A NEW GENERALISATION OF SAM-SOLAI’S MULTIVARIATE ADDITIVE F- DISTRIBUTION OF KIND-2

Research Journal of Pure Algebra -3(2), 2013, Page: 83-94

Available online through www.rjpa.info ISSN 2248–9037

Research Journal of Pure Algebra-Vol.-3(2), Feb. – 2013 83

A NEW GENERALISATION OF SAM-SOLAI’S MULTIVARIATE ADDITIVE

F- DISTRIBUTION OF KIND-2*

Dr. G. S. David Sam Jayakumar* Assistant Professor, Jamal Institute of Management, Jamal Mohamed College,

Tiruchirappalli–620 020, South India, India

Dr. A. Solairaju Associate Professor, Dept. of Mathematics, Jamal Mohamed College,

Tiruchirappalli – 620 020, South India, India

A. Sulthan Research scholar, Jamal Institute of Management, Tiruchirappalli – 620 020 South India, India

(Received on: 06-02-13; Accepted on: 15-02-13)

ABSTRACT This paper proposed a new generalization of Sam-Solai’s Multivariate additive F- distribution of Kind-2 from the uni-variate case. Further, we find its Marginal, Multivariate Conditional distributions, Multivariate Generating functions, Multivariate survival, hazard functions and also discussed its special cases. The authors derived the generating functions of this distribution in terms of Whittaker-M function and complex Meijer-G function. The special cases includes the transformation of Sam-Solai’s Multivariate additive F- distribution of Kind -2 into Multivariate additive Beta distribution of Kind-1 of Type-B, Multivariate Beta-distribution of Kind-2 of Type-B, Multivariate Fisher’s Z-distribution of Kind-2 and Multivariate Logistic-F distribution of kind-2. Moreover, it is found that the bi-variate correlation between two F- variables purely depends on the d.f and we simulated and established selected standard bi-variate F- correlation bounds from 10,000 different combination of d.f. The simulation results shows, the correlation between any two F- variables bounded from -1 to +1 for certain combination of fractional d.f. Keywords: Sam-Solai’s Multivariate additive F- distribution of Kind-2, Transformation, Multivariate additive Beta distribution of Kind-1 of Type-B, Multivariate Beta-distribution of Kind-2 of Type-B, Multivariate Fisher’s Z-distribution of Kind-2, Multivariate Logistic-F distribution of kind-2 ,Correlation bounds, Fractional d.f Mathematics Subject Classification: Primary 62H10; Secondary 62E15. INTRODUCTION The pioneering work of R.A. Fisher and George W. Snedecor gave a birth of F-distribution to the statistical literature and the origin of F- distribution, its multivariate generalization was extensively studied because of the wide applications of the distribution in various fields for the past five decades. Many authors attempted to give an alternate form of multivariate generalization of F-distribution. At first, Leo Aroian [1941] studied the F-distribution and its relationship with Fisher’s Z distribution and Seymour Geisser et al. [1958] analyzed the results proposed by Box and extend the results to study the application of F-distribution in Multivariate analysis. Similarly Nico Laubscher [1960] proposed the procedure of normalizing the Non-central t and F-distributions and Tiao et al. [1965] emphasized that the F-distribution is a special case of inverted Dirichlet distribution. Moreover, Konno [1988], Fang et al. [1990], Yasunori Fujikoshi [1993] studied the exact moments, symmetricity, Quantile approximation with special reference to the F-distribution respectively. On the other hand, Leung [1994] found the identity for the non-central wishart distribution which includes the F-distribution and Jones [2001] explored the association among the Multivariate Beta, t and F-distributions. Likewise, Pham Gia et al. [2003] and Arnold et al. [2006] studied the application of F-distribution in statistical modeling and the Rosenblatt construction of multivariate family of distributions which includes F-distribution respectively.Finally,El-Bassiouny et al [2007] proposed the bivariate –F distribution with special marginals and Nadarajah [2007] explored the Compound F-distribution and discussed its properties. From the in-depth reviews of F- distribution, this paper proposed an alternate form of multivariate F- distribution and its structure, form were discussed in the next section.

*Corresponding author: Dr. G.S. David Sam Jayakumar* Assistant Professor, Jamal Institute of Management, Jamal Mohamed College,

Tiruchirappalli–620 020, South India, India

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Dr. G.S. David Sam Jayakumar*, Dr. A. Solairaju and A. Sulthan/ A NEW GENERALISATION OF SAM-SOLAI’S MULTIVARIATE ADDITIVE F- DISTRIBUTION OF KIND-2*/ RJPA- 3(2), Feb.-2013.

