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eISSN: 2378-315X

Biometrics & Biostatistics International Journal

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Received: January 01, 1970 | Published: ,

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Abstract

In this paper,we introduce a new circular distribution called Wrapped Holla and Bhattacharya’s (HB) Skewed Laplace distribution.We obtain explicit form for the density and derive expressions for distribution function, characteristic function and trigonometric moments. Method of maximum likelihood estimation is used for estimation of parameters. The methods are applied to a data set which consists of the wind directions of a Black Mountain ACT and compared with that of wrapped variance gamma distribution (WvG) and generalised von Mises (GvM) distribution.

Keywords: circular distribution, laplace distribution, hb skewed laplace, trigonometric moments, meteorology, wind direction

Introduction

Circular or directional data arise in different ways. The two main ways correspond to the two principal circular measuring instruments, the compass and the clock. Typical observations measured by the compass include wind directions and directions of migrating birds. Data of a similar type arise from measurements by spirit level or protractor. Typical observations measured by the clock include the arrival times (on a 24-hour clock) of patients at a casualty unit in a hospital. Data of a similar type arise as times of year or times of month of appropriate events. A circular observation can be regarded as a point on a circle of unit radius, or a unit vector (i.e. a direction) in the plane. Once an initial direction and an orientation of the circle have been chosen, each circular observation can be specified by the angle from the initial direction to the point on the circle corresponding to the observation.

Circular data are usually measured in degrees. Sometimes it is useful to measure in radians. For more details see Mardia & Jupp.1 Circular Statistics is the branch of statistics that addresses the modeling and inference from circular or directional data, i.e. data with rotating values. Many interesting circular models can be generated from known probability distributions by either wrapping a linear distribution around the unit circle or transforming a bivariate linear r.v. to its directional component. Two dimensional directions can be represented as angles measured with respect to some suitably chosen ”zero direction”. Since a direction has no magnitude, it can be conveniently represented as points on the circumference of a unit circle centered at the origin or as unit vectors connecting the origin to these points. Because of this circular representation, observations on such two dimensional directions are called circular data Jammalamadakka & Sen Guptha.2 Directional data have many unique and novel features both in terms of modeling and in their statistical treatment. In meteorology, wind directions provide a natural source of circular data. A distribution of wind directions may arise either as a marginal distribution of the wind speed and direction, or as a conditional distribution for a given speed. Other circular data arising in meteorology include the times of day at which thunderstorms occur and the times of year at which heavy rain occurs.

Recently, statisticians have shown active interest in the study of circular distribution because of their wide applicability. Jammalamadakka & Kozubowski3 discussed circular distributions obtained by wrapping the classical exponential and Laplace distributions on the real line around the circle. Rao et al.,4 derived new circular models by wrapping the well known life testing models like log normal, logistic, Weibull and extreme-value distributions. Roy & Adnan5 developed a new class of circular distributions namely wrapped weighted exponential distribution. In another work, Roy & Adnan6 explored wrapped generalized Gompertz distribution and discussed its application to Ornithology. Recently, Jacob & Jayakumar7 derived a new family of circular distribution by wrapping geometric distribution and studied its properties. Rao et al.,8 discussed the characteristics of wrapped Gamma distribution. Adnan & Roy9 derived wrapped variance Gamma distribution and showed its applicability to wind direction. Joshi & Jose10 introduced a wrapped Lindley distribution and applied it for a biological data. Recently, Jammalamadakka & Kozubowski11 introduced a general approach to obtain wrapped circular distributions through mixtures. The present paper is arranged as follows. In section 1, we give the general introduction about circular distribution. Section 2 explains the basic theory of circular distribution. Section 3 explains about the importance of Laplace distribution. In section 4, we discuss about skewed Laplace distribution. In section 5, we consider Holla and Bhattacharya’s (HB) Skewed Laplace (HBSL) distribution. In section 6, the Wrapped Holla and Bhattacharya’s (HB) Skewed Laplace (WHBSL) distribution is introduced and the probability density function (pdf) is obtained. In section 7, we derive the cumulative distribution function (cdf) and in section 8, the characteristic function and trigonometric moments are obtained. In Section 9, maximum likelihood estimation method is discussed to estimate the model parameter. In section 10, we apply the proposed model to a data on wind direction compared with that of wrapped variance gamma distribution (WvG) and generalised von Mises (GvM) distribution by using Akaike information criterion (AIC), Bayesian information criterion (BIC) and log-likelihood. Finally, Section 11 summarizes the findings.

Theory of circular distribution

A circular distribution is a probability distribution whose total probability is concentrated on the circumference of a unit circle.Since each point on the circumference represents a direction, such a distribution is a way of assigning probabilities to different directions or defining a directional distribution. The range of a circular random variable (rv) θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXbaa@38C4@ , measured in radians, may be taken to be [0, 2 π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabec8aWbaa@38CB@ ) or [- π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabec8aWbaa@38CB@ , π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabec8aWbaa@38CB@ ). Circular distributions are essentially of two types: discrete and continuous. In continuous case, a probability density function (pdf) f( θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXbaa@38C4@ )exists and has the following basic properties:

(i)f(θ)0; θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaaGikai aadMgacaaIPaGaamOzaiaaiIcacqaH4oqCcaaIPaGaeyyzImRaaGim aiaaiUdaaeaacqaH4oqCaaaa@4269@

(ii) 0 2π fθdθ=1. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiIcacaWGPb GaamyAaiaaiMcadaWdXaqabSqaaiaaicdaaeaacaaIYaGaeqiWdaha niabgUIiYdGccaaIGaGaamOzaiabeI7aXjaadsgacqaH4oqCcaaI9a GaaGymaiaai6caaaa@47DA@

(iii)f(θ)=f(θ+k.2π) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiIcacaWGPb GaamyAaiaadMgacaaIPaGaamOzaiaaiIcacqaH4oqCcaaIPaGaaGyp aiaadAgacaaIOaGaeqiUdeNaey4kaSIaam4Aaiaai6cacaaIYaGaeq iWdaNaaGykaaaa@4913@

for any integer k.

