4.3 Over-dispersion
(a) Genesis
The over-dispersion occurs as
VarY   EY  . This phenomenon
can arise in a number of different ways. We describe these examples
of over-dispersion.
Example 1:
Let N ~ P  . Then,
Y  Z1  Z 2    Z N
has a compound Poisson distribution, where N is independent of
Z1 , Z 2 ,,
Important Result 1:
Let
N , Z1, Z2 ,
be independent random variables, where N
has a Poisson distribution with expectation
Z1, Z2 ,
 0
and the
are identically distributed with common moment
generating function M Z t  . Let
moment generating function of Y is
Y  Z1  Z 2    Z N
M Y t   exp M Z t   1.
[proof:]
1
. The
  N 
M Y t   E exp tY   E exp  t  Z i 
  i1 
  j


  E exp  t  Z i  I N  j  
j 0
  i1 


 independen ce of



   j


 
j



  E exp  t  Z i  E I N  j  





I
and
exp
t
Z
j 0  
 i 1  
 N  j 
  i  
 i 1  

j

j exp   
  M Z t 
j!
j 0


exp   M Z t 

j!
j 0


j
exp  M Z t M Z t 

exp M Z t    
j
!
j 0

j
 exp  M Z t   1
where I N  j  is the indicator function for the event
N  j.
Important Result 2:
E Y   E N E Z i   E Z i  ,
 .
Var Y   E Z i2
[proof:]
The cumulant generating function for Y is
kY t    M Z t   1 .
Then,
2
◆
 k t 
E Y    Y 
 M Z' 0  E Z i 
 t  t 0
and
  2 kY t 
''
2


Var Y   


M
0


E
Z
Z
i

2
 t
 t 0
 . ◆
Based on the above results, over-dispersion occurs as
 
E Z i2  E Z i  .
Note:
As Z i  1 , then Y  N .
Note:
Let
Y  Z1  Z 2    Z N , where N is distributed as some random
variable (not necessary Poisson) with moment generating function
M N t  . Then,
M Y t   M N log M Z t .
Example 2:
Let Y ~ P  , where

and
index
Var    


is a gamma random variable with mean

.
Thus,
E    
and
. The marginal distribution of Y is negative
3
binomial distribution,
  y    
PY  y  
, y  0, 1, 2,
y 
y!  1   
Thus,
E Y    , Var Y  
Therefore,
.
 1   
.

VarY   EY  .
Note:
In the absence of knowing the precise mechanism that produces the
over-dispersion or under-dispersion is known, it is convenient to
assume as an approximation that
constant
2.
Var Y    2 
for some
This assumption can and should be checked, but
even relatively substantial errors in the assumed functional form of
VarY 
generally have only a small effect on the conclusions.
(b) Asymptotic theory
If the eigenvalues of the fisher information matrix increase to infinite
as the sample size increases to infinite, the usual asymptotic results
concerning consistency and asymptotic Normality of
That is,
ˆ    N 0,  2 I 1   ,
  2l
where I   
.
 t 
2
can be estimated by
4
̂
are valid.
X2
1 n  yi  ˆ i 
2
~
 

 ˆ
n  p n  p i 1
i
2
.
Note:
The condition (the eigenvalues of the fisher information matrix tend
to infinite as the sample size increases to infinite) is usually satisfied if
one of the following conditions is satisfied
 p is fixed and n tends to infinity
 for fixed n and p,  i   for every i.
5