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SECTION-1: SAM-SOLAI’S MULTIVARIATE ADDITIVE F- DISTRIBUTION Definition 1.1: Let

1 2 3, , , pF F F F

are the random variables followed Continuous uni-variate F- distribution with

( , )i im n degrees of freedom for all i (i=1 to p), then the Multivariate Sam-Solai’s additive F- distribution of Kind-2 and its density function is defined as

12

1 2 31 11 12 2

,1( / )(( / ) )12( , , , ) ( 1)

, (1 ( / ) ) ,1 (1 ( / ) )2 2 2

i

i i

mipp

i i i i ip n m

i ii i ii i i i i i

mBm n m n F

g F F F F pm n mB m n F B m n F

−

− += =

= − − + +

∑ ∏ (1)

where 0 iF , , 0i im n Theorem 1.2: The cumulative distribution function of the Sam-Solai’s Multivariate additive F- distribution is defined by

31 21

2

1 2 31 11 10 0 0 0 2 2

( ,1) ( / )(( / ) )12( , , , ) ( 1), (1 ( / ) ) ,1 (1 ( / ) )

2 2 2

ip

i i

miFFF F ppi i i i i

p in mi ii i i

i i i i i i

mB m n m n uG F F F F p du

m n mB m n u B m n u

−

− += =

= − − + +

∑ ∏∫ ∫ ∫ ∫ (2)

where,0 i iu F , , 0i im n

12

0 2 2

1 2 311 2

10 2

( / )(( / ) )

, (1 ( / ) )( / )2 2( , , , ) ( 1)

( / )(( / ) )

,1 (1 ( / ) )2

ii

i i

ii

i

mF

i i i i iim n

i ii i ip

i ip m

Fii i i i i

imi

i i i

m n m n u dum nB m n u

m nG F F F F pm n m n u dumB m n u

−

+

−=

+

+ = − − +

∫

∑∫

12

11 0 2

(( / ) )

,1 (1 ( / ) )2

ii

i

mFp

i i iim

i ii i i

m n u dumB m n u

−

+= +

∏∫

1 2 31 1

( ; , )( , , , ) ( 1) ( ; , )( ; , )

ppi i i i

p i i i ii ii i i i

F m nG F F F F p F m nF m n

φ ωω= =

= − − ∑ ∏

where 1

2 2

0 2 2

( / )( ; , )

, (1 ( / ) )2 2

i ii

i i

m mF

i i ii i i i im n

i ii i i

m n uF m n du

m nB m n uφ

−

+=

+

∫ and 1

2 2

10 2

( / )( ; , )

,1 (1 ( / ) )2

i ii

i

m mF

i i ii i i i im

ii i i

m n uF m n du

mB m n uω

−

+=

+

∫ are the

lower incomplete Beta integrals of ith F- variable. Theorem 1.3: The Probability density function of Sam-Solai’s Multivariate additive Conditional F- distribution of

1F on 2 3, , pF F F

is

1

1

12

1 1 1 1 1

111 1 221 1 1

1 2 3

12 2

( ,1) ( / )(( / ) )12 ( 1), ,1 (1 ( / ) )(1 ( / ) )

2 2 2( / , , )

( ,1) 12

, (1 ( / ) )2 2

i

i

mip

n mi i i

i i i

p

ip

ni i i

i i i

mB m n m n Fpm n mB B m n Fm n F

g F F F FmB

m nB m n F

−

+−=

−=

− − ++ =

+

∑

∑

( 2)p

− −

(3)

where 10 F , , 0i im n

Proof: It is obtained from

1 2 31 2 3

2 3

( , , , )( / , , )

( , , )p

pp

g F F F Fg F F F F

g F F F



Theorem 1.4: Mean and Variance of Sam - Solai’s Multivariate additive Conditional F- distribution are

1 1

121 1 11 1 2

1 2 3

,11, 11 1 122 2 ( 1)

( / ) , ,1 , (1 ( / ) )2 2 2 2 2

( / , , ),1

12

, (1 ( /2 2

i

ip

ni i i

i i i

pi

i ii i

mm n BBp

m n m m nm n B B B m n FE F F F F

mB

m nB m n

−=

+ − + − − + =

+

∑

12 2

( 2)) )

i

p

ni

i

pF

−=

− −

∑

(4)

2 2

1 2 3 1 2 3 1 2 3( / , , ) ( / , , ) ( ( / , , ))p p pV F F F F E F F F F E F F F F

(5)

where

1 1 1

121 1 11 1 2

21 2 3

,12, 2 2, 11 122 2 2 ( 1)

( / ) , ,1 , (1 ( / ) )2 2 2 2 2

( / , , ),1

2

,2 2

i

ip

k ni i i

i i i

pi

i i

mm n m BB Bp


mB

m nB

−=

+ − + − + − − + =

∑

12 2

1 ( 2)(1 ( / ) )

i

p

ni

i i i

pm n F

−=

− − +

∑

Proof: The thk order moment of the distribution is

1 2 3 1 1 2 3 10

( / , , ) ( / , , )k kp pE F F F F F g F F F F dF

1

1

12

1 1 1 1 1

111 1 221 1 1

1 2 3 1

,11 ( / )(( / ) )2 ( 1)

