statistique, théorie et gestion de portefeuille - Docs at ISFA

5 Comparison of the descriptive power of the different families
As we have seen by comparing the Anderson-Darling statistics corresponding to the four parametric families
(22-25), the best model in the sense of minimizing the Anderson-Darling distance is the Stretched-
Exponential distribution.
We now compare these four distributions with the comprehensive distribution (20) using Wilks’ theorem
(Wilks 1938) of nested hypotheses to check whether or not some of the four distributions are sufficient
compared with the comprehensive distribution to describe the data. We then turn to the Wald encompassing
test for non-nested hypotheses which provides a pairwise comparison of the different models.
5.1 Comparison between the four parametric families and the comprehensive distribution
According to Wilk’s theorem, the doubled generalized log-likelihood ratio Λ:
85
Λ = 2 log maxL(CD,X,Θ)
, (34)
maxL(z,X,θ)
has asymptotically (as the size N of the sample X tends to infinity) the χ 2 -distribution. Here L denotes the
likelihood function, θ and Θ are parametric spaces corresponding to hypotheses z and CD correspondingly
(hypothesis z is one of the four hypotheses (22-25) that are particular cases of the CD under some parameter
relations). The statement of the theorem is valid under the condition that the sample X obeys hypothesis z
for some particular value of its parameter belonging to the space θ. The number of degrees of freedom of
the χ 2 -distribution equals to the difference of the dimensions of the two spaces Θ and θ. Since dim(Θ) = 3
and dim(θ) = 2 for the Stretched-Exponential and Incomplet Gamma distributions; dim(θ) = 1 for the
Pareto and the Exponential distributions, we have one degree of freedom for the formers and two degrees of
freedom for the laters. The maximum of the likelihood in the numerator of (34) is taken over the space Θ,
whereas the maximum of the likelihood in the denominator of (34) is taken over the space θ. Since we have
always θ ⊂ Θ, the likelihood ratio is always larger than 1, and the log-likelihood ratio is non-negative. If the
observed value of Λ does not exceed some high-confidence level (say, 99% confidence level) of the χ 2 , we
then reject the hypothesis CD in favour of the hypothesis z, considering the space Θ redundant. Otherwise,
we accept the hypothesis CD, considering the space θ insufficient.
The doubled log-likelihood ratios (34) are shown in figures 8 for the positive and negative branches of the
distribution of returns of the Nasdaq and in figures 9 for the Dow Jones. The 95% χ 2 confidence levels for
1 and 2 degrees of freedom are given by the horizontal lines.
For the Nasdaq data, figure 8 clearly shows that Exponential distribution is completely insufficient: for all
lower thresholds, the Wilks log-likelihood ratio exceeds the 95% χ2 1 level 3.84. The Pareto distribution is
insufficient for thresholds u1 − u11 (92.5% of the ordered sample) and becomes comparable with the Comprehensive
distribution in the tail u12 − u18 (7.5% of the tail probability). It is natural that two-parametric
families Incomplete Gamma and Stretched-Exponential have higher goodness-of-fit than the one-parametric
Exponential and Pareto distributions. The Incomplete Gamma distribution is comparable with the Comprehensive
distribution starting with u10 (90%), whereas the Stretched-Exponential is somewhat better (u9 or
u8 , i.e., 70%). For the tails representing 7.5% of the data, all parametric families except for the Exponential
distribution fit the sample distribution with almost the same efficiency. The results obtained for the Dow
Jones data shown in figure 9 are similar. The Stretched-Exponential is comparable with the Comprehensive
distribution starting with u8 (70%). On the whole, one can say that the Stretched-Exponential distribution
performs better than the three other parametric families.
21