Geometric stable distribution

Geometric stable
Parameters	α ∈ (0,2] — stability parameter β ∈ [−1,1] — skewness parameter (note that skewness is undefined) λ ∈ (0, ∞) — scale parameter μ ∈ (−∞, ∞) — location parameter
Support	x ∈ R, or x ∈ [μ, +∞) if α < 1 and β = 1, or x ∈ (−∞,μ] if α < 1 and β = −1
PDF	not analytically expressible, except for some parameter values
CDF	not analytically expressible, except for certain parameter values
Median	μ when β = 0
Mode	μ when β = 0
Variance	2λ² when α = 2, otherwise infinite
Skewness	0 when α = 2, otherwise undefined
Ex. kurtosis	3 when α = 2, otherwise undefined
MGF	undefined
CF	$\!{\Big [}1+\lambda ^{\alpha }\|t\|^{\alpha }\omega -i\mu t]^{-1}$ , where $\omega ={\begin{cases}1-i\tan {\tfrac {\pi \alpha }{2}}\beta \,\operatorname {sign} (t)&{\text{if }}\alpha \neq 1\\1+i{\tfrac {2}{\pi }}\beta \log \|t\|\,\operatorname {sign} (t)&{\text{if }}\alpha =1\end{cases}}$

A geometric stable distribution or geo-stable distribution is a type of leptokurtic probability distribution. Geometric stable distributions were introduced in Klebanov, L. B., Maniya, G. M., and Melamed, I. A. (1985). A problem of Zolotarev and analogs of infinitely divisible and stable distributions in a scheme for summing a random number of random variables.^[1] These distributions are analogues for stable distributions for the case when the number of summands is random, independent of the distribution of summand, and having geometric distribution. The geometric stable distribution may be symmetric or asymmetric. A symmetric geometric stable distribution is also referred to as a Linnik distribution.^[2] The Laplace distribution and asymmetric Laplace distribution are special cases of the geometric stable distribution. The Laplace distribution is also a special case of a Linnik distribution. The Mittag-Leffler distribution is also a special case of a geometric stable distribution.^[3]

The geometric stable distribution has applications in finance theory.^[4]^[5]^[6]^[7]

Characteristics

For most geometric stable distributions, the probability density function and cumulative distribution function have no closed form. But a geometric stable distribution can be defined by its characteristic function, which has the form:^[8]

\varphi (t;\alpha ,\beta ,\lambda ,\mu )=[1+\lambda ^{\alpha }|t|^{\alpha }\omega -i\mu t]^{-1}

where $\omega ={\begin{cases}1-i\tan {\tfrac {\pi \alpha }{2}}\beta \,\operatorname {sign} (t)&{\text{if }}\alpha \neq 1\\1+i{\tfrac {2}{\pi }}\beta \log |t|\operatorname {sign} (t)&{\text{if }}\alpha =1\end{cases}}$

$\alpha$ , which must be greater than 0 and less than or equal to 2, is the shape parameter or index of stability, which determines how heavy the tails are.^[8] Lower $\alpha$ corresponds to heavier tails.

$\beta$ , which must be greater than or equal to −1 and less than or equal to 1, is the skewness parameter.^[8] When $\beta$ is negative the distribution is skewed to the left and when $\beta$ is positive the distribution is skewed to the right. When $\beta$ is zero the distribution is symmetric, and the characteristic function reduces to:^[8]

\varphi (t;\alpha ,0,\lambda ,\mu )=[1+\lambda ^{\alpha }|t|^{\alpha }-i\mu t]^{-1}

The symmetric geometric stable distribution with $\mu =0$ is also referred to as a Linnik distribution.^[9]^[10] A completely skewed geometric stable distribution, that is with $\beta =1$ , $\alpha <1$ , with $0<\mu <1$ is also referred to as a Mittag-Leffler distribution.^[11] Although $\beta$ determines the skewness of the distribution, it should not be confused with the typical skewness coefficient or 3rd standardized moment, which in most circumstances is undefined for a geometric stable distribution.

$\lambda >0$ is the scale parameter and $\mu$ is the location parameter.^[8]

When $\alpha$ = 2, $\beta$ = 0 and $\mu$ = 0 (i.e., a symmetric geometric stable distribution or Linnik distribution with $\alpha$ =2), the distribution becomes the symmetric Laplace distribution with mean of 0,^[9] which has a probability density function of:

f(x\mid 0,\lambda )={\frac {1}{2\lambda }}\exp \left(-{\frac {|x|}{\lambda }}\right)\,\!

The Laplace distribution has a variance equal to $2\lambda ^{2}$ . However, for $\alpha <2$ the variance of the geometric stable distribution is infinite.

