Timeline for Which binomial prediction interval works well for tail probabilities, i.e. $\hat{p}=1/n$ for large $n$
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| Jun 11, 2020 at 14:32 | history | edited | CommunityBot |
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| Mar 7, 2017 at 20:49 | vote | accept | Sycorax♦ | ||
| Mar 7, 2017 at 20:07 | comment | added | whuber♦ | I believe so. Indeed, according to the third bullet of the question you don't really know what $F(k)$ is for any $k$--the best you could do (if you had to) is to estimate it. | |
| Mar 7, 2017 at 19:54 | comment | added | Sycorax♦ | I'm stumbling over part of the reasoning. The probability that the next draw from $F$ is at or below some $k$ is $F(k)$. Across $m$ iid draws, the number of draws below $k$ has a binomial $m, F(k)$ distribution. Is it the case that the distinction between your answer and this binomial model is that the binomial model supposes $k$ is fixed up front, whereas in my problem, we are interested in $x_{(1)}$? | |
| Mar 7, 2017 at 0:08 | comment | added | Sycorax♦ | It's an introductory mathematical statistics text, so I think the procedure is outlined for primarily pedagogical reasons. Your point about exact quantities and inversion is well-taken. Thank you for this well-considered answer. | |
| Mar 6, 2017 at 23:39 | comment | added | whuber♦ | That's conceivable. (I'm not familiar with H, McK, and C.) But if that's all the bootstrap is doing, you ought to consider obtaining exact answers (with much less computation) using the combinatorial formulas. They have the advantage of letting you invert the problem in order to find sample sizes to achieve any desired size in a PL, for instance. | |
| Mar 6, 2017 at 23:36 | comment | added | Sycorax♦ | Extending this line of reasoning farther, this must be exactly how we get to the two-sample bootstrap procedure to estimate quantiles outlined in Hogg McKean and Craig: the bootstrap approximates the more elaborate combinatorial result. | |
| Mar 6, 2017 at 23:28 | history | answered | whuber♦ | CC BY-SA 3.0 |