Applications of generalized H-function in Bivariate distributions

              Rev. Acad. Canar. Cienc., XIV (Núms. 1-2), 111-120 (2002)
Applications of Generalized H-function in Bivariate Distributions
*R.K.Saxena, *Chena Ram and **S.L.Kalla
*Department of Mathematics and Statistics
Jai Narain Vyas University, Jodhpur - 342 005, India
**Department of Mathematics and Computer Science
Kuwait University, P.O. Box 5969, Safat 13060, KUWAIT
Corresponding author: kalla@mcs.sci.kuniv.edu.kw
Abstract. Certain properties of a new probability density function defined in terms of generalized
H-function due to Inayat-Hussain are investigated. The generalized phase and radial density
functions are derived by making use ofthe weighted mixture representation, where the independent
x and y component distributions are not the same. The results derived provide generalization and
unification of the results given by severa! authors including McNolty and Tomsky , Saxena and
Sethi and Srivastava.
AMS Subject classification 33C40, 33C90, 60E05
Key words: Bivariate distributions, Phase and radial distributions, H-function,
H -function, confluent hypergeornetric distribution.
l. Introduction
In certain military research problern, McNolty and Tornsky [14] expressed the input of a
square law detector systern as
s(t) = u(t) + v(t) = A(t)[coswc(t) + ~(t)] = A(t)c(t),
where u(t) denotes the vectorial surn of severa! voltage returns excluding receiver noise,
v(t) any narrow band rnodulation of the carrier frequency Wc , and x and y cornponents of
the arnplitude A(t) ofthe phase carrier C(t) are connected by the relation
A2(t) = x2(t) + y(t). (1.1)
They further discussed the distributions of the randorn variables ~(t) and A(t) and
investigated the case when x and y cornponents of A(t) are not the sarne and are even
functions of the variates. Saxena and Sethi [19] introduced sorne statistical distributions
associated with generalized hypergeornetric functions which provide extension of the
results of McNolty and Tornsky [14]. Srivastava [20] considered a general case of
McNolty and Tomsky and Saxena and Sethi's problern, when x and y cornponents contain
sorne generalized probability distribution associated with Fox's H-function. In the present
. -
paper the authors introduce a generalized probability density in terms of the H -function
due to Inayat - Hussain [6] and derive its irnportant properties which provide unification
and extension of the work on bivariate distributions due to McNolty and Tornsky
[14],Saxena and Sethi [19] and Srivastava [20].
Since there are a number of parameters in the H -function defined in the next section,
without loss of generality, it is assumed that the various functions which are used in
defining the density functions in the following analysis are all non-negative.
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Applications of the generalized hypergeometric functions in the theory of statistical
distributions are available from the two monographs written by Mathai and Saxena [12 and
13]. Recently, Kalla et al [7] , Matera et al [11], Ben Nakhi and Kalla [2], Ghitany et al [5],
Al-Saqabi et al [1] and others have used generalized hypergeometric functions to define
and study a number of density functions.
2. The Generalized H-function
The H -function, introduced by Inayat - Hussain [6], in terms of Mellin- Barnes type
contour integral, is defined by
H mC,,Dn [ z 1 (aj,Aj;aj)l,n•(aj,Aj)n+l,C] =1- 1"' X (s )z sds
(/3j,B)1,m•(/3j,Bj;bj)m+1,D 2m i«>
(2.1)
flqpj -BjsYfI {r(l-aj + Ajs)}ª¡
where x(s) = ~l j=l e '
f1 {r(l-/31 +Bjs)}b1 f1r<aj-Ajs)
(2.2)
j=m+I j=n+I
which contains fractional powers of sorne of the r-functions. Here z may be real or
complex but is not equal to zero and an empty product is interpreted as unity; m, n, C and D
are integers such that1:5m:5D,0:5n:5C; Aj >0(j=1, ... ,C), Bj >0 (j = l, ... ,D) and
ªi (j = 1, ... ,C) and Pi (j = l, ... ,D) are complex numbers. The exponents ªi (j = l, ... ,n) and
bi (j = m + 1, ... ,D) take on non-integer values.
Also, from lnayat Hussain [ 6 ], it follows that
-m n [ [/3·]] H ' [z] = O(lzlg) for srnall z, where g = rnin Re -' .
