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I am reading a book on time series analysis and I am having problems understanding the section about outlier detection.

The authors say that when you want to know whether at a certain time $T$ there was an outlier, you should use a certain test statistic and a test with size less than $\alpha$. But when you don't know where an outlier could be and you have a time series of size $n$ then you should use the same test statistic for each point but you should use tests of size $\alpha/n$. They say that this is an application of the conservative Bonferroni correction.

I just don't understand this. Doesn't this mean that there will be lots of outliers that you detect in short time series but don't detect in large ones? After all, spam filters don't have stronger spam criteria for people with more incoming email, right?

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If you do $n$ tests of size $\alpha/n$, then $\alpha$ is the Bonferroni bound on at least one of the tests succeeding. It is conservative because it is the worst possible bound without any further information about dependency between the tests. It is only exact if the tests are disjoint (i.e. at most one can be true at once).

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Brendan and mine Bonferroni connection goes a long way... – Gil Kalai Sep 6 '11 at 15:46
Thank you for the clarifications on the Bonferroni bound! But still: would you say it is correct to use a test of size $0.5/n$, say, to find outliers in time series of size $n$? I would say it is wrong to divide by $n$ (as it would be wrong to use stronger spam criteria for people with more email traffic). – Frank Sep 7 '11 at 8:24
If your customers insist that they want to lose at most one valid email per week on average due to false spam positives, you do indeed need to apply a stronger test to those who get more email. If instead they demand that at most some fraction $p$ of their valid emails are rejected on average, then the volume of email is irrelevant. So it depends on what you are trying to achieve. – Brendan McKay Sep 8 '11 at 4:59

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