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B Asymptotic distribution of the sum of Weibull variables with a Gaussian
copula.
We assume that the marginal distributions are given by the modified Weibull distributions:
Pi(xi) = 1
2 √ π
c
χ c/2
i
|xi| c/2−1 e −|x i |
χ ic
413
(149)
and that the χi’s are all equal to one, in order not to cumber the notation. As in the proof of corollary 2, it
will be sufficient to replace wi by wiχi to reintroduce the scale factors.
Under the Gaussian copula assumption, we obtain the following form for the multivariate distribution :
c
P (x1, · · · , xN) =
N
2Nπ N/2√ N
x
det V
c/2−1
⎡
i exp ⎣−
V −1
ij xc/2
i xc/2
⎤
⎦
j . (150)
Let
i=1
f(x1, · · · , xN) =
i,j
i,j
V −1
ij xi c/2 xj c/2 . (151)
We have to minimize f under the constraint wixi = S. As for the independent case, we introduce a
Lagrange multiplier λ which leads to
c
j
V −1
jk x∗ j c/2 x ∗ k c/2−1 = λwk . (152)
The left-hand-side of this equation is a homogeneous function of degree c − 1 in the x∗ i ’s, thus necessarily
where the σi’s are solution of
j
x ∗ i =
1
λ c−1
· σi, (153)
c
V −1
jk σj c/2 σk c/2−1 = wk . (154)
The set of equations (154) has a unique solution due to the convexity of the minimization problem. This
set of equations can be easily solved by a numerical method like Newton’s algorithm. It is convenient to
simplify the problem and avoid the inversion of the matrix V , by rewritting (154) as
Using the constraint wix ∗ i
so that
k
= S, we obtain
Vjk wk σk 1−c/2 = σ c/2
j . (155)
1
λ c−1
c
x ∗ i =
=
S
, (156)
wiσi
σi
· S. (157)
wiσi
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