Distribution function

One way of specifying a distribution on the unit circle is by means of its distribution function. Let f( θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXbaa@38C4@ ) be the probability density function of a continuous random variable Θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfI5arbaa@3885@  i.e.f( θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXbaa@38C4@ ) is a non-negative 2 π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabec8aWbaa@38CB@  periodic function such that f( θ+2π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXjabgU caRiaaikdacqaHapaCaaa@3C1F@ ) = f( θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXbaa@38C4@ ) and 0 2π f(θ)dθ=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaapedabeWcba GaaGimaaqaaiaaikdacqaHapaCa0Gaey4kIipakiaaiccacaWGMbGa aGikaiabeI7aXjaaiMcacaWGKbGaeqiUdeNaaGypaiaaigdaaaa@4546@ . The distribution function F( θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXbaa@38C4@ ) can be defined over any interval ( θ 1 , θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXnaaBa aaleaacaaIXaaabeaakiaaiYcacqaH4oqCdaWgaaWcbaGaaGOmaaqa baaaaa@3D09@ ) by F( θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXnaaBa aaleaacaaIYaaabeaaaaa@39AC@ )-F( θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXnaaBa aaleaacaaIXaaabeaaaaa@39AB@ )= θ 1 θ 2 f(θ)dθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaapedabeWcba GaeqiUde3aaSbaaeaacaaIXaaabeaaaeaacqaH4oqCdaWgaaqaaiaa ikdaaeqaaaqdcqGHRiI8aOGaaGiiaiaadAgacaaIOaGaeqiUdeNaaG ykaiaadsgacqaH4oqCaaa@45B6@ . Suppose that an initial direction and orientation of the unit circle have been chosen (generally 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIWa qcfa4aaWbaaSqabKqaGeaajugWaiaaicdaaaaaaa@3A0C@ direction and anticlockwise orientation). Then F( θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXbaa@38C4@ ) is defined as F(θ)= 0 θ f(ϕ)dϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeacaaIOa GaeqiUdeNaaGykaiaai2dadaWdXaqabSqaaiaaicdaaeaacqaH4oqC a0Gaey4kIipakiaaiccacaWGMbGaaGikaiabew9aMjaaiMcacaWGKb Gaeqy1dygaaa@47D2@  obviously it holds that, F(2 π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabec8aWbaa@38CB@ ) = 1.

Laplace distribution

The Laplace distribution was first introduced by Pierre Simon Laplace in 1774 and is often called the ‘first law of errors’. A continuous random variable X is said to follow classical Laplace distribution with the parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXbaa@38C4@  and λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSbaa@38C2@ , if its pdf is given by Johnson et al.12

f X (x)= 1 2λ e |xθ| λ ; <x< ;θ ε R ;λ>0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaWgaa WcbaGaamiwaaqabaGccaaIOaGaamiEaiaaiMcacaaI9aWaaSaaaeaa caaIXaaabaGaaGOmaiabeU7aSbaacaaIGaGaamyzaiaaiccadaWcaa qaaiabgkHiTiaaiYhacaWG4bGaeyOeI0IaeqiUdeNaaGiFaaqaaiab eU7aSbaacaaIGaGaaG4oaiaaiccacaaIGaGaaGiiaiaaiccacaaIGa GaeyOeI0IaeyOhIuQaaGipaiaadIhacaaI8aGaeyOhIuQaaGiiaiaa iUdacqaH4oqCcaaIGaGaeqyTduMaaGiiaiaadkfacaaIGaGaaG4oai abeU7aSjaai6dacaaIWaGaaGOlaaaa@6178@ (1)

The random variable X following Laplace distribution (1) is denoted by L(θ, λ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYeacaaIOa GaeqiUdeNaaGilaiaaiccacqaH7oaBcaaIPaaaaa@3E0E@ , where the constants θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXbaa@38C4@ and λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSbaa@38C2@ are known as the location and scale parameter respectively. The corresponding cdf is obtained as

F X (x)={ 1 2 e |xθ|/λ ; xθ, 11/2 e |xθ|/λ ; x>θ. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaWgaa WcbaGaamiwaaqabaGccaaIOaGaamiEaiaaiMcacaaI9aWaaiqaaeaa faqaaeWadaaabaWaaSaaaeaacaaIXaaabaGaaGOmaaaacaaIGaGaam yzamaaCaaaleqabaGaeyOeI0IaaGiFaiaadIhacqGHsislcqaH4oqC caaI8bGaaG4laiabeU7aSbaaaOqaaiaaiUdaaeaacaWG4bGaeyizIm QaeqiUdeNaaGilaaqaaaqaaaqaaaqaaiaaigdacqGHsislcaaIXaGa aG4laiaaikdacaaIGaGaamyzamaaCaaaleqabaGaeyOeI0IaaGiFai aadIhacqGHsislcqaH4oqCcaaI8bGaaG4laiabeU7aSbaaaOqaaiaa iUdaaeaacaWG4bGaaGOpaiabeI7aXjaai6caaaaacaGL7baaaaa@6321@ (2)

A standard form of the pdf (1) is obtained by putting θ=0, λ=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXjaai2 dacaaIWaGaaGilaiaaiccacqaH7oaBcaaI9aGaaGymaaaa@3EDB@  and is given by

f Y (y)= 1 2 e |y| ; <y<. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaWgaa WcbaGaamywaaqabaGccaaIOaGaamyEaiaaiMcacaaI9aWaaSaaaeaa caaIXaaabaGaaGOmaaaacaaIGaGaamyzamaaCaaaleqabaGaeyOeI0 IaaGiFaiaadMhacaaI8baaaOGaaG4oaiaaiccacqGHsislcqGHEisP caaI8aGaamyEaiaaiYdacqGHEisPcaaIUaaaaa@4C00@ (3)

This form is sometimes called ‘Poisson’s first law of error’. Alternatively, the standard density function (3) is obtained by considering the transformation.

Y= Xθ λ . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMfacaaI9a WaaSaaaeaacaWGybGaeyOeI0IaeqiUdehabaGaeq4UdWgaaiaai6ca aaa@3EAF@ (4)

Skewed version of Laplace distribution

Skewed - Laplace distributions are considered as a generalization of the symmetric Laplace distribution. The discrete analogue of the symmetric Laplace distribution has been derived recently Kozubowski et al.,11 as in the case of normal distribution. The traditional models based on normal distribution are very often not supported by real life data in many studies because of long tails and asymmetry present in these data. The symmetric Laplace distribution(being heavier tailed than the normal) and the asymmetric Laplace distribution can account for leptokurtic and skewed data. Thus Laplace distribution is a natural, often superior alternative to the normal law. The Laplace distributions find applications in diversified fields. The classical symmetric Laplace distribution is regarded as an appropriate model in areas such as ocean engineering and fracture problems. It has also been found to be a suitable model for the formation of sand dunes. The asymmetric Laplace distributions are being used in modeling of financial data. There is a growing interest in stochastic modeling with Laplace distribution and its asymmetric extensions. The Laplace distribution is commonly encountered in image and speech compression applications. This distribution is also applicable for wind shear data, error distributions in navigation, encoding and decoding of analog signals, inventory management and quality control, astronomy and the biological and environment sciences.

In the last several decades various forms of skewed Laplace distributions have appeared in the literature. Kozubowski & Nadarajah13 studied different forms of skewed Laplace distributions. Mc Gill,14 Holla & Bhattacharya,15 Thomas Agnan & Poiraud-Casanoa,16 Fernandez & Steel17 and Yu & Zhang18 proposed different skewed Laplace distributions. Here we consider Holla and Bhattacharya’s Skewed Laplace distribution and derived its circular density function, distribution function, characteristic function and trigonometric moments.