Relationship to stable distributions

A stable distribution has the property that if $X_{1},X_{2},\dots ,X_{n}$ are independent, identically distributed random variables taken from a stable distribution, the sum $Y=a_{n}(X_{1}+X_{2}+\cdots +X_{n})+b_{n}$ has the same distribution as the $X_{i}$ s for some $a_{n}$ and $b_{n}$ .

Geometric stable distributions have a similar property, but where the number of elements in the sum is a geometrically distributed random variable. If $X_{1},X_{2},\dots$ are independent and identically distributed random variables taken from a geometric stable distribution, the limit of the sum $Y=a_{N_{p}}(X_{1}+X_{2}+\cdots +X_{N_{p}})+b_{N_{p}}$ approaches the distribution of the $X_{i}$ s for some coefficients $a_{N_{p}}$ and $b_{N_{p}}$ as p approaches 0, where $N_{p}$ is a random variable independent of the $X_{i}$ s taken from a geometric distribution with parameter p.^[5] In other words:

\Pr(N_{p}=n)=(1-p)^{n-1}\,p\,.

The distribution is strictly geometric stable only if the sum $Y=a(X_{1}+X_{2}+\cdots +X_{N_{p}})$ equals the distribution of the $X_{i}$ s for some a.^[4]

There is also a relationship between the stable distribution characteristic function and the geometric stable distribution characteristic function. The stable distribution has a characteristic function of the form:

\Phi (t;\alpha ,\beta ,\lambda ,\mu )=\exp \left[~it\mu \!-\!|\lambda t|^{\alpha }\,(1\!-\!i\beta \operatorname {sign} (t)\Omega )~\right],

where

\Omega ={\begin{cases}\tan {\tfrac {\pi \alpha }{2}}&{\text{if }}\alpha \neq 1,\\-{\tfrac {2}{\pi }}\log |t|&{\text{if }}\alpha =1.\end{cases}}

The geometric stable characteristic function can be expressed in terms of a stable characteristic function as:^[12]

\varphi (t;\alpha ,\beta ,\lambda ,\mu )=[1-\log(\Phi (t;\alpha ,\beta ,\lambda ,\mu ))]^{-1}.

References

↑ Theory of Probability & Its Applications, 29(4):791–794.
↑ D.O. Cahoy (2012). An estimation procedure for the Linnik distribution. Statistical Papers. 53. pp. 617–628.
↑ D.O. Cahoy V.V. Uhaikin W.A. Woyczyński (2010). Parameter estimation for fractional Poisson processes. Journal of Statistical Planning and Inference. 140. pp. 3106–3120.
1 2 Rachev, S.; Mittnik, S. (2000). Stable Paretian Models in Finance. Wiley. pp. 34–36. ISBN 978-0-471-95314-2.
1 2 Trindade, A.A.; Zhu, Y.; Andrews, B. (May 18, 2009). "Time Series Models With Asymmetric Laplace Innovations" (PDF). pp. 1–3. Retrieved 2011-02-27.
↑ Meerschaert, M.; Sceffler, H. "Limit Theorems for Continuous Time Random Walks" (PDF). p. 15. Archived from the original (PDF) on 2011-07-19. Retrieved 2011-02-27.
↑ Kozubowski, T. (1999). "Geometric Stable Laws: Estimation and Applications". Mathematical and Computer Modelling. 29: 241–253. doi:10.1016/S0895-7177(99)00107-7. Retrieved 2015-12-29.
1 2 3 4 5 Kozubowski, T.; Podgorski, K.; Samorodnitsky, G. "Tails of Lévy Measure of Geometric Stable Random Variables" (PDF). pp. 1–3. Retrieved 2011-02-27.
1 2 Kotz, S.; Kozubowski, T.; Podgórski, K. (2001). The Laplace distribution and generalizations. Birkhäuser. pp. 199–200. ISBN 978-0-8176-4166-5.
↑ Kozubowski, T. (2006). "A Note on Certain Stability and Limiting Properties of ν-inﬁnitely divisible distribution" (PDF). Int. J. Contemp. Math. Sci. 1 (4): 159. Retrieved 2011-02-27.
↑ Burnecki, K.; Janczura, J.; Magdziarz, M.; Weron, A. (2008). "Can One See a Competition Between Subdiffusion and Lévy Flights? A Care of Geometric Stable Noise" (PDF). Acta Physica Polonica B. 39 (8): 1048. Archived from the original (PDF) on 2011-06-29. Retrieved 2011-02-27.
↑ "Geometric Stable Laws Through Series Representations" (PDF). Serdica Mathematical Journal. 25: 243. 1999. Retrieved 2011-02-28.

Probability distributions
List
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped

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Geometric stable distribution

Characteristics

Relationship to stable distributions

See also

References