, ,1 (1 ( / ) )(1 ( / ) )2 2 2( / , , ),1

2

i

mip

n mi ii

i i ik k

pi

mBm n m n Fp

m n mB B m n Fm n FE F F F F F

mB

10

12 2

1 ( 2), (1 ( / ) )2 2

i

p

ni ii

i i i

dF

pm nB m n F

1 1 1

121 1 11 1 2

1 2 3

,1, ,11 122 2 2 ( 1)

( / ) , ,1 , (1 ( / ) )2 2 2 2 2

( / , , )

i

ip

k ni i i

i i ik

p

mm n m BB k k B k kp


12 2

( ,1) 12 ( 2), (1 ( / ) )

2 2

i

ip

ni i i

i i i

mBp

m nB m n F

If k=1, then the Conditional expectation is

1 1

11 21 11 1 2

1 2 3

2

1, 1 ( ,1)1 1 12 2 2 ( 1)( / ) ( ,1), , (1 ( / ) )22 2 2 2

( / , , ),1

12

, (1 ( / ) )2 2

i

i

ip

ni i i

i i i

pi

ni i

i i i

m n mB Bpmm n m nm n BB B m n F

E F F F FmB

m nB m n F

−=

−

+ − + − − + =

+

∑

12( 2)

p

ip

=

− −

∑



If k=2, then the second order moment is

1 1 1

121 1 11 1 2

21 2 3

,12, 2 2, 11 122 2 2 ( 1)

( / ) , ,1 , (1 ( / ) )2 2 2 2 2

( / , , ),1

2

,2 2

i

ip

k ni i i

i i i

pi

i i

mm n m BB Bp


mB

m nB

−=

+ − + − + − − + =

∑

12 2

1 ( 2)(1 ( / ) )

i

p

ni

i i i

pm n F

−=

− − +

∑

The conditional variance of the distribution is obtained by Substituting the first and second moments in (5). Theorem 1.5: If there are p = (q + k) random variables, such that q random variables

1 2 3, , , qF F F F

conditionally

depends on the k variables1 2 3, , ,q q q q kF F F F

,then the density function of Sam-Solai’s multivariate additive

conditional F- distribution is

12

1 12

1 2 3 1 2 3

( ,1) ( / )(( / ) )12 ( 1)(1 ( / ) ), ,1 (1 ( / )

2 2 2( , , , / , , , )

i

i

miq k

i i i i ini

i i ii i i i i

q q q q q k

mB m n m n Fq km n mm n FB B m n

g F F F F F F F F

1 12

1 12

)

,112 ( 1)

(1 ( / ) ),2 2

i

i

q

mi

i

i

q k

ni qi i

i i i

F

mBk

m n m n FB

(6)

where 0 iF , , 0i im n Proof: Let the multivariate conditional law for q random variables

1 2 3, , , qF F F F

conditionally depending on the k

variables 1 2 3, , ,q q q q kF F F F

is given as

1 2 3 1 2 31 2 3 1 2 3

1 2 3

( , , , , , , , )( , , , / , , , )

( , , , )q q q q q k

q q q q q kq q q q k

g F F F F F F F Fg F F F F F F F F

g F F F F

12

1 12

1 2 3 1 2 3

,1( / )(( / ) )12 ( 1)

(1 ( / ) ), ,1 (2 2 2

( , , , / , , , )

i

i

i m

q ki i i i i

nii i i

i i i

q q q q q k

mBm n m n Fq k

m n mm n FB Bg F F F F F F F F

1 12

12

11 1 12 2

1 ( / ) )

( ,1) ( / )(( / ) )12 ( 1)(1 ( / ) ), ,1 (1 ( / ) )

2 2 2

i

i

i i

q k

mi

i i i

miq k q i i i i i

n miii i i

i i i i i i

m n F

mB m n m n Fq km n mm n FB B m n F

10 0 0

k q

iidF

12

1 12

1 2 3 1 2 3

( ,1) ( / )(( / ) )12 ( 1)(1 ( / ) ), ,1 (1 ( / )

2 2 2( , , , / , , , )

i

i

miq k i i i i i

nii i i

i i i i i

q q q q q k

mB m n m n Fq km n mm n FB B m n

g F F F F F F F F

1 12

1 12

)

( ,1) 12 ( 1)(1 ( / ) ),

2 2

i

i

q

mi

i

iq k

ni qi i

i i i

F

mBk

m n m n FB

where 0 iF , , 0i im n



SECTION 2: CONSTANTS OF SAM-SOLAI’S MULTIVARIATE ADDITIVE F- DISTRIBUTION Theorem 2.1: The Marginal product moments, Co-variance and Correlation Co-efficientbetween the random variables

1F and 2F are given as

1 21 2

1 2

1 1 1( )2 2 2 2

n nE F Fn n

= + − − −

(7)

1 2 1 21 2

1 2

6 1( , )2 ( 2)( 2) 2

n n n nCOV F Fn n

+ −= − − −

(8)

1 2 1 2

1 21 1 2 2

1 2 1 2

2 2 ( 2)( 2) 12( , )

