I have a univariate time series which is an index of susceptibility to failure, and a binary variable which indicates whether a failure actually occurred in a given time window or not. I want to carry out a statistical test that would quantify the performance of the susceptibility index (e.g., compared to a null model where there is no correlation between the time series and the binary variable, or to another competing index). I am looking for methods to achieve this goal.
I understand that if the failure is a very rare event obtaining a statistical significance may be hard, but even before that, my main problem is that the data points I sample from the time series are not independent (e.g., if the index is very high at a given time period, it's very likely high also in the next period). Therefore the length of the time window that I employ should be important. Any ideas?