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probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, the "standard"
Cauchy distribution The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fu ...
is the
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
whose
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
(pdf) is :f(x) = for ''x''
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (201 ...
. This has median 0, and first and third quartiles respectively −1 and +1. Generally, a Cauchy distribution is any probability distribution belonging to the same location-scale family as this one. Thus, if ''X'' has a standard Cauchy distribution and ''μ'' is any real number and ''σ'' > 0, then ''Y'' = ''μ'' + ''σX'' has a Cauchy distribution whose median is ''μ'' and whose first and third quartiles are respectively ''μ'' − ''σ'' and ''μ'' + ''σ''. McCullagh's parametrization, introduced by
Peter McCullagh Peter McCullagh (born 8 January 1952) is a Northern Irish-born American statistician and John D. MacArthur Distinguished Service Professor in the Department of Statistics at the University of Chicago. Education McCullagh is from Plumbridge ...
, professor of statistics at the
University of Chicago The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private university, private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park, Chicago, Hyde Park neighborhood. The University of Chic ...
, uses the two parameters of the non-standardised distribution to form a single complex-valued parameter, specifically, the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
''θ'' = ''μ'' + ''iσ'', where ''i'' is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition a ...
. It also extends the usual range of scale parameter to include ''σ'' < 0. Although the parameter is notionally expressed using a complex number, the density is still a density over the real line. In particular the density can be written using the real-valued parameters ''μ'' and ''σ'', which can each take positive or negative values, as :f(x) = \,, where the distribution is regarded as degenerate if ''σ'' = 0. An alternative form for the density can be written using the complex parameter ''θ'' = ''μ'' + ''iσ'' as :f(x) = \,, where \Im = \sigma. To the question "Why introduce complex numbers when only real-valued
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
s are involved?", McCullagh wrote:
To this question I can give no better answer than to present the curious result that :Y^* = \sim C\left(\right) for all real numbers ''a'', ''b'', ''c'' and ''d''. ...the induced transformation on the parameter space has the same fractional linear form as the transformation on the sample space only if the parameter space is taken to be the complex plane.
In other words, if the random variable ''Y'' has a Cauchy distribution with complex parameter ''θ'', then the random variable ''Y'' * defined above has a Cauchy distribution with parameter (''aθ'' + ''b'')/(''cθ'' + ''d''). McCullagh also wrote, "The distribution of the first exit point from the upper half-plane of a Brownian particle starting at ''θ'' is the Cauchy density on the real line with parameter ''θ''." In addition, McCullagh shows that the complex-valued parameterisation allows a simple relationship to be made between the Cauchy and the "circular Cauchy distribution". Using the complex parameter also let easily prove the invariance of f-divergences (e.g., Kullback-Leibler divergence, chi-squared divergence, etc.) with respect to real linear fractional transformations (group action of SL(2,R)), and show that all f-divergences between univariate Cauchy densities are symmetric.


References

*
Peter McCullagh Peter McCullagh (born 8 January 1952) is a Northern Irish-born American statistician and John D. MacArthur Distinguished Service Professor in the Department of Statistics at the University of Chicago. Education McCullagh is from Plumbridge ...

"Conditional inference and Cauchy models"
''
Biometrika ''Biometrika'' is a peer-reviewed scientific journal published by Oxford University Press for thBiometrika Trust The editor-in-chief is Paul Fearnhead ( Lancaster University). The principal focus of this journal is theoretical statistics. It was ...
'', volume 79 (1992), pages 247–259
PDF
from McCullagh's homepage. * Frank Nielsen and Kazuki Okamura
"On f-divergences between Cauchy distributions"
'' arXiv 2101.12459'' (2021). {{ProbDistributions, continuous-infinite Continuous distributions