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90 3. Distributions exponentielles étirées contre distributions régulièrement variables
A Maximum likelihood estimators
In this appendix, we give the expressions of the maximum likelihood estimators derived from the four
distributions (22-25)
A.1 The Pareto distribution
According to expression (22), the Pareto distribution is given by
u
b Fu(x) = 1 − ,
x
x ≥ u (38)
and its density is
Let us denote by
L PD
fu(x|b) = b ub
x b+1
T (ˆb) = max
b
T
∑
i=1
(39)
ln fu(xi|b) (40)
the maximum of log-likekihood function derived under hypothesis (PD). ˆb is the maximum likelihood estimator
of the tail index b under such hyptothesis.
The maximum of the likelihood function is solution of
which yields
ˆb =
1
T
T
∑
i=1
lnxi − lnu
1 1
+ lnu −
b T ∑lnxi = 0, (41)
−1
, and
1
T LPD T (ˆb) = ln ˆb
u −
1 + 1
. (42)
ˆb
Moreover, one easily shows that ˆb is asymptotically normally distributed:
√ N(ˆb − b) ∼ N (0,b). (43)
A.2 The Weibull distribution
The Weibull distribution is given by equation (23) and its density is
fu(x|c,d) = c
d c · e( u d ) c
x c−1 · exp
The maximum of the log-likelihood function is
L SE
T (ĉ, d) ˆ = max
T ∑Ti=1 T
∑
i=1
c,d
T
∑
i=1
−
x
d
c
, x ≥ u. (44)
ln fu(xi|c,d) (45)
Thus, the maximum likehood estimators (ĉ, d) ˆ are solution of
1
c =
1
T ∑T xi c xi
i=1 u ln
u
1 xi c −
u − 1 1 T
T ∑ ln
i=1
xi
d
,
u
(46)
c = uc
xi
c
− 1.
T u
(47)
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