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Geometric stable distribution
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Geometric stable distribution : ウィキペディア英語版
Geometric stable distribution
, or ''x'' ∈ (−∞,μ] if and
| pdf = not analytically expressible, except for some parameter values
| cdf = not analytically expressible, except for certain parameter values
| mean =
| median = ''μ'' when
| mode = ''μ'' when
| variance = 2''λ''2 when , otherwise infinite
| skewness = 0 when , otherwise undefined
| kurtosis = 3 when , otherwise undefined
| entropy =
| mgf = undefined
| char = \!\Big(\omega - i \mu t )^,
where \omega = \begin 1 - i\tan\tfrac \beta\, \operatorname(t) & \text\alpha \ne 1 \\
1 + i\tfrac\beta\log|t| \, \operatorname(t) & \text\alpha = 1 \end
}}
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, Theory of Probability & Its Applications, 29(4):791–794. These distributions are analogues for stable distributions for the case when the number of summands is random, independent on 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. The Laplace distribution is a special case of the geometric stable distribution and of a Linnik distribution. The Mittag–Leffler distribution is also a special case of a geometric stable distribution.
The geometric stable distribution has applications in finance theory.〔
==Characteristics==

For most geometric stable distributions, the probability density function and cumulative distribution function have no closed form solution. But a geometric stable distribution can be defined by its characteristic function, which has the form:
:
\varphi(t;\alpha,\beta,\lambda,\mu) =
(\omega - i \mu t )^

where \omega = \begin 1 - i\tan\tfrac \beta \, \operatorname(t) & \text\alpha \ne 1 \\
1 + i\tfrac\beta\log|t| \operatorname(t) & \text\alpha = 1 \end
\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.〔 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.〔 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:〔
:
\varphi(t;\alpha, 0, \lambda,\mu) =
(- i \mu t )^

The symmetric geometric stable distribution with \mu=0 is also referred to as a Linnik distribution. 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. 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.〔
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,〔 which has a probability density function of:
:f(x|0,\lambda) = \frac \exp \left( -\frac \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.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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