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A Remark on the Convolution of the Generalised
Logistic Random Variables
Matthew Oladejo Ojo
∗
Department of Mathematics,
Obafemi Awolowo University,
Ile-Ife, Nigeria.
Abstract
In this paper, the distribution of the sum of independent random variables
from the generalized logistic distribution is determined by using a much simpler
and shorter method than had earlier been used.
1.
INTRODUCTION
A generalization of the logistic random variable whose probability density function
and the characteristic function are given respectively as
f (x) =
∗
Γ(p + q)
epx
,
Γ(p)Γ(q) (1 + ex )( p + q)
− ∞ < x < ∞, p > 0, q > 0
(1.1)
Research supported by the University Research Committee of Obafemi Awolowo University, Ile-
Ife, Nigeria.
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and
Ψ(s) =
Γ(p + is)Γ(q − is)
Γ(p)Γ(q)
(1.2)
has earlier been considered by George and Ojo (1980). They evaluated the cumulants
and showed how its distribution function can be approximated by the t-distribution.
The importance of the sampling distribution of the sample mean of a population is
well known. On the basis of this, the distribution of the sum of independent and
identically distributed random variables from a generalised logistic population has been
determined in an earlier paper Ojo and Adeyemi, (1989) by using the Mittag-Leffler
expansion of the characteristic function. In this paper a much simpler and shorter
method is used to obtain the same distribution.
2.
THE PROBABILITY DENSITY FUNCTION OF THE CONVO-
LUTION OF THE GENERALISED RANDOM VARIABLES
Let X1 , ..., Xn be n independent and identically distributed random variables each
having the distribution (1.1). Let
Y =
n
X
Xi
i=1
We want to determine the distribution of Y . The characteristic function of Y is given
as
Ψn (s) =
(Γ(p + is))n (Γ(q − is))n
(Γ(ip))n (Γ(q))n
and by inversion formula, the density function of Y is given as
1 Z +∞ −isy
e
Ψn (s)ds
fn (y) =
2π −∞
−n
= (Γ(p))
−n
(Γ(q))
1 Z +∞ −isy
e
(Γ(p + is))n (Γ(q − is))n ds
2π −∞
(2.1)
Now if we let p + is = −z, we have
py
−n
fn (y) = e (Γ(p))
−n
(Γ(q))
1 Z −p−i∞ py
e (Γ(−z))n (Γ(z + p + q))n dz
2πi −p+i∞
2
(2.2)
If g(z) denotes the integrand in (2.2). It can be shown that |z m g(z)| → 0 as z → ∞ so
that the integral converges and then
fn (y) = epy (Γ(p))−n (Γ(q))−n
1 Z py
×
e (Γ(−z))n (Γ(z + p + q))n dz
2πi C
(2.3)
where C is the contour bounded by the line x = −p and that part of the circle |z| =
m + 21 ,
m → ∞ which lies to the right of the straight line. The contour is then
transversed in a counter-clockwise direction.
Now
Γ(−z) =
−π
(Γ(z + 1)sinπz)
So that (2.3) becomes
fn (y) = −epy π n (Γ(p))−n (Γ(q))−n (−1)n
×
1 Z
(Γ(z + 1))−n (Γ(z + p + q))n (sinπz)−n eyz dz
2πi C
The poles of the integrand in (2.4) are of nth order and are those of Γ(−z), viz, z =
r, r = 0, 1, 2, ... and so the integrand in (2.4)is 2πi times the sum of the residues of the
poles within the contour C. Thus the density function of Y is given as
fn (y) = epy (Γ(p))−n (Γ(q))−n
×
∞
X
(−1)n+nr+1 dn−1
r=0
(n − 1)! dz
[eyz
n−1
(Γ(z + p + q))n
]z=r
(Γ(z + 1))n
This density function can be evaluated for fixed values of p and q. For example when
p = q = 1, we get
fn (y) =
∞
X
(−1)n+nr+1
r=0
dn−1 yz
e [ n−1 e (z + 1))n ]z=r
(n − 1)!
dz
y
When n = 1, the density function (2.4) reduces to
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fn (y) =
ey
(1 + ey )2
which is to be expected. For higher values of n and various combinations of p and q,
the infinite series form obtained for the density function can be expressed as a finite
term by using the following lemma.
Lemma 2.1
Let m be a positive integer. Then
∞
X
(a)
k m eky =
k=1
∞
X
(b)
ey
dm
(
)
dy m (1 − ey )
(−1)k−1 k m eky =
dm
ey
(
)
dy m (1 + ey )
k m e−ky = (−1)m
dm
e−y
(
)
dy m (1 − e−y )
k=1
∞
X
(c)
k=1
(d)
∞
X
(−1)k−1 k m e−ky = (−1)m
k=1
dm
e−y
)
(
dy m (1 + e−y )
Proof: The proof of the lemma follows immediately by differentiating m times the
following identities
∞
X
(i)
eky =
k=1
∞
X
(ii)
ey
(1 − ey )
(−1)k−1 eky =
k=1
(iii)
∞
X
−ky
e
k=1
(iv)
∞
X
e−y
=
(1 − e−y )
k−1 −ky
(−1)
e
k=1
ey
(1 + ey )
e−y
=
(1 + e−y )
The distribution function of Y can be evaluated for various combinations of p and q.
For instance when p = q = 1, the distribution function is given by


(−1)n+1  n − 1 
j!
Fn (y) =
(j − n + k + 1)!
k
j=0 k=0 (n − 1)!
n−1
X n−1
X
4
×
∞
X
(−1)nr rj+k+1−n e(r+1)y
r=0
k
X
(−1)s k!y k−s
s=0
(k − s)!
When n = 1, the distribution function reduces to
Fn (y) =
ey
(1 + ey )
which is to be expected.
References
[1] George, E.O. and Ojo, M.O. (1980) “A generalization of the logistic distribution”,
Annals of the Institute of Statistical Mathematics, vol. 32, no. 2A, 161-169.
[2] Ojo, M.O. and Adeyemi, H.O. (1989) “On the convolution of generalised logistic
random variable” Statistica SLIX, no. 3, 453-459.
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