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).

8 Responses to “complex Cauchys”

  1. Georges Henry Says:

    Franchement, montrer que $C=X/Y$ est Cauchy standard quand $X$ et $$ sont iid N(0,1) est completement trivial car $C$ est symetrique et E(|C|^s) se calcule trivialement pour $-1<\Re s<1.$

  2. Emmanuel Charpentier Says:

    Ahem… Cette paramétrisation complexe est citée dans la page Wikipedia [idoine](https://en.wikipedia.org/wiki/Cauchy_distribution#Characterisation) (4° alinéa)…

    Par ailleurs, j’ai eu le plus grand mal à trouver une référence pour l’article de Geary, et ne suis pas arrivé à trouver une référence pour l’article de Feller. Vous serait-il possible de citer un DOI ou, si celui-ci n’est pas disponible, une référence quand vous renvoyez à un papier important ?

  3. Georges Henry Says:

    bon cette idee de parametrer les Cauchy par les complexes n’a quand meme pas ete inventee par McCullagh puisqu’elle est essentielle dans l’article ‘Which functions preserve Cauchy laws?’ Proceedings AMS 1977 pages 277-286.

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