~D ~~ ~
(2.3)
- m n h { [ª· -1]] and H ' [z] = O(jzj) forlargez,where h=max R a1 -'- .
C, D 1s;Sn A1
(2.4)
Buschman and Srivastava [ 3, p. 4708 ] have shown that the sufficient condition for
absolute convergence ofthe contour integral (2.1) is given by
m n D C
0 =IIB1l+Ila1A1I- L lb1B1I- IIA1I >O (2.5)
j=I j=I j=m+I j=n+I
This condition evidently provides exponential decay of the integrand in (2.1) and the region
of(absolute) convergence in (2.1) is iarg zl < .!._Jr n . (2.6)
2
Abelian theorems and complex inversion formulas for the distributional H -function
transformation are established by Saxena and Gupta [ 16, 17]. Two new characterizations for
the distributional H -function transformation are investigated by Saxena and Gupta [18].
Functional relations for the H -function are derived by Saxena [15].
When the exponents a¡ = bj =1 V i and j, the H -function reduces to the familiar Fox's H­function
defined by Fox [4]; (see also Mathai and Saxena [13]) in the following form:
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(2.7)
m n rr f(/Jj -Bjs)IT f(l-aj + Ajs)
(2.8)
J=m+ l j=n+l
an empty product is interpreted as unity; the integers m, n, p, q satisfy the inequalities
O ~ n ~ p and 1 ~ m ~ q, the coefficients Aj> O (j = 1, ... ,p), Bj >O (j = 1, ... ,q) and the
complex parameters, Clj and l3i are such that the poles of the integrand in (2.8) are simple.
Further L is a suitable contour of Mellin - Bames type in complex s-plane, separating the
poles off (l3i - Bis) for j = 1, ... ,m from those of r (1- Clj + Ajs) for j = 1, ... ,n. The integral
in (2.7) converges absolutely and defines the H-function analytic in the sector
1
!arg zl <-:¡"A , (2.9)
(2.10)
J=I J=m+I j=I j=n+l
the point z = O being tacitly excluded.
A detailed account of the H-function and its various properties are available from the
monograph of Mathai and Saxena [13). A comprehensive account of the asymptotic
expansion of the H-function is recently given by Kilbas and Saigo [8,9), making
improvements on the earlier derived results on the H-function.
Saxena [15, p. 127] has shown that H (z) makes sense and defines an analytic function of z
in the following two cases:
l. 'I' > O and O < !zl < oo ,
m D n C
where 'I' = L IB 11 + L IB 1 fJ 1 1- L IA p 1 1- L IA 1 I· (2.11)
J=I j =m +I j=I j=n+I
11. 'I' = o and o < 1 z 1 < e .¡ holds,
where () = {ri (B ) - B¡ }{rr (A) A¡a¡ }{ (I (A) A1 }{ .rr (B J )-B1P1}
¡ = I ¡=I ¡=n+I ¡=m+I
(2.12)
The importance of the results obtained in this paper lies in the fact that besides the special
functions which are special cases of Fox's H-function, the H -function also contains the
Polylogarithm of a complex order and the exact partition function of the Gaussian Model in
Statistical Mechanics.
A relation connecting Lv (z), the polylogarithm of complex order v, and the H -function is
recently derived by Saxena [15,p.127, eq. (1.12)) as
C(z) = ii: :~[zl (l,l;v) . ] (2.13)
( Ü , 1 ), ( Ü , 1 , V - 1 )
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An account of Lv (z) , the polylogarithm of complex order v is available from the book by
Marichev [10].
3. Computable representation of the H -function
'" When the poles of TI í(p1 - B1s) are simple, that is, when Bh ( !Ji + A. ) * Bi ( !Jh + v ) for
j=I
j * h ; j, h = l, ... ,m; A., v =O, 1, 2, ... ; then there holds [ 15, p. 128, eq. (3.1) ] the
following series representation ofthe H -function
'" '° (-l)k
H~:~(z) = ¿ L (k)'.Bh z(p)zP,
h=I k=O
(3.1)
p = p h + k (3.2)
Bh
which exists for O < 1 z 1 < oo , if \Ji > O or \Ji= O and O < 1 z 1 < e ·1, where \Ji is defined
in (2.11) ande in (2.12).
