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1
C h a r a c t e r i z a t i o n
T o g g l e C h a r a c t e r i z a t i o n s u b s e c t i o n
1 . 1
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T o g g l e t h e t a b l e o f c o n t e n t s
q - e x p o n e n t i a l d i s t r i b u t i o n
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A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Originally proposed by the statisticians George Box and David Cox in 1964,[2] and known as the reverse Box–Cox transformation for
q
=
1
−
λ
,
{\displaystyle q=1-\lambda ,}
a particular case of power transform in statistics.
Characterization
[ edit ]
Probability density function
[ edit ]
The q -exponential distribution has the probability density function
(
2
−
q
)
λ
e
q
(
−
λ
x
)
{\displaystyle (2-q)\lambda e_{q}(-\lambda x)}
where
e
q
(
x
)
=
[
1
+
(
1
−
q
)
x
]
1
/
(
1
−
q
)
{\displaystyle e_{q}(x )=[1+(1-q)x]^{1/(1-q)}}
is the q -exponentialif q ≠ 1 . When q = 1 , e q (x ) is just exp(x ).
Derivation
[ edit ]
In a similar procedure to how the exponential distribution can be derived (using the standard Boltzmann–Gibbs entropy or Shannon entropy and constraining the domain of the variable to be positive), the q -exponential distribution can be derived from a maximization of the Tsallis Entropy subject to the appropriate constraints.
Relationship to other distributions
[ edit ]
The q -exponential is a special case of the generalized Pareto distribution where
μ
=
0
,
ξ
=
q
−
1
2
−
q
,
σ
=
1
λ
(
2
−
q
)
.
{\displaystyle \mu =0,\quad \xi ={\frac {q-1}{2-q}},\quad \sigma ={\frac {1}{\lambda (2-q)}}.}
The q -exponential is the generalization of the Lomax distribution (Pareto Type II), as it extends this distribution to the cases of finite support. The Lomax parameters are:
α
=
2
−
q
q
−
1
,
λ
L
o
m
a
x
=
1
λ
(
q
−
1
)
.
{\displaystyle \alpha ={\frac {2-q}{q-1}},\quad \lambda _{\mathrm {Lomax} }={\frac {1}{\lambda (q-1)}}.}
As the Lomax distribution is a shifted version of the Pareto distribution , the q -exponential is a shifted reparameterized generalization of the Pareto. When q >1 , the q -exponential is equivalent to the Pareto shifted to have support starting at zero. Specifically, if
X
∼
q
-
E
x
p
(
q
,
λ
)
and
Y
∼
[
Pareto
(
x
m
=
1
λ
(
q
−
1
)
,
α
=
2
−
q
q
−
1
)
−
x
m
]
,
{\displaystyle X\sim \operatorname {{\mathit {q}}-Exp} (q,\lambda ){\text{ and }}Y\sim \left[\operatorname {Pareto} \left(x_{m}={\frac {1}{\lambda (q-1)}},\alpha ={\frac {2-q}{q-1}}\right)-x_{m}\right],}
then
X
∼
Y
.
{\displaystyle X\sim Y.}
Generating random deviates
[ edit ]
Random deviates can be drawn using inverse transform sampling . Given a variable U that is uniformly distributed on the interval (0,1), then
X
=
−
q
′
ln
q
′
(
U
)
λ
∼
q
-
E
x
p
(
q
,
λ
)
{\displaystyle X={\frac {-q'\ln _{q'}(U )}{\lambda }}\sim \operatorname {{\mathit {q}}-Exp} (q,\lambda )}
where
ln
q
′
{\displaystyle \ln _{q'}}
is the q -logarithm and
q
′
=
1
2
−
q
.
{\displaystyle q'={\frac {1}{2-q}}.}
Applications
[ edit ]
Being a power transform , it is a usual technique in statistics for stabilizing the variance, making the data more normal distribution-like and improving the validity of measures of association such as the Pearson correlation between variables.
It has been found to be an accurate model for train delays.[3]
It is also found in atomic physics and quantum optics, for example processes of molecular condensate creation via transition through the Feshbach resonance.[4]
See also
[ edit ]
Notes
[ edit ]
^ Tsallis, C. Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years. Braz. J. Phys. 2009, 39, 337–356
^ Box, George E. P. ; Cox, D. R. (1964). "An analysis of transformations". Journal of the Royal Statistical Society, Series B . 26 (2 ): 211–252. JSTOR 2984418 . MR 0192611 .
^ Keith Briggs and Christian Beck (2007). "Modelling train delays with q -exponential functions". Physica A . 378 (2 ): 498–504. arXiv :physics/0611097 . Bibcode :2007PhyA..378..498B . doi :10.1016/j.physa.2006.11.084 . S2CID 107475 .
^ C. Sun; N. A. Sinitsyn (2016). "Landau-Zener extension of the Tavis-Cummings model: Structure of the solution". Phys. Rev. A . 94 (3 ): 033808. arXiv :1606.08430 . Bibcode :2016PhRvA..94c3808S . doi :10.1103/PhysRevA.94.033808 . S2CID 119317114 .
Further reading
[ edit ]
External links
[ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Q-exponential_distribution&oldid=1081124706 "
C a t e g o r i e s :
● S t a t i s t i c a l m e c h a n i c s
● C o n t i n u o u s d i s t r i b u t i o n s
● P r o b a b i l i t y d i s t r i b u t i o n s w i t h n o n - f i n i t e v a r i a n c e
● T h i s p a g e w a s l a s t e d i t e d o n 5 A p r i l 2 0 2 2 , a t 1 3 : 2 0 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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