complex Cauchys

During a visit of Don Fraser and Nancy Reid to Paris-Dauphine where Nancy gave a nice introduction to confidence distributions, Don pointed out to me a 1992 paper by Peter McCullagh on the Cauchy distribution. Following my recent foray into the estimation of the Cauchy location parameter. Among several most interesting aspects of the Cauchy, Peter re-expressed the density of a Cauchy C(θ¹,θ²) as

f(x;θ¹,θ²) = |θ²| / |x-θ|²

when θ=θ¹+ιθ² [a complex number on the half-plane]. Denoting the Cauchy C(θ¹,θ²) as Cauchy C(θ), the property that the ratio aX+b/cX+d follows a Cauchy for all real numbers a,b,c,d,

C(aθ+b/cθ+d)

[when X is C(θ)] follows rather readily. But then comes the remark that

“those properties follow immediately from the definition of the Cauchy as the ratio of two correlated normals with zero mean.”

which seems to relate to the conjecture solved by Natesh Pillai and Xiao-Li Meng a few years ago. But the fact that  a ratio of two correlated centred Normals is Cauchy is actually known at least from the1930’s, as shown by Feller (1930, Biometrika) and Geary (1930, JRSS B).

Death sequence

August is not looking kindly at statisticians as I have now learned (after ten days of disconnection) of both Arnold Zellner and John Nelder passing away, on Aug. 11 and 15, respectively. Following this close the death of Julian Besag, this is a sad series of departures of leading figures in the fields of statistics and econometrics. Arnold was 83 and, although I had met him in several Valencia meetings—including one in Alicante where we sat together for breakfast with Persi Diaconis and  where an irate [and well-known ] statistician came to Arnold demanding apologies about comments made late the night before!—, I only had true interactions with him during the past years, over the Jeffreys reassessment I conducted with Judith Rousseau and Nicolas Chopin. On this occasion, Arnold was very kindly helpful, pointing out the volume that he had edited on Jeffreys and that I overlooked, discussing more philosophical points about the early part of Theory of Probability, and making a very nice overview of it at the O’Bayes 09 meeting. Always in the kindest manner. Sid Chib wrote an obituary of Arnold Zellner on the ISBA website (Arnold was the first ISBA president). Andrew Gelman also wrote some personal recollections about Arnold. A memorial site has been set up in his honour.

John Nelder was regularly attending the Read Paper sessions at the RSS and these are the only times I met him. He was an impressive figure in many ways, first and foremost for his monumental Generalised Linear Models with Peter McCullagh, a (difficult and uncompromising) book that I strongly recommend to (i.e. force upon!) my PhD students for its depth. I also remember being quite intimidated the first time I talked with him, failing to understand his arguments so completely that I dreaded later discussions… John Nelder was at  Fisher’s Rothamsted Experimental Station for most of his career and was certainly one of the last genuine Fisherians (despite a fairly rude letter of Fisher to him!).