Download Convergence Gamma Distribution to Normal

Convergence Gamma Distribution to Normal
Distribution by Using Differential Geometry
Wasan Abbas Jasim
Department of Mathematics, College of Education, University of Babylon
Abstract
In this research, we use the differential geometry to show that Gamma distribution converges to
normal distribution by connecting between differential geometry and statistics.
We apply some formulas to calculate the Gaussian curvature (k ) for Gamma distribution in the
case of parametric lines are not orthogonal and show that if it is convergent to normal distribution by
comparing the value of Gaussian curvature for Gamma distribution with the value of Gaussian
curvature for the normal distribution.
‫الخالصة‬
‫ي‬
‫ تييا اخييت داا احهندخييا احتضليييلتا حت تييل ا توزيييك مل ييل ت ت ي ا احييع احتوزيييك اح تس ي و حييب ثلخييت داا اح ي‬، ‫ف ي ه ي ا احث ي‬
.‫احهندخا احتضليلتا واال صلء‬
‫تسل ييد وتويييت فت ييل ا ا ه ي ا‬
‫ييوت اح سل تييا ا ي‬
‫لحييا اح‬
‫ييض ثسييغ احصييتت ح خييلا ت ييوع مييلوع حتوزيييك مل ييل ف ي‬
‫تييا ت‬
. ‫احتوزيك ت ل ا احع احتوزيك اح تس ث ل نا اح ت ا حت وع ملوع حتوزيك مل ل ك اح ت ا حت وع ملوع حلتوزيك اح تس‬
Introduction
This research present some interesting connections between statistics and
differential geometry. Also, it contains the result of computing the Gaussian curvature
of Gamma distribution by using different formulas. If the Gaussian curvature of
Gamma distribution approaches to the Gaussian curvature of the normal distribution,
we say that the distribution converges to normal distribution.
There are some researchers who worked in this field in end of twenty century
and beginning of twenty one century, including (Kass, 1997) used the following
formula
1    1  G    1  E  

 


EG  u  E u  v  G v  
to compute the Gaussian curvature (K) of trinomial and t families, (Chen, 1999) used
the formula
2
R1212
(12,12)
m
(
12
,
12
)

R

R121
g m2
where


1212
EG  F 2 EG  F 2
m 1
 1
 1
1
1
1
, sum on m,
Rijk

ik 
 jk  ikm mj
  jkm mi
u j
ui
to compute the Gaussian curvature (K) for the distributions (normal, Cauchy and t
1
family). He showed that in normal distribution Gaussian curvature K   , and in
2
Cauchy distribution K  2 , while in t family distribution with r degrees of freedom,
r 3
he got K  
and (Gruber, 2003) used the following formula
2r
E F G

1
Eu Fu Gu

2 2
4
(
EG

F
)

Ev Fv Gv
to compute the Gaussian curvature of gamma family of distributions in the case of
parametric lines are orthogonal (F=0) and showed that Gamma distribution converges
to the normal distribution.
  Gu  Fv
 Fu  Ev



2 EG  F 2  u EG  F 2 v EG  F 2
1
1. Metric Tensor for Gamma Distribution (Wasan, 2008)
In the case of parametric lines  and v are not orthogonal ( F  0 ).
We know that
v 1
v x
f ( x ;  , v)  
e  x
x  0,
, v  0
(v)
be probability density function of Gamma distribution.
Then the logarithm of likelihood function of family Gamma, can be written as
follows:
(1)
ln f  v ln   (v  1) ln x  x  ln (v)
From equation (1), we can derive the first and second partial derivatives with
respect to the parametric lines  and v .
 2 ln f
v
 2
2


 2 ln f
1

v

(2)

  (v) 
 2 ln f

 
v 2
  (v ) 
For the Gamma family of distributions, the coefficients of the first fundamental
form:
  2 ln f  v
 2
E   E
2 
   
  2 ln f 
1
  
F   E

 v 
  2 ln f 
   (v)
G   E
2
 v 
 (v)
where  (v) 
is the digamma function.
 (v )
1 
 v

 2

 


Metric Tensor = 

1
 (v) 

 

