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?



Bonferroni for outlier detection?
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).


