Timeline for What is a rigorous, mathematical way to obtain the shortest confidence interval given a confidence level?
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| Jul 19, 2020 at 20:24 | comment | added | Sextus Empiricus | It should be stressed that the tails considered here are the tails of the distribution of the data conditional on the parameter to be estimated. For instance we can use 2.5% tails and symmetry on both ends (north south) like estimating the parameter of a weird distribution here$${\hat\theta \sim \mathcal{N}(\mu=\theta, \sigma^2=1+\theta^2/3)}$$ but it does not necessarily give symmetric confidence intervals. In that particular example of the link one may end up with one of the boundaries at infinity. | |
| Jul 19, 2020 at 0:18 | comment | added | BruceET | Addendum to my Answer shows various lengths of the CI for $\sigma$ depending on the part of the 5% 'error probability' is allocated to the lower tail of the chi-sq dist'n. | |
| Jul 18, 2020 at 23:58 | history | edited | BruceET | CC BY-SA 4.0 |
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| Jul 18, 2020 at 21:42 | comment | added | BruceET | I did not mean my examples to give the impression that lengths of CIs are 'not much different' depending on allocation of 5% probability to the two tails. // Roughly speaking, shorter CIs result when cut-off points straddle the part of a dist'n with highest density (clearly the case of the CIs for $\mu.)$ But I guess, for my CIs for $\sigma,$ the relevant dist'n may be inverse chi-sq, not chi-sq. | |
| Jul 18, 2020 at 21:32 | history | edited | BruceET | CC BY-SA 4.0 |
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| Jul 18, 2020 at 20:13 | comment | added | Victor Luu | Based on @BruceET answer, it seems that there is no much difference between symmetric and non-symmetric tails. But are there some cases when there is a big difference in CI lengths which may call for the need of non-symmetric tails? | |
| Jul 18, 2020 at 19:59 | comment | added | Victor Luu | Very detailed and clear answer, thanks. | |
| Jul 18, 2020 at 19:44 | history | answered | BruceET | CC BY-SA 4.0 |