DDated 30 November 2007.
Geometric Gamma Max-Infinitely Divisible Models
Satheesh S
NEELOLPALAM, S. N. Park Road Trichur – 680 004,
India. [email protected]
Sandhya E
Department of Statistics, Prajyoti Niketan College Pudukkad, Trichur – 680 301,
India. [email protected]
Abstract.
A transformation of gamma max-infinitely divisible laws viz . geometric gamma max-infinitely divisible laws is considered in this paper. Some of its distributional and divisibility properties are discussed and a random time changed extremal process corresponding to this distribution is presented. A new kind of invariance (stability) under geometric maxima is proved and a max-AR(1) model corresponding to it is also discussed.
AMS (2000) Classifications:
Keywords and phrases.
Max-infinite divisibility, geometric-max-infinite divisibility extremal processes, max-autoregressive process.
1. Introduction.
Parallel to the classical notions of infinitely divisible (ID) laws and its subclass geometrically ID (GID) laws we have max-infinitely divisible (MID) and geometric-MID (G-MID) laws in the maximum setup which are discussed in Balkema and Resnick (1977), Rachev and Resnick (1991), Mohan (1998) and Satheesh (2002). For d.f s )( )( x exF ψ− = that are MID (which is always true in R ) distributions with d.f s of the form ))(1( 1 x ψ+ are referred to as G-MID laws. Processes related to these distributions are extremal processes, Rachev and Resnick (1991), Pancheva, et al . (200) and max-AR(1) processes, Satheesh and Sandhya (2006). Satheesh (2002) introduced φ -MID laws with d.f ϕ { − log F ( x )} for a Laplace transform (LT) φ and a d.f F . φ -MID laws can also be seen as the d.f obtained by randomizing the parameter c >0 in the first Lehman alternative F c obtained from F , by a distribution with LT φ . Setting − log F ( x ) = ψ ( x ) and taking φ to be the LT of a gamma( β ) law we get the d.f β ψ ))(1( 1 x + which we will refer to as the d.f of a gamma-MID law. When F is max-semi-stable and φ is exponential we get exponential max-semi-stable laws characterized in a max-AR(1) set up in Satheesh and Sandhya (2006). A gamma-max-semi-stable law was also discussed therein to illustrate the derivation of max-semi-selfdecomposable laws. 2 Pillai (1990) had introduced geometric exponential (G-exponential) laws having Laplace transform (LT) )1log(1 1 λ++ . Similarly, the LT β λ )1log(1 1 ++ , β >0 will be called as that of geometric gamma( β ) (G-gamma( β )) law. Motivated by this construction, and writing the d.f of gamma-MID( β ) laws as ))(1log( x e ψβ +− , β >0, we get the d.f of geometric-gamma-MID (G-gamma-MID( β )) laws as )}(1log(1{ 1 x ψβ ++ . Geometric generalized gamma laws were introduced and studied in Sandhya and Satheesh (2007). The purpose of this note is to discuss certain properties of gamma-MID and G-gamma-MID models. Potential applications of these models are in finance, insurance and stock market; see Kaufman (2001), Rachev (1993) and Mittnick and Rachev (1993). Distributional properties of these models including a new kind of invariance (stability) under geometric( p )-maximum, are presented in section.2. In section.3 we discuss extremal processes and a max-AR(1) model related to them. The support of the distributions discussed here is R , and by a geometric( p ) law we mean a geometric law on {1,2,…} with mean p a = .
2. Divisibility properties of gamma-MID and G-gamma-MID laws.
With the above terminologies we have:
Theorem.2.1
G-gamma-MID laws are G-MID.
Proof . We know that a d.f F ( x ) is G-MID iff }1{ )(1 −− xF e is MID, Rachev and Resnick (1991). From our construction we have the d.f ))(1log( x e ψβ +− , β >0 that is always MID. Setting )}(1log{1)(1 xxF ψβ +=− we have ))(1log(1 1 x ψβ ++ = )( xF is G-MID. Theorem.2.2 A d.f β ψ ))(1log(1 1)( xxF ++= , β >0 is G-gamma-MID iff >+ βψ β x is a d.f . This is clear from their construction above. Remark.2.1
It is interesting to note that )())(log1( 1 xGxF =− is a d.f for a given d.f F ( x ) and in turn ))(log1( 1 xG − is also a d.f and this operation can be repeated to obtain new d.f s. Theorem.2.3
Every G-gamma-MID( β ) distribution is the limit distribution of geometric )( n -max of i.i.d gamma-MID )( n β variables as ∞→ n . 3 Proof . Let )( xF n denote the d.f of a geometric )( n -max of i.i.d gamma-MID )( n β variables. Thus, nn xxxF nnnn ββ ψψ −− − +− += ))(1(1 ))(1()( = )1())(1( 1 −−+ nxn n β ψ = }1))(1{(1 1 −++ n xn β ψ . Hence, = ∞→ )(lim xF nn ))(1log(1 1))(1log(1 1 xx ψβψ β ++=++ , proving the assertion. Theorem.2.4
The limit of n -max of G-gamma-MID )( n β laws is gamma-MID( β ) as ∞→ n . Proof . Since βψββ ψψ ))(1( 1))(1log(1 1lim ))(1log( xex xnnn +== ++ +−∞→ , the claim is proved. Theorem.2.5
