THE INSTITUTE
OF STATISTICS
THE CONSOLIDATED UNIVERSITY
OF NORTH CAROLINA
i.·
:f
g
ABEL DISTRIBUTlrnS
by
N .L. Johnson
University of North carolina at Chapel Hill
Institute of Statistics Mimeo Series #1289
O::tober 1980
DEPARTMENT OF STATISTICS
Chapel Hill, North Carolina
ABEL DISfRIBUfIONS
N.L. .Johnson
University of North Carolina at O1ape1 Hill
1.
l);finition
IMass
(l97~))
utilized the identity
(a+!~)(n) =
Jl
I
(n)a Cx)f3(n-x)
x
:x=O
(1)
(\.mere a Cb ) = a(a-1) ... (a-b+1)) to define various
distributiolls.
"generalized binomial"
Note that u and f3 need not be positive; by reversing signs
of both a and B in (1), the equivalent identity
(a+B) [n]
n
= I
(~)a[x]f3[n-x]
(1) ,
x=O
is obtained, wh,~re a[b]
= a(a+1) ... (a+b-1).
In a similar way, Abel's identity (e.g. Riordan (1979))
(2)
can be used to define distributions
Pr [X=x]
provided a and
positive P.
x
~)
(3)
are such that Px 2:. 0 for x
=
0,1, ... ,n, with at least one
Such values of a and f3 are
S>
(a)
a > 0 ,
(0)
a < -n, S < 0
n
We may call these distributions Abel distributions (with parameters n;a;S).
From here all, we suppose (a) holds.
2
2.
Sorre Proper! ies
An altemative fonn for (3) is
(3) ,
e = a(o.+S) -1.
with
As a, S
-+
00
binomial with parameters n, 8.
and
0.
=
with n arid
e
fixed, the distributicn tends to
On the other hand, if n,S -+
00
with
ns- I
= AQ
AI/A O fixed then
P -+ e
x
-A AX
1
x.
J
(1
(4)
The distribution tends to a Lagrange dotblc Poisson (generalized BorelTanner) distributicn.
Jain and Consul (1971) have shown that the Lagrange
double negative binomial
p
tends
N
(N+er+ l)X)Q-N(~)x
x - N+(r+l)x
"<
x
ex
to Lagrange double Poisson as N -+00, r
-+
= 0,1, ... ; P = Q-l)
0, P
-+
0 with NP
= AI'
rP • A '
Z
but (4) is not the same as the Lagrange double binomial of Consul and Shenton
(1972) .
The expected value of the distribution of
(Riordan (1979), Table 1.2)
.:i-
n
1 eo.+S+F)n::
L (n)
(a+x)x(S-x)n-x
x
x=0
k
\\There F is interpreted as k!.
E[a+X]
whence
= o.(a+S)
This gives
n
-n \'
L
n
(·)j!ea+S)
j=O J
.
n-J
:x can be obtained from the formula
.
3
=
(2)
)
na{1 + n -1 + ~ +... + (n -I! }.
a+B
a+B
(a+8)
(a+8)n-l
Also
var(X)
= var(a+X) = a
(S}
I n(j)(a+S)-j(j+1)(a+~j)
j=O
2 n C)
. 2
- a { l n J (a+S)-J} .
j=O
(6)
MOTe compact fonnulas can be obtained fOT expected values of other ftnctions
Differentiating (2) s times with respect to 8, we obtain
of X.
whence
(FOT
S
>
Tl,
(s
= 1,2, ... ,n)
(s
= 1,2, ... ,n-l)
(7)
•
both sides of (7) are zero.)
Differentiating with respect to a leads to
•
(8)
More general formulas, such as
E[
(n-X) (s)
(R-X)s
(X-I) (t)
•
t]
(a+X)
=
dt
n(s)a
(a+8)
n
-t(
cia
(a+8)n-s
a
)
arc easily derived.
3.
Maximum Likelihood Estimation.
For a random sample Xl' .•. '~ of size
L
=N
N
log a -
Nn log(a+S) +
-
N, the log-likelihood is
N
.
n
10g(x ) + 1. (X.-1)log(a+X.)
i=l
' i
i=l 1
1
N
I (n-X.)log(S-X.) .
i=l
1
1
I
(9)
4
The equations for the maxinn.un likelihood estimators are
a log
oa
L
A
= N _ Nn
A
d log L
"-
as
A
N X-1
A
a
a+S
=-
Nn
-+
" "a+8
2...- = 0
+ \'
L
A
i=l a+X.1
N n-X.
1: ~ = o .
i=l 8-X.1
Combining these equations
n-X.
L ~.
i=l 8-X.
1
N
The left-hand side can be written
"- N
l+a 1:
&
X.
_1_
i=l &+X.
1
and is a decreasing Met ion of
&.
Also
N
Nn
_ NE[ X-I ]
2
- ")"" + ("'+0)2
'"
'" tJ
(a+ X)
=-
Nna
] = - -----,Zor-..:...-.
(8-X)2
(a+8) (l:s-n+l)
NE[ n-X
The asymptotic value of N var(&,S) is
(a+6) 2(a+2)(S-n+l) [ 6-:+1
Zn(n-l)
1
6-n+l+21la -1 )
a+2
(a+8)2
= 2n Cn-I)
a(a+2)
[ (a+2) (8-n+1)
(a+2) (8-n+l)
(8-n+l) (S-n+l+28a- 1)
].