Holla and Bhattacharya’s skewed Laplace distribution

Mc Gill14 considered distributions with the pdf given by

f(x)={ 1 2ψ exp ( xβ ψ ) , if x β, 1 2ϕ exp ( βx ϕ ) , if x > β. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgacaaIOa GaamiEaiaaiMcacaaI9aWaaiqaaeaafaqaaeGadaaabaWaaSaaaeaa caaIXaaabaGaaGOmaiabeI8a5baacaaIGaGaciyzaiaacIhacaGGWb GaaGiiamaabmaabaWaaSaaaeaacaWG4bGaeyOeI0IaeqOSdigabaGa eqiYdKhaaaGaayjkaiaawMcaaaqaaiaaiYcaaeaacaaMi8UaamyAai aadAgacaaMi8UaaGiiaiaadIhacaaIGaGaeyizImQaaGiiaiabek7a IjaaiYcaaeaadaWcaaqaaiaaigdaaeaacaaIYaGaeqy1dygaaiaaic caciGGLbGaaiiEaiaacchacaaIGaWaaeWaaeaadaWcaaqaaiabek7a IjabgkHiTiaadIhaaeaacqaHvpGzaaaacaGLOaGaayzkaaaabaGaaG ilaaqaaiaayIW7caWGPbGaamOzaiaayIW7caaIGaGaamiEaiaaicca caaI+aGaaGiiaiabek7aIjaai6caaaaacaGL7baaaaa@71B3@ (5)

where <x<; <β<; ϕ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiabg6 HiLkaaiccacaaI8aGaamiEaiaaiYdacqGHEisPcaaI7aGaaGiiaiab gkHiTiabg6HiLkaaiccacaaI8aGaeqOSdiMaaGipaiabg6HiLkaaiU dacaaIGaGaeqy1dyMaaGOpaiaaicdaaaa@4BDE@  and ψ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI8a5jaai6 dacaaIWaaaaa@3A5E@ . These distributions are also known as the two-piece double exponential. Holla and Bhattacharya’s skewed Laplace distribution is a variation of Mc Gill’s skewed Laplace distribution studied by Holla & Bhattacharya.15 The pdf of this distribution is given by

f(x)={ pc exp{c(βx)}; if xβ, (1p)cexp{c(xβ)}; if x<β. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgacaaIOa GaamiEaiaaiMcacaaI9aWaaiqaaeaafaqaaeWacaaabaGaamiCaiaa dogacaaIGaGaciyzaiaacIhacaGGWbGaaG4EaiaadogacaaIOaGaeq OSdiMaeyOeI0IaamiEaiaaiMcacaaI9bGaaG4oaaqaaiaayIW7caWG PbGaamOzaiaayIW7caaIGaGaamiEaiabgwMiZkabek7aIjaaiYcaae aaaeaaaeaacaaIOaGaaGymaiabgkHiTiaadchacaaIPaGaam4yaiGa cwgacaGG4bGaaiiCaiaaiUhacaWGJbGaaGikaiaadIhacqGHsislcq aHYoGycaaIPaGaaGyFaiaaiUdaaeaacaaMi8UaamyAaiaadAgacaaM i8UaaGiiaiaadIhacaaI8aGaeqOSdiMaaGOlaaaaaiaawUhaaaaa@6F47@ (6)

 where <x< MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaey OhIuQaaGipaiaadIhacaaI8aGaeyOhIukaaa@3C4E@ , <β<;c>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiabg6 HiLkaaiYdacqaHYoGycaaI8aGaeyOhIuQaaG4oaiaadogacaaI+aGa aGimaaaa@4139@ . and 0<p<1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaicdacaaI8a GaamiCaiaaiYdacaaIXaaaaa@3B04@ . Holla and Bhattacharya used this distribution as the compounding distribution of the expected value of a normal distribution.

The mean is obtained as

E(X)=β+ 1 c (2p1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadweacaaIOa GaamiwaiaaiMcacaaI9aGaeqOSdiMaey4kaSYaaSaaaeaacaaIXaaa baGaam4yaaaacaaIOaGaaGOmaiaadchacqGHsislcaaIXaGaaGykaa aa@43D5@

 The variance is

V(X)= 34p(p+1) c 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAfacaaIOa GaamiwaiaaiMcacaaI9aWaaSaaaeaacaaIZaGaeyOeI0IaaGinaiaa dchacaaIOaGaamiCaiabgUcaRiaaigdacaaIPaaabaGaam4yamaaCa aaleqabaGaaGOmaaaaaaaaaa@4427@

 The n th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaCaaaleqaba GaamiDaiaadIgaaaaaaa@3921@  moment is

μ n '= k=0 n ( n k ) β nk × k! c k [ p+ (1) k (1p) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTnaaBa aaleaacaWGUbaabeaakiaaiEcacaaI9aWaaabCaeqaleaacaWGRbGa aGypaiaaicdaaeaacaWGUbaaniabggHiLdGccaaIGaWaaeWaaeaafa qabeGabaaabaGaamOBaaqaaiaadUgaaaaacaGLOaGaayzkaaGaaGii aiabek7aInaaCaaaleqabaGaamOBaiabgkHiTiaadUgaaaGccqGHxd aTdaWcaaqaaiaadUgacaaIHaaabaGaam4yamaaCaaaleqabaGaam4A aaaaaaGcdaWadaqaaiaadchacqGHRaWkcaaIOaGaeyOeI0IaaGymai aaiMcadaahaaWcbeqaaiaadUgaaaGccaaIGaGaaGikaiaaigdacqGH sislcaWGWbGaaGykaaGaay5waiaaw2faaaaa@5D05@     (7)

Particular case

When p= 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadchacaaI9a WaaSaaaeaacaaIXaaabaGaaGOmaaaaaaa@3A51@ , Holla and Bhattacharya’s skewed Laplace distribution reduces to classical Laplace distribution. The pdf of Holla and Bhattacharya’s skewed Laplace distribution is given by (7)

For p= 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadchacaaI9a WaaSaaaeaacaaIXaaabaGaaGOmaaaaaaa@3A51@ , (7) becomes,

f(x)={ 1 2 cexp{c(βx)}; if β, (1 1 2 )cexp{c(xβ)}; if x<β. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgacaaIOa GaamiEaiaaiMcacaaI9aWaaiqaaeaafaqaaeWacaaabaWaaSaaaeaa caaIXaaabaGaaGOmaaaacaWGJbGaciyzaiaacIhacaGGWbGaaG4Eai aadogacaaIOaGaeqOSdiMaeyOeI0IaamiEaiaaiMcacaaI9bGaaG4o aaqaaiaayIW7caWGPbGaamOzaiaayIW7caaIGaGaeyyzImRaeqOSdi MaaGilaaqaaaqaaaqaaiaaiIcacaaIXaGaeyOeI0YaaSaaaeaacaaI XaaabaGaaGOmaaaacaaIPaGaam4yaiGacwgacaGG4bGaaiiCaiaaiU hacaWGJbGaaGikaiaadIhacqGHsislcqaHYoGycaaIPaGaaGyFaiaa iUdaaeaacaaMi8UaamyAaiaadAgacaaMi8UaaGiiaiaadIhacaaI8a GaeqOSdiMaaGOlaaaaaiaawUhaaaaa@6EC4@

 It can be simplified as

f(x)={ 1 2 cexp{c|xβ|} }. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgacaaIOa GaamiEaiaaiMcacaaI9aWaaiWaaeaadaWcaaqaaiaaigdaaeaacaaI YaaaaiaadogaciGGLbGaaiiEaiaacchacaaI7bGaam4yaiaaiYhaca WG4bGaeyOeI0IaeqOSdiMaaGiFaiaai2haaiaawUhacaGL9baacaaI Uaaaaa@4BE0@

 where <x<; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiabg6 HiLkaaiYdacaWG4bGaaGipaiabg6HiLkaaiUdaaaa@3E2B@   <β<; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiabg6 HiLkaaiYdacqaHYoGycaaI8aGaeyOhIuQaaG4oaaaa@3ECF@   c>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogacaaI+a GaaGimaaaa@3978@ , which is the classical Laplace distribution.