0r
v
2
ds 2  2 (d ) 2  ddv   (v) (dv) 2


(3)
2.The Gaussian curvature of the Probability Distribution (Chen, 1999)
First, we define the six well known Christoffel symbols as:
111 
GEu  2FFu  FEv
,
2( EG  F 2 )
2 EFu  EEv  FEu
,
2( EG  F 2 )
GEv  FGu
121 
,
2( EG  F 2 )
112 
122 
EGu  FEv
,
2( EG  F 2 )
2GFv  GGu  FGv
,
2( EG  F 2 )
EGv  2FFv  FGu
222 
.
2( EG  F 2 )
1
22

(4)
Since E,F and G are functions of parameters (u,v) and are continuously twice
differentiable Eu, Ev, Fu, Fv, Gu and Gv all exists and are all well defined. Additionally,
no assumption is made regarding F=0, and so the parametric lines are not necessarily
orthogonal. However, if F=0, the six Christoffel symbols can be greatly simplified.
Now, we select four formulas that can be used to compute the Gaussian
curvature of the distributions:
( A) K 
   1  G    1  E 
 
 

EG  u  E u  v  G v  
1
  Gu  Fv
 Fu  Ev
( B) K  


2 EG  F 2  u EG  F 2 v EG  F 2
1
(C ) K 
E F G

1
Eu Fu Gu

2 2
4
(
EG

F
)

Ev Fv Gv
1    D 2    D 2  1    D 1    D 1 
 11    12     22    12 
D  v  E
 u  E
 D  u  G
 v  G

D 2  EG  F 2 where
( D) K 
R1212
(12,12)

2
EG  F
EG  F 2
2
m
g m2
where (12,12)  R1212   R121
m 1
, sum on m,
1
Rijk

 1
 1
1
1
ik 
 jk  ikm mj
  jkm mi
u j
ui
where the quantities of Rijk1 are components of a tensor of the fourth order. This tensor
is called the mixed Riemann curvature tensor. Notice that g11, g12, and g22 are simply
tensor notation for E, F and G.
Clearly formula (A) is a special case that is valid only when the parametric lines
are orthogonal.
Remark:
We can not use the formula (A) to compute the Gaussian curvature of Gamma
distribution because we take the case that the parametric lines μ, v of this distribution
are not orthogonal(F≠0).
3.The Gaussian Curvature for Gamma Distribution
3.1 We use the formula (B) to compute the Gaussian curvature of this distribution.
E F G
  G   Fv
 F  E v 
1
K 

E  F G 


2 2
2 EG  F 2   EG  F 2 v EG  F 2  4( EG  F )
E v Fv Gv
where
v
1

 (v)
2


E F G
 v (v) 2v (v)  (v) 
2v 1
E  F G    3
0



2
4


4
 4 
 
E v Fv Gv
1
0
 (v)
2
1


v (v)   (v)
4
.
Now, we can say that
E F G
1
K 
E  F G 
4( EG  F 2 ) 2
E v Fv Gv


where
 v (v)  (v) 


4(v (v)  1) 
4

4
2
v (v)   (v)
,
4(v (v)  1) 2
 (v ) 
(v)
is the digamma function
 (v )
 (1 / 2v 2  o(1 / v 3 ))
1
lim K (v)  lim

v 
v  4((1 / 4v 2 )  o (1 / v 3 ))
2
lim K (v)  
v 0
1
4
3.2 We use the formula (C) to compute the Gaussian curvature of this distribution.
We can find that
111 
GE  2FF  FEv
2( EG  F 2 )
 2v (v) 2
1
 3 3
3
 2v (v)  1
2





v (v) 1
2v (v)  2
3
2(

)
2
2



1  2v (v)
2 (v (v)  1)
 (v)
GEv  FG
2
121 

2( EG  F 2 ) 2v (v)  2
2
 (v)
2
 (v)


2
 2v (v)  2 2(v (v)  1)
1
22

2GFv  GG  FGv
2( EG  F 2 )
 (v)
 (v)
2
 (v)




2(v (v)  1)
 2(v (v)  1) 2(v (v)  1)
2

2v
112 
2 EF  EEv  FE
2( EG  F 2 )


4

v

4

2v
4
2(v (v)  1)
2
v
2
v
 2(v (v)  1) 2 (1  v (v))
1
EG  FEv
3
122 

2( EG  F 2 ) 2(v (v)  1)


