Intuition behind a 0% central/equal-tailed confidence interval At a vague intuitional level, I feel like, if we generated a 0%, two-tailed, truly central (or "equal-tailed") confidence interval (CI), the number it would wrap around would be a median-unbiased estimate of the population parameter in question.  Am I confused, or is this right?  Is there another intuition of it that is better?
I can't seem to find anything about it when I google, but my intuition rests on the idea of the CI as the inversion of a p-value.  I think, this CI would also match one end of a one-sided 50% CI, which would also be the inversion of a 0.5 one-sided p-value.  And a 0.5 one sided p-value is saying "if this were our parameter, then 50% of datasets would be look to be 'higher' and 50% of them 'lower'".  So I feel like my intuition is right, but I can't quite formalise it, or find anything online to confirm or deny this.
 A: A frequentist 0% confidence interval can be any point within the parameter space. One might prefer to choose a point that is near to the maximum likelihood estimate, but any other point will be just as validly a 0% confidence interval.
Typical 95% confidence intervals are usually at least roughly centred around the maximum likelihood estimate because of a preference for a shorter interval over a longer one, not because of any definitional requirement. With a 0% interval all potential intervals have the same length!
Consider that a 95% confidence interval is an interval derived from a method that will in the long run yield an interval that covers the true value of the parameter on 95% of occasions (when the model is appropriate to the data generating system). Then it is clear that any point within the parameter space will cover the true value on 0% of occasions in the long run and will thus be a valid 0% confidence interval. (For continuous parameter space.)
A: A zero-level confidence interval can be seen as an estimator. Indeed, it has been advocated by Skovgaard (1989) "A review of higher-order likelihood inference". Bull. Int. Statist. Inst., 53, 331–351, in a class of two-sided equal-tailed confidence intervals, and is defined as the intersection of all confidence intervals at all levels.
An implementation of this idea needs estimating functions based on pivotal quantities for the parameter of interest. However, unless the pivot is exact and has a symmetric distribution around the true parameter (s.t. in the case of the Student's $t$ statistic), the median unbiasedness property for this kind of estimator does not necessarily hold.
One way to get such an estimator with low median bias is to use a higher-order pivotal quantity such as the $r^*$ of Barndorff- Nielsen (1986) "Inference on full or partial parameters based on the standardized signed log-likelihood ratio". Biometrika, 73, 307-322.
A: You are close
For a continuous distribution, the 0% equal-tail CI occurs at the point corresponding to the median of the true distribution of the pivotal quantity that is used in constructing the CI.  It is not always possible to invert the pivotal quantity in a way that yields an unbiased estimator of a corresponding median (of what exactly?).