( 2)( 2)8

( 4)( 4)

n n n nF F

m n m nm m n n

ρ+ − − − −

=+ − + −

− −

(9)

where

1 21 ( , ) 1F F for certain d.f (see Result3.4) Proof: Assume that

1F and 2F are random variables from Sam-Solai’s multivariate additive F- distribution. Let the

product moment of the distribution is

1 2 1 2 1 2 310 0 0

( ) ( , , , )p

p ii

E F F F F g F F F F d F

Its Co-variance is

1 2 1 2 1 2( , ) ( ) ( ) ( )COV F F E F F E F E F (10)

Then

12

1 2 1 21 111 2 20 0 0

,1( / )(( / ) )12( ) ( 1)

, (1 ( / ) ) ,1 (1 ( / ) )2 2 2

i

i i

mip p

i i i i in m

ii i i ii i i i i i

mBm n m n FE F F F F p d

m n mB m n F B m n F

1

p

ii

F

By evaluation, it follows that 1 21 2

1 2

1 1 1( )2 2 2 2

n nE F Fn n

= + − − −

.The marginal expectation of the random variables

1F and 2F are 1 1 1( ) / ( 2)E F n n= − and 2 2 2( ) / ( 2)E F n n= − respectively. The marginal Product moment for

1 2( )E F F is obtained by substituting the above marginal expectations for 1F and

2F in (10).

Thus 1 2 1 21 2

1 2

6 1( , )2 ( 2)( 2) 2

n n n nCOV F Fn n

+ −= − − −

(11)

Correlation coefficient of a distribution is

1 2 1 2 1 2( , ) ( , ) /F F COV F F (12a)

It observes that 1 1 1 1 1 1 1( / ( 2)) 2( 2) / ( 4)n n m n m nσ = − + − − and 2 2 2 2 2 2 2( / ( 2)) 2( 2) / ( 4)n n m n m nσ = − + − − (12b) From (11), (12a) and (12b), it follows that

1 2 1 21 2

1 1 2 2

1 2 1 2

2 2 ( 2)( 2) 12( , )

( 2)( 2)8

( 4)( 4)

n n n nF F

m n m nm m n n

ρ+ − − − −

=+ − + −

− −

(13)

where

1 21 ( , ) 1F F for certain d.f Remark 2.1: The Product moments, Co-variance and Correlation Co-efficient between the thi and thj F- variables are

given as



1 1 1( )2 2 2 2i j

i ji j

n nE F F

n n

= + − − − (14)

6 1( , )2 ( 2)( 2) 2i j i j

i ji j

n n n nCOV F F

n n + −

= − − − (15)

2 2 ( 2)( 2) 12( , )

( 2)( 2)8

( 4)( 4)

i j i ji j

i i j j

i j i j

n n n nF F

m n m nm m n n

ρ+ − − − −

=+ − + −

− −

(16)

where, i j , 1 21 ( , ) 1F F for certain d.f

Theorem 2.2: The Moment generating function of Sam-Solai’s Multivariate additive F- distribution is

1 2, 1 211

( ; , )( , ) ( ; , ) ( 1)( ; , )P

p pi i i

x x x P i i iii i i i

t m nM t t t t m n pt m n

φϕϕ==

= − −

∑∏

(17)

Proof: Let the moment generating function of a Multivariate distribution is given as

11 2 3,, , 1 2 3, 1 2 3

10 0 0

( , , ) ( , , , )

pt Fi i

ip

p

F F F F p p ii

M t t t t e f F F F F dF

11 2 3,

12

, , 1 2 3,11 20 0 0

,1( / )(( / ) )12( , , ) ( 1)

, (1 ( / ) ) ,12 2 2

p it Fi i

ip i

mip

i i i i iF F F F p n

i i i ii i i

mBm n m n FM t t t t e p

m n mB m n F B

11 12(1 ( / ) )i

p p

imi i

i i i

dFm n F

(17a)

From (17a), it observed

1/2 /2 /2 42

210 2 2

( / ) ( / )1 ( ; , )( / 2) ( / 2)( 4), (1 ( / ) )

2 2

ii i i i i

i i

i i

mm m t n m

t F i i i i i ii r i i im n

ri i i i i i ii i i

m n F t n m ee dF t m nm n t n m n nB m n F

γ−∞ − −

+ =

−=

Γ Γ − +

∑∫

[ ]1/2 2 2

10 2

( / ) ( / ) 11 [0],[] , , ,[] ,( / 2) 2 2,1 (1 ( / ) )

