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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 ddv (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.