Which Outlier Detection Method? Why?
You can justify your choices by using data.

Treat the anomaly detection like a supervised learning problem where the concept is being anomaly.
Then you'll be able to present - for each method - its confusion matrix.
Not only it will be a good justification, it will enable to understand the expected results.

Many times, we have models and we wonder which confidence threshold we should use for alerting. In the supervised learning framework you'll be able to do trade-off like "increasing the confidence to X will lead to a better precision Y yet a decrease of the recall to Z".