Inverse Gaussian Distribution

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1 5 Kauhisa Matsuda All rights reserved. Iverse Gaussia Distributio Abstract Kauhisa Matsuda Departmet of Ecoomics The Graduate Ceter The City Uiversity of New York 65 Fifth Aveue New York NY March 5 This paper presets the basic kowledge of the iverse Gaussia distributio. 5 Kauhisa Matsuda All rights reserved.

2 5 Kauhisa Matsuda All rights reserved. [] Iverse Gaussia Distributio: JKB (994) Parameteriatio There are may differet parameteriatios of the iverse Gaussia distributio which ca be really cofusig to begiers. I this sectio basic properties of the iverse Gaussia distributio is preseted followig Johso Kot ad Balakrisha (994) s parameteriatio of the equatio (5.4a). The probability desity fuctio of the iverse Gaussia distributio is a two parameter family: λ / λ IG(; µλ ) = exp ( µ ) x> () π µ where µ R ad λ R. By Fourier trasformig the IG probability desity () its characteristic fuctio φ ( ω ) is calculated as: iω φ ( ω ) F [ IG ( ; µ λ)]( ω) e IGd ( ) iω λ λ φ ( ω) = e IG( ) d = exp i µ λ µ ω. () Simplifyig the equatio () yields: ( ) exp λ λ i exp λ φ ω λ ω µ λ = = λiω µ µ µ λ µ λ µ λ µ φ ( ω) = exp λiω µ λ µ λ λ µ iω φ ( ω) = exp µ λ. () The characteristic expoet (i.e. cumulat geeratig fuctio) ψ ( ω ) of the IG distributio is: λ ψ ( ω) l φ( ω) = µ λ µ iω. (4)

3 5 Kauhisa Matsuda All rights reserved. ψ ( ω) Usig (4) the first four cumulats defied by cumulat( ) ω= i ω calculated as the follows: are cumulat = µ cumulat = µ λ / cumulat µ / λ 7 cumulat 4 = 5 µ / λ. 5 = Usig the above cumulats the mea variace skewess ad excess kurtosis of the IG radom variable are obtaied as (cosult Table 4. of Matsuda (4)): E [ ] = µ (5) Variace[ ] = µ / λ Skewess[ ] = µ / λ Excess Kurtosis[ ] = 5 µ / λ. The momet geeratig fuctio M ( ω ) of the IG distributio ca be expressed as: ω ( ) ω M ω e IG ( ; µ λ) d= e IGd ( ) ( ) M ( ω) = exp ξ ( ω) where the Laplace expoet ξ ( ω ) is give by: ξ ( ω) = µ λ λ µω. (6) Usig the momet geeratig fuctio M ( ω ) with (6) first four raw momets (i.e. r M ( ω) ω ω= ) of the IG distributio are computed as: r E[ ] = µ µ r E[ ] = µ λ 4 5 r µ µ E[ ] = µ λ λ µ 5µ 5µ r4 E[ ] = µ. λ λ λ

4 5 Kauhisa Matsuda All rights reserved. Note tha t the form of cetered momets of (5) tells us that the IG probability desity is always positively skewed ad the excess kurtosis is always positive. Figure illustrates the shape of the IG distributio with varyig parameters. I Pael A as λ icreases its variace ske wess ad excess kurtosis decreases. I Pael B as µ rises holdig λ costat all momets rise. Probability Desit y.5.5 λ=. λ=.5 λ= λ= λ=4 λ=8 λ=6 A) µ = ad varyig λ.5.5 Probability Desit y.5.5 µ=.5 µ= µ= µ=4 µ=8 µ=6 λ= B) Varyig µ ad λ = 4 5 Figure Plot of IG probability desity 4

