βalldata) with ˆ β calculated by
all data and ˆ λ +,− (ˆν, ˆ βSI2) with ˆ β calculated by second β-smile condition:
� Y ≥ Y (k) � by applying the non-parametric approach according
to Sornette & Malevergne to 1000 synthetic samples consisting of
N = 2507 data points. The bias is calculated by comparing mean
values ˆ λ +,−
(ˆν, ˆ βalldata) and ˆ λ +,−
(ˆν, ˆ βSI 2) to original values denoted by
ˆλ +,−
(α, βorig) and for the estimation of error bars standard deviation
’std’ and 95% quantiles of the estimates are provided. ∗ means that
relative bias (’rel.bias’) calculated as percentage of ˆ λ +,−
4.21 Comparison of estimates by non-parametric estimators ˆ λ·,m, ˆ λ
(βorig) ≥ 100%. 153
·,m
cording to Schmidt & Stadtmüller, and ˆχ +,− according to Poon, Rockinger,
and Tawn with ˆ λ +,− (βalldata) according to Sornette & Malevergne by
applying the different concepts to 1000 synthetic samples consisting of
N = 2507 data points created by different βorig. For the estimation of
error bars standard deviation ’std’ and 95% quantiles of the estimates
are calculated and for ˆχ +,− the proposed standard deviation ˆσ is given
for comparison. Tail indexes α were estimated by the Hill estimator. . . 155
EV T
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