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A note of the convolution of a generalised F-distribution
Statistics & Probability North-Holland
Letters 12 (1991) 315-316
October
A note of the convolution F-distribution Harold
1991
of a generalised
Exton
“Nyuggel’: Lunabister, Dunrossness, Shetland ZE2 9JH, United Kingdom Received October 1990 Revised November 1990
Abstract: The distribution function of the sum of two independent generalised F variates has been obtained by Dyer (1982) in terms of a triple hypergeometric function. In this note, this function is reduced to a well-known hypergeometric function of two variables. This offers distinct advantages, in particular with regard to numerical implementation. Keywords:
Generalized
F-distribution.
The ratio of two independent gamma variates is often referred to as the generalised F variate. Random variables written in terms of the sum of two independent generalised F variates arise in several rather different contexts. Examples are (a) a certain type of two-component series system serviced by a single repairman, (b) a test of hypothesis on the mean vector of the covariance matrix of a multivariate normal distribution with equal variances and covariances, and (c) the distribution of a predictive region for a future vector observation associated with Bayesian analysis of growth curves. In the course of a study of the convolution of generalised F distributions, Dyer (1982) has encountered the triple hypergeometric function D(+i,
G2) = Ml
+A
-ml,
Pl+P2+1,
pzT p2; pl,
1 +p2 - m2, pl;
pl+P2+lr
Pl+P2+1;
919
~2~1-o-M1-+2,))~
(1)
where &(a, =
b, b; d, c, d; e, e, e; x, Y, z)
E
(a, m)(b, n +p)(d, m +P)(c, (e,
m,n,p=o
n)xmynzP
m + n +p)m!n!p!
(2)
,
and, as usual, (a,
m)=a(a+l)(a+2)...
(a+m-l)=T(a+m)/T(a),
(a,
0) = 1.
(3)
See Exton (1976, p. 68), Saran (1954) and Srivastava and Karlsson (1985, p. 43). The function FT possesses a transform of Euler type, so that (2) is equal to (1-y)-hF,(a,
0167-7152/91/$03.50
b, b; d, e-c-d,
0 1991 - Elsevier Science Publishers
d; e, e, e; x, -&,
B.V. All rights reserved
z),
(4 315
Volume 12, Number 4
STATISTICS & PROBABILITY
LETTERS
October 1991
(5.14) of Saran (1954, p. 89), in which yz should read yi. Hence, (1) may be written in the
see equation
form N49,
h)
=
(1-
~2)-P2C-
1 +pl -ml,
p2, p2; pl,
p1+p2+l,
p,+p2+1,
m2,
PI;
( p1+p2+1;
91,
.
&!+Jl
(5)
i It will be seen that the first and third arguments of (5) are equal and that (P2, q&Y
$2)
=
(1
-dP2
5
nzO
xF,(
The
4(m2,
n)
(p*+p2+L
n)n!
pl, 1 +pl
(P2 ( 92-l
n 1
- ml, p2 + n; PI
+P2
+
1+
n;
+I,
+I).
(6)
Appell function of the first kind, F,, is reducible, and we have F,(p,,l+p,-m,,
p2+n;
P,+P2+1+n,
=2E;(p,,l+p,+pZ-ml+n;
+1,+1)
P1+P2+l+n;
+,I.
(7)
Appell and Kampe de Ftriet (1926, p. 34). The Euler transform for the Gauss may now be applied, and (6) becomes
See
(1 - c?l
(
Ply m,;
p,+p,+l+n;
G2) = (I-
+1)-“‘(1
-
function
43
91-l
See ErdClyi (1953, p. 109). If this expression symmetrical representation D(Gl,
-
hypergeometric
i .
is inserted into (5), we have, after a little reduction,
&-‘*F3
pl,
p2,
ml,
m2;
PI +P2
+ 1;
&,
.
j$
1
i
the
(9)
The Appell function of the third kind, F3, is given by F,(a,
a’,
b, b’; c; x, y) =
E
(a,
m,n=O
m)(a’,
n)(b, m)(b’, (c, m + n)m!n!
~)x”Y”
See Appell and KampC de F&et (1926, p. 14). The representation (9) is much more tractable than Dyer’s original expression (1). Additional relevent results are to be found in Olsson (1977) and Buschman (1987) for example.
References Appell, P. and J. Karnpt de F&et (1926) Fonctions HypergPometriques et Hypkrsphkriques (Gauthier Villars, Paris). Buschman, R.G. (1987), Contiguous relations for Appell functions, Indian J. Math. 29, 165-171. Dyer, D. (1982), The convolution of generalized F distributions, J. Amer. Statist. Assoc. 77, 184-189. Erdelyi, A. (1953). Higher Transcendental Functions, Vol. I (McGraw-Hill, New York). 316
Exton, H. (1976) Multiple Hypergeometric Functions and Applications (Halsted Press, New York). Olsson, P.O.M. (1977) On the integration of the differential equations of five-parametric double hypergeometric functions of second order, J. Math. Phys. 18, 1285-1294. Saran, S. (1954), Hypergeometric functions of three variables, Ganita 5, 77-91. Srivastava, H.M. and P. Karlsson (1985). Multiple Gaussian Hypergeometric Series (Halsted Press, New York).