Wrapped Holla and Bhattacharya’s(HB) skewed Laplace distribution

Let X follows Holla and Bhattacharya’s skewed Laplace distribution with pdf (7).Then the wrapped Holla and Bhattacharya’s skewed Laplace random variable is defined as θ=X(mod2π) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXjaai2 dacaWGybGaaGikaiaad2gacaWGVbGaamizaiaaikdacqaHapaCcaaI Paaaaa@4115@ , such that θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXbaa@38C4@   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIGiodaa@3892@  [0, 2 π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabec8aWbaa@38CB@ ).The pdf of the corresponding wrapped distribution is given by

f(θ)= m= f(θ+2πm) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgacaaIOa GaeqiUdeNaaGykaiaai2dadaaeWbqabSqaaiaad2gacaaI9aGaeyOe I0IaeyOhIukabaGaeyOhIukaniabggHiLdGccaWGMbGaaGikaiabeI 7aXjabgUcaRiaaikdacqaHapaCcaWGTbGaaGykaaaa@4C03@

={ pc exp{c(βθ)} m=β exp(2mπc); if θ+2πmβ, (1p)cexp{c(xβ)} m= β1 exp(2mπc); if θ+2πm<β. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaai2dadaGaba qaauaabaqadiaaaeaacaWGWbGaam4yaiaaiccaciGGLbGaaiiEaiaa cchacaaI7bGaam4yaiaaiIcacqaHYoGycqGHsislcqaH4oqCcaaIPa GaaGyFamaaqahabeWcbaGaamyBaiaai2dacqaHYoGyaeaacqGHEisP a0GaeyyeIuoakiGacwgacaGG4bGaaiiCaiaaiIcacqGHsislcaaIYa GaamyBaiabec8aWjaadogacaaIPaGaaG4oaaqaaiaayIW7caWGPbGa amOzaiaayIW7caaIGaGaeqiUdeNaey4kaSIaaGOmaiabec8aWjaad2 gacqGHLjYScqaHYoGycaaISaaabaaabaaabaGaaGikaiaaigdacqGH sislcaWGWbGaaGykaiaadogaciGGLbGaaiiEaiaacchacaaI7bGaam 4yaiaaiIcacaWG4bGaeyOeI0IaeqOSdiMaaGykaiaai2hadaaeWbqa bSqaaiaad2gacaaI9aGaeyOeI0IaeyOhIukabaGaeqOSdiMaeyOeI0 IaaGymaaqdcqGHris5aOGaciyzaiaacIhacaGGWbGaaGikaiaaikda caWGTbGaeqiWdaNaam4yaiaaiMcacaaI7aaabaGaaGjcVlaadMgaca WGMbGaaGjcVlaaiccacqaH4oqCcqGHRaWkcaaIYaGaeqiWdaNaamyB aiaaiYdacqaHYoGycaaIUaaaaaGaay5Eaaaaaa@9997@   (8)

According to Jammalamadakka and Kozubowski (2004, 2017), we can obtain the pdf of WHB skewed Laplace distribution as

f(θ)= pcexp[c(βθ+2π2βπ)]+(1p)cexp[c(θβ+2βπ)] [exp(2πc)1] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgacaaIOa GaeqiUdeNaaGykaiaai2dadaWcaaqaaiaadchacaWGJbGaciyzaiaa cIhacaGGWbGaaG4waiaadogacaaIOaGaeqOSdiMaeyOeI0IaeqiUde Naey4kaSIaaGOmaiabec8aWjabgkHiTiaaikdacqaHYoGycqaHapaC caaIPaGaaGyxaiabgUcaRiaaiIcacaaIXaGaeyOeI0IaamiCaiaaiM cacaWGJbGaciyzaiaacIhacaGGWbGaaG4waiaadogacaaIOaGaeqiU deNaeyOeI0IaeqOSdiMaey4kaSIaaGOmaiabek7aIjabec8aWjaaiM cacaaIDbaabaGaaG4waiGacwgacaGG4bGaaiiCaiaaiIcacaaIYaGa eqiWdaNaam4yaiaaiMcacqGHsislcaaIXaGaaGyxaaaaaaa@7277@  (9)

where θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXbaa@38C4@   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIGiodaa@3892@  [0, 2 π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabec8aWbaa@38CB@ ), <β< MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiabg6 HiLkaaiYdacqaHYoGycaaI8aGaeyOhIukaaa@3E0A@ , 0<p<1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaicdacaaI8a GaamiCaiaaiYdacaaIXaaaaa@3B04@  and c>0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogacaaI+a GaaGimaiaai6caaaa@3A30@

The density plot for different values of the parameters are given in the following Figure 1–4.


Figure 1 Density for p=0.5, c=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogacaaI9a GaaGymaaaa@3978@ , β=0.25 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aIjaai2 dacqGHsislcaaIWaGaaGOlaiaaikdacaaI1aaaaa@3D50@ ; Density for p=0.5, c=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogacaaI9a GaaGymaaaa@3978@ , β=0.25 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aIjaai2 dacaaIWaGaaGOlaiaaikdacaaI1aaaaa@3C63@ .


Figure 2 Density for p=0.5, c=1.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogacaaI9a GaaGymaiaai6cacaaI1aaaaa@3AEF@ , β=0.2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aIjaai2 dacaaIWaGaaGOlaiaaikdaaaa@3BA4@ ; Density for p=0.5, c=1.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogacaaI9a GaaGymaiaai6cacaaI1aaaaa@3AEF@ , β=0.2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aIjaai2 dacqGHsislcaaIWaGaaGOlaiaaikdaaaa@3C91@ .


Figure 3 Density for p=0.75, c=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogacaaI9a GaaGymaaaa@3978@ , β=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aIjaai2 dacaaIWaaaaa@3A30@ ; Density for p=0.1, c=0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogacaaI9a GaaGimaiaai6cacaaI1aaaaa@3AEE@ , β=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aIjaai2 dacqGHsislcaaIXaaaaa@3B1E@ .


Figure 4 Density for p=0.5, c=1.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogacaaI9a GaaGymaiaai6cacaaI1aaaaa@3AEF@ , β=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aIjaai2 dacaaIWaaaaa@3A30@ ; Density for p=0.5, c=0.2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogacaaI9a GaaGimaiaai6cacaaIYaaaaa@3AEB@ , β=0.25 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aIjaai2 dacaaIWaGaaGOlaiaaikdacaaI1aaaaa@3C63@ .

