Outlier detection for bivariate bimodal distributions

I have been trying various methods to detect outliers in a bivariate dataset using Mahalanobis in R, but I am unsure about how correct it is since both of my variable vectors are not normally distributed, instead they are bimodal. If what I've understood so far is correct, Mahalanobis distances to detect outliers only works on normally distributed data.

Perhaps my solution is Stahel-Donoho outlyingness (SDO), which in principle is implemented in the package mrfDepth, but since I'm not a statistician I'm not sure if that's the right way to proceed.

I found this thread which talks about the same problem, but I'm still stumped.

My scientific question is as follows:

I have two sets of continuous bimodal variables, with a correlation coefficient of 0.83. In the scatterplot of the two variables, I want to detect the outliers, for example the ones which look very interesting to me are the two small triangles perpendicular to the imaginable regression line.

For bi/multivariate data with non-normal(bimodal in this case) distributions, is there a robust way to proceed to find outliers?

Thanks!

• This can depend on what constitutes an outlier for your problem. One method is to look at how a change in a data point affects the estimate of a model parameter that is important to you. This is based on the use of the influence function for that parameter. – Michael Chernick Jun 10 at 14:42
• @MichaelChernick thanks for your reply. For this problem, an outlier would perhaps be any point which on its removal, increases the correlation coefficient by a significant amount. Can that be a model parameter? – rishi Jun 10 at 15:01
• Yes in fact you might want read my paper "The influence function and its application to data validation" in the American Journal of Mathematics and Management Science 1979. In that paper I look at how the influence function for bivariate correlation and the multiple correlation coefficient can be used to detect outlier. There is also discussion of influence functions in Gnanadesikan's book on Multivariate Data and Barnett and Lewis' book on Outliers.. – Michael Chernick Jun 10 at 16:17