Letters 12 (1991) 315-316
October
A note of the convolution F-distribution Harold
1991
of a generalised
Exton
“Nyuggel’: Lunabister, Dunrossness, Shetland ZE2 9JH, United Kingdom Received October 1990 Revised November 1990
Abstract: The distribution function of the sum of two independent generalised F variates has been obtained by Dyer (1982) in terms of a triple hypergeometric function. In this note, this function is reduced to a well-known hypergeometric function of two variables. This offers distinct advantages, in particular with regard to numerical implementation. Keywords:
Generalized
F-distribution.
The ratio of two independent gamma variates is often referred to as the generalised F variate. Random variables written in terms of the sum of two independent generalised F variates arise in several rather different contexts. Examples are (a) a certain type of two-component series system serviced by a single repairman, (b) a test of hypothesis on the mean vector of the covariance matrix of a multivariate normal distribution with equal variances and covariances, and (c) the distribution of a predictive region for a future vector observation associated with Bayesian analysis of growth curves. In the course of a study of the convolution of generalised F distributions, Dyer (1982) has encountered the triple hypergeometric function D(+i,
G2) = Ml
+A
-ml,
Pl+P2+1,
pzT p2; pl,
1 +p2 - m2, pl;
pl+P2+lr
Pl+P2+1;
919
~2~1-o-M1-+2,))~
(1)
where &(a, =
b, b; d, c, d; e, e, e; x, Y, z)
E
(a, m)(b, n +p)(d, m +P)(c, (e,
m,n,p=o
n)xmynzP
m + n +p)m!n!p!
(2)
,
and, as usual, (a,
m)=a(a+l)(a+2)...
(a+m-l)=T(a+m)/T(a),
(a,
0) = 1.
(3)
See Exton (1976, p. 68), Saran (1954) and Srivastava and Karlsson (1985, p. 43). The function FT possesses a transform of Euler type, so that (2) is equal to (1-y)-hF,(a,
0167-7152/91/$03.50
b, b; d, e-c-d,
0 1991 - Elsevier Science Publishers
d; e, e, e; x, -&,
B.V. All rights reserved
z),
(4 315
Volume 12, Number 4
STATISTICS & PROBABILITY
LETTERS
October 1991
(5.14) of Saran (1954, p. 89), in which yz should read yi. Hence, (1) may be written in the
see equation
form N49,
h)
=
(1-
~2)-P2C-
1 +pl -ml,
p2, p2; pl,
p1+p2+l,
p,+p2+1,
m2,
PI;
( p1+p2+1;
91,
.
&!+Jl
(5)
i It will be seen that the first and third arguments of (5) are equal and that (P2, q&Y
$2)
=
(1
-dP2
5
nzO
xF,(
The
4(m2,
n)
(p*+p2+L
n)n!
pl, 1 +pl
(P2 ( 92-l
n 1
- ml, p2 + n; PI
+P2
+
1+
n;
+I,
+I).
(6)
Appell function of the first kind, F,, is reducible, and we have F,(p,,l+p,-m,,
p2+n;
P,+P2+1+n,
=2E;(p,,l+p,+pZ-ml+n;
+1,+1)
P1+P2+l+n;
+,I.
(7)
Appell and Kampe de Ftriet (1926, p. 34). The Euler transform for the Gauss may now be applied, and (6) becomes
See
(1 - c?l
(
Ply m,;
p,+p,+l+n;
G2) = (I-
+1)-“‘(1
-
function
43
91-l
See ErdClyi (1953, p. 109). If this expression symmetrical representation D(Gl,
-
hypergeometric
i .
is inserted into (5), we have, after a little reduction,
&-‘*F3
pl,
p2,
ml,
m2;
PI +P2
+ 1;
&,
.
j$
1
i
the
(9)
The Appell function of the third kind, F3, is given by F,(a,
a’,
b, b’; c; x, y) =
E
(a,
m,n=O
m)(a’,
n)(b, m)(b’, (c, m + n)m!n!
~)x”Y”
See Appell and KampC de F&et (1926, p. 14). The representation (9) is much more tractable than Dyer’s original expression (1). Additional relevent results are to be found in Olsson (1977) and Buschman (1987) for example.
References Appell, P. and J. Karnpt de F&et (1926) Fonctions HypergPometriques et Hypkrsphkriques (Gauthier Villars, Paris). Buschman, R.G. (1987), Contiguous relations for Appell functions, Indian J. Math. 29, 165-171. Dyer, D. (1982), The convolution of generalized F distributions, J. Amer. Statist. Assoc. 77, 184-189. Erdelyi, A. (1953). Higher Transcendental Functions, Vol. I (McGraw-Hill, New York). 316
Exton, H. (1976) Multiple Hypergeometric Functions and Applications (Halsted Press, New York). Olsson, P.O.M. (1977) On the integration of the differential equations of five-parametric double hypergeometric functions of second order, J. Math. Phys. 18, 1285-1294. Saran, S. (1954), Hypergeometric functions of three variables, Ganita 5, 77-91. Srivastava, H.M. and P. Karlsson (1985). Multiple Gaussian Hypergeometric Series (Halsted Press, New York).