5 5 Kauhisa Matsuda All rights reserved. Probably the most importat property of the IG distributio is its ifiite divisibility. Let be a IG radom variable with µ R ad λ R. The there exist pieces of iid... radom variable... each from the IG distributio with µ / R ad λ / R such that: d... which is the defiitio. of Matsuda (5). This idicates that the IG distributio geerates a class of icreasig Lévy processes (subordiators). [] Iverse Gaussia Distributio: Bardorff-Nielse (998) Parameteriatio I this sectio basic properties of the iverse Gaussia distributio is preseted followig Bardorff-Nielse (998) s parameteriatio. Reparameterie the IG probability desity () usig µ = δ / γ ad λ = δ : IG(; µλ ) = λ / λ exp ( µ ) x> π µ IG(; δγ ) = δ / δ δ exp π ( δ / γ) γ δ / γ δ δ IG(; δγ ) = exp π γ γ δ γ δ π / (; δγ ) = exp δγ x> IG IG δ = π ( δ γ ) / (; δγ ) exp δγ x> x> x> (7) where δ R ad γ R. By Fourier trasformig the IG probability desity (7) its characteristic fuctio φ ( ω ) is calculated as: iω φ ( ω) F [ IG ( ; δ γ)]( ω) e IGd ( ) ( ) iω φ ( ω) = e IG( ) d = exp δγ δ γ iω. (8) 5

6 5 Kauhisa Matsuda All rights reserved. The characteristic expoet (i.e. cumulat geeratig fuctio) ψ ( ω ) of the IG distributio is: ψ ω φ ω = δγ δ γ ω. (9) ( ) l ( ) i ψ ( ω) Usig (9) the first four cumulats defied by cumulat( ) ω= i ω calculated as the follows: are cumulat cumulat = δ / γ = δ / γ cumulat δ / γ 7 cumulat4 = 5 δ / γ. 5 = Usig the above cumulats the mea variace skewess ad excess kurtosis of the IG radom variable are obtaied as (cosult Table 4. of Matsuda (4)): E [ ] = δ / γ () Variace[ ] = δ / γ Skewess[ ] = δγ 5 Excess Kurtosis[ ] =. δγ The momet geeratig fuctio M ( ω ) of the IG distributio ca be expressed as: ω ( ) ω M ω e IG ( ; δ γ) d= e IGd ( ) ( ) M ( ω) = exp ξ ( ω) where the Laplace expoet ξ ( ω ) is give by: ξ ω = δγ δ γ ω. () ( ) Usig the momet geeratig fuctio M ( ω ) with () first four raw momets (i.e. r M ( ω) ω ω= ) of the IG distributio are computed as: r E[ ] = δ / γ 6

7 5 Kauhisa Matsuda All rights reserved. δ ( δγ ) r E[ ] = γ δ ( δγ δ γ ) E[ ] 5 r = γ 4 δ (5 5δγ 6 δ γ δ γ ) 4 E[ ] 7 r =. γ Note that the form of cetered momets of () tells us that the IG probability desity is always positively skewed ad the excess kurtosis is always positive. Figure illustrates the shape of the IG distributio with varyig parameters. I Pael A as γ icreases holdig δ costat all the stadardied momets decrease. I Pael B as δ rises holdig γ costat the mea ad variace rise while the skewess ad excess kurtosis fall. Probability Desit y γ=. γ=.5 γ= γ= γ=4 γ=6 γ=8 A) δ = ad varyig γ

8 5 Kauhisa Matsuda All rights reserved. Probability Desit y δ=. δ=.5 δ= δ= δ=4 δ=6 δ=8 B) Varyig δ ad γ = Figure Plot of IG probability desity Probably the most importat property of the IG distributio is its ifiite divisibility. Let be a IG radom variable with µ R ad λ R. The there exist pieces of iid... radom variable... each from the IG distributio with µ / R ad λ / R such that: d... which is the defiitio. of Matsuda (5). This idicates that the IG distributio geerates a class of icreasig Lévy processes (subordiators). 8

9 5 Kauhisa Matsuda All rights reserved. Refereces Bardorff-Nielse O. E. 998 Processes of Normal Iverse Gaussia Type Fiace ad Stochastics Johso N. L. Kot S. ad Balakrisha N. 994 Cotiuous Uivariate Distributios Volume Secod Editio Joh Wiley & Sos. Matsuda K. 4 Itroductio to Optio Pricig with Fourier Trasform: Optio Pricig with Expoetial Lévy Models Workig Paper Graduate School ad Uiversity Ceter of the City Uiversity of New York. Matsuda K. 5 Itroductio to the Mathematics of Lévy Processes Vol. of Ph.D. thesis expected to be filed o May 6 Graduate School ad Uiversity Ceter of the City Uiversity of New York. 9

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