Limit Property of a Compound of the Generalized Negative Binomial and Beta Distributions

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

FO L IA O ECO N O M ICA 225, 2009

Tadeusz Gerstenkorn*

LIMIT PROPERTY OF A COMPOUND OF THE GENERALIZED

NEGATIVE BINOMIAL AND BETA DISTRIBUTIONS

A b stract

In Central European Journal o f M athematics (CEJM ) 2(4) 2004, 527-537 T. Gerstenkorn published a probability distribution as a result o f compounding o f the generalized negative binominal distribution with the generalized beta distribution. Assuming that a param eter o f that distribution (w) tends to infinity one obtains a new limit distribution, interesting also in some special cases.

Key w ords: generalized negative binomial distribution, generalized beta dis-tribution, compound disdis-tribution, limit theorems o f the compound distribution.

1. G en eralized probab ility d istribu tion s

In “Central European Journal o f Mathematics” (CEJM) 2(4) 2004, 527-537 T. G e r s t e n k o r n published the paper A compound o f the generalized negative binomial distribution with the generalized beta distribution.

In 1971 G. C. J a i n and P. C. C o n s u l published a generalized negative binomial distribution. After a correction o f W. D y c z k a in 1978 it can be written in the form

GNBD(x; n , p , ß ) = Pp{x\ n, p ) = - n + ß x p x{ \ - p r ßx-x, x = 0,1 ,2 ,...

v x

n + ß x where:

0 < р < 1 , n > 0, ß p <1 and ß > \ (1)

* Professor o f the Faculty o f M athematics o f the University o f Łódź: Professor o f the Academy o f Social and M edia Culture in Toruń.

or

О < p < 1, и e N, / 7 = 0

₍

₎

If ß = 1 one obtains the negative binomial distribution. In the case (2) one obtains the binomial distribution and for n e N , ß = 1 the Pascal distribution.

The generalized beta distribution (GBD) is defined as follows:

GB D (y; a ,b ,w ,r) = ay (bw)r a B( r / a , w) 0 f « v - 1 1 - ^ - if 0 < y < ( b w ) ' ln, if у < 0 or у > (ftw)'/0 where a, b , r , w > 0 and B(r/a, w) is a beta function.

This distribution is a special case o f a Bessel distribution investigated by T. Ś r ó d k a (1973). It was also analysed by J. S e w e r y n (1986) and W. O g i ń s k i (1979) and applicated in reliability theory.

2. T he com pound d istribu tion

By compounding these two distributions in respect o f Y, that is

GNBD л GBD

Y

one obtains the distribution

PflGB(x) = D Y J( - c ) k' n + ß x - x X , (bw)“ В- n ( x + r + k---,w

4

0,

1,

0 < c y < l , . n > 0 ,

1.

If ß = 0 (binomial distribution) one obtains a distribution given in 1982 by T. G e r s t e n k o r n , i.e. P0GB(x) = V*/ cx(bw)x/a B ( r / a 9 w) _k=0 ' n - x ' к ' x + r + k N < k , (.bw)a В ,w , x < a , x = 0, 1 ,2 ...

When /7 = 1 and n = 1, 2, ..., one obtains from PpGB(x) a compound distribution Pascal л generalized beta in the form

P,GB{x) = n + x - 1 4 * J Ä £ ( - c) * r B ( r /a ,w ) fť o U . * ' x + r + k N (bw)“ В ,w

U J

from which one also can obtain interesting special cases, as it was with the binomial distribution.

In the paper treated b y T . G e r s t e n k o r n (2004) there were also given other special cases. In that paper there is also to find (Theorem 4.2, formula (15)) a formula for the factorial moment o f order I o f the compound distribution o f the negative binomial distribution with the generalized beta one, i.e.

s b-g b n ^ c ' j b w y ^ „л L , 0 B ( r / a , w ) t к ' I + r + к N (~c)k (bw)" В ,w <k ; \ a where = n(n + 1)(л + 2)...(« + / -1 ).

