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Recently, I found out that the Confidence Intervals on some parameter estimate does not necessarily need to be "symmetric".

By this I mean, suppose we estimate some parameter to be 0.3 - generally, we will say something like "there is a 0.95 probability that true value of this parameter might fall anywhere between (0.25,0.35)". Loosely put, we can say that this Confidence Interval is "symmetric" around 0.3. This being said, I was under the impression that Confidence Intervals must always be symmetric.

However, recently I found out that this is not always the case - in some instances, Confidence Intervals can be "asymmetric" (e.g. The parameter corresponding to the "Probability of Success" from a Binomial Distribution). As an example, the parameter might be estimated as 0.3 but a 95% Confidence Interval might fall anywhere between (0.28, 0.35) - in this case, the Confidence Interval is not "symmetric".

This leads me to my question:

  • Do we know why Confidence Intervals can sometimes be "asymmetric"? It it because the underlying Probability Distribution might be "asymmetric"? Or perhaps this has something to do with sample size?
  • A priori, is it possible to determine if the Confidence Interval for a parameter being estimated from a specific Probability Distribution might end up being "asymmetric"? As an example, are there certain instances where we can mathematically show that the Confidence Interval will always be symmetric?
  • And finally, in the case of Maximum Likelihood Estimation (MLE) - parameters estimated using MLE are said to be "Asymptotically Normal". As I understand, this means that as the sample size grows bigger and bigger, the distribution of the estimates from MLE will become closer and closer to a Normal Distribution. I have heard that Confidence Intervals for parameter estimates generally exploit this fact - as an example, $$\bar{x} \pm z_{\alpha/2} \sqrt{\frac{s^2}{n}}$$ Here, the "z-alpha" term is related to the (Standard) Normal Distribution. Thus, given that estimates from MLE are Asymptotically Normal and the Normal Distribution is said to be symmetric - how can the Confidence Intervals for parameters estimated under MLE ever be asymmetric?

Thanks!

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In a handwaving sense, a confidence interval is an interval produced by a method which is designed to give a probability (often $95\%$ is used lazily but it could be any desired level of confidence) that the interval will cover the parameter of interest. There is no particular need for the interval to be symmetric about some central estimate of the parameter. Here are four types of confidence interval methods (others are possible), where the first will ensure symmetry by construction, the second and third may or may not be symmetric depending on the particular question and the fourth will not be symmetric:

  • make a central estimate of the parameter, and then chose the width of the confidence interval on each side so that overall the probability of covering the true parameter is $95\%$ (or whatever). Your $\bar{x} \pm z_{\alpha/2} \sqrt{\frac{s^2}{n}}$ might be an example of this.
  • construct the confidence interval so the likely width of the interval is in some sense minimised while ensuring the probability of covering the true parameter is $95\%$ (or whatever)
  • construct the confidence interval so the probability of the interval being completely below the true value is equal to the probability of the interval being completely above the true value - each $2.5\%$ (or whatever)
  • construct a one-tailed confidence interval so, if covering the lower tail and stretching down to the minimum possible value of the parameter, the probability of the interval being completely below the true value is $5\%$ (or whatever); similarly you could produce a one-tailed confidence interval covering the upper tail

Each has its advantages and disadvantages: for example the third and fourth intervals work well when you transform the parameter with a monotonic increasing function, since you can apply the same function to the ends of the interval, but these two methods need not have the useful properties of the first two methods.

For some questions such as a confidence interval for the unknown mean of a normal distributed random variable, the first three may produce the same results and so a symmetric confidence interval, but not for example when estimating the unknown mean of an exponential distribution.

So the answers to your detailed questions depend on the method used for the construction of the confidence interval and the particular question you are trying to answer.

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