Distribution function

One way of specifying a distribution on the unit circle is by means of its cumulative distribution function (cdf). Let f( θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXbaa@38C4@ ) be the probability density function of a continuous random variable Θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfI5arbaa@3885@ . Hence f( θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXbaa@38C4@ ) is a non-negative periodic function with period 2 π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabec8aWbaa@38CB@  such that f( θ+2π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXjabgU caRiaaikdacqaHapaCaaa@3C1F@ ) = f( θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXbaa@38C4@ ) and 0 2π f(θ)dθ=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaapedabeWcba GaaGimaaqaaiaaikdacqaHapaCa0Gaey4kIipakiaaiccacaWGMbGa aGikaiabeI7aXjaaiMcacaWGKbGaeqiUdeNaaGypaiaaigdaaaa@4546@ . The distribution function F( θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXbaa@38C4@ ) can be defined over any interval( θ 1 , θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXnaaBa aaleaacaaIXaaabeaakiaaiYcacqaH4oqCdaWgaaWcbaGaaGOmaaqa baaaaa@3D09@ ) by F( θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXnaaBa aaleaacaaIYaaabeaaaaa@39AC@ ) - F( θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXnaaBa aaleaacaaIXaaabeaaaaa@39AB@ ) = θ 1 θ 2 f(θ)dθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaapedabeWcba GaeqiUde3aaSbaaeaacaaIXaaabeaaaeaacqaH4oqCdaWgaaqaaiaa ikdaaeqaaaqdcqGHRiI8aOGaaGiiaiaadAgacaaIOaGaeqiUdeNaaG ykaiaadsgacqaH4oqCaaa@45B6@ . Suppose that an initial direction and orientation of the unit circle have been chosen (generally 0 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIWa qcfa4aaWbaaSqabKazba4=baqcLbkacaaIWaaaaaaa@3BAF@ direction and anticlockwise orientation). Then F( θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXbaa@38C4@ ) is defined as

F(θ)= 0 θ f(θ)dθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeacaaIOa GaeqiUdeNaaGykaiaai2dadaWdXaqabSqaaiaaicdaaeaacqaH4oqC a0Gaey4kIipakiaaiccacaWGMbGaaGikaiabeI7aXjaaiMcacaWGKb GaeqiUdehaaa@47AE@

Obviously it follows that, F(2 π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWdahaaa@37B3@ ) = 1

The cdf of (10) is given by

F(θ)= 0 θ f(θ)dθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeacaaIOa GaeqiUdeNaaGykaiaai2dadaWdXaqabSqaaiaaicdaaeaacqaH4oqC a0Gaey4kIipakiaaiccacaWGMbGaaGikaiabeI7aXjaaiMcacaWGKb GaeqiUdehaaa@47AE@

= 0 θ pcexp[c(βθ+2π2βπ)]+(1p)cexp[c(θβ+2βπ)] [exp(2πc)1] dθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaai2dadaWdXa qabSqaaiaaicdaaeaacqaH4oqCa0Gaey4kIipakmaalaaabaGaamiC aiaadogaciGGLbGaaiiEaiaacchacaaIBbGaam4yaiaaiIcacqaHYo GycqGHsislcqaH4oqCcqGHRaWkcaaIYaGaeqiWdaNaeyOeI0IaaGOm aiabek7aIjabec8aWjaaiMcacaaIDbGaey4kaSIaaGikaiaaigdacq GHsislcaWGWbGaaGykaiaadogaciGGLbGaaiiEaiaacchacaaIBbGa am4yaiaaiIcacqaH4oqCcqGHsislcqaHYoGycqGHRaWkcaaIYaGaeq OSdiMaeqiWdaNaaGykaiaai2faaeaacaaIBbGaciyzaiaacIhacaGG WbGaaGikaiaaikdacqaHapaCcaWGJbGaaGykaiabgkHiTiaaigdaca aIDbaaaiaadsgacqaH4oqCaaa@75B4@

 On simplification, we get

F(θ)= pexp[c(β+2π2βπ)][1exp(cθ)]+(1p)exp[c(2βπβ)][exp(cθ)1] [exp(2πc)1] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeacaaIOa GaeqiUdeNaaGykaiaai2dadaWcaaqaaiaadchaciGGLbGaaiiEaiaa cchacaaIBbGaam4yaiaaiIcacqaHYoGycqGHRaWkcaaIYaGaeqiWda NaeyOeI0IaaGOmaiabek7aIjabec8aWjaaiMcacaaIDbGaaG4waiaa igdacqGHsislciGGLbGaaiiEaiaacchacaaIOaGaeyOeI0Iaam4yai abeI7aXjaaiMcacaaIDbGaey4kaSIaaGikaiaaigdacqGHsislcaWG WbGaaGykaiGacwgacaGG4bGaaiiCaiaaiUfacaWGJbGaaGikaiaaik dacqaHYoGycqaHapaCcqGHsislcqaHYoGycaaIPaGaaGyxaiaaiUfa ciGGLbGaaiiEaiaacchacaaIOaGaam4yaiabeI7aXjaaiMcacqGHsi slcaaIXaGaaGyxaaqaaiaaiUfaciGGLbGaaiiEaiaacchacaaIOaGa aGOmaiabec8aWjaadogacaaIPaGaeyOeI0IaaGymaiaai2faaaaaaa@80DD@ (10)

The survival function and hazard rate function can be obtained from the corresponding density function and distribution function.

Characteristic function and trigonometric moments

The characteristic function of a random angle θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXbaa@38C4@ is the doubly-infinite sequence of complex numbers { ϕ p :p=0,±1,...} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiUhacqaHvp GzdaWgaaWcbaGaamiCaaqabaGccaaI6aGaamiCaiaai2dacaaIWaGa aGilaiabgglaXkaaigdacaaISaGaaGOlaiaai6cacaaIUaGaaGyFaa aa@4584@ given by

ϕ t =E( e itθ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew9aMnaaBa aaleaacaWG0baabeaakiaai2dacaWGfbGaaGikaiaadwgadaahaaWc beqaaiaadMgacaWG0bGaeqiUdehaaOGaaGykaaaa@41B9@

= 0 2π e itθ dF(θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaai2dadaWdXa qabSqaaiaaicdaaeaacaaIYaGaeqiWdahaniabgUIiYdGccaaIGaGa amyzamaaCaaaleqabaGaamyAaiaadshacqaH4oqCaaGccaWGKbGaam OraiaaiIcacqaH4oqCcaaIPaaaaa@4773@