4. Integrals Involving H -function
In view of the well-known Mellin inversion theorem, the Mellin transform of the
function follows from the definition (2.1). We have
H-
"'J s-IH'"·" [ 1 (aJ,AJ;aJ)¡ ,.,(aJ,AJ)•+t,c ld / C ,D a/ / = a-s z(-S) ,
o ( p 1 • B 1) 1,m ' ( p J • B J; b J),. + 1,D
a-·(J r(pj + Bjs)IT {r(l-a j - Ajs)}ªj
}=I j=I
}=m+I j=n+I
p . 1 ª 1
where - min Re(-1 ) <Re( s) < --max Re(-) ,
l :Sj:Sm Bj Aj l:Sj:Sn A¡
1
\Ji ~ o' iarg a 1 < -1[ n' n > o and n is defined in (2.5) and \Ji in (2.11).
2
114
(4.1)
(4.2)
(4.3)
(4.4)
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where Re[A. + lm;>j;i>nrn (p1 / B1 )] > O,Re( b) > o,¡arg xi <.!_1dl, '1' ~o.n >O. 2
From ( 4.4), it follows that
bA. j¡x¡u-1e-bx2 H~:~[ax2 (a¡,A¡;a¡)1,n•(a¡,A¡)n+1,c ]dx
-co (p }'B )1,m '(pi' B ¡;b ¡)m+l,D
-m,n+I [ª 1 {1- A.,1), (a 1,A 1;a 1) 1,n,(a 1,A 1)n+l,C] = H C+l,D - '
b <P1,B1)1,m•<P1,B1;bj)m+l,D
where-oo<X<oo, Re(b)>O, Re(l..)>O,
Re[A. + rrrin (p1 I B)J >O, jarg aj < _!_ tr n, 'I' ~O, i;;¡;;m 2
n > o and n is defined in (2.5) and 'I' in (2.11 ).
Now we establish the integral
"'J t A-1 e -btn"e' ·n[ 1 (a1,AJ;a1)1,n•(a¡,A1)n+1,c] ,D xt
o <P1,B¡)1,rn•<P1,B1;bj)rn+l,D
n"'•·n• [ 1 (a~,A;;a~) 1,n,•(a~,A~)n,+i,c, ]dt
x c,.v, yt (P' B') (P' B' b )
}' j l,rn 1 ' }' l' j rn 1+l,D1
ft í(p; - B~t) fI {r(l - a~ + A;t) }ª1
'VI (t) = -''--·=_I ----"'-J=_l -------
A D, ~ ' 11 {r(l - p; + B;t) }bi 11 í(a~ - A~t)
j=m1+1 j=n1+J
Re[A.+rrrin (p1 !B1 )+ rnin (p¡1B;)]>o, l;>JSm ISkSm 1
(4.5)
(4.6)
(4.7)
(4.8)
Re(b)>O, larg xJ<.!_tr O, 1 arg y l<.!_tr 0 1 , '1'~0, '1'1~0; when '1'=0, then o<i~<e-1
2 2
and when '1'1 =O, then O< lzl < B1- 1;where n, 'l',B are defined in (2.5), (2.11) and
(2.12) respectively. Here
(4.9)
(4.10)
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(4.11)
-m n
( 4.6) can be established by writing the series expansion for H P 1 (yt) with the help of
C1,D1
(3.1) and integrating term by term with the help of ( 4.4).
(4.12)
where Re(.A.)>0,Re[μ+min(/Jj/Bj)]>o, largxl< !nn, 0>0 and 'I' ~O.
lSjSm 2
The next formula to be proved here is
Jr<-1(1-t)"-1 H~:~[x(l -t)I (aj, Aj;a¡)1,n• (aj'.Aj)n+1,c ldt
o (/Jj, Bj)l,m• (/Jj,Bj,bj)m+l,D
Hm,,n, [ 1 (a;,A;;a;)1,.,,(a~,A;).,+i,c, ld
X c,.v, yt (/3' B') (/3' B' b ) t
f' j l,m1 ' 'j> }' ¡ m1+1,D1
(4.13)
( 4.13) can be established by writing the series expansion of ¡¡ m,' n, (yt) with the help of
C,,D,
(3 .1) and integrating term by term using ( 4.12)
(4.13) holds for [A,+ min (/3 ~ / B ~ )] > O,
IS¡Sm 1
Re[μ+ m~n (/3¡ / Bj)] >O, iarg YI < _!_Jl' n 1 and larg xi< _!_Jl' n 'where
IS ¡ Sm 2 2
p, , x, , n and n 1 are respectively defined in ( 4. 7), ( 4.8), (2.5) and ( 4.1 O).
