# Are confidence intervals open or closed intervals?

I have a question about confidence intervals.

In general, are confidence intervals open or closed?

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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 asumptions, etc.) and if we are talking a continuous variable then the probability of getting an exact value (the bound equalling 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.

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 +1 for the first sentence! – Dilip Sarwate Aug 9 '12 at 2:55

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 the confidence interval would contain all its limit points (i.e. the endpoints are inclusive).

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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.

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The confidence interval is usualy defined as 2,5% and 97,5% quantiles, so in that case the it must be closed by definition.

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Please explain the downvotes: this should be part of good culture here... – Tomas Mar 20 '12 at 23:21
1) no, the confidence interval is not defined by those quantiles specifically. a 90% confidence interval has the 5% and 95% quantiles of the sampling distribution of the test statistic, 2) the (meta) distribution from which the quantiles are generated is more important than their actual rank values, 3) your description hitherto has in no way invoked the definition of being closed, i.e. containing limit points on the real line. – AdamO Jun 20 '12 at 15:49