5
a and 8 is
The limiting correlation between
j
4.
a+2
1
Ct+2S(S-n+l)
Multivariate Abel Distributions.
From the identity (Riordan (1979), p. 25)
m
1 m
( IT a.)- ( L a.+S)n =
j=l J
j=l J
L ... I(
Ex. ':5n
J
n
_~
xl"" ,xm,n L.lX j
)
rn
x.-l
rn
n-Llx.
x { IT (a.+x.) J }(S- I x.)
J
j=l J J
j=l J
m
(10)
we define the rn-variate Abel distribution, with parameters n;2;8
m
Pr[ n (X.=x.)] :: P
j=l J J
~
=(
~
a.+(3)
j=l J
L
-n
(
n
m )
x1,···,~,n-Elxj
x.-l
m n-~.
a. (a.+x.) J }(S- LX.)
J
j=l J J J
j=l J
m
x {IT
m
(0 ~ x.; L x. s n).
J j=l J
To ensure Px
~
0 we take
Ct
j
> 0 (j = 1, .•• ,TIl); B > n.
(11)
Formula (11) can be
written
x.
P
=
x
m+1 G. J
rn
1
x.-l
{n! IT ~}{ IT (l+a : x .) J }(1
J J
j=l x j ' j=1
-
s-
n-E mx.
I x.) 1 J
j=l J
1 rn
(11) ,
~rn
-1.
~rn
-1
where 8 i = ':.1 i (Lj=la j +S)
(1 = 1, .. "rn) and em+1 = S(Lj=lC1 j +S) , so that
1
~rn
Li=la i :: 1; and ~l = n - Lj=l Xj'
,m+
Similarly, as in the l..D1ivariate case, differentiating (10) s times with
respect to 3 leads to
(s'" 1,2, ... ,n) •
(12)
6
Comparmg this with (7) it follows that Ij=lXj has a tmivariate Abel distributim
with parameters n; Lj=la ; 13.
j
A similar argunent leads to the conclusion that
the margmal distribution of Xl is Abel with parameters n; ClI ; Ij=2Clj+13.
For the conditional distribution of Xl' given X. = x. (j = 2, •.. ,m) we have
J
J
m
m
m
n-L: 2x.
xl-l
m
n-L:Zx,-x
l
J
Pr[XI=x11 n (X.=x.)] a: (
J)Cl1 (al+xl )
(13 - I x.-xI )
j=2 J J
Xl
j=2 J
(13)
This is an Pillel distribution
If 2,6
-r ~,
00
with n,
= l, ... ,~)
.2 ccnstant, the distribution (11)
I
tends to multi=
, -1
with nB
= A and Cl = Aj/A
O
O
j
fixed, the distribution tends. to that of mdependent Lagrange
nomial with parameters
(j
with parameters n - Ij=2 X j; ~; 13 - Ij=zx j .
n,~.
If n,B
-r
00
double Poisson variables Witll parameters AO,A
j
TIle marginal joint distribution of Xl and
parameters n; aI' a Z;
Lj=3aj+B.
(j = l, ... ,m).
Xz
is bivariate Abel with
Also
(14)
The parallelism between the structures of multivariate Abel and multinomial distributions - in regard to marginal and ccnditicnal distributions,
[or example - is noteworthy.
However, the regression of one variable en the
others is not linear for the multivariate Abel.
S.
A Related Distribution
I f X has distribution (3), then a-X has distribution
7
P
x
=
a
n
x
n-x-l
(x)((3-n+x) (a+n-x)
(a+f3)n
(x
=
(15)
0,1, ... ,n).
This is not an Abel distribution as defined in (3), although it is sanewhat
similar to an Abel distribution with parameters n, f3-n, a+n.
It might be
called a reverse Abel distribution.
Replacing (S-n) by a, and (a+n) by (3 we have
whence
n
\'L (n)
x (r-<+x)x(B_x)n-x-1
u.
x=O
==
(
a+B)
f3-n
n
(x
= 0 "1 ••• ,n;
B > n, a > 0) •
(16)
The identity (16) can be used to extend further the expected value fonnulas
obtained in Section 2.
For example, inurediately fran (16), if X has distribution
(3)
(17)
differentiating s times with respect to a
E[
(5)
X
]
(a+X)s-I(B-X)
=
n (5)
_
(18)
(a+B)s(s-n)
These fonnulas could, of course, be derived from (2) with patience.
REFERENaS
Consul, P. C. and Shenton, L. R. (1972). Use of Lagrange expansion for generating
discrete generalized probability distributions, SI~1 J. Appl. Math., ~,
239-248.
LWass, 1'-1. (1979).
A generalized binomial distribution, Amer. Statist.,
.21,
86-87.
Jain, C.C. and Consul, P.C. (1971). A generalization of negative binomial
distributions, SIAM J. App1. ~~th., £1, 501-513.
Riordan, J. (1979).
Combinatorial Idcnti ties.
Krieger: Huntingtm, New York.
8
Appendix
For rurposes of calculation, the following fonnulae are useful
E [a+X] = fa
(cf. (5))
,n (a+f3)
(cf. (6))
(cf. (14))
where
f r n CY)
•
~(y)
f 0 ,0(y) = 1
,
f O n CY) = 1 + ny
•
-1
fa ,n-1(Y)
f r n (y) = f r - 1 ,n(Y) + ny
•
-1
f r,n- 1 (y)
(r
= 1,2)
.