Wrapped Geometric Stable Distributions

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1 Wrapped Geometric Stable Distributions Sophy Jacob Study on circular distributions Thesis. Department of Statistics, University of Calicut, 0

2 Chapter 3 Wrapped Geometric Stable Distributions 3. Introduction Geometric stable family of distributions arise as the limiting class in the random summation scheme, when the number of terms is geometrically distributed. It is a four parameter family denoted by GS α (σ, β, µ) and is usually expressed in terms of its characteristic function. If the linear random variable X GS α (σ, β, µ), then its characteristic function is given by Φ(t) = [ + σ α t α ω α,β (t) iµt] (3..) iβ sign(x) tan( πα ), ifα where ω α,β (x) = (3..) + iβ( ) sign(x) ln x, if α = π The parameter α (0, ] is called the index which determines the tail of the distribution P (Y > y) Cy α ( as y ) for 0 < α <. The parameter β [, ] is the 77

3 skewness parameter, µ R and σ 0 controlls the location and scale respectively. Geometric stable laws include Linnik distribution where β = 0 and µ = 0 (see, Linnik (963)), and Mittag- Leffler distributions, which are geometric stable with β = and either α = and σ = 0 (exponential distribution) or 0 < α < and µ = 0. The Mittag- Leffler distributions are the only non negative geometric stable distributions (see, Pillai (990), Jayakumar and Pillai (993)). The practical importance of wrapped geometric stable distributions lies in the fact that the Fourier coefficients of a wrapped model correspond to the characteristic function at integer values of the unwrapped model. So we can deduce many of the important properties of linear geometric stable distributions, for example, infinite divisibility, geometric infinite divisibility etc.the currently used von Mises distribution and the wrapped normal and Laplace distributions do not provide sufficient degree of flexibility within the class of symmetric distributions. So in section, we introduce and study wrapped geometric stable distribution. Special cases of wrapped geometric stable distributions are studied in Section 3. Trigonometric moments are determined in Section 4. Section 5 deals with estimation of parameters. 3. Definition and Fundamental Properties of Wrapped Geometric Stable Distributions Consider a linear random variable X GS α (σ, β, µ) and define the circular random variable Θ = X(mod( π)) [0, π) (3..) Since the Fourier coefficients for a wrapped circular random variable corresponds to the characteristic function at integer values for the unwrapped random variable (see, 78

4 Mardia (97)), Φ Θ (p) : p = 0, ±, ±,... of the characteristic function of Θ are given by Φ Θ (p) = Φ X (p). (3..) Therefore, from (3..), the characteristic function corresponding to the wrapped geometric stable angular random variable is Φ Θ (p) = [ + σ α p α ω α,β (p) iµ p] (3..3) iβ sign(x) tan( πα ), ifα where ω α,β (x) = (3..4) + iβ( ) sign(x) ln x, if α = π and µ = µ mod( π) [0, π) (3..5) Thus for p =,,..., we have [ + σ α p α ( iβ tan πα Φ(p) = ) iµ p ] [ + σ α p α ( + iβ ln(p)) π iµ p ] if α if α =. We shall use the notation Θ W GS α (σ, β, µ ) to denote that Θ is distributed according to the wrapped geometric stable distribution under this parametrization. Definition 3... An angular random variable Θ is said to follow wrapped geometric stable distribution with parameters α, σ, β, µ if its characteristic function is Φ(p) = [ + σ α p α ω α,β (p) iµ p] (3..6) iβsign(x)tan( πα where ω α,β (x) = ) ifα (3..7) + iβ( )sign(x)ln x ifα = π The probability density function of the wrapped geometric stable angular random 79

5 variable Θ [0, π), when α is given by f w (θ) = π = π p= p= Φ(p)e ipθ [ + σ α p α ( iβ sign(p) tan( πα ] )) iµ p e ipθ (3..8) On simplification, we get [ f w (θ) = + π ( ( + σ α p α ) cos(pθ) + (σ α p α β tan( πα) + ) pµ ) sin(pθ) ] p= ( + σ α p α ) + (σ α p α β tan( πα ) + pµ ) [ = + π ] (α p cos(pθ) + β p sin(pθ)) p= (3..9) where α p = and β p = ( + σ α p α ) ( + σ α p α ) + (σ α p α β tan( πα) + pµ ) (σ α p α β tan( πα) + pµ ) ( + σ α p α ) + (σ α p α β tan( πα) +, p =,,... pµ ) (see, Sophy and Jayakumar(0a)). Following are the graphs of wrapped geometric stable distributions for various values of the parameters 80