A distribution is invariant under geometric( p )-max up to a scale-change iff it is geometric max-semi-stable with d.f )bx(a)x( ψψ +=+ a >1 and ),1()1,0( ∞∪∈ b . Proof . See theorem.3.2 in Satheesh and Sandhya (2006) and its proof which is formulated in the max-AR(1) set up and under the terminology of exponential-max-semi-stable law. If b >1 the geometric-max-semi-stable law is Frechet type and if b <1 it is Weibull type, see Satheesh and Sandhya (2006). More generally we also have: Theorem.2.6
For a d.f of the form )x()x(H ψ+= xa ψ+ is a d.f of a geometric( p )-max for any a >0. Proof . )(1 1 xa ψ+ = )(11 − + xaa ψ = − −+ )x(Haa = − −+ a )x(aHa)x(H)x(Ha = − −− )x(Ha)x(Ha
4 = ∑ ∞= − − k kk )]x(H[aa = ( ) xXXXMaxP pN ≤ ),....,( )(21 , a p = , where N ( p ) is a geometric( p ) r.v and X i ’s are i.i.d with d.f H ( x ) proving the assertion. This is the max-analogue of lemma.3.1 in Pillai (1990). In particular we have: Theorem.2.7
Geometric( p )-max of i.i.d G-gamma-MID )( β variables is G-gamma-MID )( p β for any p ∈ (0,1) where the geometric( p ) r.v is independent of the components. Proof . The d.f of geometric( p )-max of i.i.d G-gamma-MID )( β variables is given by; ))}(1log(1/{)1(1 ))(1log(1/{ xp xp ψβ ψβ ++−− ++ ))(1log(1 1))(1log( xxp p p ψψβ β ++=++= , which is the d.f of a G-gamma-MID )( p β law. Remark.2.2
The invariance property under geometric( p )-max described above is new and is different from the invariance property under geometric( p )-max up-to a scale-change in theorem.2.5 characterizing geometric-max-semi-stable laws. In theorem.2.7 it is invariance up-to a change of shape parameter.
3. Processes related to gamma-MID and G-gamma-MID laws. 3.1 Random Time Changed Extremal Processes.
Extremal processes (EP) are processes with increasing right continuous sample paths and independent max-increments. The univariate marginals of an EP determine its finite dimensional distributions. Also, the max-increments of an EP are MID. The EPs will be referred to by the distribution of their max-increments. Pancheva, et al. (2006) has discussed random time changed or compound EPs and their theorem.3.1 together with property.3.2 reads: Let { Y ( t ), t ≥
0} be an EP having homogeneous max-increments with d.f F t ( y ) = exp { − t µ ([ λ , y ] c )}, y ≥ λ , λ being the bottom of the rectangle { F >0} and µ the exponential measure of Y (1), that is, µ ([ λ , y ] c ) = − log F ( y ). Let { T ( t ), t ≥
0} be a non-negative process independent of Y ( t ) having stationary, independent and additive increments with Laplace transform (LT) ϕ t . If { X ( t ), t ≥
0} is the compound EP obtained by randomizing the time parameter of Y ( t ) by T ( t ), then X ( t ) = Y ( T ( t )) and its d.f is: 5 P { X ( t )< x } = { ϕ ( µ ([ λ , x ] c ))} t . Pancheva, et al. (2006) also showed that in the above setup Y ( T ( t )) is also an EP. Extending the discussion in Pancheva, et al. (2006) Satheesh and Sandhya (2006) showed that the EP obtained from a random time changed (by compounding a) max-semi-stable EP is max-semi-selfdecomposable( b ) if the compounding process is semi-selfdecomposable. Here we have some more results in this direction. Theorem.3.1
The EP obtained by compounding a gamma-MID )( β EP having homogeneous max-increments is G-gamma-MID )( β if the compounding process is unit exponential. Proof . If { Y ( t )} is a gamma-MID( β ) EP and { T ( t )} is unit exponential with d.f G then: ))x(log( ψβ ++
11 1 = )t(dGe ))x(log(t ∫ ∞ +− ψβ , which proves the assertion. Theorem.3.2
The EP { Y ( T ( t ))} obtained from a random time changed EP { Y ( t )} having homogeneous max-increments with d.f )( xH e − is G-gamma-MID )( β if the compounding process { T ( t )} is G-gamma )( β . Proof . We know that the LT of a G-gamma )( β law is β λ )1log(1 1 ++ . If G denote its d.f then; ))x(log( ψβ ++
11 1 = ∫ ∞ − )t(dGe )x(t ψ , proving the assertion. Now consider a first order max-autoregressive (max-AR(1)) model described as below. Here a sequence of r.v s { X n , n >0 integer} defines the max-AR(1) scheme if for some 0< p <1 there exists an innovation sequence { ε n } of i.i.d r.v s such that; X n = X n -1 , with probability p = X n -1 ∨ ε n , with probability (1- p ). (1) In terms of d.f s and assuming stationarity this is equivalent to; F ( x ) = F ( x ){ p + (1- p ) F ε ( x )}. That is; )()1(1 )()( xFp xpFxF εε −−=
6 Hence { X n } is a geometric( p )-max of the innovation sequence { ε n }. Invoking theorem.2.7 we have proved; Theorem.3.3
In the max-AR(1) structure (1) the sequence { X n } and the innovation sequence { ε n } are related as follows for any p ∈ (0,1). { X n } is G-gamma-MID )( β variables iff { ε n } is G-gamma-MID )( p β . Equivalently, Theorem.3.4
A necessary and sufficient condition for an max-AR(1) process { X n } with the structure in (1) is stationary Markovian for any p ∈ (0,1) with G-gamma-MID )( β distribution is that the innovation’s are G-gamma-MID )( p β distributed. References.
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