Tadeusz Gerstenkorn*

ACTA
UNIVERSITATIS
LODZIENSIS
FO LIA O ECONOM ICA 225, 2009
Tadeusz Gerstenkorn*
LIMIT PROPERTY OF A COMPOUND OF THE GENERALIZED
NEGATIVE BINOMIAL AND BETA DISTRIBUTIONS
A bstract
In Central European Journal o f Mathematics (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 distribution, compound distribution, limit theorems o f the compound distribution.
1. G eneralized probability distributions
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 +ßx v
n + ßx
p x{ \ - p r ßx-x,
x = 0,1,2,...
x
where:
0<р<1,
n > 0, ß p <1 and ß > \
(1)
*
Professor o f the Faculty o f Mathematics o f the University o f Łódź: Professor o f the
Academy o f Social and Media Culture in Toruń.
[Ill]
or
О < p < 1, и e N, / 7 = 0
(2)
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:
f
ay
GBD(y; a,b,w ,r) = (bw)r aB( r / a, w)
«v - 1
1 -^ if 0 < y < ( b w ) ' ln,
0
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 distribution
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
4
'n +ß x - x X ,
- n( x +r +k
(bw)“В
--------,w
where:
n + fix
D =n + fix
cx(bw)a
B( r / a, w)
while
a , b , w , r > 0,
ficy < 1,
0<cy<l,.n>0,
f i > 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.
cx(bw)
x/a
P0GB(x) = V*/
B ( r / a 9w)
к
' x +r +k N
,w , x = 0, 1 ,2 ...
(.bw)a В
<
a
< k ,
'n -x'
k=0
When /7 = 1 and n = 1, 2, ..., one obtains from PpGB(x) a compound
distribution Pascal л generalized beta in the form
n + x -1
P,GB{x) =
4
*
*
' x +r +k
N
,w
Ä
£ ( - c) * r (bw)“ В
U J.
К
a
/
U
J B (r /a ,w )fťo
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
of the negative binomial distribution with the generalized beta one, i.e.
sb
-gb
к
' I +r + к N
(~c)k (bw)" В
,w
B ( r / a , w„л) L
t ,0 <k ;
\
a
n ^ c 'jb w y
^
where
= n(n + 1)(л + 2)...(« + / -1 ).
3. L im it distributions
For the presented compound distribution PpGB(x) we obtain an interesting
limit distribution. Namely, we have
T heorem 1. If w -> oo, then
oo
k_
lim PßGB(x) = LP„GB(x) = Z>, У b" (-c)*
to
' n +ß x - x '
ч
k
,
г1
' x + r + k^\
I a J
or, if n, ß x e N , then
n+ßx-x k
LPpGB(x) = Z), X
b '{ -c )k
' n +ß x-x"
V
к* О
*
^ x + r +к Л
1г
a
J
> I
where
x_ŕ..
n +, ßox..\
n c 'b “
Ч * У
D\ =r
(л + Дх)Г
Proof. Taking in account the following limit properties
i r—
»lim
v—
»XfJ f/ W)
. -уф7К = 1,
lim
w
—
>00м / •B(y, w) = Г(>>)
(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
n + ßx
k
f
я:
к
lim PaGB(x) = / \ Х > ч -с)
ľ
(n + ß x ) ■Г
\ и/
íу
\
—+ w
lim-
х+ г+ к
■w 0
T(w )-\va
В
( х +г +к
----------, w = LPpGB(x).
\
a
Regarding the limit process ( w -» oo) we also obtain the following
T heorem 2. The factorial moment o f order / of the limit distribution is
expressed by
L(NB-CB)
ч'>
/
к
„ i'. - Vc °a « S*P ' - 0
Г/ + г + АЛ
_ n
(-с)*
ba
■
Г
(Г
„ \ 2-j
к-0 <
kJ
V«
I « J
R cferenccs
A n t o n i e w i c z J. (1980), Tablice fu nkcji dla inżynierów (Tables o f fu nctions fo r engineers),
2nd edition, W arszawa-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,
Matematyka 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. Grossmann, 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 compound 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 binomial 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, Matematyka 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, Matematyka 19,
69-87.
Ś r o d k a 'Г. (1973), On some generalized Bessel-type probability distribution, „Zeszyty
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5-31.
Tadeusz Gerstenkorn
W łasność graniczna złożonego rozkładu uogólnionego
ujem nego dw um ianow ego z uogólnionym beta
W czasopiśmie „Central European Journal o f Mathematics (CEJM )”, 2(4) 2004, 527-537
T. Gerstenkorn podal rozkład prawdopodobieństwa, który jest wynikiem złożenia uogólnionego
ujemnego rozkladu dwumianowego z uogólnionym beta.
Zakładając, że jeden z parametrów rozkładu (w) dąży do nieskończoności, otrzymuje się
nowy rozkład graniczny, ciekawy także w przypadkach szczególnych.