6 Figure 3.. α = 0.8, β = 0.5, σ = Figure 3.. α =, β = 0.5, σ = Figure 3.3. α = 0.8, β = 0.5, σ = Figure 3.4. α = 0.8, β = 0, σ = Proposition 3... If Θ,..., Θ n are n independent and identically distributed wrapped geometric stable angular random variables with µ = β = 0 then n α ( n j= Θ j) mod( π) follows wrapped symmetric α- stable distribution as n and σ p < n α. Proof. Since Θ i W GS α (σ, 0, 0), Φ Θi (p) = + σ α p α Let S n = (Θ Θ n ) mod( π) Then Φ Sn n α (p) = Φ Sn ( p = = n j= n α ) [ + σ α p n α ] [ + σα p α n. n α] 8

7 As n, Φ Sn n α (p) e σα p α, which is the characteristic function of wrapped symmetric α stable distributions. When α =, the above expression reduces to e σ p which is the characteristic function of wrapped normal distribution with mean direction zero and variance σ. 3.. Wrapped Strictly Geometric Stable Distributions All strictly geometric stable(sgs) distributions have characteristic function Φ(t) = [ + λ t α e iπ α τ sign(t)] (3..0) where 0 < α, λ 0, τ min(, α ) (see, Kozubowski and Rachev(999)). Therefore, the corresponding wrapped SGS distributions have characteristic function Φ(p) = [ + λ p α e iπ α τ sign(p)], for p = 0, ±, ±,... (3..) The probability distribution of wrapped SGS angular random variable Θ is given by f w (θ) = π = π On simplification, it reduces to, [ f w (θ) = + π p= Φ(p)e ipθ [ + λ p α e iπ α τ sign(p) ] e ipθ. (3..) p= ( ( + λp α cos( πατ p= )) cos(pθ) + (λpα sin( πατ ( + λp α cos( πατ )) + (λp α sin( πατ )) ) )) sin(pθ) ] 8

8 [ = + π ] (α p cos(pθ) + β p sin(pθ)), (3..3) p= where α p = and β p = ( + λp α cos( πατ )) + λ p α + λ p α cos( πατ ) λp α sin( πατ ) + λ p α + λ p α cos( πατ, p =,,... ) We shall denote the angular random variable Θ obeying wrapped SGS distribution by W SGS α(λ, τ) 3.3 Special Cases Densities and distribution functions of wrapped geometric stable angular random variables are usually not in closed forms. But consider the case when σ = 0 in (3..3).Then we have, Φ p = iµ p, (3.3.) which is the characteristic function of wrapped exponential distribution with mean direction µ. When σ = µ = 0 in (3..3) we will get the characteristic function of a circular point distribution concentrated at Θ = 0. Consider the case when β = µ = 0 in (3..6), then Φ p = + σ α p α, (3.3.) reduces to the characteristic function of wrapped Linnik distribution. 83

9 When β = µ = 0, σ = and α = in (3..3), we have Φ p = + p, (3.3.3) which is the characteristic function of wrapped Laplace distribution. When α =, we have ω α,β (p) = (in that case β is irrelevant). Therefore, Φ p = + σ p iµ p, (3.3.4) which is the characteristic function of wrapped asymmetric Laplace distribution. Proposition If the angular random variable Θ follows strictly geometric stable distribution,w SGS α(, τ) then λ α (Θ) W SGS α(λ, τ) Proof. The W SGS α(, τ) have characteristic function Φ p = [ + λ p α e iπ α τ sign(p) ] Putting η = iπ α τ, we have Φ p = [ + λ p α e iη sign(p) ] Therefore, Φ λ α Θ (p) = E[e ip(λ α Θ) ] = [ + λ p α e iη sign(p) ] which is the characteristic function of wrapped strictly geometric stable distribution, W SGS α(λ, τ). 84

10 3.4 Trigonometric Moments The sequence of trigonometric moments of Θ is the characteristic function of Θ. Hence the p th trigonometric moment of Θ is defined by Φ(p) = E[e ipθ ] = E[cos(pΘ)] + ie[sin(pθ)] = a p + ib p where a p = E[cos(pΘ)] and b p = E[sin(pΘ)]. Thus for α, we have a p = and b p = ( + σ α p α ) ( + σ α p α ) + (σ α p α β tan( πα) + pµ ) (σ α p α β tan( πα) + pµ ) ( + σ α p α ) + (σ α p α β tan( πα) +, pµ ) for p =,,... For α =, we have a p = and b p = + σp ( + σp) + (pµ pσβ π ln(p)) pµ pσβ ln(p) π ( + σp) + (pµ pσβ, for p =,,... ln(p)) π 85

