The likelihood 
of an observed path
was given by Eschenbach and Winkler (1975) as
where 
is
and the sum is over all 
-tuples 
of nonnegative integers 
such that 
and 
. This is equivalent to summing over all
partial family trees for which the first 
generation sizes take on the values
.
Denote the number of such trees by 
.
Further denote by 
the number of individuals
of the 
'th generation with exactly 
offspring in the 
'th of these
trees. Let 
.
Dion et al. (1982)
prove a representation theorem for the nonparametric mle of an offspring
distribution, supported on 
for known 
:
Notice that 
is a weighted average of the 
, with the weights being the estimated relative likelihood of
the 'th tree. Harris (1948)
showed that if one can observe the entire family tree, the mle of 
is just 
. The Proposition suggests a recursive algorithm
for computing the mle's. However, this requires the enumeration of all trees
that could have yielded the sample. This may be quite a formidable task (see
Dion et al. for a discussion).
A computationally more feasible algorithm for computing the
mle of the offspring distribution also uses Harris' form of the estimator,
but instead of weighting the Harris estimators with respect to the relative
likelihood of different possible trees, we compute the conditional
expectation of 
, given 
.
Let 
.
Write
This is a function of 
. Select a starting
value 
.
Now compute 
as if 
were the true
distribution. Use the to compute a new estimate
by the Harris formula,
That is, replace the true value 
by your best current guess, 
,
and use that guess to reestimate the probabilities as if we knew the
true . Iterate this procedure until it converges. This algorithm is
a special case of the strategy called the EM-algorithm
(Baum, ?, Dempster et al., 1977).
The algorithm is somewhat sensitive to starting values, but
it appears to work well to start
from a Poisson distribution with parameter 
, truncated above at
.
It is easy to see that
This formula is easily computed using a discrete convolution algorithm.
Define
Then 
is a maximum likelihood estimate of the offspring
variance. There are other estimates in the literature. Dion (1975)
suggested
where 
. A similar estimate, with 
replacing 
, was proposed by Heyde (1974). Both Dion's and
Heyde's estimates
are consistent, conditional upon nonextinction at time .
Duby and Roualt (1982) use a local limit theorem to compute a
normal approximation to the likelihood when
both 
and get large, and obtain an approximation which is
uniform in a class of supercritical offspring distributions with lattice
size one and finite fourth moment. The estimators obtained by maximizing this
approximate likelihood are and 
, respectively.
The work of Guttorp (1991) uses a similar result in a similar class of
distributions, but uses a better estimate of the approximation error to deduce
the consistency of the mle . In the next section we develop a
new local limit theorem which does not require the assumption that the lattice
size is the same for all offspring distributions under consideration.