The Theil-Sen estimator is a really nifty algorithm that produces a regression line that is relatively insensitive to outliers both in the response variable and the predictor variable.
I've been wondering what a parametric "analogue" to the non-parametric Theil-Sen estimator would be. Or if there is no strict analogue, what would be a good example of a parametric model with properties similar to the Theil-Sen estimator?
With parametric analogue I mean something in the same sense as the Laplace distribution is the model that "gives you" the median. My hunch is that a bivariate Laplace distribution would have properties similar to the Theil-Sen estimator, but this is just a hunch...
As mentioned by Wilcox (2010) it does not seems like a linear regression with an error distribution that is insensitive to outliers will have similar properties as the Theil-Sen estimator. While both Theil-Sen and, for example, LAD are relative robust against outliers on the Y-axis, LAD is sensitive to outliers on the X-axis while Theil-Sen is not. For an example of this see the figure from Wilcox (2010) below:
Wilcox, R. R. (2010). Fundamentals of modern statistical methods: Substantially improving power and accuracy. Springer.