Some properties and applications of the stuttering generalized waring distribution

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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 Stuttering Generalized Waring Distribution arises in connection with sampling from an urn that contains balls of two colours (black and white) and it can be thought of as an intermingling of generalized Waring streams (Panaretos and Xekalaki [4]).

Because

of its application potential a study of its properties would be worthwhile. In this paper it is shown that it can be obtained

as a mixture of the generalized Poisson distribution. It is also demonstrated that, in an urn scheme, increasing the number of balls in the urn in an appropriate fashion one can end up with a Poisson type or a negative blnomial

type

sampling distribution as an approximation to the stuttering generalized Waring distribution.

Keywords and Phrases: Generalized aring Distribution, Generalized Poisson

Distribution, Mixtures of Distributions, Urn Models, Accident Theory, Hypergeormetric Function.

AMS 1980 Subject Classification.

Primary: 62E20, Secondary: 60E06, 60F99, 62EI0, 62P2S

1. INTRODUCTION

With the aim of preventing accidents, accident theory has received

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that have been developed the generalized Waring distribution (GWD) was obtained as the distribution of accidents (see e.g. Irwin [2], Xekalaki [S], [8], [7]). Generalizing this distribution

Panaretos

and Xekalaki [4] introduced the stuttering generalized Warlng distribution (SGWD) in the context of an un scheme

This is an intermingling of generalized Waring streams and is defined by the probability function

(p.f)

P(X=x)=

C(zm

’Zxt

k (m)(x

X

k

"+c+Yml

(Yx) i:l

]

J=l

(1.1) where

a)denotes

the ratio

FC+)/F(a),

>0,

R

x=O,l,2,..

The probability generating function

(p.g.f)

of this distribution

is given by

C(Zm

G(s)=

F

D(;m

mk;+mi+c’s,s

s

+c _(Zm)

(1.2)

where

F

D denotes Lauricella’s hypergeometric series of

type

D defined by

FD(CX;

1,

8

2,

k; X+-i+’;

S S2,

Sk

=-

k

r =o r =o

(+Xi+)(Zr)

"

k

S k k

s s i=I k. For k=l one obtains the GWD. So the

definition of this distribution enhances 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 the fact that (1.1) can be employed in situations

where single

events,

pairs of events, triplets of events k-plets of events can be thought of as beeing jointly distibuted according to the

k-variate GWD.

In

the context of car accident statistics this implies that the SGWD would be expected to describe the distibution of the total number of cars involved in accidents if it is reasonable to assume

that the joint distribution of the numbers X X

2,

Xk

of accidents

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The ordinary GWD (case k=l) cRn be obtained through mixin

E

from a Poisson distribution.

In

particular, it cn arise as a mixture of a

Poisson distribution whose parameter

A

is itself a random variable that follows a distribution which is a scale mixture of gamma distributions.

Moreover,

the GWD tends to a Polsson distribution for certain limiting values of its

parameters.

One would therefore

expect

that a similar connection exists between

its generalization as given by (1.1) and the Polsson or the generalized Poisson distribution. Indeed it has been shown (Panaretos [3]) that the SGWD can be obtained as a mixture of generalized Polsson distibutlons when the mixing distribution is a scale mixture of gamma distlbutions.

In

the

Panaretos’s

[3] result is restated and then some limiting cases of the SGWD are examined. Specifically, it is shown that for certain limiting values of its

parameters

the SGWD tends

to

a generalized Poisson distribution as well as to a negative binomial

type

of distribution. Finally, it is demonstrated in section 3 that the SGWD can arise in the context of an accident proneness hypothesis.

"2. SOME

PROPERTIES

OF

THE SGWD.

As

is well known, a generalized Poisson distribution is a distribution whose p.g.f, can be

put

in the form

G(s)

exp{A(g(s)-l)},

A>O (2.1)

where

g(s)

is a valid p.g.f.. It can be shown (Feller,

[I],

p.291)

that G(s) in (2.1) can alternatively be represented by

G(s) exp

A

i(si-1)

(2.2)

where

A

=AE

(I) (O)/i!, m

U

{+

m},

i.e. by the p.g.f, of the random variable (r.v.)

Z

iZ!

with

Zl,Z

2

Z,

as independent Poisson

(A)

I=1

variables.

Theorem 2.1 (Panaretos [3]) Let

XI(A

)k

k)

be a non-negative integer valued r.v. whose distribution contitional on A

,A2,

,Ak

is the generalized Poisson distribution with p.g.f, given by (2.2) for

are independent gsmma r.v. s m=k<+m Assume that

A

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with probability density functions (p.d.f.)

h m-1

f

CA

A

e-A

lhi

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) ha-* (l+h)-(a+c) a,c > 0 r(a)r(c)

Then the distribution of

X

is the SGWD given by (I.I).