2

mii

ii i

i

mm

t F i i i i i i i i i i i iim

i i i ii i i

m n F t n t n m m m t ne dF Gm m m mB m n F

− −∞

+

− − − = − Γ +

∫

Thus, by integration 1 2, 1 2

11

( ; , )( , ) ( ; , ) ( 1)( ; , )P

p pi i i

x x x P i i iii i i i

t m nM t t t t m n pt m n

φϕϕ==

= − −

∑∏

where /2 /2 4

21

( / )( ; , ) ( ; , )( / 2) ( / 2)( 4)

i i i im t n mi i i

i i i r i i iri i i i i

t n m et m n t m nt n m n n

φ γ− −

=

−=

Γ Γ − ∑ and

/2 /41

1( ; , ) ( / ) ( / ) ( ( / 2) ( / 2))( 2)( 2) 1, 1 ,2 4 2 2

i im n i i i i ii i i i i i i i i i i i i i

i

m n n t nt m n m t n m t n m m n n m Mm

γ = − − − Γ Γ + + + + − −

( )/2 /42

1( ; , ) ( / ) ( / ) ( ( / 2) ( / 2))( 2)( 2) 1, 1 ,2 4 2 2

i i im n n i i i i ii i i i i i i i i i i i i i i i

i

m n n t nt m n m t n m t n m m n n n m n Mm

γ + = − Γ + Γ − − + + + + +

/2 /4 23

1( ; , ) ( / ) ( / ) ( ( / 2) ( / 2))( 2)( ( 2 ) ) , 1 ,2 4 2 2

i im n i i i i ii i i i i i i i i i i i i i i i

i

m n n t nt m n t n m t n m m n n n m t m Mm

γ = − Γ Γ + − + + − −

( )/2 /4 24

1( ; , ) ( / ) ( / ) ( ( / 2) ( / 2))( 2)( 2 ) , 1 ,2 4 2 2

i i im n n i i i i ii i i i i i i i i i i i i i i i

i

m n n t nt m n t n m t n m m n n n m n t Mm

γ + = − − Γ + Γ − − − + +



Similarly, [ ]2( / ) 1( ; , ) [0],[] , , ,[] ,

( / 2) 2 2

mi

i i i i i i i i ii i i

i i i

t n t n m m m t nt m n Gm m m

ϕ−

− − − = − Γ , M and G are the

Whittaker-M function and Meijer-G function respectively. Theorem 2.3: The Cumulant of the Moment generating function of the Sam-Solai’s Multivariate additive F- distribution is

1 2, 1 21 1

( ; , )( , ) log( ( ; , )) log ( 1)( ; , )P

p pi i i

x x x P i i ii i i i i

t m nC t t t t m n pt m n

φϕϕ= =

= + − −

∑ ∑

(18)

Proof: It is found from

1 2 3, 1 2 3,, , 1 2 3, , , 1 2 3,( , , ) log( ( , , ))p pF F F F p F F F F pC t t t t M t t t t

Theorem 2.4: The Characteristic function of the Sam-Solai’s Multivariate additive F- distribution is

1 2, 1 211

( ; , )( , ) ( ; , ) ( 1)

( ; , )P

p pj j j

x x x P j j jjj j j j

t m nt t t t m n p

t m nω

φ ψψ==

= − −

∑∏

(19)

Proof: Let the characteristic function of a multivariate distribution is given as

1

1 2 3,, , 1 2 3, 1 2 310 0 0

( , , ) ( , , , )

pt Fj j

j

p

i p

F F F F p p jj

t t t t e g F F F F dF

1

1 2 3,

12

, , 1 2 3,110 0 0 2

,1( / )(( / ) )2 1( , , ) ( 1)

, (1 ( / ) ) ,2 2 2

p jt Fj j

j

p j

mji p

j j j j iF F F F p n

j j j jj j j

mB

m n m n Ft t t t e p

m n mB m n F B

11 121 (1 ( / ) )j

p p

jmj j

j j j

dFm n F

(19a)

From (19a), it observed 1/2 /2 /22 4

210 2 2

( / ) ( / )1 ( ; , )( / 2) ( / 2)( 4)

, (1 ( / ) )2 2

jj j j j j

j j

j j

mm m it n m

it F j j j j j jj r j j jm n

rj j j j j j jj j j

m n F it n m ee dF t m n

m n it n m n nB m n F

γ− − −∞

+ =

−=

Γ Γ − +

∑∫

[ ]1/2 2 2

10 2

( / ) ( / ) 11 [0],[] , , ,[] ,( / 2) 2 2

,1 (1 ( / ) )2

j jj

j j

j

m mm

it F j j j j j j j j j j j jjm

j j j jj j j

m n F it n it n m m m it ne dF G

m m m mB m n F

− −∞

+

− − − = − Γ +

∫

Thus, by integration 1 2, 1 2

11

( ; , )( , ) ( ; , )( ( 1))

( ; , )P

p pj j j

x x x P j j jjj j j j

t m nt t t t m n p

t m nω

φ ψψ==

= − −∑∏

where /2 /2 4

21

( / )( ; , ) ( ; , )

