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80 3. Distributions exponentielles étirées contre distributions régulièrement variables
the (SE) model goes to the Pareto model. Indeed, we can write
c
dc · xc−1
· exp − xc − uc dc
u
c = c ·
d
xc−1
u
c
x
c
exp − · − 1 ,
uc d u
β · x −1
u
c exp −c · ln
d
x
, as c → 0
u
β · x −1
exp −β · ln x
,
u
β uβ
, (27)
xβ+1 which is the pdf of the (PD) model with tail index β. The condition (26) comes naturally from the properties
of the maximum-likelihood estimator of the scale parameter d given by equation (47) and underlined by
equation (90) in the appendices at the end of the paper. It implies that, as c → 0, the characteristic scale d
of the (SE) model must also go to zero with c to ensure the convergence of the (SE) model towards the (PD)
model.
This shows that the Pareto model can be approximated with any desired accuracy on an arbitrary interval
(u > 0,U) by the (SE) model with paramters (c,d) satisfying equation (26) where the arrow is replaced by an
equality. Although the value c = 0 does not give strickly speaking a Stretched-Exponential distribution, the
limit c → 0 provides any desired approximation to the Pareto distribution, uniformly on any finite interval
(u,U). This deep relationship between the SE and PD models allows us to understand why it can be very
difficult to decide, on a statistical basis, which of these models fits the data best.
4.2 Methodology
We start with fitting our two data sets (DJ and ND) by the five distributions enumerated above (20) and (22-
25). Our first goal is to show that no single parametric representation among any of the cited pdf’s fits the
whole range of the data sets. Recall that we analyze separately positive and negative returns (the later being
converted to the positive semi-axis). We shall use in our analysis a movable lower threshold u, restricting by
this threshold our sample to observations satisfying to x > u.
In addition to estimating the parameters involved in each representation (20,22-25) by maximum likelihood
for each particular threshold u 7 , we need a characterization of the goodness-of-fit. For this, we propose
to use a distance measure between the estimated distribution and the sample distribution. Many distances
can be used: mean-squared error, Kullback-Liebler distance 8 , Kolmogorov distance, Spearman distance (as
in Longin (1996)) or Anderson-Darling distance, to cite a few. We can also use one of these distances
to determine the parameters of each pdf according to the criterion of minimizing the distance between the
estimated distribution and the sample distribution. The chosen distance is thus useful both for characterizing
and for estimating the parametric pdf. In the later case, once an estimation of the parameters of particular
distribution family has been obtained according to the selected distance, we need to quantify the statistical
significance of the fit. This requires to derive the statistics associated with the chosen distance. These
statistics are known for most of the distances cited above, in the limit of large sample.
We have chosen the Anderson-Darling distance to derive our estimated parameters and perform our tests
of goodness of fit. The Anderson-Darling distance between a theoretical distribution function F(x) and its
7 The estimators and their asymptotic properties are derived in appendix A.
8 This distance (or divergence, strictly speaking) is the natural distance associated with maximum-likelihood estimation since
it is for these values of the estimated parameters that the distance between the true model and the assumed model reaches its
minimum.
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