4
2
2

1
2
 2(v (v)  1)
3

1
2 (v (v)  1)
222 
EGv  2FFv  FG
2( EG  F 2 )
v (v)
2
v (v)


2
2(v (v)  1) 2(v (v)  1)

We take the left hand side of formula (C)
K

1    D 2    D 2 
 11  
 12 
D  v  E
   E


EG  F 2 2 
(
11 )

E
EG  F 2  v

1
   v (v)  1  2

v

 

v 2 2 (v (v) 
v (v)  1  v 
1 
1


(v (v)  1) 2 

2 v (v)  1  v

v (v)   (v)
v (v)   (v)


3/ 2
4(v (v)  1) 2
4 v (v)  1 (v (v)  1)


lim K (v)  
1
2
lim K (v)  
1
4
v 
v 0
Now, we take the right hand side of formula (C)
K
1    D 1    D 1 
 22    12 
D    G
 v  G

   EG  F 2

 
121 

G
EG  F 2  v 

   v (v)  1 1


 (v)


 





v


(
v
)
2
(
v

(
v
)

1
)

v (v)  1  

1



(v (v)  1) 1 / 2 

2 v (v)  1  v



1
v (v)   (v)
v (v)   (v)

3/ 2
4(v (v)  1) 2
4 v (v)  1 (v (v)  1)
lim K (v)  
v 
1
2
lim K (v)  
v 0
1
4
3.3 We use the formula (D) to compute the Gaussian curvature of this distribution.
R1212
K
EG  F 2
1
2
R1212  R121
g12  R121
g 22 ,
where
 1
 1
11 
21  11m m1 2  21m m1 1
v


 1
1
1
1
 111 
21  111 121  112 22
 21
111  212 21
v



v (v)   (v)
v (v)
 (v)



2 (v (v)  1) 2 4 (v (v)  1) 2 4 (v (v)  1) 2
1
R121

2v (v)  2 (v)  v (v)   (v)
4 (v (v)  1) 2
v (v)   (v)

4 (v (v)  1) 2
v (v)   (v)
1
R121
g12   2
4 (v (v)  1) 2

 2
2
R121
 112 
21  11m m2 2  21m m21
v


 2
1
 112 
21  111 122  112 222  21
112  212 212
v

2
1  v  (v)
1
1  2v (v)



2
2
2
2 (v (v)  1)
2 (v (v)  1) 4 2 (v (v)  1) 2


v 2 (v)
v (v)
1


2
2
2
2
2
4 (v (v)  1)
4 (v (v)  1)
4 (v (v)  1) 2

2v 2 (v)  v 2 (v)  v (v)
4 2 (v (v)  1) 2

v 2 (v)  v (v)
4 2 (v (v)  1) 2
v 2 (v) (v)  v  2 (v)
4 2 (v (v)  1) 2
v 2 (v) (v)  v  2 (v)  v (v)   (v)

4 2 (v (v)  1) 2
2
R121
g 22 
R1212
K (v) 
v 2 (v) (v)  v 2 (v)  v (v)  (v)
2
4 2 (v (v)  1) 2
v (v)  1
v (v)v (v)  1   (v)v (v)  1
4(v (v)  1) 3
v (v)   (v)

4(v (v)  1) 2

1
1
lim K (v)  
v

0
2
4
As v→ ∞ the curvature of the Gamma family of distributions tends toward -1/2
the curvature of the normal family of distributions.
This result shows that Gamma distribution converges to the normal distribution.
lim K (v)  
v 
References
Chen, W.W.S. (1999), "On Computing Gaussian Curvature of Some Well Known
Distributions".
http://www.irs.gov/pub/irs-soi/gauscurv.pdf.
Gruber, M.H.J. (2003), "Some Interesting Connections Between Statistics and
Differential Geometry".
http://www.rit.edu/~mjgsma/maatalk/geometrystatistics.html
Kass, R.E. and Vos, P.W. (1997) "Geometrical Foundations of Asymptotic
Inference", John Wily & Sons, Inc.
Wasan, A.J. (2008), "Metric Tensor for Two Probability Distributions Gamma and
Beta", Accepted in Journal of Babylon University.