Limit Property of The Compound Distribution Binomial-Generalized Two-Parameter Gamma

A C T A U N I V E R S I T A T I S L O D Z I E N S I S

Ta deu s z G e r s t e n k or n *

LIM IT PRO PERTY OF TH E C O M P O U N D D IST R IBU TIO N

B IN O M IA L -G E N E R A L IZ E D TW O -PARAM ETER G AM M A

Abstract. In 1920 M . G reenwood and G. U. Yule presented a com pound distribution Poisson-tw o-param eter gamm a and gave some interesting applications o f this compounding. In 1973 H . Jakuszenkow published a com pound o f the Poisson distribution with the generalized tw o-param eter gam m a one. In 1982 T. G erstenkorn dem onstrated a com pound binomial- generalized beta distribution and in the limit procedure received a com pound binomial- generalized three-param eter gam m a one. Now there is presented a limit property of that distribution when one o f its param eters takes a special constant value, i.e. if we have a particular tw o-param eter generalized gamma, giving the theorem o f Jakuszenkow.

Key words: limit distributions, com pound distributions, generalized gam m a distribution.

1. IN TRO D U C TIO N

In 1920 M. Greenwood and G. U . Yule presented a com pound o f the

Poisson distribution

P ( X = x; Л) = ^ exp( — A),

Я > 0 , x = 0 , 1 , 2 , . . .

(1)

with the two-parameter gamma one:

a'

,

=

Л

exP ( - ДЯ), 0 < Я < oo,

v, a > 0,

obtaining the negative binomial distribution

* Professor, University o f T rade in Łódź, Fac. o f M anagem ent; Prof. emeritus, o f the Łódź Univ., Fac. o f M athem atics.

\pxq \

x =

,

,

,...,

v >

,

where p = ---,

q =

— p.

ľ

+ a

It was the first paper in the field o f compound probability distributions

demonstrating also some interesting applications o f the given method. Since

that time there have been published many papers having respect to the

com pounding problem. For the m ost part they are mentioned in the com ­

prehensive elaborations, as e.g. Patil et al. (1968), Johnson et al. (1969),

(1992), (1997), Wimmer and Altmann (1999).

In this paper we refer to the paper by Jakuszenkow (1973) who gave

a com pounding o f the Poisson (1) and generalized two-parameter gamma

distributions, the last given in the form

where x = 0 ,1 ,2 ,..., a, p > 0, and Dp(z) is the so-called parabolic cylinder

function, i.e. it is a particular solution o f the differential equation

In 1982 T. Gerstenkorn published a com pound binomial-generalized beta

distribution. It is known that if one parameter o f the generalized beta

distribution tends to infinity then one obtains the so-called generalized

three-parameter gamma distribution:

0

< p < + oo,

a, p >

,

(

)

obtaining the distribution

P { X = x ) =

fl

T

-Г(р + х )е х р

axp

/ ( * ; a, b, p) = “'"'-/Л ' exp f

x , a , b , p > 0,

(4)

b i r ®

^ b>

(cf. Stacy 1962, p. 1187; Środka 1973, p. 77).

T he com pound binomial - (4) distribution

f>(x; a, b . c . n, p) = ß )

‘

- * y - l ) * ( b c - ) ^ r [ L t í ± í ] . (5)

where a, b, p, c >

, n - a fixed natural number, is obtained in the same

limit procedure.

For the aim o f further discussion, that is for a com parison with the

result o f H. Jakuszenkow, we simplify that distribution by one parameter,

assuming a —

, i.e. taking

л « „ ь , „ Р ) = ( ; )

■

( «

V 2 /

The gamma function occurring in (

) is next presented in the integral form

for convenience o f further transformations and we obtain

t

N ext by putting у

= í, we have

Г = 2 J t ' +*+ł-1e"'*A .

о

X

Then w e assume n = - , where Я is a finite constant greater than zero

с

We notice that

and

lim ('n 7 JiV

„-00 W

X-

«—Л k J

ŕ

/с!'