( / 2) ( / 2)( 4)

j j j jm it n mj j j

i i i r j j jrj j j j j

it n m et m n t m n

it n m n nω γ

− −

=

−=

Γ Γ − ∑ and

/2 /41

1( ; , ) ( / ) ( / ) ( ( / 2) ( / 2))( 2)( 2) 1, 1 ,2 4 2 2

j jm n j j j j jj j j j j j j j j j j j j j

j

m n n it nt m n m it n m it n m m n n m M

mγ

= − − − Γ Γ + + + + − −

( )/2 /42

1( ; , ) ( / ) ( / ) ( ( / 2) ( / 2))( 2)( 2) 1, 1 ,2 4 2 2

j j jm n n j j j j jj j j j j j j j j j j j j j j j

j

m n n it nt m n m it n m it n m m n n n m n M

mγ +

= − Γ + Γ − − + + + + +

/2 /4 23

1( ; , ) ( / ) ( / ) ( ( / 2) ( / 2))( 2)( ( 2 ) ) , 1 ,2 4 2 2

j jm n j j j j jj j j j j j j j j j j i j j j j

j

m n n it nt m n it n m it n m m n n n m it m M

mγ

= − Γ Γ + − + + − −

( )/2 /4 24

1( ; , ) ( / ) ( / ) ( ( / 2) ( / 2))( 2)( 2 ) , 1 ,2 4 2 2

j j jm n n j j j j jj j j j j j j j j j j j j j j j

j

m n n it nt m n it n m it n m m n n n m in t M

mγ +

= − − Γ + Γ − − − + +



Similarly, [ ]2( / ) 1

( ; , ) [0],[] , , ,[] ,( / 2) 2 2

jm

j j j j j j j j jj j j

j j j

it n it n m m m it nt m n G

m m mψ

− − − −

= − Γ , M and G are the

complex Whittaker-M function and complex Meijer-G function respectively. Theorem 2.5: The survival function of the Sam-Solai’s Multivariate additive F- distribution is

1 2 31 1

( ; , )( , , , ) 1 ( 1) ( ; , )

( ; , )

ppi i i i

p i i i ii ii i i i

F m nS F F F F p F m n

F m nφ

ωω= =

= − − − ∑ ∏ (20)

Proof: Let the survival function of a multivariate distribution is given as

1 2 3 1 2 3( , , , ) 1 ( , , , )p pS F F F F G F F F F= −

31 21

2

1 2 31 11 10 0 0 0 2 2

( ,1) ( / )(( / ) )12( , , , ) 1 ( 1)( , ) (1 ( / ) ) ( ,1)(1 ( / ) )2 2 2

ip

i i

miFFF F ppi i i i i

p in mi ii i i

i i i i i i

mB m n m n uS F F F F p du

m n mB m n u B m n u

−

− += =

= − − − + +

∑ ∏∫ ∫ ∫ ∫

1

2

0 2 2

1 2 311 2

10 2

( / )(( / ) )

, (1 ( / ) )(2 2( , , , ) 1 ( 1)

( / )(( / ) )

,1 (1 ( / ) )2

ii

i i

ii

i

mF

i i i i iim n

i ii i ip

ip m

Fii i i i i

imi

i i i

m n m n u dum nB m n u

mS F F F F p

m n m n u dumB m n u

−

+

−=

+

+ = − − − +

∫

∑∫

12

11 0 2

/ )(( / ) )

,1 (1 ( / ) )2

ii

i

mFp

i i i iim

i ii i i

n m n udu

mB m n u

−

+= +

∏ ∫

1 2 31 1

( ; , )( , , , ) 1 ( 1) ( ; , )

( ; , )

ppi i i i

p i i i ii ii i i i

F m nS F F F F p F m n

F m nφ

ωω= =

= − − − ∑ ∏

where 1

2 2

0 2 2

( / )( ; , )

, (1 ( / ) )2 2

i ii

i i

m mF

i i ii i i i im n

i ii i i

m n uF m n du

m nB m n uφ

−

+=

+

∫ and 1

2 2

10 2

( / )( ; , )

,1 (1 ( / ) )2

i ii

i

m mF

i i ii i i i im

ii i i

m n uF m n du

mB m n uω

−

+=

+

∫ are the

lower incomplete Beta integrals of ith F-variables Theorem 2.6: The hazard function of the Sam-Solai’s Multivariate additive F- distribution is

12

1 11 12 2

1 2 3

1 1

( ,1) ( / )(( / ) )12{( ) ( 1)}( , ) (1 ( / ) ) ( ,1)(1 ( / ) )2 2 2( , , , )

( ; , )1 {( ) ( 1)} ( ; , )( ; , )

i

i i

mipp

i i i i in m

i ii i ii i i i i i

p ppi i i i

i i i ii ii i i i

mB m n m n Fpm n mB m n F B m n Fh F F F F

F m n p F m nF m n

φ ωω

−

− += =

= =

− −+ +

=− − −

∑ ∏

∑ ∏ (21)

Proof: It is obtained from

1 2 31 2 3

1 2 3

( , , , )( , , , )

( , , , )p

pp

g F F F Fh F F F F

S F F F F=

and 1 2 3 1 2 3( , , , ) 1 ( , , , )p pS F F F F G F F F F= −

Theorem 2.7: The Cumulative hazard function of the Sam-Solai’s Multivariate additive F- distribution is

