Wrapped exponential distribution

Wrapped Exponential
Probability density function The support is chosen to be [0,2π]
Cumulative distribution function The support is chosen to be [0,2π]
Parameters	$\lambda >0$
Support	$0\leq \theta <2\pi$
PDF	${\frac {\lambda e^{{-\lambda \theta }}}{1-e^{{-2\pi \lambda }}}}$
CDF	${\frac {1-e^{{-\lambda \theta }}}{1-e^{{-2\pi \lambda }}}}$
Mean	$\arctan(1/\lambda )$ (circular)
Variance	$1-{\frac {\lambda }{{\sqrt {1+\lambda ^{2}}}}}$ (circular)
Entropy	$1+\ln \left({\frac {\beta -1}{\lambda }}\right)-{\frac {\beta }{\beta -1}}\ln(\beta )$ where $\beta =e^{{2\pi \lambda }}$ (differential)
CF	${\frac {1}{1-in/\lambda }}$

In probability theory and directional statistics, a wrapped exponential distribution is a wrapped probability distribution that results from the "wrapping" of the exponential distribution around the unit circle.

Definition

The probability density function of the wrapped exponential distribution is^[1]

f_{{WE}}(\theta ;\lambda )=\sum _{{k=0}}^{\infty }\lambda e^{{-\lambda (\theta +2\pi k)}}={\frac {\lambda e^{{-\lambda \theta }}}{1-e^{{-2\pi \lambda }}}},

for $0\leq \theta <2\pi$ where $\lambda >0$ is the rate parameter of the unwrapped distribution. This is identical to the truncated distribution obtained by restricting observed values X from the exponential distribution with rate parameter λ to the range $0\leq X<2\pi$ .

Characteristic function

The characteristic function of the wrapped exponential is just the characteristic function of the exponential function evaluated at integer arguments:

\varphi _{n}(\lambda )={\frac {1}{1-in/\lambda }}

which yields an alternate expression for the wrapped exponential PDF in terms of the circular variable z=e^{i (θ-m)} valid for all real θ and m:

{\begin{aligned}f_{WE}(z;\lambda )&={\frac {1}{2\pi }}\sum _{n=-\infty }^{\infty }{\frac {z^{-n}}{1-in/\lambda }}\\[10pt]&={\begin{cases}{\frac {\lambda }{\pi }}\,{\textrm {Im}}(\Phi (z,1,-i\lambda ))-{\frac {1}{2\pi }}&{\text{if }}z\neq 1\\[12pt]{\frac {\lambda }{1-e^{-2\pi \lambda }}}&{\text{if }}z=1\end{cases}}\end{aligned}}

where $\Phi ()$ is the Lerch transcendent function.

Circular moments

In terms of the circular variable $z=e^{i\theta }$ the circular moments of the wrapped exponential distribution are the characteristic function of the exponential distribution evaluated at integer arguments:

\langle z^{n}\rangle =\int _{\Gamma }e^{{in\theta }}\,f_{{WE}}(\theta ;\lambda )\,d\theta ={\frac {1}{1-in/\lambda }},

where $\Gamma \,$ is some interval of length $2\pi$ . The first moment is then the average value of z, also known as the mean resultant, or mean resultant vector:

\langle z\rangle ={\frac {1}{1-i/\lambda }}.

The mean angle is

\langle \theta \rangle ={\mathrm {Arg}}\langle z\rangle =\arctan(1/\lambda ),

and the length of the mean resultant is

R=|\langle z\rangle |={\frac {\lambda }{\sqrt {1+\lambda ^{2}}}}.

and the variance is then 1-R.

Characterisation

The wrapped exponential distribution is the maximum entropy probability distribution for distributions restricted to the range $0\leq \theta <2\pi$ for a fixed value of the expectation $\operatorname {E}(\theta )$ .^[1]

References

1 2 Jammalamadaka, S. Rao; Kozubowski, Tomasz J. (2004). "New Families of Wrapped Distributions for Modeling Skew Circular Data" (PDF). Communications in Statistics - Theory and Methods. 33 (9): 2059–2074. doi:10.1081/STA-200026570. Retrieved 2011-06-13.

Probability distributions
List
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped

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