Internat. J. Math. & Math. Sci.
VOL. 12 NO. 3 (1989) 531-538
531
SOME PROPERTIES AND APPLICATIONS OF THE STUTTERING
GENERALIZED WARING DISTRIBUTION
J. PANARETOS
School of Engineering
Division of Applied Mathematics
University of Patras
P.O. Box 1325- Patras,
GREECE
(received July 27, 1987 and in revised form March 3, 1989)
ABSTRACT
The
in
connection
urn that contains balls of two colours
(black and
Stuttering Generalized Waring Distribution arises
with sampling from an
white) and it can be thought of as an intermingling of generalized Waring
streams (Panaretos
Because
would
of
and Xekalaki [4]).
its
be worthwhile.
as a
mixture
of
application
In this
the
demonstrated that, in
potential
a study of its properties
paper it is shown that
it can
to
stuttering
the
Distribution,
in the
Poisson type or a
as
an approximation
Waring distribution.
generalized
Keywords and Phrases:
distribution
sampling
also
is
an urn scheme, increasing the number of balls
type
blnomial
It
generalized Poisson distribution.
urn in an appropriate fashion one can end up with a
negative
be obtained
Generalized aring Distribution, Generalized Poisson
Mixtures of Distributions,
Urn Models,
Accident Theory,
Hypergeormetric Function.
AMS 1980 Subject Classification.
Primary:
62E20,
Secondary: 60E06, 60F99, 62EI0, 62P2S
1.
With
the aim
much attention.
INTRODUCTION
of preventing
In the framework
accidents, accident theory has
of some of the
various
received
hypotheses
J. PANARETOS
532
have
that
was obtained
of
as the distribution
Xekalaki
[S],
[8],
Xekalaki
[4]
introduced the
[7]).
distribution (GWD)
the generalized Waring
developed
been
(see
accidents
Generalizing
this
Irwin [2],
e.g.
Panaretos and
distribution
stuttering generalized Warlng distribution
(SGWD) in the context of an un scheme
of generalized Waring streams and is defined by
is an intermingling
This
the probability function (p.f)
k
’Zxt
C(zm
(m)
(x
P(X=x)=
"+c+Yml
k
(Yx)
X
i:l
]
J=l
(1.1)
where
a)denotes
The
the ratio FC+)/F(a), >0,
probability
function
generating
R
x=O,l,2,..
(p.g.f)
of this distribution
is given by
C(Zm
G(s)=
F D(;m
+c
where F
(1.2)
s
mk;+mi+c’s,s
(Zm)
denotes Lauricella’s hypergeometric series of
D
FD(CX;
1, 8 2,
=-
k; X+-i+’; S
Sk
2,
S
k
r =o
k
definition
of
For
k.
i=I
this
one obtains
k=l
distribution
k
k
"
(+Xi+)(Zr)
r =o
s
s
S
type D defined by
enhances
the
GWD.
So
the
the application potential
of the GWD as the underlying mechanism causing accidents as well as various
other phenomena in many diverse areas ranging from linguistcs to inventory
control.
The
reason lies in
where single events,
events can be
k-variate
GWD.
of
that
the
involving
cars
(1.1) can be employed in situations
pairs of events,
thought of as
In the context
that the SGWD would
number
the fact that
triplets of events
beeing jointly distibuted
of car
in
accidents
if
the distibution
this implies
of the total
is reasonable
it
joint distribution of the numbers
one, two
according to the
accident statistics
be expected to describe
involved
k-plets of
X
X 2,
Xk
to
of
k cars simultaneously is the k-variate GWD.
assume
accidents
GENERALIZED WARING DISTRIBUTION
The
Poisson
(case
GWD
ordinary
k=l)
cRn be obtained
In particular,
distribution.
a distribution which is a
through mixinE from a
cn arise as a mixture of a
it
Poisson distribution whose parameter A is
follows
533
itself a random
variable that
scale mixture of gamma
distributions.
Moreover, the GWD tends to a Polsson distribution for
certain limiting values
of its parameters.
One
would
expect
therefore
that a similar connection exists between
its generalization as given by (1.1) and the Polsson
can be obtained as a mixture of generalized
when the mixing distribution is a scale mixture
In the next section Panaretos’s [3]
as to a
Finally,
the
distlbutions.
and then some
Specifically, it is
generalized Poisson distribution as well
distribution.
gamma
of
shown that
parameters the SGWD
for certain limiting values of its
SGWD can
generalized
Polsson distibutlons
result is restated
SGWD are examined.
limiting cases of the
of
the
Indeed it has been shown (Panaretos [3]) that
Poisson distribution.
SGWD
or
negative
it is demonstrated in
tends
to a
binomial
type
section 3
that the
context of an accident proneness hypothesis.
arise in the
SOME PROPERTIES OF THE SGWD.
"2.
As is well known,
a generalized Poisson distribution is a distribution
whose p.g.f, can be put in the form
g(s) is
where
G(s)
that
(2.1)
G(s)
A
where
=AE
(I)
can
A
exp
U
Z
I=1
iZ!
shown (Feller,
can be
[I],
i(si-1)
{+ m},
with
p.291)
be represented by
alternatively
(O)/i!, m
variable (r.v.)
It
p.g.f..
a valid
in
(2.1)
exp{A(g(s)-l)}, A>O
G(s)
(2.2)
i.e.
by the
Z,
Zl,Z2
p.g.f,
of the random
as independent Poisson (A)
variables.
