  The number of elements (N) and types (S) sampled from an ecological system
are among the most powerful constraints on observations of abundance,
distribution, and diversity. Together, N and S determine sets of possible forms
(i.e., feasible sets) for the species abundance distribution (SAD). There are
three approaches to the description of the null SAD (= the average feasible
SAD). The first approach is based on the random uniform sampling of
surjections. I calculate the probability of a given SAD, given N and S, under
this approach (Eq. 4). The second approach is based on the random sampling of
compositions. I calculate the probability of a given SAD, given N and S, under
this approach (Eq. 8). The third approach is based on the random uniform
sampling of partitions. I review the approach, which was developed by Locey &
White (2013), and provide some asymptotic results useful for ecologists. The
center of a feasible set is a null expectation, which should deviate enough
from the alternative models before invoking for biological or ecological
mechanisms underlying the SAD. Here, I integrate the feasible set approach with
the typical framework of inference in ecology (goodness-of-fit, null hypothesis
testing, model comparison, null modelling). I describe how to perform numerical
simulations to describe expectations under different approaches to the feasible
set. I develop objective or fitness functions to allow the estimation of the
most likely SAD using numerical optimization. I provide tools to compare null
expectations based on the feasible set approach with the observations, in the
context of model comparison and null hypothesis testing.
