Timeline for Two definitions of p-value: how to prove their equivalence?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Oct 25, 2015 at 15:59 | comment | added | math | @Glen_b Thanks for the comment. Indeed, my previous comment does violate the definition. Thanks for pointing it out. | |
| Oct 21, 2015 at 21:03 | comment | added | Glen_b | The given definition of p-value explicitly requires the test statistic for the sample to be in the rejection region. You are not free to change that part of the definition of p-value. | |
| Oct 16, 2015 at 17:50 | comment | added | math | Many thanks for your quick answer and in advance for your updated version. What I meant was the following: We reject $H_0$ if $T(x_n)\in R_\alpha$, where $x_n$ is the observed sample. Say I'm very extreme and choose $R_\alpha$ very small, so that for the given sample $T(x_n)\notin R_\alpha$ which just means we DONT reject $H_0$. So a small $R_\alpha$ isnt apriori a bad thing. Clearly, at one point it is so small, that's very very very unlikely to observe a sample belonging to $R_\alpha$. Again, thanks for your patience / help. really appreciated! | |
| Oct 16, 2015 at 17:45 | comment | added | heropup | Yes. The test statistic $T$ is a predetermined fixed function of the sample, where "fixed" in this sense means that the form of the function does not change for any $\alpha$. The value it takes on may (and should) depend on the sample. Your statement "we don't reject $H_0$" reveals why your disagreement is incorrect: by definition, $R_\alpha$ comprises the set of all values for which the test statistic leads to rejection of the null. That's why it's labeled $R$--for "R"ejection. I will post an update to my answer to explain in more detail. | |
| Oct 16, 2015 at 17:37 | comment | added | math | I'm still a little bit confused. So first, in definition $2$ is the statistic $T$ fixed for all $\alpha$? I disagree with your statement: "...at some point, $R_\alpha$ will be so small that it will exclude (i.e., fail to contain) the event we observed." Perfectly fine, if $R_\alpha$ is so small that it doesnt contain the observed sample, we dont reject $H_0$. What is the problem with this? thanks for you help / patience | |
| Oct 15, 2015 at 19:38 | history | answered | heropup | CC BY-SA 3.0 |