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78 3. Distributions exponentielles étirées contre distributions régulièrement variables
dard extreme value estimators, the sequel of this paper is devoted to the investigation of a parametric approach
in order to decide which class of extreme value distributions, rapidly versus regularly varying, accounts
best for the empirical distributions of returns.
4 Fitting distributions of returns with parametric densities
Since our previous results lead to doubt the validity of the rejection of the hypothesis that the distribution of
returns are rapidly varying, we now propose to pit a parametric champion for this class of functions against
the Pareto champion of regularly varying functions. To represent the class of rapidly varying functions, we
propose the family of Stretched-Exponentials. As discussed in the introduction, the class of stretched exponentials
is motivated in part from a theoretical view point by the fact that the large deviations of multiplicative
processes are generically distributed with stretched exponential distributions (Frisch and Sornette 1997).
Stretched exponential distributions are also parsimonious examples of sub-exponential distributions with fat
tails for instance in the sense of the asymptotic probability weight of the maximum compared with the sum
of large samples (Feller 1971). Notwithstanding their fat-tailness, Stretched Exponential distributions have
all their moments finite 6 , in constrast with regularly varying distributions for which moments of order equal
to or larger than the index b are not defined. This property may provide a substantial advantage to exploit in
generalizations of the mean-variance portfolio theory using higher-order moments (Rubinstein 1973, Fang
and Lai 1997, Hwang and Satchell 1999, Sornette et al. 2000, Andersen and Sornette 2001, Jurczenko and
Maillet 2002, Malevergne and Sornette 2002, for instance ). Moreover, the existence of all moments is
an important property allowing for an efficient estimation of any high-order moment, since it ensures that
the estimators are asymptotically Gaussian. In particular, for Stretched-Exponentially distributed random
variables, the variance, skewness and kurtosis can be well estimated, contrarily to random variables with
regularly varying distribution with tail index in the range 3 − 5.
4.1 Definition of a general 3-parameters family of distributions
We thus consider a general 3-parameters family of distributions and its particular restrictions corresponding
to some fixed value(s) of two (one) parameters. This family is defined by its density function given by:
A(b,c,d,u) x
fu(x|b,c,d) =
−(b+1) exp −
x c
d if x u > 0
(20)
0 if x < u.
Here, b,c,d are unknown parameters, u is a known lower threshold that will be varied for the purposes of
our analysis and A(b,c,d,u) is a normalizing constant given by the expression:
A(b,c,d,u) =
db c
Γ(−b/c,(u/d) c , (21)
)
where Γ(a,x) denotes the (non-normalized) incomplete Gamma function. The parameter b ranges from
minus infinity to infinity while c and d range from zero to infinity. In the particular case where c = 0,
the parameter b also needs to be positive to ensure the normalization of the probability density function
(pdf). The interval of definition of this family is the positive semi-axis. Negative log-returns will be studied
by taking their absolute values. The family (20) includes several well-known pdf’s often used in different
applications. We enumerate them.
6 However, they do not admit an exponential moment, which leads to problems in the reconstruction of the distribution from the
knowledge of their moments (Stuart and Ord 1994).
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