1 2 31 1

( ; , )( , , , ) lo g1 ( 1) ( ; , )

( ; , )

ppi i i i

p i i i ii ii i i i

F m nH F F F F p F m n

F m nφ

ωω= =

= − − − − ∑ ∏ (22)



Proof: Let the Cumulative hazard function of a multivariate distribution is given as

1 2 3 1 2 3( , , , ) log(1 ( , , , ))p pH F F F F G F F F F= − −

1 2 3 1 2 3( , , , ) log( ( , , , ))p pH F F F F S F F F F= −

1 2 31 1

( ; , )( , , , ) lo g1 ( 1) ( ; , )

( ; , )

ppi i i i

p i i i ii ii i i i

F m nH F F F F p F m n

F m nφ

ωω= =

= − − − − ∑ ∏

SECTION 3: SOME SPECIAL CASES Results 3.1: The uni-variate marginal of the Sam-Solai’s multivariate additive F- distribution of kind-2 are the uni-variate F-distributions. Result 3.2: From (1) and if P=1, then the Sam-Solai’s multivariate additive F- density is reduced to density of univariate F-distribution. Result 3.3: From (1) and if P=2, then the density of Sam-Solai’s Multivariate F- distribution of Kind-2 was reduced into

1 1 2 2

1 2 1

1 2 1 12 2 2 2

1 1 1 2 2 21 2

1 1 11 1 2 2 1 22 2 21 1 1 2 2 2 1 1 1 2 2

,1 ,1( / ) ( / )2 2( , ) 1

, (1 ( / ) ) , (1 ( / ) ) ,1 (1 ( / ) ) ,1 (1 ( / )2 2 2 2 2 2

m m m m

n n m

m mB Bm n F m n Fg F F

m n m n m mB m n F B m n F B m n F B m n

− −

− − +

= + −

+ + + +

2 12

2 )m

F+

(23)

where 1 20 ,F F , 1 2 1 2, , , 0m m n n

This is called Sam-Solai’s Bi-variate additive F- distribution of kind-2

Result 3.4: The table 1 and Bi-variate probability surface for (23) shows the selected simulated standard Bi-variate (Negative) correlations between two F-variables which are bounded between 0 and -1 calculated from 10,000 different combinations for 1 1 2 2, , ,m n m n d.f.

Table 1: Simulation of Bi-variate F- correlations for different combinations of d.f

Runs m1 n1 m2 n2 1 2( , )x xρ 3416 0.50 1.70 0.10 2.90 -0.9941 8582 3.30 1.70 0.10 2.50

1139 2.10 0.50 0.90 1.70 -0.9029 6648 3.70 0.90 2.10 0.10 -0.8000 4278 2.90 0.10 2.10 0.50 -0.7006 8267 1.70 0.50 0.50 3.30 -0.6013 6888 3.30 1.70 0.50 2.10 -0.5004 3146 0.90 1.30 3.30 0.10 -0.4003 9226 2.90 3.30 1.30 0.90 -0.3001 3954 2.90 0.50 0.90 3.30 -0.2001 8483 3.30 0.50 2.90 2.90

-0.1000 8603 2.90 0.90 0.90 3.70 8103 0.10 3.70 3.70 3.70 -0.0020



Result:3.5 From(1) and if ' 1 /i iF F= ,then the Sam-solai’s Multivariate additive F- distribution of Kind-2 with

( , )i im n d.f transformed into same Sam-solai’s Multivariate additive 'F -distribution of Kind-2 with ( , )i in m d.f and its density function is given as

1' 2

1 2 31 11 1' '2 2

,1(( / ) ) ( / )2( , , , ) ( 1)

, (1 ( / ) ) ,1 (1 ( / ) )2 2 2

i

i i

nipp

i i i i ip n m

i ii i ii i i i i i

mBn m F n m

g F F F F pm n mB n m F B n m F

−

− += =

= − − + +

∑ ∏ (24)

where '0 iF , , 0i in m Result:3.6 From(1) and if ( / ) / (1 ( / ) )i i i i i i ix m n F m n F= + , 2i im a , 2i in b then the density of Sam-Solai’s Multivariate additive F- distribution of Kind-2 transformed into Sam-Solai’s Multivariate additive Beta distribution of Kind-1 of Type-B with shape parameters ( , )i ia b and its density function is given as

1 1

1 2 31 1

(1 ) ( ,1)( , , , ) ( 1)

( , ) ( ,1)

i ib appi i i

pi ii i i

x B a xg x x x x p

B a b B a

− −

= =

− = − − ∑ ∏ (25)

where 0 1ix , 0i ia b Result:3.7 From (1) and if /i i i ix m F n= , 2i im a , 2i in b ,then the Sam-solai’s Multivariate additive F- distribution of Kind-2 transformed into Sam-solai’s Multivariate additive Beta distribution of Kind-2 of Type-Awith shape parameters ( , )i ia b and its density function is given as

