Robust significance test for rare events

Let us assume that we're sampling species and observe the sample of say
N=1000
N
events. The real distribution of species frequencies shows that they are extremely low, e.g. a log-normal distribution around
p=
10
−8
p
spanning two orders of magnitude, or an empirical distribution to be inferred from the fact that we've observed
m=10
m
events after sampling
M=
10
9
M
times. Any event that we observe will (likely) have a frequency of
1/1000
1
, and, according to Binomial or Poisson distribution we can conclude that the observed frequency is significant. My question is to how perform an appropriate statistical testing for such events that will be in line with basic reasoning and filter events that are expected to have
1
1
or
2
2
observations simply by chance. I would suggest that some form of multiple testing correction can fix the things, but I don't know if its the right way to go. Other suggestion is to go with Beta Binomial distribution or to utilize models of unseen species introduced by Chao et al, but perhaps there are some previous works dedicated to this issue that offer a ready-to-use statistical framework?