truncated Gumbels

As I had to wake up pretty early on Easter morning to give my daughter a ride, while waiting I came upon this calculus question on X validated of computing the conditional expectation of a Gumbel variate, conditional on its drifted version being larger than another independent Gumbel variate with the same location-scale parameters. (Just reminding readers that a Gumbel G(0,1) variate is a double log-uniform, i.e., can be generated as X=-log(-log(U)).) And found after a few minutes (and a call to Wolfram Alpha integrator) that

\mathbb{E}[\epsilon_1|\epsilon_1+c>\epsilon_0]=\gamma+\log(1+e^{-c})

which is simple enough to make me wonder if there is a simpler derivation than the call to the exponential integral Ei(x) function. (And easy to check by simulation.)

Incidentally, I discovered that Emil Gumbel had applied statistical analysis to the study of four years of political murders in the Weimar Republic, demonstrating the huge bias of the local justice towards right-wing murders. When he signed the urgent call [for the union of the socialist and communist parties] against fascism in 1932, he got expelled from his professor position in Heidelberg and emigrated to France, which he had to leave again for the USA on the Nazi invasion in 1940. Where he became a professor at Columbia.

2 Responses to “truncated Gumbels”

  1. telescoper Says:

    That’s actually a Gumbel Type 1 distribution, to be precise.

    Not that I’m pedantic or anything..

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