A Generalization of the Marshal-Olkin Scheme and Related Processes
aa r X i v : . [ m a t h . S T ] D ec A Generalization of the Marshal-OlkinScheme and Related Processes
Satheesh S , Sandhya E and Prasanth C B Department of Applied Sciences, Vidya Academy ofScience and Technology, Thalakkottukara, Thrissur-680 501, India. Department of Statistics, Prajyoti Niketan CollegePudukkad, Thrissur-680 301, India. Department of Statistics, St. Thomas College, Pala-686 574, India.e-mail ids: [email protected]; [email protected] [email protected] Abstract
A generalization of the Marshal-Olkin parametrization scheme is devel-oped and stochastic models related to it are discussed here.
Key words and Phrases : geometric, Harris, extremal processes, autoregressive model, stable, infinitely divisible. AMS(2010) Subject Classification Numbers:
Given a distribution, the Marshal-Olkin parametrization scheme (M-Oscheme) gives a generalization of it in terms of its survival function ( s.f )by adding a parameter to it, thus providing more flexibility, Marshal andOlkin (1997). For a given s.f F , the M-O scheme is described by the s.f G ( x, α ) = αF ( x )1 − (1 − α ) F ( x ) , x ∈ R, α > . (1.1) atheesh and Nair (2004) observed that the M-O scheme has es-sentially a geometric-minimum structure. Their investigation also sug-gested a generalization of the M-O scheme as follows. For a given s.f F , this is described, for k > a >
0, by the s.f H ( x, a ) = ( F k ( x ) a − ( a − F k ( x ) ) /k , x ∈ R. (1.2)This scheme thus adds two parameters to make it more flexible.When k = 1, (1.2) reduces to the M-O scheme. This generalization hasa Harris-minimum structure since replacing F ( x ) by s ∈ (0 ,
1) and re-stricting a >
1, the RHS of (1.2) is the probability generating functionof a Harris( a, k ) distribution studied in detail by Sandhya et al . (2008).With reference to the M-O scheme it is to be noted that if F isused instead of F in (1.1) we will get G ( x, /α ) = 1 − G ( x, α ). In otherwords, the distribution function ( d.f ) corresponding to the s.f definedin (1.1) is similar to the s.f but for a change in the parameter α asgiven above. But, this is not true for the generalised schemes (1.2) and(2.1) discussed in this paper. Hence there is relevance in studying theseschemes seperately.The purpose of this note is to generalize the M-O scheme on thelines of (1.2) in terms of d.f and discuss stochastic models related to it.Possible applications of this model are in reliability studies of parallelsystems. We first show that (1.2) is indeed a s.f . Notice that for any s.f F ( x ), F ν ( x ) , ν > s.f . Thus H ( x, α ) = αF k ( x ) / { − (1 − α ) F k ( x ) } is a s.f by the M-O scheme, with ν = k integer. Since H /k ( x, α ) is alsoa s.f , the generalization at (1.2) is a s.f where a = α . Further, thisscheme is closed under Harris( b, k )-minimum ( b > b, k )-maximum, Satheesh and Nair (2004). ince for any d.f F ( x ), F ν ( x ) , ν > d.f we have thefollowing parametrization scheme for d.f s, for k > a > H ( x, α ) = (cid:26) F k ( x ) a − ( a − F k ( x ) (cid:27) /k , x ∈ R. (2.1)Notice that (2.1) is closed under Harris( b, k )-maximum ( b > b, k )-minimum, Satheesh and Nair (2004). Theorem 2.1 If H ( x ) = ψ ( x ) is a d.f , then n aψ ( x ) o /k , k > a > d.f . Proof . Since n aψ ( x ) o /k = { ψ ( x ) } − /k n aψ ( x )1+ ψ ( x ) o /k = { ψ ( x ) } − /k n aψ ( x )+ a − a ψ ( x ) o /k , we have (cid:26)
11 + aψ ( x ) (cid:27) /k = (cid:26) H ( x ) a − ( a − H ( x ) (cid:27) /k . (2.2)Since for any d.f H ( x ), H /k ( x ) is a d.f , the RHS in (2.2) is theparametrization (2.1) for H /k ( x ), proving the assertion. Remark 2.1
This theorem suggests a convenient way to construct d.f sthat are invariant under Harris( b, k )-maximum since for any d.f H ( x )we can take ψ ( x ) = − logF ( x ) in { x : F ( x ) > } . See also remark 3.1. Corollary 2.1
Since for any d.f H ( x ), H k ( x ) is a d.f , aψ ( x ) is also a d.f . Corollary 2.2
When a >
1, we get the Harris( a, k )-maximum of H /k ( x ). Remark 2.2
In particular, if ψ ( x ) = aψ ( cx ), for some 0 < c < < a , then theorem 2.1 shows that the d.f H /k ( x ) is invariant un-der Harris( a, k )-maximum upto a scale change c . We may call such d.f sHarris-max-semi-stable( a, c, k ).In terms of CFs, theorem 2.1 generalizes lemma 3.1 in Pillai (1990).In general, for a characteristic function (CF) f , f ν , ν > b, k )-sum. If we re considering semi-stable CFs, then they are ID and an analogue oftheorem 2.1 implies that the CF f ( t ) = n ω ( t ) o /k , ω ( t ) = aω ( ct ), forsome 0 < c < < a , is invariant under Harris( a, k )-sum. Here we discuss two stochastic processes where the parameterization(2.1) or d.f s of the form n aψ ( x ) o /k , k > a >
