6

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.

Peter Mortensen
  • 343
  • 3
  • 10
justme
  • 775
  • 2
    A 0% confidence interval usually equals the observed effect estimate (see this recent paper and the references therein). Visually, this becomes obvious in the framework of $p$-value functions (for a recent, gentle introduction, see this paper). – COOLSerdash Nov 20 '22 at 20:03
  • @COOLSerdash Thank you! These p-value functions are a great idea that I hadn't seen before and might have to start using! I appreciate that it is customary to put the point estimate at the centre of the CI, but my question is specifically about a central CI... Another way to phrase it is, "if the 0% CI is not a median unbiased estimate, can it still be a central CI?" – justme Nov 20 '22 at 20:29
  • what do you mean by "central confidence interval" ? – utobi Nov 20 '22 at 20:38
  • @utobi as I understand it, a central confidence interval is one that chops the same amount off each end (e.g. a two-sided 95% interval loses 2.5% at each end). I think an intuitive way of thinking of that, is that it is as likely to undershoot the target as it is to overshoot (though I might have that wrong). E.g. with a CI for a mean with known variance, you look at the 2.5th and 97.5th percentiles of the sampling distribution, but you could equally (though in this case it would be odd) have a CI that was lopsided off to one side. With skewed sampling distributions I think you get a ... – justme Nov 20 '22 at 20:51
  • ...shorter CI if it is non-central. – justme Nov 20 '22 at 20:53
  • @justme ok, that's sometimes called equal-tailed CI. – utobi Nov 20 '22 at 20:55
  • 1
    @utobi Ahh thank you! Maybe that is what I should have called it. I'll edit my post to include both. – justme Nov 20 '22 at 20:57

3 Answers3

8

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.)

Michael Lew
  • 15,102
  • 1
    Thank you! This makes a lot of sense, but does it still hold if we are talking about a central confidence interval? In that case I tend to think of it as a limiting case that is pinned down in its location by symmetry of the central CI. (To put it another way, if it is not a median-unbiased estimate, is the CI really central?) – justme Nov 20 '22 at 20:27
  • (I've edited the title to add the emphasis on the central / equal-tailed aspect) – justme Nov 20 '22 at 20:59
7

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.

utobi
  • 11,726
6

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?).

Ben
  • 124,856
  • 1
    Ahhhh yes, that makes a lot of sense! Thank you. I didn't think of that. -- Regarding the median of what -- I mean it's a median unbiased estimator of the population value you are constructing the CI of. But I didn't think about the impact of constructing the pivotal quantity: I think you're right on that. – justme Nov 20 '22 at 22:21
  • Usually the population quantity of interest is an unknown parameter, and within the classical statistical framework this is considered to be an "unknown constant" rather than a random variable. What would it mean to refer to the "median" of such a value? – Ben Nov 20 '22 at 22:23
  • It would (as I understand it) mean that for any given fixed setting of that parameter theta, the sampling distribution of these 0% CIs would have theta as its median. So, you're not treating theta as a random variable, but rather the different estimates of theta that might arise from the sampling process -- and the median of those should equal theta. (Just the same as saying that a 95% CI includes theta in 95% of samples for a given theta). – justme Nov 20 '22 at 22:32
  • (To add a little bit of colour: part of what troubles me is that we build CIs often around mean-unbiased estimates, but it feels to me that a CI is inherently a percentile-based quantity, so it feels like they should really be centred on median-unbiased estimates to be really clean) – justme Nov 20 '22 at 22:38
  • Okay, I see what you mean. That property won't hold in general, but it might hold for certain kinds of pivotal quantities (e.g., when built on a symmetric distribution). – Ben Nov 20 '22 at 22:40
  • I still don't see a reason why it wouldn't hold in the single parameter case (so that it isn't necessary to make the quantity pivotal). A central (equal tailed) CI should (I think?) miss the target "below" vs "above" an equal amount of the time, so the 0% case must surely be median unbiased? I do agree with your point with regard to e.g. studentizing / other distortions based on nuisance parameters to make it pivotal, though... – justme Nov 20 '22 at 22:44