Theorem
integer
is
the
m=k<+m
(Panaretos
2.1
valued
r.v.
generalized
Assume that
whose
Let
[3])
XI(A
)k
k)
on
distribution contitional
Poisson distribution with p.g.f,
A 1, A.,
be a non-negative
A
k
are
A
,A2,
,Ak
given by (2.2) for
independent
gsmma
r.v. s
J. PANARETOS
534
with probability density functions (p.d.f.)
h
m-1
CA
f
e-A lhi
A
(2.3)
r(m
where m > O, i=l 2
k and h is itself a. r v. with p d f.
r(a+c)
f(h)
h
a-*
-(a+c)
(l+h)
a,c > 0
(2.4)
r(a)r(c)
Then the distribution of X is the SGWD given by (I.I).
Theorem
2.2 The
SGWD with
parameters k a, m
k
the
distribution
of
>
Y 1’"
where
iY
1"
m
m
2
Y
k
a.nd c tends to
independent
k
nega.t ive
i=1
binomial
r.v.’s
with parameters
so tha.t a/(a.+c)<+m.
Proof: Let lim
stand for limit
H
Then, using (1.2) we ha.ve
.
limG(s)
C(Zm
.
FD
lim
(a.+c)
(a.;m
mk;a.+m+c;s,s
m2,
(m)
r ,...,r
]---[
[ C/
[
s
k
(Zm)
Zm
k
2
Ca+c)
m
(r
r !...r
(r
k
s
Eir
k
k
]
(m)
i
r! --o
+ (Jc)(l-s )
[a./ ]
Ca.+c)
(r
(s l)
r
}
]-m
(2. S)
i=1
Hence the result.
Theorem 2.3 The SGWD with parameters k,
mx,mm,...,mk
generalized Poisson distribution with p.g.f, given
A =am /(c+a),
i=1,2
am /(a.+C)<+m
and
Proof:
Let
lim
k if
stand
a.->+m
for
m ->+m
limit
by
and c tends to the
(2.2)
i=1,2
k
as
a+m
m +m
from
(1.2)
where m=k<+m,
c->+m
so that
i=1,2
H’
J(a+c)->co,
am /(a+c)<+m
Then
we
obtain
lim G(s)
H’
k,
GENERALIZED WARING DISTRIBUTION
c()’:.m
FD(a;ml,m2,... mk;a+Em +c;s,s
H’
lira
tim
H’
(a+cl).:m)
2
535
s
k
k
im
.’
r
a+c
(a+c)(y.m)
i"
"I’’’’’"
Observe that
C(Em
m
lim
.’ (a+c) (Zm)
a+c
H
-m
In(c/(a+c)
lime
H’
However, since
m In
-<
-ml
-
follows that
m
Zm in
lim
a+c
s
a+c
a+c
H’
Hence
E+c
-Zm
-1
-m
a+c
-m
lim
e
a+c
H
which implies that
lim G(s)
e
H
H
a+c
rl...,r k
=e
-ami /(a+c)
k
If’
i=l
exp {-
Z
ri=O
aJa S
ac
|"
m
This establishes the proof of the theorem.
ACCIDENT THEORY AND THE STI/ITERING GENERALIZED WARING
3.
DI STRI BUTI ON.
One of the various
of
accident
accident
exposure
individual.
arise
as
analysis
the
is
the
of
hypotheses that have been
is
that
of accident proneness-accident
yield of factors that ceun
individual
to
external
risk
the
developed in
risk:
distribution
of accidents
An
be attributed to chance,
nd
psychology
In this context the ordinary CWD was shown (Irwin,
the
area
incurred by
of
[2])
the
to
an accident prone
J. PANARETOS
536
forms
the
concerning
In particular
parameters.
the distribution
of
on
arises
it
assumptions
external risk on suitable
exposed to varying
population
of
proneness
the
risk
and
assumption that for a given
the
individual of proneness h the accident experience is Poisson with parameters
(Alh)
if
Alh
and
Alh
where
and h
refers to the effect of the individual risk exponure.
from individual to
vary
a beta distribution
arrived at
therefore,
It
gamma
GWD is
becomes obvious,
cun be put
of theorem 2.1
results
the
the
respectively,
as the distribution of accidents.
that
to a
individual according
second kind
of the
Then
in a similar
perspective thus leading or a generalization of Irwin’s accident proneness
hypothesis giving rise to the SGWD as an accident distribution
Consider a population of individuals
of accidents
enviromental
experienced
risk
effects
Poisson
differences
in
effected
with p.d.f,
exposure
risk
through
beta
differences
distribution
accidents
of hazards.
with
from
proneness
defined
h exposed
to an
A
k
)[h)
p.f.
X[(,h)
that
given
by
to
multivariate
(2.2)
follows a
and
are
individual
gsauna
that
distribution
Then for individuals of the same proneness h
accidents is given
in
number
(A[h)=((A 1’ A2
Assume
individual
an uncorrelated
given by (2.3).
be the
may be considered to be reflecting the
Ak
distribution
the distribution of
that
parameter vector
(%1,2
of different types
generalized
X[(,h)
by an individual of proneness
indexed by a
the paraeters
where
and let
by
by (2. S).
If we further
manifest themselves in the
(2.4),
then
the
final
assume
form of the
distribution
of
will be the SGWD.
ACKNOWLEDGEMENT
The work in this paper was partially supported by a grant from the Ministry
of Research and Technology of Greece.
GENERALIZED WARING DISTRIBUTION
537
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J.
Roy. Star.
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2.
On the Relationship of the Stuttering Generalized
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