Q: Is there an example of the delta method being used in applied statistics where the sample size needed for a good approximation is practically unfeasible?
I am looking for an example from applied statistics where the delta method works poorly. By that I mean the sample size needs to be very 'large' (in some sense) for the asymptotic distribution to be a good approximation (I do not have a specific criterion in mind, anything reasonable OK) of the sampling distribution of the statistic in question. I am not looking for a theoretical 'failure'; the analytical asymptotics should hold.
The ideal answer would include an example from applied work where the delta method has been used, or at least been considered, but it has also (likely later in that case) been shown that the sample size needed for a good approximation is practically unfeasible. If there are no such examples, I would also be happy with a somewhat contrived example. The closer to a real setting, the better. I may use the example in teaching if I find it illustrative.