- -m,n [ 1 (aj,A j; a j )ln'(a j,Aj)n+IC l . In what follows, the H -function, He .v x (/3 . B ) (./3. B .. b .) · w1ll be
;' J 1,m' J' J' J m+l ,D
-m n
represented in the contracted notation by H ' ( x )
C,D
5. A Generalized density function
,< 1 12.<-I bx' -m,n [ 2] Theorem l. Let f (x) = Rb x e- H e.o ax (5.1)
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where -oo<x<oo, Re(b)>Re(A.)>O, q.i;;:::o, R{,1, +min(/Jj / Bj)J >0, iarg~ <_!.;r O, O> O,
ISJSm 2
n is defined in (2.5) and q.i in (2.11);
R = H C+ l , D -
_ 1 -m,n+ I [ª 1 (l-A-,l),(a 1,A1;a 1)1,n,(a 1,A)n+l,C]
b (f31,B)1,m'(j3J,BJ;bJ)m+1,D
(5.2)
then f(x) defined by ( 5.1) is a probability density function as it satisfies all the conditions of
a density function by virtue of the integral (4.5) and non-negativeness of f(x) already
assumed.
Now, if we denote the probability density of the x-component of the amplitude A (t) by
the relation
f(x) = Rb ..t lxl2He-bx' H~:;[ax21 (aJ,AJ;a 1) 1,n'(a J'.AJ)n+1,c ] (5.3)
(/3 j' B j )1,m' (/3 j' B j 'b j) m+l,D
then the density of the y-component of A (t) may be defined and represented by
!( ) = Sb"'I JH'-1 -by'Hm,n [ 2 1 (a~ , A~;a~)1,n, (a~,A~)n+l,C] (5.4)
y y e c,D ay (/3' , ) (/3' , b')
i' BJ 1,m' i' BJ; j m+l,D
S _1 = Hm,n+I [!:_I (l-A-',1),(a~,A;;a~) 1,n,(a~,A;).+1,c]
C+l,D b (/3' B') (/3' B' b') j' j l,m' j' j; j m+l,D
(5.5)
The distribution for u= x2 and v = y2 is clearly
g1 (u ) -- Rb..t u ..t-1 e -buHmc,,nD [ au 1 (aJ,AJ;aj)l,n'(aj,Aj)n+l,C] ,u_>O
(/Jj, Bj )1,m, (/Jj, Bj;bj )m+l,D
(5.6)
and ( )- Sb..t' _.._, -bv Hm,n [ 1 (a~,A~;a~),,.,(a~,A~).+1,c] >O (5.7)
g2 v - v e c,D av (/3' B') (/3' B'·b') ,v_
j' j l,m' j' j' j m+l,D
Theorem 2. If the distributions of u and v are defined by (5.6) and (5.7), then the
distribution for z = u/v is given by
J; (z) = RSb ..t+..t'z..t-i [b(l + z)j<..t+..t') f f (-l)~' z1 (p1) a:'
h,=o v,=o (v,). Bh,
(5.8)
(5.9)
(5.8) holds for
p, = l</3~ + v1)/ B~ j,z >0 ,Re[,1, + m#t(/31 I B)+m#t(/3; I B;)] >O, Re(b)> O,
1 1 IS;Sm IS;Sm
1 1
q¡;;:::o, \}'¡;;:::o, iarg azl<-;r O, 0>0,¡arg al <-;r O*;
2 2
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m n D C
and n· = L:JB;J- L:la~A;I- ¿ Jb;B;J- L:IA;l>o (5.10)
j=l j=l j=m+I j=n+I
f1(z) =O, elsewhere.
