Wrapped Skew Laplace Distribution on Integers : A New Probability Model for Circular Data

In this paper we propose a new family of circular distributions, obtained by wrapping discrete skew Laplace distribution on Z = 0, ±1, ±2, around a unit circle. In contrast with many wrapped distributions, here closed form expressions exist for the probability density function, the distribution function and the characteristic function. The properties of this new family of distribution are studied.


Introduction
Circular data arise in various ways.Two of the most common correspond to circular measuring instruments, the compass and the clock.Data measured by compass usually include wind directions, the direction and orientations of birds and animals, ocean current directions, and orientation of geological phenomena like rock cores and fractures.Data measured by clock includes times of arrival of patients at a hospital emergency room, incidences of a disease throughout the year, where the calendar is regarded as a one-year clock.Circular or directional data also arise in many scientific fields, such as Biology, Geology, Meteorology, Physics, Psychology, Medicine and Astronomy [1].
Study on directional data can be dated back to the 18th century.In 1734 Daniel Bernoulli proposed to use the resultant length of normal vectors to test for uniformity of unit vectors on the sphere [2].In 1918 von Mises introduced a distribution on the circle by using characterization analogous to the Gauss characterization of the normal distribution on a line [2].Later, interest was renewed in spherical and circular data by [3][4][5].
Circular distributions play an important role in modeling directional data which arise in various fields.In recent years, several new unimodal circular distributions capable of modeling symmetry as well as asymmetry have been proposed.These include, the wrapped versions of skew normal [6], exponential [7] and Laplace [8].
Wrapped distributions provide a rich and useful class of models for circular data.
The special cases of the wrapped normal, wrapped Pois-son, wrapped Cauchy are discussed in [9].We give a brief description of circular distribution in Section 2. In Section 3 we introduce and study Wrapped Discrete Skew Laplace Distribution.Section 4 deals with the estimation of the parameters using the method of moments.

Circular Distributions
A circular distribution is a probability distribution whose total probability is concentrated on the circumference of a circle of unit radius.Since each point on the circumference represents a direction, it is a way of assigning probabilities to different directions or defining a directional distribution.The range of a circular random variable Θ measured in radians, may be taken to be  0,2π or   π, π  .Circular distributions are of two types: they may be discrete -assigning probability masses only to a countable number of directions, or may be absolutely continuous.
In the latter case, the probability density function   f θ exists and has the following basic properties.

Wrapped Distributions
One of the common methods to analyze circular data is known as wrapping approach [10].In this approach, given a known distribution on the real line, we wrap it around the circumference of a circle with unit radius.Technically this means that if X is a random variable on the real line with distribution function and the distribution function of X is given by By this approach, we are accumulating probability over all the overlapping points So if represents a circular density and the density of the real valued random variable, we have By this technique, both discrete and continuous wrapped distributions may be constructed.In particular, if X has a distribution concentrated on the points and m is an integer, the probability function of X is given by where "p" is the probability function of the random variable X.

Discrete Skew Laplace Distribution
Discrete Laplace distribution was introduced by [11] following [12], who defined a discrete analogue of the normal distribution.The probability mass function of a general Discrete Normal random variable Y can be written in the form where "f" is the probability density function of a normal distribution with mean µ and variance  [13].
Using Equation ( 5), for any continuous random variable X on R, we can define a discrete random variable Y that has integer support on Z.When the skew Laplace density The characteristic function of Y is given by In this paper, we study the probability distribution obtained by wrapping discrete skew Laplace distribution on around a unit circle.  As we know, reduction modulo 2π wraps the line onto the circle, reduction modulo 2πm (if m is a positive integer) wraps the integers onto the group of root of 1, regarded as a subgroup of the circle.That is, if X is a random variable on the integers, then Θ, defined by , on the circle.The probability function of Θ is given by Equation (4).
In particular, if X has a discrete skew Laplace distribution with parameters and , then the probability and we denote it by WDSL Following are the graphs of wrapped discrete skew Laplace distribution for various values of κ, σ and m.In Figure 1, the graph of the pdf of wrapped discrete skew Laplace distribution for κ = 0.25, σ = 1 and for m = 5, 10, 20, 30, 40, 50 and 100 are given. In

Special Cases
When either "p" or "q" converges to zero, we obtain the following two special cases: with Hence represents a probability distribution.
w Definition 3.2 An angular random variable "Θ" is said to follow wrapped skew Laplace distribution on integers with parameters p, q and m if its probability mass function is given by is a wrapped geometric distribution with probability mass function while  with is a wrapped geometric distribution with probability mass function Cop OJS when p = q, we have which is the probability mass function of wrapped discrete Laplace distribution.

Distribution Function of WDSL
The distribution function, is given by , , p q m

Probability Generating Function and
Characteristic Function of WDSL


The probability generating function of WDSL m m s q p p q p q sp q q sp q p q s q q s sp q Also, we have .
is the characteristic function of a linear random variable X, then the characteristic function of the wrapped random variable, w π 2

1
(1 . Thus for the wrapped discrete skew Laplace distribution, we have On simplification it reduces to Again, we have and where where 1 Θ and 2 Θ are two independently distributed d g wrappe eometric random variables with probability mass functions Proof.e,

 
We hav We know that the geometric mass function,   (1 ) , 0,1, ution is infinitely divisible.Hence Θ ~( , , ) WDSL p q m is infinitely divisible.By the well kn n properties of geometric law [14], Θ ~ ( , , ) WDSL p q m admits a representation involving wr omial distribution, so wrapped geom ow appe 2 , be identically and independently distribut gular ran ean Let 1 ,   ed an dom variables following WDSL ( , , )  p q m and u N is a geometric random variable with m 1 u itio the nd .Cond ning on the distribution of u N we can write characteristic function of the right ha side of the above equation as (Equation (20)) Now we show that the above function coincides with the characteristic function of

 
, , WDSL s r m distribution with s and r as given above.Setting Equation (20) equal to (1 e ) w ld hold for each p, q.This will happen when the following two equations hold simultaneously.
Dividing Equation (23) by Equation (24), w s .q r p  Substituting the value of r in Equation ( 23) we get On simplification, Equation (25) reduces to   , and Therefore,   s f admits a unique solution in the interval   and is given by q p p q u q p p q u qp q p q p p q u q p p q u qp te skew Laplace distribuar random varidivisible random 2  Remark 3.1.Wrapped discre tion is infinitely divisible since a circul able obtained by wrapping an infinitely variable is infinitely divisible by [1,11] .

Trigonometric Moments
The th n trigonometric moment of the WDSL   , , p q m is given by The above expression can also be expressed in the form is the p th mean resultant length and The length of the mean resultant vector, The measure of skewness, The measure of kurtosis,

Method of Moments
Let , , k Using Equations ( 33)-( 34) and for a fixed value of "m" we can find estimates for "p" and "q".We have, which gives Substituting the value of "p" in terms of "q" in Equation (33) or in Equation (34) we will get an equation in "q" and solving that we can find the estimate of "q" and thus "p".

( 6 )Definition 3 . 1 A
where,   , are inserted into Equation (5), the probability mass function of the resulting discrete distribution takes an explicit form in terms of the parameters p * = e random variable Y has a discrete skew Laplace distribution with parameters and denoted by DSL , if

Figure 1 .Figure 2 .Figure 3 .
Figure 1.Wrapped discrete skewed Laplace distribution for , and for different values of "m".= 1 σ = 0.25 κ sample of size n taken from L ( , , ) p q m distribution with parameters p, q,

Stability with Respect to Geometric Summation
u N j j