= 0 2π e itθ pcexp[c(βθ+2π2βπ)]+(1p)cexp[c(θβ+2βπ)] [exp(2πc)1] dθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaai2dadaWdXa qabSqaaiaaicdaaeaacaaIYaGaeqiWdahaniabgUIiYdGccaaIGaGa amyzamaaCaaaleqabaGaamyAaiaadshacqaH4oqCaaGcdaWcaaqaai aadchacaWGJbGaciyzaiaacIhacaGGWbGaaG4waiaadogacaaIOaGa eqOSdiMaeyOeI0IaeqiUdeNaey4kaSIaaGOmaiabec8aWjabgkHiTi aaikdacqaHYoGycqaHapaCcaaIPaGaaGyxaiabgUcaRiaaiIcacaaI XaGaeyOeI0IaamiCaiaaiMcacaWGJbGaciyzaiaacIhacaGGWbGaaG 4waiaadogacaaIOaGaeqiUdeNaeyOeI0IaeqOSdiMaey4kaSIaaGOm aiabek7aIjabec8aWjaaiMcacaaIDbaabaGaaG4waiGacwgacaGG4b GaaiiCaiaaiIcacaaIYaGaeqiWdaNaam4yaiaaiMcacqGHsislcaaI XaGaaGyxaaaacaWGKbGaeqiUdehaaa@7BDF@

On simplification, we get

ϕ t = pcexp[c(β+2π2βπ)][exp[(itc)2π]1] (itc)[exp(2πc)1] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew9aMnaaBa aaleaacaWG0baabeaakiaai2dadaWcaaqaaiaadchacaWGJbGaciyz aiaacIhacaGGWbGaaG4waiaadogacaaIOaGaeqOSdiMaey4kaSIaaG Omaiabec8aWjabgkHiTiaaikdacqaHYoGycqaHapaCcaaIPaGaaGyx aiaaiUfaciGGLbGaaiiEaiaacchacaaIBbGaaGikaiaadMgacaWG0b GaeyOeI0Iaam4yaiaaiMcacaaIYaGaeqiWdaNaaGyxaiabgkHiTiaa igdacaaIDbaabaGaaGikaiaadMgacaWG0bGaeyOeI0Iaam4yaiaaiM cacaaIBbGaciyzaiaacIhacaGGWbGaaGikaiaaikdacqaHapaCcaWG JbGaaGykaiabgkHiTiaaigdacaaIDbaaaaaa@6D9B@

+ (1p)cexp[c(β+2βπ)][exp[(it+c)2π]1] (it+c)[exp(2πc)1] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgUcaRmaala aabaGaaGikaiaaigdacqGHsislcaWGWbGaaGykaiaadogaciGGLbGa aiiEaiaacchacaaIBbGaam4yaiaaiIcacqGHsislcqaHYoGycqGHRa WkcaaIYaGaeqOSdiMaeqiWdaNaaGykaiaai2facaaIBbGaciyzaiaa cIhacaGGWbGaaG4waiaaiIcacaWGPbGaamiDaiabgUcaRiaadogaca aIPaGaaGOmaiabec8aWjaai2facqGHsislcaaIXaGaaGyxaaqaaiaa iIcacaWGPbGaamiDaiabgUcaRiaadogacaaIPaGaaG4waiGacwgaca GG4bGaaiiCaiaaiIcacaaIYaGaeqiWdaNaam4yaiaaiMcacqGHsisl caaIXaGaaGyxaaaaaaa@6B3D@  (11)

By the definition of trigonometric moments, we have

ϕ p = α p +i β p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew9aMnaaBa aaleaacaWGWbaabeaakiaai2dacqaHXoqydaWgaaWcbaGaamiCaaqa baGccqGHRaWkcaWGPbGaeqOSdi2aaSbaaSqaaiaadchaaeqaaaaa@4224@ ; p=±1,±2,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadchacaaI9a GaeyySaeRaaGymaiaaiYcacqGHXcqScaaIYaGaaGilaiaai6cacaaI UaGaaGOlaaaa@41B1@

α p =E(cospθ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHnaaBa aaleaacaWGWbaabeaakiaai2dacaWGfbGaaGikaiGacogacaGGVbGa ai4CaiaadchacqaH4oqCcaaIPaaaaa@424C@

= 0 2π cospθf(θ)dθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaai2dadaWdXa qabSqaaiaaicdaaeaacaaIYaGaeqiWdahaniabgUIiYdGccaaIGaGa ci4yaiaac+gacaGGZbGaamiCaiabeI7aXjaadAgacaaIOaGaeqiUde NaaGykaiaadsgacqaH4oqCaaa@4A09@

= 0 2π cospθ pcexp[c(βθ+2π2βπ)]+(1p)cexp[c(θβ+2βπ)] [exp(2πc)1] dθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaai2dadaWdXa qabSqaaiaaicdaaeaacaaIYaGaeqiWdahaniabgUIiYdGccaaIGaGa ci4yaiaac+gacaGGZbGaamiCaiabeI7aXnaalaaabaGaamiCaiaado gaciGGLbGaaiiEaiaacchacaaIBbGaam4yaiaaiIcacqaHYoGycqGH sislcqaH4oqCcqGHRaWkcaaIYaGaeqiWdaNaeyOeI0IaaGOmaiabek 7aIjabec8aWjaaiMcacaaIDbGaey4kaSIaaGikaiaaigdacqGHsisl caWGWbGaaGykaiaadogaciGGLbGaaiiEaiaacchacaaIBbGaam4yai aaiIcacqaH4oqCcqGHsislcqaHYoGycqGHRaWkcaaIYaGaeqOSdiMa eqiWdaNaaGykaiaai2faaeaacaaIBbGaciyzaiaacIhacaGGWbGaaG ikaiaaikdacqaHapaCcaWGJbGaaGykaiabgkHiTiaaigdacaaIDbaa aiaadsgacqaH4oqCaaa@7C9F@

On simplification, we get

α p = c 2 p 2 + c 2 {pexp[(12π)βc]+(1p)exp[(2π1)βc]} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHnaaBa aaleaacaWGWbaabeaakiaai2dadaWcaaqaaiaadogadaahaaWcbeqa aiaaikdaaaaakeaacaWGWbWaaWbaaSqabeaacaaIYaaaaOGaey4kaS Iaam4yamaaCaaaleqabaGaaGOmaaaaaaGccaaI7bGaamiCaiGacwga caGG4bGaaiiCaiaaiUfacaaIOaGaaGymaiabgkHiTiaaikdacqaHap aCcaaIPaGaeqOSdiMaam4yaiaai2facqGHRaWkcaaIOaGaaGymaiab gkHiTiaadchacaaIPaGaciyzaiaacIhacaGGWbGaaG4waiaaiIcaca aIYaGaeqiWdaNaeyOeI0IaaGymaiaaiMcacqaHYoGycaWGJbGaaGyx aiaai2haaaa@6280@  (12)

β p =E(sinpθ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aInaaBa aaleaacaWGWbaabeaakiaai2dacaWGfbGaaGikaiGacohacaGGPbGa aiOBaiaadchacqaH4oqCcaaIPaaaaa@4253@

= 0 2π sinpθf(θ)dθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaai2dadaWdXa qabSqaaiaaicdaaeaacaaIYaGaeqiWdahaniabgUIiYdGccaaIGaGa ci4CaiaacMgacaGGUbGaamiCaiabeI7aXjaadAgacaaIOaGaeqiUde NaaGykaiaadsgacqaH4oqCaaa@4A0E@