0, appearnaturally.Pancheva et al. (2006) discussed random time changed or compoundextremal process (EP) and their theorem 3.2 together with Property 3.2reads: Let { Y ( t ) , t ≥ } be an EP with homogeneous max-incrementsand d.f F t ( y ) = exp {− tµ [( λ, y ) c ] } , y ≥ λ, λ > { F > } and µ the exponential measure of Y (1), thatis, µ [( λ, y ) c ] = − logF ( y ). Let { T ( t ) , t ≥ } be a non-negative processindependent of { Y ( t ) } having stationary, independent and additive in-crements with Laplace transform ϕ t . If { X ( t ) , t ≥ } is the compoundEP obtained by randomizing the time parameter of { Y ( t ) } by { T ( t ) } then X ( t ) = Y ( T ( t )). Its d.f is P { X ( t ) ≤ x } = { ϕ [ µ ( λ, x ) c ] } t which is ϕ -max-ID, Satheesh et al . (2008). Pancheva et al . (2006) also showedthat in this set up { Y ( T ( t )) } is also an EP. We now have Theorem 3.1
The EP obtained by compounding a homogeneous EP isgamma-max-ID if the compounding process is gamma( α, β ). Proof . If { Y ( t ) } is an EP with homogeneous max-increments and d.f e − ξ ( x ) , { T ( t ) } a gamma( α, β ) process with stationary, independent andadditive increments and d.f G , then the d.f of the process { Y ( T ( t )) } isgiven by R ∞ e − tξ ( x ) dG ( t ) = n αξ ( x ) o β , proving the assertion. Remark 3.1
Clearly this is a gamma mixture. When β = k , this d.f is the same as the one in theorem 2.1 and is closed under Harris( b, k )-maximum ( b > a > d.f is itself a Harris( a, k )-maximum. nother stochastic model is the max-AR(1) process described by i.i.d r.v s { Y i,n } and innovations ( i.i.d r.v s) { ǫ i,n , i = 1 , , ..., k } for afixed positive integer k . Such generalized models were considered bySatheesh et al. (2008). k _ i =1 Y i,n = (W ki =1 ǫ i,n , with probability p, nW ki =1 Y i,n − o W nW ki =1 ǫ i,n o , with probability (1 − p ).(3.1)In terms of d.f s and assuming stationarity this reads F k ( x ) = pF kǫ ( x ) + (1 − p ) F k ( x ) F kǫ ( x ), or F k ( x ) = pF kǫ ( x )1 − (1 − p ) F kǫ ( x ) .That is, F ( x ) = n F kǫ ( x ) a − ( a − F kǫ ( x ) o /k , a = p . Hence we have, Theorem 3.2 (Satheesh et al. (2008)) A d.f F ( x ) can model the gen-eralised stationary max-AR(1) scheme (3.1) for some p ∈ (0 ,
1) if it isHarris( a, k )-maximum, a = p , and the distribution of the innovations isthat of the components and conversely. Theorem 3.3 (Satheesh et al. (2008)) A d.f F ( x ) can model the gen-eralised stationary max-AR(1) scheme (3.1) for every p ∈ (0 ,
1) (or as p ↓
0) if it is Harris( a, k )-max-ID and conversely.
Remark 3.2
We saw that for k > a > H ( x, α ) = n F k ( x ) a − ( a − F k ( x ) o /k , x ∈ R , is closed under Harris( b, k )-maximum. Hencewe can use the above to model the innovations in (3.1). The closureproperty implies also that if we repeat this operation with the same pa-rameter b > b, k )-maximum and a passageto the limit shows that the limit is Harris-max-ID.Now consider a variation of this max-AR(1) scheme described by i.i.d r.v s { Y i,n } and innovations { ǫ i,n , i = 1 , , ..., k } for a fixed positiveinteger k and some c > k _ i =1 Y i,n = (W ki =1 1 c Y i,n − , with probability p, nW ki =1 1 c Y i,n − o W nW ki =1 ǫ i,n o , with probability (1 − p ).(3.2) n terms of d.f s and assuming ǫ i, = Y i, and stationarity of { Y i,n } ,this reads F k ( x ) = pF k ( cx ) + (1 − p ) F k ( cx ) F k ( x ), or F k ( x ) = pF k ( cx )1 − (1 − p ) F k ( cx ) . That is, F ( x ) = n F k ( cx ) a − ( a − F k ( cx ) o /k , a = p . Hence by induction we have Theorem 3.4
For the stationary max-AR(1) model (3.2) to hold as-suming ǫ i, = Y i, , for some p ∈ (0 , d.f F ( x ) (of Y i,n ) mustbe Harris( a, c, k )-max-semi-stable and if we demand (3.2) is to be truefor every p ∈ (0 , F ( x ) is Harris( a, k )-max-stable and conversely. Remark 3.3
By remark 2.2 we can construct Harris( a, c, k )-max-semi-stable d.f s. Acknowledgement
Authors thank the referee for asking to clarify a point that resulted ina better discussion in the paper.
References
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