11 The p th mean resultant length, ρ p = = a p + b p { }, for α ( + σ α p α ) + (σ α p α β tan( πα) + pµ ) { }, for α = ( + σp) + (pµ pσβ π ln(p)) (3.4.) The p th mean direction, µ p [0, π) is given by ( ) µ p = tan bp = a p { σ α p α β tan( πα tan ) + } pµ (mod (π)), if α { + σ α p α pµ pσβ tan ln(p) } π (mod (π)), if α = + σp (3.4.) The mean direction, µ = µ [0, π) is { µ + σ α β tan( πα tan ) } (mod (π)), if α µ = { } + σ α µ tan (mod (π)), if α = + σ (3.4.3) The p th cosine and sine moments about the mean direction µ, a p = E[cos p(θ µ)] (3.4.4) and b p = E[sin p(θ µ)], (3.4.5) 86

12 can be represented, for p =,,... as [ ( { µ a p = ρ p cos p θ tan + σ α β tan ( )})] πα + σ α [ { })] µ b p = ρ p sin p (θ tan if α = + σ if α The circular variance of Θ,denoted by V 0, is given by V 0 = ρ = E[cos(θ µ)] { = ( + σ α ) + (σ α β tan( πα ) + µ ) } (3.4.6) The circular standard deviation, σ 0, is given by σ 0 = ln( V 0 ) ln[( + σα ) + (σ α β tan( πα = ) + µ ) ] if α ln[( + σ) + (µ ) ] if α = (3.4.7) The circular measures of skewness and kurtosis are respectively given by γ 0 = β V 3 0 (3.4.8) and γ 0 = α ρ 4 ( ρ) (3.4.9) 3.5 Estimation Since the densities and distribution functions of wrapped geometric stable distributions are generally not in closed forms, the problem of estimation of their parameters 87

13 is bit difficult. Here we consider the method of moments 3.5. Method of Moments-Type Estimators Let Θ,..., Θ n be independent and identically distributed angular random variables with characteristic function Φ given by (3..6)and (3..7). Consider the case where α. Let us put λ = σ α. Then the empirical characteristic function is defined by ˆΦ(p) = n j= e ipθ j n. (3.5.) (see, Kozubowski and Rachev (999)). Let Re(z) and Im(z) denote the real and imaginary parts of a complex number z, respectively. Since, Φ p = + λ p α iµ p, we have v(p i ) = λ p α i =, (3.5.) where v(p) = Re( Φ p ) and p p, are both greater than 0. Solving equations (3.5.) for α and λ, we have v(p ) = λp α (3.5.3) v(p ) = λp α (3.5.4) Dividing (3.5.3) by (3.5.4) and solving for α,we get α = log(v(p )) log(v(p )) log(p ) log(p ) (3.5.5) log(v(p )) = logλ + αlog p (3.5.6) Again, log(v(p )) = logλ + αlog p (3.5.7) 88

14 Dividing (3.5.6) by (3.5.7) and solving for λ, we get log(λ) = α[log p log(v(p )) log p log(v(p ))] log(v(p )) log(v(p )) (3.5.8) Substituting (3.5.5) in (3.5.8), we have λ = e [log p log[v(p )] log p log(v(p ))] log[v(p )] log[v(p )] (3.5.9) Substituting the sample characteristic function ˆΦ n (p) for Φ(p) into (3.5.5) and (3.5.9) we get ˆα = log[ˆv n(p )] log[ˆv n (p )] log(p ) log(p ) log p log[ˆv n (p )] log p log[ˆv n (p )] and ˆλ = e log(p ) log(p ) ( ) where ˆv n = Re ˆΦ n (p) (3.5.0) (3.5.) (3.5.) To estimate β and µ, we consider the imaginary part of Φ p. Let u(p) = Im[ Φ(p) ] for any p 3 p 4 both greater than 0. Then we get the system of equations, u(p i ) = µ p i + λ p i α β tan( πα )sign(p i) p =, (3.5.3) Solving u(p 3 ) and u(p 4 ) for β and µ, we get µ = u(p 4) p 3 α u(p 3 ) p 4 α p 4 p 3 α p 3 p 4 α (3.5.4) and β = u(p 3)p 4 u(p 4 )p 4 p 4 p 3 α p 3 p 4 α λ cot(πα ) (3.5.5) 89

15 Substituting û n (p) = Im[ ˆΦ n(p) ] and ˆα and ˆλ leads to estimators of µ and β. In a similar way we find estimators in the case of α =. 90