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86 3. Distributions exponentielles étirées contre distributions régulièrement variables
We should stress that each log-likelihood ratio represented in figures 8 and 9, so-to say “acts on its own
ground,” that is, the corresponding χ 2 -distribution is valid under the assumption of the validity of each
particular hypothesis whose likelihood stands in the numerator of the double log-likelihood (34). It would
be desirable to compare all combinations of pairs of hypotheses directly, in addition to comparing each
of them with the comprehensive distribution. Unfortunately, the Wilks theorem can not be used in the
case of pair-wise comparison because the problem is not more that of comparing nested hypothesis (that
is, one hypothesis is a particular case of the comprehensive model). As a consequence, our results on the
comparison of the relative merits of each of the four distributions using the generalized log-likelihood ratio
should be interpreted with a care, in particular, in a case of contradictory conclusions. Fortunately, the main
conclusion of the comparison (an advantage of the Stretched-Exponential distribution over the three other
distribution) does not contradict our earlier results discussed above.
5.2 Pair-wise comparison of the Pareto model with the Stretched-Exponential model
In order to compare formally the descriptive power of the Stretched-Exponential distribution (the best twoparameters
ditribution) with that of the Pareto distribution (the best one-parameter distribution), we need to
use the methods for testing non-nested hypotheses. There are in fact many ways to perform such a test (see
Gouriéroux and Monfort (1994) for a review). Concerning the log-likelihood ratio test used in the previous
section for nested-hypotheses testing, a direct generalization for non-nested hypotheses has been provided
by Cox’s test (1961, 1962). However, such a test requires that the true distribution of the sample be nested in
one of the two considered models. Our previous investigations have shown that it is not the case, so we need
to use another testing procedure. Indeed, when comparing the Pareto model with the Stretched-Exponential
model, we have found that using the methodology of non-nested hypothesis leads to inconsistencies such
as negative variances of estimators. This is another (indirect) confirmation that neither the Pareto nor the
Stretched-Exponential distributions are the true distribution.
In the case where none of the tested hypotheses contain the true distribution, it can be useful to consider the
encompassing principle introduced by Mizon and Richard (1986). A model, (SE) say, is said to encompass
another model, (PD) for instance, if the representative of (PD), which is the closest to the best representative
of (SE), is also the best representative of (PD) per se. Here, the best representative of a model is the
distribution which is the nearest to the true distribution for the considered model. The detailled testing
procedure is based on the Wald and Score encompassing tests (Gouriéroux and Monfort 1994), which are
detailed in appendix D.
Table 13 presents the results of the tests for the null hypothesis “(SE) encompasses (PD)”. In every cases,
the null hypothesis cannot be rejected at the 95% significance level for quantiles higher than q6 = 0.5 and at
the 99% significance level for quantiles higher than q10 = 0.9. The unfilled entries for the largest quantiles
correspond to MLE giving c → 0. In this case, as shown in Appendix D.1.2, b † has a non-trivial and welldefined
limit which is nothing but the true value ˆb. Thus, the Wald tests is automatically verified at any
confidence levels. Thus, in the tail, the Stretched-Eponential model encompasses the Pareto model. We can
then conclude that it provides a description of the data which is at least as good as that given by the later.
In order to test whether the (SE) model is superior to the Pareto model, we should perform the reverse
test, namely the encompassing of the former model into the later. This task is difficult since the pseudotrue
values of the parameters (c,d) are not always well-defined as exposed in appendix D. Thus, Wald
encompassing test cannot be performed as in the previous case. As a remedy and alternative, we propose
a test of the null hypothesis H0 that the Pareto distribution is the true underlying distribution. This test is
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