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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 $N$.

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"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.

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    $\begingroup$ +1 For this literature compilation. Some of them are frequentist, rather than classical (Fisherian), but still very interesting. Are you suggesting to "change the approach" in order to solve "what can we do"? $\endgroup$
    – user10525
    May 7, 2012 at 6:25
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    $\begingroup$ @Procrastinator, I think I'm not alone in not viewing "classical" as a synonym for "Fisherian". But yes, there are several approaches other than profile likelihood, that one might motivate by either frequentist or Bayesian methods. These methods may be better than profile likelihood, in situations where the extreme intervals you describe occur frequently. $\endgroup$
    – guest
    May 8, 2012 at 2:29

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