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For anomaly detection, people often assume Gaussian distribution and then given a test sample, calculate it's p-value. If that value is less than significance level, we claim it as an anomaly.

However, what if my underlying null hypothesis distribution is significantly off from Gaussian? Consider the 1-D simple case, enter image description hereI have sampled data from that distribution by MC sampling and I use kernel density estimator to approximate the distribution (looks like a two mode distribution with gap between, see the black line in the figure above)

Then for the test data, one is the green star and it is reasonable to think it comes from the null-distribution. Another one is the read star, it is reasonable to consider it as anomaly just by looking at this graph. But how should I quantify this? Is there a corresponding p-value in this case? I think I should not just make decision based on the value of pdf valued at that given point because the value of pdf does not represent probability... but I am very confused is there any other quantity can be used to measure the abnormality of this.

Thanks!

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