Profile likelihood of N in binomial model

Question

Let $(n_1,...,n_k)$ be a sample from a Binomial$(N,p)$ where both parameters are unknown. In many cases, the profile likelihood of $N$ is asymptotic in the sense that it never decays to $0$. An example of this is given here. The following code shows the profile likelihood of $N$ for this data set.

data = c(16, 18, 22, 25, 27)
prof = function(n) sum(dbinom(data,n,mean(data)/n,log=TRUE))
n = points = seq(25,500,1)
for(i in 1:length(n)) points[i] = exp(prof(n[i]))
m = max(points)
plot(n,points/m)

What can we do from a classical viewpoint in such scenario?

Shall we just say that a likelihood-confidence interval is $(N_0,\infty)$?

Note: Using the Bayesian approach, the posterior of $N$ is going to decay to $0$ for any prior, due to its integrability. The issue here is that the inferences are strongly influenced by the prior given that, in vague words, the likelihood has little information about $N$.

Answer 1

"What can we do" has been asked by many researchers; there is a long literature on this problem, too much to describe here. One recipe for an interval is here, it would also be worth reading this, and a landmark paper in the area is this one. A review of why the problem is hard is here. For a full list of "what can we do" you could search for papers that cite these.