(2.3)

(2.4)

Theorem 2.2 The SGWD with parameters k a, m m

2 m a.nd c tends to

1" k k

the distribution of

>

iY where

Y

Y independent

nega.t

ive 1’" k

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

C(Zm

2 k

(a.;m

m2,

mk;a.+m+c;s,s

s

limG(s)

.

lim

.

FD

(a.+c) (Zm)

(m) m

k

Zm

(r (r

r !...r

r ,...,r k

k

Eir s

k

[[C/

]

i

[a./

]

(m)

}

(r

]---[

Ca+c) Ca.+c) (s

l)

r --o

r!

[

+ (Jc)(l-s

)

]-m

i=1

(2. S)

Hence

the result.

Theorem 2.3 The SGWD with parameters k,

mx,mm,...,mk

and c tends to the

generalized Poisson distribution with p.g.f, given by (2.2) where m=k<+m, A =am/(c+a), i=1,2 k if a.->+m m ->+m i=1,2 k c->+m so that am /(a.+C)<+m and

Proof: Let lim stand for limit as a+m m +m i=1,2 k,

H’

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c()’:.m

tim

H’

(a+cl).:m)

2 k

lira

FD(a;ml,m2,...

mk;a+Em

+c;s,s s

H’

k

r im

a+c i"

.’

(a+c)(y.m)

"I’’’’’"

Observe that

C(Em

lim

.’

(a+c) (Zm)

However,

since

m

H a+c

-m

In(c/(a+c)

lime

H’

-<

m

In s -1

-m

-ml

a+c

E+c

follows that

m

lim

Zm

in

-

-Zm

a+c a+c a+c

H’

Hence

-m

lim e

H a+c

which implies that

lim G(s) e

H

a+c

H

rl...,r

k

/(a+c) k aJa S

-ami

Z

=e

If’

ac

|"

i=l

ri=O

exp {-

m

This establishes the proof of the theorem.

3. ACCIDENT THEORY

AND

THE

STI/ITERING GENERALIZED WARING

DI

STRI

BUTI

ON.

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population exposed to varying external risk on suitable assumptions concerning the forms of the distribution of the

proneness

and risk

parameters.

In

particular it arises on the assumption that for a given individual of proneness h the accident experience is Poisson with

parameters

(Alh)

where

Alh

refers to the effect of the individual risk exponure. Then if

Alh

and h vary from individual to individual according to a gamma and a beta distribution of the second kind respectively, the GWD is arrived at as the distribution of accidents.

It

becomes obvious, therefore, that the results of theorem 2.1 cun be put 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 and let

X[(,h)

be the number of accidents experienced by an individual of proneness h exposed to an

enviromental risk indexed by a

parameter

vector

(A[h)=((A

A2

A

)[h)

1’ k

where the paraeters

(%1,2

Ak

may be considered to be reflecting the

effects of different types of hazards. Assume that

X[(,h)

follows a

generalized Poisson distribution with p.f. given by (2.2) and that differences in risk exposure from individual to individual are effected through an uncorrelated multivariate gsauna distribution with p.d.f, given by (2.3). Then for individuals of the same proneness h the distribution of accidents is given by (2. S). If we further assume

that differences in proneness manifest themselves in the form of the beta distribution defined by (2.4), then the final distribution of

accidents will be the SGWD.

ACKNOWLEDGEMENT

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REFERENCES

I. FELLER, W. An Introduction to Probbl ty Theory and Its Applications, Vol.l,

(---ised

printing of the third

edition).-Wily,

New

York, 1970.

2.

IRWIN,

J.O. The Generalized Waring Distribution.

J.

Roy.

Star.

Soc., A

138, 18-31 (Part

I),

204-22? (Part

II),

374-384 (PPt III), 1975. 3. PANARETOS, J. On the Relationship of the Stuttering Generalized

Waring Distribution

to

the Generalized Poisson Distribution. Proceedings of the 4?th Session of the International Statistical Institute, Tokyo,

Japan,

1987, 341-342.

4. PANARETOS,

J.

and

XEKALAKI,

E. The

Stutterin

Generalized Waring Distribution. Statist. and Probab. Letters, 4(6), 1986, 313-318.

5.

XEKALAKI,

E. Chance Mechanisms for the Univarlate Generalized

WarinE

Distribution and Related Characterizations, Statistical Distributions in Scientific Work, Vol.4, (Models, Structures and Characterizations, 1981, (C. Taillie, G.P. Patll and B. Baldessari eds),

D.

Reidel, Holland, 157-171.

6.

XEKALAKI,

E. The Univariate Generalized Waring Distribution in Relation

to

Accident Theory:

Proneness,

spells of contagion? Biometrics, 39, 1983, 887-895.