Page 38 - Textos de Matemática Vol. 47

28 E. GONC¸ALVES AND C. M. MARTINS
We also note that the kurtosis of the processes X and " are related by 6 2 1 2
So, the process X is leptokurtic if the generator process, ", is leptokurtic. Moreover, this equality shows that KX is an increasing function of if K" < 3, and a decreasing function if K" > 3. Also, when tends to 1, KX tends to 3, independently of the value of K".
Examples
We evaluate now the presence of the Taylor property in model (2.1) con- sidering generator processes with non-negative distributions and weight tails significantly di↵erent. We take into account the necessary and su cient condi- tion of the presence of the Taylor property, written in the form
T ( )=1 2m Cov Xt,Xt2 >0, (2.2) L 1 V ( X t2 )
where L denotes the marginal law of the generator process, ". Generator process with platykurtic distribution
We consider the process X defined in (2.1) in cases where "t is distributed
according to a platykurtic Beta probability law, Beta(↵, ✓), whose density has
the form f1(x) = (↵+✓) x↵ 1(1 x)✓ 1 I]0,1[(x), ↵ > 0, ✓ > 0. More precisely, (↵) (✓)
KX( )=1+ 2 +1+ 2K".
we analyze the cases ✓ = ↵ (symmetrical distribution), ✓ = 2↵ (right-skewed distribution) and ✓ = ↵ (left-skewed distribution). Concerning the values of
2
thekurtosis,wehaveK" =31+2↵ when✓=↵,K" =32+3↵ when✓= ↵ and
3+2↵ 4+3↵ 2
K" =31+3↵ when✓=2↵.Inthecases✓=↵and✓=2↵,Corollary2.3
2+3↵
allows us to conclude that the Taylor property is present. In both cases, the
functions KX( ) and TL( ) are also functions of ↵ and the presence of the Taylor property becomes, in general, weaker when KX increases.
Comparing now the cases ✓ = ↵ and ✓ = 2↵, it is easy to verify that, for each ↵, the kurtosis is greater for the right-skewed distribution.
In Figure 1 we present the graphs of the functions KX( ) related to the cases Beta(↵,↵) and Beta(↵,2↵), as well as the graphs of TBeta(↵,↵)( ) and TBeta(↵,2↵)( ), 0< <1, 0<↵<5.