statistique, théorie et gestion de portefeuille - Docs at ISFA

92 3. Distributions exponentielles étirées contre distributions régulièrement variables
The maximum of the log-likelihood function is reach at
and is given by
T
d ˆ = 1
T ∑ xi − u, (55)
i=1
1
T LED T ( d) ˆ = −(1 + lnd). ˆ
(56)
The random variable √ N( ˆ
d − d) is asymptotically normally distributed with zero mean and variance d 2 /N.
A.4 The Incomplete Gamma distribution
The expression of the Incomplete Gamma distribution function is given by (25) and its density is
fu(x|b,d) =
d b
Γ −b, u
d
· x −(b+1)
exp −
x
, x ≥ u. (57)
d
Let us introduce the partial derivative of the logarithm of the incomplete Gamma function:
Ψ(a,x) = ∂
∞ 1
lnΓ(a,x) = dt lnt t
∂a Γ(a,x) x
a−1 e −t . (58)
The maximum of the log-likelihood function is reached at the point (ˆb, ˆ
d) solution of
and is equal to
1
T
T
∑
i=1
1
T
1
T LIG
T (ˆb, d) ˆ = −lndˆ − lnΓ
ln xi
d
T
xi
∑
i=1 d =
−b, u
d
= Ψ
−b, u
d
1
Γ −b, u
d
+ (b + 1) · Ψ
28
, (59)
u
−b e
d
− u d − b, (60)
−b, u
d
+ b −
1
Γ −b, u
d
u
−b e
d
− u d . (61)