= 0 2π sinpθ pcexp[c(βθ+2π2βπ)]+(1p)cexp[c(θβ+2βπ)] [exp(2πc)1] dθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaai2dadaWdXa qabSqaaiaaicdaaeaacaaIYaGaeqiWdahaniabgUIiYdGccaaIGaGa ci4CaiaacMgacaGGUbGaamiCaiabeI7aXnaalaaabaGaamiCaiaado gaciGGLbGaaiiEaiaacchacaaIBbGaam4yaiaaiIcacqaHYoGycqGH sislcqaH4oqCcqGHRaWkcaaIYaGaeqiWdaNaeyOeI0IaaGOmaiabek 7aIjabec8aWjaaiMcacaaIDbGaey4kaSIaaGikaiaaigdacqGHsisl caWGWbGaaGykaiaadogaciGGLbGaaiiEaiaacchacaaIBbGaam4yai aaiIcacqaH4oqCcqGHsislcqaHYoGycqGHRaWkcaaIYaGaeqOSdiMa eqiWdaNaaGykaiaai2faaeaacaaIBbGaciyzaiaacIhacaGGWbGaaG ikaiaaikdacqaHapaCcaWGJbGaaGykaiabgkHiTiaaigdacaaIDbaa aiaadsgacqaH4oqCaaa@7CA4@

On simplification, we get

β p = pc p 2 + c 2 {(p1)exp[(2π1)βc]pexp[(12π)βc]} MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aInaaBa aaleaacaWGWbaabeaakiaai2dadaWcaaqaaiaadchacaWGJbaabaGa amiCamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadogadaahaaWcbe qaaiaaikdaaaaaaOGaaG4EaiaaiIcacaWGWbGaeyOeI0IaaGymaiaa iMcaciGGLbGaaiiEaiaacchacaaIBbGaaGikaiaaikdacqaHapaCcq GHsislcaaIXaGaaGykaiabek7aIjaadogacaaIDbGaeyOeI0IaamiC aiGacwgacaGG4bGaaiiCaiaaiUfacaaIOaGaaGymaiabgkHiTiaaik dacqaHapaCcaaIPaGaeqOSdiMaam4yaiaai2facaaI9baaaa@628F@  (13)

According to Jammalamadaka and SenGupta (2001), an alternative expression for the PDF of the wrapped distribution using the trigonometric moments is given by

f(θ)= 1 2π ( 1+2 p=1 α p cos(pθ)+ β p sin(pθ) );θ[0,2π) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgacaaIOa GaeqiUdeNaaGykaiaai2dadaWcaaqaaiaaigdaaeaacaaIYaGaeqiW dahaamaabmaabaGaaGymaiabgUcaRiaaikdadaaeWbqabSqaaiaadc hacaaI9aGaaGymaaqaaiabg6HiLcqdcqGHris5aOGaeqySde2aaSba aSqaaiaadchaaeqaaOGaam4yaiaad+gacaWGZbGaaGikaiaadchacq aH4oqCcaaIPaGaey4kaSIaeqOSdi2aaSbaaSqaaiaadchaaeqaaOGa am4CaiaadMgacaWGUbGaaGikaiaadchacqaH4oqCcaaIPaaacaGLOa GaayzkaaGaaG4oaiabeI7aXjabgIGiolaaiUfacaaIWaGaaGilaiaa ikdacqaHapaCcaaIPaaaaa@66FB@  (14)

Substituting the values of α p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaadchaaeqaaaaa@38B6@  and β p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aInaaBa aaleaacaWGWbaabeaaaaa@39D0@  given in (13) and (14) in (15), we get an alternative expression for the PDF of the wrapped HB skewed Laplace distribution.

Maximum likelihood estimation

In this section, the maximum likelihood estimators of the unknown parameters (p,c,β) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiIcacaWGWb GaaGilaiaadogacaaISaGaeqOSdiMaaGykaaaa@3D5D@  of the WHB skewed Laplace distribution are derived. Let θ 1 , θ 2 , θ 3 ,... θ n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXnaaBa aaleaacaaIXaaabeaakiaaiYcacqaH4oqCdaWgaaWcbaGaaGOmaaqa baGccaaISaGaeqiUde3aaSbaaSqaaiaaiodaaeqaaOGaaGilaiaai6 cacaaIUaGaaGOlaiabeI7aXnaaBaaaleaacaWGUbaabeaaaaa@4625@  be a random sample of size n from WHB skewed Laplace distribution, then the the likelihood function is

L= i=1 n f( θ i ;b,σ,β) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYeacaaI9a WaaebCaeqaleaacaWGPbGaaGypaiaaigdaaeaacaWGUbaaniabg+Gi vdGccaaIGaGaamOzaiaaiIcacqaH4oqCdaWgaaWcbaGaamyAaaqaba GccaaI7aGaamOyaiaaiYcacqaHdpWCcaaISaGaeqOSdiMaaGykaaaa @4A95@

= i=1 n [ pcexp[c(β θ i +2π2βπ)]+(1p)cexp[c( θ i β+2βπ)] [exp(2πc)1] ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaai2dadaqeWb qabSqaaiaadMgacaaI9aGaaGymaaqaaiaad6gaa0Gaey4dIunakiaa iccadaWadaqaamaalaaabaGaamiCaiaadogaciGGLbGaaiiEaiaacc hacaaIBbGaam4yaiaaiIcacqaHYoGycqGHsislcqaH4oqCdaWgaaWc baGaamyAaaqabaGccqGHRaWkcaaIYaGaeqiWdaNaeyOeI0IaaGOmai abek7aIjabec8aWjaaiMcacaaIDbGaey4kaSIaaGikaiaaigdacqGH sislcaWGWbGaaGykaiaadogaciGGLbGaaiiEaiaacchacaaIBbGaam 4yaiaaiIcacqaH4oqCdaWgaaWcbaGaamyAaaqabaGccqGHsislcqaH YoGycqGHRaWkcaaIYaGaeqOSdiMaeqiWdaNaaGykaiaai2faaeaaca aIBbGaciyzaiaacIhacaGGWbGaaGikaiaaikdacqaHapaCcaWGJbGa aGykaiabgkHiTiaaigdacaaIDbaaaaGaay5waiaaw2faaaaa@78F4@

