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98 3. Distributions exponentielles étirées contre distributions régulièrement variables
so that the binding function can be expressed as
b † (c,d) = c · e −( u d ) c
−1 u
c Γ 0, , when c > 0 (89)
d
In the case when c goes to zero and c and d are related by equation (47), we can still calculate b † . Indeed,
we can first show that
u
c ∼
d
1
1
c T
T
ln xi
−1 =
u
1
c · ˆb. (90)
Now, using the asymptoptic relation
we conclude that
∑
i=1
e x · Γ(0,x) ∼ x −1 , as x → ∞, (91)
b † (ĉ, ˆ
d) = ˆb, as ĉ → 0. (92)
This result is in fact natural because the PD model can be seen formally as the limit of the SE model for
c → 0 under the condition
as previously exposed in section 4.1
u
c c · → β,
d
as c → 0 , (93)
Now, following Mizon and Richard (1986), the model (SE) with pdf f1 is said to encompass the model
(PD) with pdf f2 if the best representative of (PD) -with parameter b ∗ - is also the distribution nearest to the
best representative of (SE) -with parameters (c ∗ ,d ∗ ). Thus, (SE) is said to encompass (PD) if and only if
b ∗ = b † (c ∗ ,d ∗ ).
The reverse situation can be considered in order to study the encompassing of the model (SE) by the model
(PD). Such situation occurs if and only if
c ∗
D.1.3 Wald encompassing test
d ∗
c † (b∗ )
=
d † (b∗ )
. (94)
We first test the encompassing of (PD) into (SE), namely the null hypothesis H0 = {b ∗ = b † (c ∗ ,d ∗ )}.
Under this null hypothesis, it can be shown (Gouriéroux and Monfort 1994) that the random variable
√ T ˆb − b † (ĉ, ˆ
d) is asymptotically normally distributed with zero mean and variance V given by
V = K −1
C12]K −1
22 [C22 −C21C −1
11
+ K −1 −1
22
[C21C11 − ˜C21K −1
11
22 +
]C11[C −1
11 C12 − K −1
11
˜C12]K −1
22
where the expression of the coefficients involved in V will be given below. Thus, under H0, the random variable
ξT = T ˆb − b † (ĉ, d) ˆ
ˆV −1 ˆb − b † (ĉ, d) ˆ
, where ˆV −1 is a consistent estimator of V −1 , follows asymptotically
a χ2-distribution with one degrees of freedom.
The matrix K11 is given by
K11(i, j) = −E0
∂ 2 ln f1(x|c ∗ ,d ∗ )
∂αi∂α j
34
(95)
, i, j = 1,2, (96)