Proof: If we set z = u/v and 't = v, then the joint distribution of z and 't is given by
f (z, r) = g(u, v) a(u, v) = g¡ (u)g2(v)v (5.11)
a(z,r)
and hence on using (5.6) and (5.7), it gives
.,
¡;(z)= f f(z, r)dr (5.12)
o
=RSbJ.+A'zA-1 JrJ.+A'-le-b( z+I)• xH~:;[azr 1 (a¡,A¡;a ¡ )1,n•(a¡,A¡)n+1,c]
O (p¡,B)1,m•(pj ,Bj;bj )m+l,D
H m,n [ 1 (a~,A;;a~) 1 ,.,(a~,A;).+ 1 ,c ]d
x c ,v ar (p' ') CP' , , r (5.13)
¡•B¡ l,m• 1•Bj;b¡)m+1 ,v
By virtue of the formula (4.6), the result (5.8) readily follows.
Theorem 3. If the density for u and v is defined by (5.6) and (5.7), then the density of
r = ..Ju + v is given by
f2(r)=2RSbA+A'e-b'1rH+H'-1f f (-,ir• X1CP1)(ar2y•r(A.'+p1)
h1=0 V1=0 V¡ .B h1
-m,n+I [ 2 1 (1-A.,1), (a 1, A1;a 1 )1 .• 1 ,(a1, A1 )n+l,C ]
x H c+1,v+1 ar , , ,
(p¡,B¡ )1,m•(p¡,B¡;bj)m+l,D•(I-A.-A. - ppl)
(5.14)
where r >O, Pi = ( p~ +Vi J. Re[...i + mjn(/Jj I Bj )] > O,
B~ is1sm
Re[A.' + rnin (p; / B; )] >O, Jarg (ar 2 )J <_!_{minen, n•)}, Q >O, n• >O
IS}Sm 2
n· is defined in (5.10); and f2 (r) =o, elsewhere.
Proof. Ifwe use the transformation r =..Ju+ v and t = v then ¡a(u, v)I = 2r and the joint
o(r, t)
density of r and t will be given by
f(r,t) = g(u,v) ¡a(u, v) = 2g1(u)g2(v)r (5.15)
o(r,t)
(5.16)
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where O < t < r2 and O ~ r < oo .
The distribution of r is evidently the integral
r'
fi(r) = Jf( r,t)dt
o
1
= 2RSb A.+J.' e-br' rv.+v.·-i x Jtl '-i (1 - t)J.-l H ~:;[ar 2 (1 - t)] x H ~:;[<ar 2t) }it (5.17)
o
On using the result (4.13), the desired result (5.14) follows.
6. Special Cases
If we take a¡ = bi = 1 V i and j, the results of this paper give rise to the results of generalized
hypergeometric distribution associated with Fox's H-function [15) discussed earlier by
Srivastava [ 15) in the form
f(x)dx = Aw ;,lxlH-le_wx'H;:~[1cx 2 ¡(ac,ac)]dx, (6.1)
(bv,Pv)
where - oo < x < oo, Re( w) >O, Re [-t + min (b 1 /p 1 )] >O,
IS}Sm
~p. - ~ . >O A-1 =Hm,n+1l!_[(l-A.,l),(ac,ac)]
L., ¡ L., a¡ - ' C+l,D
j=I j=I X (bD,pD)
Further, ifwe use the identity [ 13, p. 159]
q rr ._1 r(b¡) [ ] (1- apl), .. ., (1- a P ,1)
,F,[a,. ... , a'; b,. ... , b,;-z] =ti H~;.., zJ(0,1), (1-b.,l), ... , (1- b,,l) '
f(a¡)
(6.2)
j=I
where p ~ q or p = q+ 1, 1 zl < 1 ;
we obtain the results for the generalized hypergeometric distribution discussed by Saxena
and Sethi [ 19, p. 174] in the form
b;, B 1 12,i-1 -bx' i f(x)dx=--x e pFq[al' ... ,aP;P1>···•Pq;kx ]dx
f(,t)
where - oo < x < oo, p :$; q, Re( A.) > O, Re( b) > O, Re( k) > O
and B-1·= p+IFq [ap···· a P ,A.; P1>···· Pq ;k 1 b],
(6.3)
which itself is a generalization of the confluent hypergeometric distribution discussed by
McNolty and Tornsky [ 14, p. 257] in the form
a V bμ _ 2 2μ+2v-I
f(x)dx = e bx lxl ,F,[v;μ + v;(b- a)x 2 ]dx
r(μ + v))
(6.4)
where - oo < x < oo, b > a ;¿>; O, v > O and μ + v > O.
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Acknowledgement
The first author expresses his gratitude to the University Grants Commission of India for
providing financia! support for the present work.
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