In this paper, we discuss statistical families
with the property that if the distribution of a random variable
X
is in
, then so is the distribution of
Z
∼Bi(
X
,
p
) for 0≤
p
≤1. (Here we take
Z
∼Bi(
X
,
p
) to mean that given
X
=
x
,
Z
is a draw from the binomial distribution Bi(
x
,
p
).) It is said that the family is closed under binomial subsampling. We characterize such families in terms of probability generating functions and for families with finite moments of all orders we give a necessary and sufficient condition for the family to be closed under binomial subsampling. The results are illustrated with power series and other examples, and related to examples from mathematical biology. Finally, some issues concerning inference are discussed.
An Invitation to Stochastic Mathematical Biology