In consequence

lim(

) = —т-r— 1

=

k‘

г ( ? ) х ! °

k-2 b 4 x “

f-

2ЬЧ* „£+?

x

Л>А2\

Д M

y r— Jip

exp( -

- At V b)dt = ~ /p\ -

r (P + * ) exP ( - g - J ö - p - * ( 'W j }

2

W

( ! >

The last relation is obtained by the known formula (Ryżyk, Gradsztejn

1964, 3. 462; Рыжик, Градшгейн 1971):

]x' ,- i e \ p ( - ß x 2 - y x ) d x = (2ß)

ß , v > 0 , by putting: v = p + x, ß = \ , у = A*Jb.

In the end, the limit o f (

) is expressed by

2A*

/ 2 \ - £ i i

/ЬА2\

/

/ b \

\ 2J

and for A = 1 we obtain

- / P \

i * m k !

V 2 7

( ? ) - 4 " r 0 , + , ) exp

REFERENCES

G erstenkorn T. (1982), “T he Com pounding o f the Binomial and Generalized Beta D ist­ ributions” . In: Grossm an W. et. al., Probability and Statistical Inference, Proceedings of the 2nd Pannom ian Conference on M athematical Statistics (Bad T atzm annsdorf, Austria, June 14-20, 1981), D. Reidel Publ. Comp., D ordrecht, H olland, 87-99.

Greenwood M., Yule G. U. (1920), “ An Inquiry into the N ature o f Frequency D istributions Representative o f M ultiple Happenings with Particular Reference to the Occurrence o f M ultiple A ttacks o f Disease or of Repeated Accidents” , Journal o f Royal Statistical Society, 83(2), 255-279.

Jakuszenkow H. (1973), “ Nowe złożenia rozkładów ” (New Com pounds o f D istributions, in Polish), Przegląd Statystyczny, 20(1), 67-73.

Johnson N. L., K o tz S. (1969), Distributions in Statistics: Discrete Distributions, H oughton Mifflin Com p., B oston-N ew York.

Johnson N. L., K otz S., K emp A. W. (1992), Univariate Discrete Distributions, 2nd ed., John Wiley and Sons, New York.

Johnson N. L., K otz S., Balakrishnan N. (1997), Discrete Multivariate Distributions, John Wiley and Sons, New York.

Patil G . P., Joshi S. W., R ao C. R . (1968), A Dictionary and Bibliography o f Discrete Distributions, Oliver and Boyd, Edinburgh.

Ryżyk J., G radsztejn I. (1964), Tablice całek, sum, szeregów i iloczynów, Państwowe W ydaw­ nictwo N aukowe, Warszawa.

Рыжик И. М ., Градш тейн И. С. (1971), Таблицы интегралов сум м рядов и произведений И зг. V, Гос. И зд. Техн.-теор. Литер., Москва.

Stacy Е. W. (1962), “ A G eneralization o f the G am m a D istribution” , Annals o f Mathematical Statistics, 33(3), 1187-1192.

Sródka T. (1973), “ On Some Generalized Bessel-type Probability D istribution” , Zeszyty Naukowe Politechniki Łódzkiej (Scient. Bull. Łódź Techn. Univ.), 179, M atem atyka, Fase. 4, 5-31.

W immer G ., Altm ann G. (1999), Thesaurus o f Discrete Probability Distributions, Stamm, Essen.

Tadeusz Gerstenkorn

W ŁASN OŚĆ GRANICZNA Z Ł O Ż O N E G O RO ZK ŁA DU

D W U M IA N O W EG O Z U O G Ó LN IO N YM D W UPA RA M ETRO W YM GAMM A (Streszczenie)

W roku 1920 M . Greenwood i G. U. Yule przedstawili złożony rozkład Poisson - dwu- param etrow y gam m a i podali interesujące zastosowanie tego złożenia. W 1973 r. H. Jakuszen­ kow podała złożenie rozkładu Poissona z uogólnionym dw uparam etrow ym rozkładem gamma. W 1982 r. T. G erstenkorn opublikował złożony rozkład dwumianowy z uogólnionym beta i w przejściu granicznym otrzym ał złożony rozkład dwumianowy z uogólnionym trzyparam et- rowym gamm a. Obecnie przedstaw iona jest własność graniczna tegoż rozkładu, dająca twier­ dzenie H. Jakuszenkow, jeśli jeden z param etrów tego rozkładu a = 2, tzn. przy ograniczeniu się do szczególnego dw uparam etrow ego uogólnionego rozkładu gamma.