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D Testing non-nested hypotheses with the encompassing principle
D.1 Testing the Pareto model against the (SE) model
D.1.1 Pseudo-true value
Let us consider the two models, stretched exponential (SE) and Pareto (P). The pdf’s associated with these
two models are f1(x|c,d) and f2(x|b) respectively . Under the true distribution f0(x), we will determine the
pseudo-true values of the maximum likelihood estimators ˆb, ĉ and d, ˆ namely, the values of these parameters
which minimize the (Kullback-Leibler) distance between the considered model and the true distribution
(Gouriéroux and Monfort 1994). Thus, the pseudo-true values b∗ , c∗ and d∗ of ˆb, ĉ and dˆ appear as the
expected values of the estimators under f0. For instance
b ∗ = arginf
b E0
ln f0(x)
, (81)
f2(x|b)
where, in all what follows, E0[·] denotes the expectation under the probability measure associated with the
true distribution f0. Thus, b∗ is simply solution of
∂
∂b E0
ln f0(x)
= 0, (82)
f2(x|b)
which yields
b ∗ = (E0[lnx] − lnu) −1 , (83)
and is consistently estimated by the maximum likelihood estimator
ˆb
1
=
T ∑lnxi
−1 − lnu . (84)
In fact, the maximum likelihood estimator ˆb converges to its pseudo-true value, with (ˆb−b ∗ ) asymptotically
normally distributed with zero mean.
Similarly, the maximum likelihood estimators ĉ and ˆ
d converge to their pseudo-true values c ∗ and d ∗ .
D.1.2 Binding functions and encompassing
Let us now ask what is the value b † (c,d) of the parameter b for which f2(x|b) is the nearest to f1(x|c,d), for
a given (c,d). Such a value b † is the binding function and is solution of
b † (c,d) = arginf
b E1
ln f1(x|c,d)
, (85)
f2(x|b)
which only involves f1 and f2 but not the true distribution f0. The binding function is given by
b †
(c,d) = E1 ln x
− ln
d
u
−1 . (86)
d
After some calculations, we find
E1 ln x
d
97
= c
dc e( u d ) c
·∞
dx ln
u
x
d xc−1 e −( x d ) c
, (87)
= ln u
d + e( u d ) c
c Γ
u
c 0, ,
d
(88)
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