1

1 2 3 1 11 1

( ,1) 1( , , , ) ( 1)( , ) (1 ) ( ,1)(1 )

i

i i

appi i

p b ai ii i i i i

B a xg x x x x pB a b x B a x

−

− += =

= − − + + ∑ ∏ (26)

where 0 ix , 0i ia b Result:3.8 From (1) and if log / 2i iZ F= , then the Sam-solai’s Multivariate additive F- distribution of Kind-1

transformed into Sam-solai’s generalization of Multivariate Fisher’s Z distribution of Kind-1 with ( , )i im n d.f and its density function is given as

2 2

1 2 31 11 12 22 2

( ,1) (( / ) )12( , , , ) ( 1) 2, (1 ( / ) ) ,1 (1 ( / ) )

2 2 2

ii

i ii i

miZpp

P i ip n m

i ii i Z Zii i i i

mB m n eg F F F F p

m n mB m n e B m n e− += =

= − − + +

∑ ∏ (27)

whereiZ , , 0i im n

Result:3.9 From(1) and if logi ix F= − , then the Sam-solai’s Multivariate additive F- distribution of Kind-1 transformed into Sam-solai’s generalization of Multivariate Logistic- F distribution of Kind-1 with parameters ( , )i im n and its density function is given as

2

1 2 31 11 12 2

,1(( / ) )12( , , , ) ( 1)

, (1 ( / ) ) ,1 (1 ( / ) )2 2 2

ii

i ii i

mixpp

i ip n m

i ii i x xii i i i i

mBm n e

g x x x x pm n mB m n e B m n e

−

− += =− −

= − − + +

∑ ∏ (28)

whereix , , 0i im n

CONCLUSION The multivariate generalization of F- distribution in an additive form of Sam-Solai’s generalization having some interesting features. At first, the marginal univariate distributions of the Sam-Solai’s Multivariate additive F- distribution are univariate and enjoyed the symmetric property. Secondly, the Correlation co-efficient between any two F-variables of the proposed distribution is bounded between -1 and +1 for certain Fractional d.f and the authors established the simulated standard bivariate correlations. Finally, the multivariate generalization of F-distribution in an



additive form open the way for the same additive form of the transformation of Sam-Solai’s Multivariate additive F- distribution of kind-2 into Multivariate additive Beta distribution of Kind-1 of Type-B, Multivariate Beta-distribution of Kind-2 of Type-B, Multivariate Fisher’s Z-distribution of Kind-2 and Multivariate Logistic-F distribution of kind-2. REFERENCES [1] Arnold, B. C., Castillo, E. and Sarabia, J. M, 2006, Families of multivariate distributions involving the Rosenblatt construction. Journal of the American Statistical Association, 101, 1652-1662. [2] El-Bassiouny, A. H. and Jones, M. C, 2007, A bivariate F distribution with marginals on arbitrary numerator and denominator degrees of freedom, and related bivariate beta and t distributions. Technical Report, The Open University, UK. [3] Fang, K. T., Kotz, S. and Ng, K. W, 1990, Symmetric Multivariate and Related Distributions. Chapman and Hall, London. [4] Konno.Y, 1988, Exact moments of the multivariate F and Beta distributions, Journal of the Japan Statistical Society, 18, 123-130. [5] Jones, M.C, 2001, Multivariate t and the beta distributions associated with the multivariate F distribution, Metrika 54 215-231. [6] Leo A. Aroian, 1941, A Study of R. A. Fisher's z- Distribution and the Related F Distribution, The Annals of Mathematical Statistics. Volume 12, Number 4, 429-448. [7] Leung, P. L., 1994, An identity for the non-central Wishart distribution with application. J. Multivariate Anal. 48, 107-114. [8] Nico F. Laubscher, 1960, Normalizing the Noncentral t and F Distributions, The Annals of Mathematical Statistics. Volume 31, Number 4, 1105-1112. [9] Pham-Gia, T. and Duong, Q.P, 2003, The Generalized Beta- and F-Distributions in Statistical Modelling. Mathematical and Computer Modelling, 12, 1613-1625. [10] Saralees Nadarajah, Samuel Kotz, 2007, The Compound F-Distribution, Missouri Journal of Mathematical Sciences. Volume 9, Issue 3, 200-212. [11] Seymour Geisser, Samuel W. Greenhouse, 1958, An Extension of Box's Results on the Use of the F- Distribution in Multivariate Analysis, The Annals of Mathematical Statistics. Volume 29, Number 3, 885-891. [12] G. C. Tiao & I. Guttman, 1965, The inverted Dirichlet distribution with applications, "Amer. Statist. Assoc., v. 60, pp. 793-805. MR 32 #3177. [13] Yasunori Fujikoshi, Satoru Mukaihata, 1993, Approximations for the quantiles of Student's t and F-distributions and their error bounds, Hiroshima Mathematical Journal. Volume 23, Number 3, 557-564.

Source of support: Nil, Conflict of interest: None Declared

A NEW GENERALISATION OF SAM-SOLAI’S MULTIVARIATE ADDITIVE F- DISTRIBUTION OF KIND-2

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