I have a question about confidence intervals.
In general, are confidence intervals open or closed?
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$\begingroup$ It might be a question of convention. APA recommends brackets, but a ti-84 calculator shows parentheses. $\endgroup$– Guillaume F.Commented Aug 1, 2023 at 5:11
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5 Answers
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The short answer is "Yes".
The longer answer is that it does not really matter that much because the ends of the intervals are random variables based on the sample (and assumptions, etc.) and if we are talking a continuous variable then the probability of getting an exact value (the bound equaling the true parameter) is 0.
Confidence intervals are the range of null values that would not be rejected, so what do you do if you compute a p-value that is exactly $\alpha$? (another probability 0 event for continuous cases). If you reject when p=$\alpha$ exactly then your CI is open, if you don't reject then the CI is closed. For practical purposes, it doesn't matter that much.
answered Sep 21, 2011 at 22:31
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3$\begingroup$ Considering that the OP asked "in general", I think this answer is wrong (and AdamO's one is right). $\endgroup$ Commented Mar 7, 2018 at 13:41
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$\begingroup$ @Semoi It doesn't matter whether the support is continuous or discrete, but whether the parameter space is continuous or discrete. Consider a Poisson distribution; the data are discrete, but the usual parameter we're interested in is the mean, which is continuous. On the other hand, if you want a confidence interval for the median of a Poisson distribution, then that would be discrete and it would indeed make a difference whether the CI is open or closed. $\endgroup$ Commented May 6, 2023 at 16:42
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$\begingroup$ @Semoi I do think that treating CIs as closed intervals is a safer general rule, but I disagree with your reasoning. In the case you describe, the sampling distribution of the statistic might be discrete. But the parameter $\lambda$ is still continuous. The fact that $\hat\lambda$ takes on discrete values (0, 1/20, 2/20, 3/20, ...) isn't immediately relevant. Say you calculate a lower CI endpoint of 2/20 = 0.1. There's no meaningful scientific difference between an open set ($\lambda$ could be 0.1000001 but not 0.1) and a closed set ($\lambda$ could be 0.1 but not 0.099999). $\endgroup$ Commented May 6, 2023 at 17:23
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1$\begingroup$ @Semoi, You mention data values being on the boundary of the interval, but that does not matter for confidence intervals (which are about the mean/parameter, no the data). Are you thinking of prediction or tolerance intervals? (which are very different). While it is true that with discrete distributions, rounding, small sample sizes etc. there is a non-zero probability of a CI bound being a specific number, those same things (plus asymptotics and approximate distributions) mean that all CIs are approximate anyways. Open or closed will both be approximately correct. $\endgroup$ Commented May 8, 2023 at 15:11
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$\begingroup$ @Greg Snow: You are absolutely right. Thank you very much! $\endgroup$– NotMeCommented May 8, 2023 at 17:11
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Depends on the support of the DF for the sampling distribution of the value you're trying to estimate. I would say that confidence intervals for binomial proportions are, in fact, closed intervals since there are only a finite number of values a statistic could achieve and the confidence interval would contain all its limit points (i.e. the endpoints are inclusive).
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$\begingroup$ It's hard to see why this argument might work, because the finite support of the statistic does not tell us anything about the topology of the confidence intervals: they are subsets of the parameter space, which is the entire interval $[0,1].$ $\endgroup$– whuber ♦Commented May 6, 2023 at 15:58
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A confidence set with confidence $1-\alpha$ for parameter $\theta$ is a set $\mathcal{S}$ for which $P(\theta\in\mathcal{S}) = 1-\alpha$. This set could be an open interval, a closed interval, or it could not even be an interval at all. I think it makes sense to call any confidence set which takes the form of either an open, closed, or half open interval a "confidence interval".
answered May 6, 2023 at 17:22
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As other answers here correctly point out, it doesn't usually matter.
If the parameter space is continuous, and the method of finding the CI is "continuous" (e.g. sample mean $\pm$ margin of error; what I have in mind are CI methods where the endpoints are not 2 of the sampled observations), then indeed it doesn't matter whether it's open or closed.
However, I'd like to make a case for why the convention should be "closed."
- If the parameter space is discrete, it does matter. Consider an integer-valued parameter, such as the unknown size of a finite population in a capture-recapture problem[1]. In that case, it feels much more natural and less confusing to say
"We are 95% confident that the population size is between 93 and 106, including the endpoints"
rather than
"...between 92 and 107, excluding the endpoints."
In other words, the closed CI [93, 106] makes more sense than an open CI such as (92, 107) or (92.9, 106.1) or whatever. - Even with a continuous parameter space, some CI calculation methods just choose two observations to be the CI endpoints. Consider a bootstrap percentile CI. Conventionally, we include those endpoints as part of the CI to ensure our estimated coverage is at least nominal: if we're trying to get a 95% CI, we include the endpoints so that at least 95% of the bootstrap statistics are in the CI. (I know bootstrap percentile CIs are not guaranteed to have the right coverage! But this is what we usually hope they will do, even if they don't necessarily succeed.)
- More generally, statisticians tend to conventionally prefer theoretical guarantees that are slightly conservative: most of us would rather have slight over-coverage than under-coverage in our CIs. In this spirit, closed intervals are slightly more appropriate.
[1] Edited to replace initial example (Poisson median, which may be questionable as per @whuber's comment below) with finite population size. Another example could be the unknown count within a finite population. If there are $N$ students in my class and I want to know $\theta$ = the number who would say "Yes" to a sensitive question, I could use randomized response to ask it in a privacy-protecting way. Then I may want a confidence interval for $\theta$ which has the discrete parameter space $\{0,1,\ldots,N\}$.
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$\begingroup$ ?? The parameter space for a Poisson problem usually is continuous: it's the sample space that is discrete. $\endgroup$– whuber ♦Commented May 8, 2023 at 21:40
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$\begingroup$ For the mean $\lambda$, yes. But if you are interested in the median, it can be discrete. There may be better examples of a discrete parameter space; feel free to suggest one. $\endgroup$ Commented May 8, 2023 at 22:08
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1$\begingroup$ That's a good point, thanks. But it's an (extremely) unusual situation in my experience. If I were truly interested in the median, I would still study the underlying Poisson parameter itself, develop uncertainty statements for it, and translate those into statements for the median. That would be more precise because it wouldn't be subject to the information loss arising from reporting only about the median. I believe most theoretical accounts rule this out by insisting that a property of a distribution family be a continuous function on that family: the Poisson median is not. $\endgroup$– whuber ♦Commented May 8, 2023 at 22:11
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2$\begingroup$ Fair enough. But my point isn't about the Poisson median as such; it's just that sometimes we could have a discrete parameter space. Thanks for pressing me to seek a better example. I'll edit my answer above to mention the population size in capture-recapture problems, where the parameter is unambiguously a discrete count. $\endgroup$ Commented May 10, 2023 at 18:52
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1$\begingroup$ That's a great example, thank you. Notice, though, that a discrete parameter space is automatically closed and therefore all confidence regions one might construct are also closed (and open -- so you could equally well say all confidence regions in such cases are open!). It's certainly illustrative and good food for thought. $\endgroup$– whuber ♦Commented May 10, 2023 at 19:28
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My answer is that it is open.
Since we have a interval from which we will get a neighbourhood value of our unknown parameter, and as we all know that this interval will give us an approximate value of the estimator, i.e., estimate then how it can be possible to declare it to be a closed interval.
One more point is that if we have a closed interval, then our estimate will be bounded fully, and we want a value that will lie between this interval only. By definition it must be closed, but in my opinion it should be open.