3. L im it d istribu tion s

For the presented compound distribution PpGB(x) we obtain an interesting limit distribution. Namely, we have

T heorem 1. If w -> o o , then oo k_ lim PßGB(x) = LP„GB(x) = Z>, У b" (-c)* t o ' n + ß x - x ' г ' x + r + k^\ ч k , 1

I a J

or, if n, ß x e N , then LPpGB(x) = Z), X b ' { - c ) k n+ßx-x k к * О ' n + ß x-x " г ^ x + r + к Л V * > 1 I a J where x_ŕ.. , o ..\ n c 'b “ n + ß x D\ =- Ч * У r (л + Дх)Г

Proof. Taking in account the following limit properties

lim f f i r— 7 = 1, lim м / • B (y, w) = Г(>>)

»v—»X J / W) . -уфК w—>00

(see, e.g. J. A n t o n i e w i c z (1969, p. 433); fonnula 8.35.4), we have

ncxbu lim PaG B (x) = -n + ß x я: k f к (n + ß x ) ■ Г / \ ľ \ и /

Х > ч - с )

Regarding the limit process ( w -» oo) we also obtain the following

T heorem 2. The factorial moment o f order / o f the limit distribution is expressed by / „i'. - V a« * L ( N B - C B ) _ n c ° S P ч'> - ( „ \ 2-j к-0 Г V« ' - 0 к (-с)* ba ■ГГ/ + г + АЛ <k J

I « J

R cferenccs

A n t o n i e w i c z J. (1980), Tablice fu n kcji dla inżynierów (Tables o f fu n c tio n s f o r engineers), 2nd edition, W arszaw a-W arsaw ( IS1 ed. 1969).

D y c z k a W. (1978), A generalized negative binomial distribution as a distribution o f PSD class, „Zeszyty Naukowe Politechniki Łódzkiej” - Scicnt. Bull. Łódź Techn. Univ., 272, M atematyka 10, 5-14.

G e r s t e n k o r n T. (1982), The compounding o f the binom ial and generalized beta distributions, [in:] Proc. 2"d Pannonian C o n f on Matliem. Statistics. Tatzmannsdorf, Austria. 1981, eds W. Grossm ann, G. Ch. Pflug and W. Wertz, D. Reidcl Publ., Dordrecht, Holland, 87-99. G e r s t e n k o r n T. (2004), A com pound o f the generalized negative binom ial distribution with

the generalized beta distribution, “Central European Journal o f M athematics (CEJM )” 2(4),

527-537.

J a i n G. С., C o n s u 1 P. C. (1971), A generalized negative binom ial distribution, SIAM J. Appl. Math, 21, 4, 507-513.

O g i ń s k i L. (1979), Zastosowanie pewnego rozkladu typu Bessela w teorii niezawodności

(Application o f a distribution o f the Bessel type in the reliability theory, in Polish), „Zeszyty

Naukowe Politechniki Łódzkiej” - Scient. Bull. Łódź Techn. Univ., 324, M atematyka 12, 31-42.

S e w e r y n J. G. (1986), Some probabilistic properties o f Bessel distributions, „Zeszyty Naukowe Politechniki Łódzkiej" - Scient. Bull. Łódź Techn. Univ., 466, M atematyka 19, 69-87.

Ś r o d k a 'Г. (1973), On some generalized Bessel-type probability distribution, „Zeszyty Naukowe Politechniki Łódzkiej” - Scicnt. Bull. Łódź. Techn. Univ., 179, Matematyka 4, 5-31.

Tadeusz Gerstenkorn

W łasn ość graniczn a złożon ego rozk ład u u ogóln ion ego

ujem n ego d w u m ianow ego z u ogóln ionym beta

W czasopiśm ie „Central European Journal o f M athematics (C EJM )”, 2(4) 2004, 527-537 T. Gerstenkorn podal rozkład prawdopodobieństwa, który jest wynikiem złożenia uogólnionego ujemnego rozkladu dwum ianowego z uogólnionym beta.

Zakładając, że jeden z param etrów rozkładu (w) dąży do nieskończoności, otrzymuje się nowy rozkład graniczny, ciekawy także w przypadkach szczególnych.