Page 40 - Textos de Matemática Vol. 47

30 E. GONC¸ALVES AND C. M. MARTINS
Generator process with leptokurtic distribution
We consider now the process X defined in (2.1) with "t following the Gamma
distribution with parameters ✓ and ↵, (✓,↵), ↵ > 0, ✓ > 0, that is, with
density f2(x) = ↵✓ e ↵xx✓ 1I]0,+1[ (x), whose kurtosis coe cient is K" = (✓)
3+6.So,K ( )= 6 2 +1 2 3+6 alsodependson✓(butnoton↵)asit ✓ X 1+ 2 1+ 2 ✓
happens with T (✓,↵)( ). The Taylor property is always present in model (2.1) as the distribution (✓, ↵) is right-skewed.
We note that when ✓ increases then T (✓,↵) ( ) approaches zero, that is, the
autocorrelation functions in study are closer as ✓ increases. This result is in
agreement with the evolution of the kurtosis, taking into account that now
the process is more leptokurtic when ✓ decreases. Analogous conclusions are
obtained when we consider "t following a Pareto distribution with parameters
↵and✓,Par(↵,✓),withdensityf (x)= ✓↵✓ I (x),↵>0,✓>4(inorder 3 x✓+1 ]↵,+1[
to assure the existence of m4). The kurtosis is K" = 3 + 6(✓3+✓2 6✓ 2) which ✓(✓ 3)(✓ 4)
is a decreasing function of ✓ that goes to 9 when ✓ tends to infinity, and to infinity when ✓ tends to 4. So, the Pareto distribution is leptokurtic, no matter the value of ✓.
These conclusions are illustrated in Figure 3, which shows the graphs of the functions KX ( ), related to the Gamma and Pareto distributions, and those of the functions T (✓,↵) ( ) and TPar(↵,✓) ( ), 0 < < 1, 4 < ✓ < 10.
4 800
K
10
Θ
0 0
4 1
T
10
Θ
0 0
ΦΦ
11
Figure 3. Graphs for K = KX( ) (on the left) and T = TL( ) (on the right) in the cases Par(↵,✓) (upper surface) and Gamma(✓,↵) (lower surface), 0 < < 1, 4 < ✓ < 10.