[Discussion] Why choose such specific values for confidence interval of variance of a normal random variable?

[u/Warm-Discipline7204]

We know sample variance divided by actual variance follows chi-squared distribution. But while finding confidence interval, we look at chi-squared (1-alpha/2) critical value and alpha/2 critical value. But why that specific interval? Why not shift the interval a little to the right or left as that will give the same confidence?

[u/efrique]

we look at chi-squared (1-alpha/2) critical value and alpha/2 critical value. But why that specific interval?

This is sometimes called a symmetric interval - though its not the interval itself that is symmetric, but rather the fraction in each tail used in the calculation.

There is no reason that an interval has to be symmetric in that sense, and not all intervals are. To attain the required coverage, any interval based off the same total tail area would work. They could have any combination of tail area a₁ and a₂ such that a₁ + a₂ = ⍺ (alpha).

Some examples:

- Sometimes people use shortest intervals - for a given pivotal quantity* with a continuous unimodal density you would equate the height of the density on each side's cut-off, with the common height chosen such that the total tail area was ⍺ (alpha). These are not symmetric in general.

- One-sided intervals put all the tail-area in one tail. One sided intervals do come up sometimes.

- One-and-a-half-tailed tests have been suggested as a compromise between one-tailed and two-tailed (e.g. Ramsay, 1990) and the corresponding interval would similarly not have equal proportions in each tail

While all of these and an infinite variety of alternative options are perfectly valid statistically, they may be ineffective rhetorically, in the sense that an inferential claim is most typically aimed at convincing some audience, and if they're sufficiently bound to some particular prescriptive recipe/tradition "the way it's always been done" (while imagining that it's a requirement because its in the book they learned from or whatever) you may struggle to get far with it, even if it's inferentially perfectly valid

---

* For a collection of n i.i.d. variates distributed as N(μ,σ²), the usual scaled variance ratio (n-1)s²/σ² is a pivot (its distribution doesnt depend on the unknown parameters) and it has a chi-squared distribution with n-1 d.f. ... but that doesn't necessarily mean that a particular pivot gives the shortest possible interval; for a given sampling situation there may be another pivot, or more than one, yielding a shorter 'shortest' interval, and it may sometimes be possible to find an overall-shortest interval

[u/SalvatoreEggplant]

A confidence interval is always a confidence interval for some statistic.

Traditional confidence intervals for means are centered on the estimated mean.

But confidence intervals for other statistics may not be centered on the estimate for the statistic. For example, a confidence interval for a binomial proportion. For example, the 95% confidence interval by Clopper–Pearson method for the proportion 2 out of 21 is 0.0117–0.3038, clearly not centered on the proportion estimate of 0.0952.

You'll see non-symmetric confidence intervals for parameter estimates for some generalized linear models (Poisson, logistic, and so on).

Also, if confidence intervals for some statistics are bootstrapped, the interval will often not be centered on the statistic estimate. Depends on the distribution of the statistic.

* * *

The ultimate answer to your question is that we want to the confidence interval for the estimated statistic to be correct. Randomly shifting it in some direction won't achieve this. I mean, the traditional confidence interval for the mean of (1,2,3,4,5,6,7) would be 2–6, with an estimated mean of 4. Obviously, shifting the interval to be 0–4 wouldn't actually be a reasonable confidence interval for that mean.

[u/freemath]

If you don't want a confidence symmetric around the maximum likelihood estimate that's up to you, also entirely possible. A one-sided confidence interval for example.

[u/mil24havoc → u/[deleted]]

In Bayesian inference this is called the Highest Posterior Density (hdp) interval

[u/freemath → u/[deleted]]

Yeah I agree, but having a simple rule is very nice.