What would be a parametric model with properties similar to the Theil-Sen estimator? 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... 
Edit:
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:

References
Wilcox, R. R. (2010). Fundamentals of modern statistical methods: Substantially improving power and accuracy. Springer.
 A: I beleive, the S estimator[1] (and it's algorithm, FastS[2]) is the closest parametric equivalent to the Theil-Sen estimator.
This is because the S estimator explicitly adds a parametric assumption on the distribution of the residuals (through the tuning constant $c$) to get better efficiency at uncontaminated samples. 
The FastS algorithm is implemented in the robustbase R package[3] distributed through CRAN.
There are some differences between the two approaches:


*

*FastS is more robust to outliers than Theil-Sen (the latter has a breakdown point of 0.29, the former 0.5)

*FastS can be computed efficiently for moderately sized dataset, including when there are more than one regressor. Theil-Sen is only defined for univariate regression.


These two differences explain why the Theil Sen estimator is essentially deprecated.


*

*Rousseeuw, P.J. and Yohai, V.J. (1984).
Robust regression by means of S-estimators, In Robust and
Nonlinear Time Series, J. Franke, W. Hardle and R. D. Martin (eds.). Lectures Notes in Statistics
26, 256--272, Springer Verlag, New York.

*Salibian-Barrera, M. Yohai, V.J. (2006).
A Fast Algorithm for S-Regression Estimates.
Journal of Computational and Graphical Statistics, Vol. 15, 414--427.

*Rousseeuw P.,  Croux C.,  Todorov V., 
Ruckstuhl A.,  Salibian-Barrera M., Verbeke T., Koller M.,
Maechler M. (2012). robustbase: Basic Robust Statistics. R
package version 0.9--5. 

A: One possibility consists of using flexible error distributions. This is, you have a model
$$y_j = x_j^{\top}\beta + \epsilon_j,$$
where $\epsilon_j\sim F$, and $F$ is a flexible distribution. So, for instance, in order to produce a model that is (relatively) robust to the presence of outliers and skewness, a possible choice for $F$ is a skew-t distribution (there are several types of this).
