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).
“It will suffice therefore, either to resolved this last equation, or at least to attribute to θ a sufficiently small value, in order to obtain a new value of u inferior to u. If the new value of u is not a minimum, one will be able to deduce, by operating always in the same manner, a third still smaller value; and, by continuing thus, one will find successively some values of u more and more small, which will converge toward a minimum value of u. If the function u, which is supposed not at all to admit negative values, offers some null values, there will be able always to be determined by the preceding method, provided that one chooses conveniently the values of x, y, z, …” Augustin Cauchy, CRAS, 25, 536–538, 1847 [translation of Richard J. Pulskamp, 2010]
