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How many way to find the outlier?

For cook's distance, which level is the cut off of outlier?

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The criterion can vary. For a clear beginning explanation you can look up "Cook's distance" in Wikipedia and look at the section entitled "Detecting Highly Influential Observations". The conservative option they discuss is basically a hypothesis test, so your threshold value is based on the significance level (i.e. value of alpha) you choose.

HTH, Jack

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You ask how many ways to find outliers? If you get the chance to flip through Deza & Deza's Dictionary of Distances 2006.

It is an eye opener. It catalogs literally hundreds of families of distance functions (I say families because within each family distinct distance functions can be obtained by parametrization). It contains a section on distance functions used in data analysis.

Various distance functions can weigh outliers very differently. I don't know specifically about Cook's distance, but the classical example that shows this distinction is regression in $L_2$ versus $L_1$ loss functions.

$L_2$ is smooth, least-mean-squared (associated with Gauss) and weighs outliers quadratically.

$L_1$ loss is robust, polyhedral least-absolute-sum (associated with Laplace) and weighs outliers linearly.

So if you fit a line through a point cloud with $L_2$ distance and move just one point, you can easily affect the slope, whereas with $L_1$ you would have to move the point much more. Further the slope may be positive with one distance but negative with the other, so handling outliers can be quite problematic.

There are robust methods such as rank-order loss functionals where if you move the furthest outlier is has no effect at all. You can Wikipedia terms like segmented regression, robust regression etc.

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