Timeline for Do the 2.5th and 97.5th percentile of the theoretical sampling distribution of a statistic always contain the true population parameter?
Current License: CC BY-SA 4.0
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| Oct 2, 2023 at 6:12 | history | bounty ended | Sextus Empiricus | ||
| Sep 28, 2023 at 12:19 | comment | added | Ute | @Christian, you may be right about the asymptotics. I constructed the example not with asymptotics in mind, but with focus on the percentile construction. There is a theoretical argument around (due to Efron?) stating that the percentile construction would work if the estimator in question can be transformed to have a normal (or symmetric) distribution. However the cases where this assumption is fulfilled are quite uninteresting imho. Nevertheless the percentile method sometimes gives good coverage close to the nominal level. I was not able to find any good explanation in the literature. | |
| Sep 25, 2023 at 16:26 | comment | added | Christian Hennig | @Ute Have you checked theory regarding when the bootstrap works? It's some time that I read this stuff (e.g. Enno Mammen "When does bootstrap work", also Davison and Hinkley discuss this), but my recollection is that bootstrap percentile intervals often need adjustment, and in any case it requires certain conditions that makes them asymptotically valid confidence intervals, maybe even something strong such as asymptotic normality of the test statistic (if with unknown asymptotic variance). This looks violated in your examples. | |
| Jul 31, 2023 at 1:55 | history | edited | Ute | CC BY-SA 4.0 |
typo
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| Jul 31, 2023 at 1:54 | comment | added | Ute | @whopper510, thank you, you are completely right, I was thinking at the sum and forgot to divide... | |
| Jul 31, 2023 at 1:01 | comment | added | whopper510 | Perhaps I misunderstand something about the counterexample given, but isn't the true mean of the Bernoulli variable $X$ equal to $p$ (not $np$)? | |
| Jul 30, 2023 at 22:05 | comment | added | Antonios Sarikas | @Ute Oops, I didn't notice the fixed size. | |
| Jul 30, 2023 at 22:03 | comment | added | Ute | @adosar, no, only if the population is normal distributed. However, if you let sample size increase, the sampling distribution will be approximately normal distributed, if the original distribution has finite mean and variance. This is due to the central limit theorem. In my example, I assumed a given sample size. The larger the sample size, the smaller gets $p = 1 -.975^{1/n}$. | |
| Jul 30, 2023 at 21:32 | comment | added | Antonios Sarikas | @Ute Isn't the sampling distribution of the sample mean normally distributed with mean equal to the population mean? As such, shouldn't the true parameter (population mean) be inside $[p_{0.025}, p_{97.5}]$? | |
| Jul 30, 2023 at 18:32 | history | edited | Ute | CC BY-SA 4.0 |
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| Jul 30, 2023 at 10:33 | comment | added | Ute | Yes, exactly @SextusEmpiricus. I just picked a very simple example. | |
| Jul 30, 2023 at 10:16 | comment | added | Sextus Empiricus | Another example is the case of asymmetric distributions, where one has to inverse the interval boundaries. Why do the bootstrap calculated p-value and the confidence intervals seem to contradict each other? | |
| Jul 30, 2023 at 2:11 | history | edited | Ute | CC BY-SA 4.0 |
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| Jul 29, 2023 at 23:47 | history | edited | Ute | CC BY-SA 4.0 |
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| Jul 29, 2023 at 23:37 | history | edited | Ute | CC BY-SA 4.0 |
Added a counter example
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| Jul 29, 2023 at 18:59 | history | edited | Ute | CC BY-SA 4.0 |
Fixed typo
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| Jul 29, 2023 at 15:13 | history | answered | Ute | CC BY-SA 4.0 |