A question on X validated on the comparative merits of L¹ versus L² estimation led me to the paper of Stephen Portnoy and Roger Koenker entitled “The Gaussian Hare and the Laplacian Tortoise: Computability of Squared-Error versus Absolute-Error Estimators”, which I had missed at the time, despite enjoying a subscription to Statistical Science till the late 90’s.. The authors went as far as producing a parody of Granville’s Fables de La Fontaine by sticking Laplace’s and Gauss’ heads on the tortoise and the hare!
I remember rather vividly going through Steve Stigler’s account of the opposition between Laplace’s and Legendre’s approaches, when reading his History of Statistics in 1990 or 1991… Laplace defending the absolute error on the basis of the default double-exponential (or Laplace) distribution, when Legendre and then Gauss argued in favour of the squared error loss on the basis of a defaul Normal (or Gaussian) distribution. (Edgeworth later returned to the support of the L¹ criterion.) Portnoy and Koenker focus mostly on ways of accelerating the derivation of the L¹ regression estimators. (I also learned from the paper that Koenker was one of the originators of quantile regression.)
