statistique, théorie et gestion de portefeuille - Docs at ISFA

empirical analog FN(x), estimated from
· a sample of N realizations, is evaluated as follows:
[FN(x) − F(x)]
ADS = N 2
dF(x) (28)
F(x)(1 − F(x))
= −N − 2
N
∑
1
{wk log(F(yk)) + (1 − wk)log(1 − F(yk))}, (29)
where wk = 2k/(2N + 1), k = 1...N and y1 ... yN is its ordered sample. If the sample is drawn
from a population with distribution function F(x), the Anderson-Darling statistics (ADS) has a standard
AD-distribution free of the theoretical df F(x) (Anderson and Darling 1952), similarly to the χ 2 for the
χ 2 -statistic, or the Kolmogorov distribution for the Kolmogorov statistic. It should be noted that the ADS
weights the squared difference in eq.(28) by 1/F(x)(1 − F(x)) which is nothing but the inverse of the variance
of the difference in square brackets. The AD distance thus emphasizes more the tails of the distribution
than, say, the Kolmogorov distance which is determined by the maximum absolute deviation of Fn(x) from
F(x) or the mean-squared error, which is mostly controlled by the middle of range of the distribution. Since
we have to insert the estimated parameters into the ADS, this statistic does not obey any more the standard
AD-distribution: the ADS decreases because the use of the fitting parameters ensures a better fit to the sample
distribution. However, we can still use the standard quantiles of the AD-distribution as upper boundaries
of the ADS. If the observed ADS is larger than the standard quantile with a high significance level (1 − ε),
we can then conclude that the null hypothesis F(x) is rejected with significance level larger than (1 − ε). If
we wish to estimate the real significance level of the ADS in the case where it does not exceed the standard
quantile of a high significance level, we are forced to use some other method of estimation of the significance
level of the ADS, such as the bootstrap method.
In the following, the estimates minimizing the Anderson-Darling distance will be refered to as AD-estimates.
The maximum likelihood estimates (ML-estimates) are asymptotically more efficient than AD-estimates for
independent data and under the condition that the null hypothesis (given by one of the four distributions (22-
25), for instance) corresponds to the true data generating model. When this is not the case, the AD-estimates
provide a better practical tool for approximating sample distributions compared with the ML-estimates.
We have determined the AD-estimates for 18 standard significance levels q1 ...q18 given in table 6. The
corresponding sample quantiles corresponding to these significance levels or thresholds u1 ...u18 for our
samples are also shown in table 6. Despite the fact that thresholds uk vary from sample to sample, they
always corresponded to the same fixed set of significance levels qk throughout the paper and allows us to
compare the goodness-of-fit for samples of different sizes.
4.3 Empirical results
The Anderson-Darling statistics (ADS) for five parametric distributions (Weibull or Stretched-Exponential,
Generalized Pareto, Gamma, exponential and Pareto) are shown in table 7 for two quantile ranges, the first
top half of the table corresponding to the 90% lowest thresholds while the second bottom half corresponds
to the 10% highest ones. For the lowest thresholds, the ADS rejects all distributions, except the Stretched-
Exponential for the Nasdaq. Thus, none of the considered distributions is really adequate to model the data
over such large ranges. For the 10% highest quantiles, only the exponential model is rejected at the 95%
confidence level. The Stretched-Exponential distribution is the best, just before the Pareto distribution and
the Incomplete Gamma that cannot be rejected. We now present an analysis of each case in more details.
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