Wrapped Cauchy distribution
In probability theory and directional statistics, a wrapped Cauchy distribution is a wrapped probability distribution that results from the "wrapping" of the Cauchy distribution around the unit circle. The Cauchy distribution is sometimes known as a Lorentzian distribution, and the wrapped Cauchy distribution may sometimes be referred to as a wrapped Lorentzian distribution. The wrapped Cauchy distribution is often found in the field of spectroscopy where it is used to analyze diffraction patterns (e.g. see Fabry–Pérot interferometer). Description. The probability density function of the wrapped Cauchy distribution is: where formula_2 is the scale factor and formula_3  is the peak position of the "unwrapped" distribution. Expressing the above pdf in terms of the characteristic function of the Cauchy distribution yields: The PDF may also be expressed in terms of the circular variable "z" = "e" and the complex parameter "ζ" = "e" where, as shown below, "ζ" = ⟨"z"⟩. In terms of the circular variable formula_6 the circular moments of the wrapped Cauchy distribution are the characteristic function of the Cauchy distribution evaluated at integer arguments: where formula_8 is some interval of length formula_9. The first moment is then the average value of "z", also known as  the mean resultant, or mean resultant vector: The mean angle is and the length of the mean resultant is yielding a circular variance of 1 − "R". Estimation of parameters. A series of "N" measurements formula_13 drawn from a wrapped Cauchy distribution may be used to estimate certain parameters of the distribution. The average of the series formula_14 is defined as and its expectation value will be just the first moment: In other words, formula_14 is an unbiased estimator of the first moment. If we assume that the peak position formula_3 lies in the interval formula_19, then Argformula_20 will be  a (biased) estimator of the peak position formula_3. Viewing the formula_22 as a set of vectors in the complex plane, the formula_23 statistic is the length of the averaged vector: and its expectation value is In other words, the statistic will be an unbiased estimator of formula_27, and formula_28 will be a (biased) estimator of formula_2. Entropy. The information entropy of the wrapped Cauchy distribution is defined as: where formula_31 is any interval of length formula_9. The logarithm of the density of the wrapped Cauchy distribution may be written as a Fourier series in formula_33: where which yields: (c.f. Gradshteyn  and Ryzhik 4.224.15) and (c.f. Gradshteyn and Ryzhik 4.397.6). The characteristic function representation for the wrapped Cauchy distribution in the left side of the integral is: where formula_39. Substituting these expressions into the entropy integral, exchanging the order of integration and summation, and using the orthogonality of the cosines, the entropy may be written: The series is just the Taylor expansion for the logarithm of formula_41 so the entropy may be written in closed form as: Circular Cauchy distribution. If "X" is Cauchy distributed with median μ and scale parameter γ, then the complex variable has unit modulus and is  distributed on the unit circle with density: where and ψ expresses the two parameters of the associated linear Cauchy distribution for "x" as a complex number: It can be seen that the circular Cauchy distribution has the same functional form as the wrapped Cauchy distribution in "z" and ζ (i.e. f(z,ζ)). The circular Cauchy distribution is a reparameterized wrapped Cauchy distribution: The distribution formula_48 is called the circular Cauchy distribution (also the complex Cauchy distribution) with parameters μ and γ. (See also McCullagh's parametrization of the Cauchy distributions and Poisson kernel for related concepts.) The circular Cauchy distribution  expressed in complex form has finite moments of all orders for integer "n" ≥ 1. For |φ| < 1, the transformation is holomorphic on the unit disk, and the transformed variable "U"("Z", φ) is distributed as complex Cauchy with parameter "U"(ζ, φ). Given a sample "z", ..., "z" of size "n" > 2, the maximum-likelihood equation can be solved by a simple fixed-point iteration: starting with ζ = 0. The sequence of likelihood values is non-decreasing, and the solution is unique for samples containing at least three distinct values. The maximum-likelihood estimate for the median (formula_53) and  scale parameter (formula_54) of a real Cauchy sample is obtained by the inverse transformation: For "n" ≤ 4, closed-form expressions are known for formula_56. The density of the maximum-likelihood estimator at "t" in the unit disk is necessarily of the form: where Formulae for "p" and "p" are available.