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".

I would add to Dan Levin's answer that when you want to justify a method, the "scientific/engineering way" is to first produce a bibliographical study, where you basically prove that your approach covers an important part of what is commonly known as state-of-the-art methods. I would resume this as follow:

1. Look for commonly used methods that are known to be efficient for outlier detection.
2. Summarize their applicability domains (medicine, biology, network security..) and try to link their strong points to your application in order to select some promising methods.
3. Try the selected methods with usual validation processes inherent to machine learning problems.

Defining what the state-of-the-art consists of a lot of work, is absolutely necessary and is very specific to your application.