= [ 1 e 2πc 1 ] n i=1 n [ pcexp[c(β θ i +2π2βπ)]+(1p)cexp[c( θ i β+2βπ)] ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaai2dadaWada qaamaalaaabaGaaGymaaqaaiaadwgadaahaaWcbeqaaiaaikdacqaH apaCcaWGJbaaaOGaeyOeI0IaaGymaaaaaiaawUfacaGLDbaadaahaa Wcbeqaaiaad6gaaaGcdaaeWbqabSqaaiaadMgacaaI9aGaaGymaaqa aiaad6gaa0GaeyyeIuoakmaadmaabaGaamiCaiaadogaciGGLbGaai iEaiaacchacaaIBbGaam4yaiaaiIcacqaHYoGycqGHsislcqaH4oqC daWgaaWcbaGaamyAaaqabaGccqGHRaWkcaaIYaGaeqiWdaNaeyOeI0 IaaGOmaiabek7aIjabec8aWjaaiMcacaaIDbGaey4kaSIaaGikaiaa igdacqGHsislcaWGWbGaaGykaiaadogaciGGLbGaaiiEaiaacchaca aIBbGaam4yaiaaiIcacqaH4oqCdaWgaaWcbaGaamyAaaqabaGccqGH sislcqaHYoGycqGHRaWkcaaIYaGaeqOSdiMaeqiWdaNaaGykaiaai2 faaiaawUfacaGLDbaaaaa@7747@

 The log likelihood function is given by

logL=nlog[ e 2πc 1]+log{ i=1 n { pcexp[c(β θ i +2π2βπ)]+(1p)cexp[c( θ i β+2βπ)] } } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYgacaWGVb Gaam4zaiaadYeacaaI9aGaeyOeI0IaamOBaiGacYgacaGGVbGaai4z aiaaiUfacaWGLbWaaWbaaSqabeaacaaIYaGaeqiWdaNaam4yaaaaki abgkHiTiaaigdacaaIDbGaey4kaSIaciiBaiaac+gacaGGNbWaaiWa aeaadaaeWbqabSqaaiaadMgacaaI9aGaaGymaaqaaiaad6gaa0Gaey yeIuoakmaacmaabaGaamiCaiaadogaciGGLbGaaiiEaiaacchacaaI BbGaam4yaiaaiIcacqaHYoGycqGHsislcqaH4oqCdaWgaaWcbaGaam yAaaqabaGccqGHRaWkcaaIYaGaeqiWdaNaeyOeI0IaaGOmaiabek7a Ijabec8aWjaaiMcacaaIDbGaey4kaSIaaGikaiaaigdacqGHsislca WGWbGaaGykaiaadogaciGGLbGaaiiEaiaacchacaaIBbGaam4yaiaa iIcacqaH4oqCdaWgaaWcbaGaamyAaaqabaGccqGHsislcqaHYoGycq GHRaWkcaaIYaGaeqOSdiMaeqiWdaNaaGykaiaai2faaiaawUhacaGL 9baaaiaawUhacaGL9baaaaa@83A0@ (15)

The partial derivatives of the log likelihood with respect to p, c and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aIbaa@38AF@  are obtained as

logL p = i=1 n [ e c(β θ i +2π2βπ) e c( θ i β+2βπ) ] i=1 n [ p e c(β θ i +2π2βπ) +(1p) e c( θ i β+2βπ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaey OaIyRaciiBaiaac+gacaGGNbGaamitaaqaaiabgkGi2kaadchaaaGa aGypamaalaaabaWaaabCaeqaleaacaWGPbGaaGypaiaaigdaaeaaca WGUbaaniabggHiLdGcdaWadaqaaiaadwgadaahaaWcbeqaaiaadoga caaIOaGaeqOSdiMaeyOeI0IaeqiUde3aaSbaaeaacaWGPbaabeaacq GHRaWkcaaIYaGaeqiWdaNaeyOeI0IaaGOmaiabek7aIjabec8aWjaa iMcaaaGccqGHsislcaWGLbWaaWbaaSqabeaacaWGJbGaaGikaiabeI 7aXnaaBaaabaGaamyAaaqabaGaeyOeI0IaeqOSdiMaey4kaSIaaGOm aiabek7aIjabec8aWjaaiMcaaaaakiaawUfacaGLDbaaaeaadaaeWb qabSqaaiaadMgacaaI9aGaaGymaaqaaiaad6gaa0GaeyyeIuoakmaa dmaabaGaamiCaiaadwgadaahaaWcbeqaaiaadogacaaIOaGaeqOSdi MaeyOeI0IaeqiUde3aaSbaaeaacaWGPbaabeaacqGHRaWkcaaIYaGa eqiWdaNaeyOeI0IaaGOmaiabek7aIjabec8aWjaaiMcaaaGccqGHRa WkcaaIOaGaaGymaiabgkHiTiaadchacaaIPaGaamyzamaaCaaaleqa baGaam4yaiaaiIcacqaH4oqCdaWgaaqaaiaadMgaaeqaaiabgkHiTi abek7aIjabgUcaRiaaikdacqaHYoGycqaHapaCcaaIPaaaaaGccaGL BbGaayzxaaaaaaaa@9321@ (16)

logL c = 2πn e 2πc ( e 2πc 1) + i=1 n p e c(β θ i +2π2βπ) [1+c(β θ i +2π2βπ)] i=1 n [ pc e c(β θ i +2π2βπ) +(1p)c e c( θ i β+2βπ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaey OaIyRaciiBaiaac+gacaGGNbGaamitaaqaaiabgkGi2kaadogaaaGa aGypamaalaaabaGaeyOeI0IaaGOmaiabec8aWjaad6gacaWGLbWaaW baaSqabeaacaaIYaGaeqiWdaNaam4yaaaaaOqaaiaaiIcacaWGLbWa aWbaaSqabeaacaaIYaGaeqiWdaNaam4yaaaakiabgkHiTiaaigdaca aIPaaaaiabgUcaRmaalaaabaWaaabCaeqaleaacaWGPbGaaGypaiaa igdaaeaacaWGUbaaniabggHiLdGccaWGWbGaamyzamaaCaaaleqaba Gaam4yaiaaiIcacqaHYoGycqGHsislcqaH4oqCdaWgaaqaaiaadMga aeqaaiabgUcaRiaaikdacqaHapaCcqGHsislcaaIYaGaeqOSdiMaeq iWdaNaaGykaaaakiaaiUfacaaIXaGaey4kaSIaam4yaiaaiIcacqaH YoGycqGHsislcqaH4oqCdaWgaaWcbaGaamyAaaqabaGccqGHRaWkca aIYaGaeqiWdaNaeyOeI0IaaGOmaiabek7aIjabec8aWjaaiMcacaaI DbaabaWaaabCaeqaleaacaWGPbGaaGypaiaaigdaaeaacaWGUbaani abggHiLdGcdaWadaqaaiaadchacaWGJbGaamyzamaaCaaaleqabaGa am4yaiaaiIcacqaHYoGycqGHsislcqaH4oqCdaWgaaqaaiaadMgaae qaaiabgUcaRiaaikdacqaHapaCcqGHsislcaaIYaGaeqOSdiMaeqiW daNaaGykaaaakiabgUcaRiaaiIcacaaIXaGaeyOeI0IaamiCaiaaiM cacaWGJbGaamyzamaaCaaaleqabaGaam4yaiaaiIcacqaH4oqCdaWg aaqaaiaadMgaaeqaaiabgkHiTiabek7aIjabgUcaRiaaikdacqaHYo GycqaHapaCcaaIPaaaaaGccaGLBbGaayzxaaaaaaaa@AA19@

+ i=1 n (1p) e c( θ i β+2βπ) [1+c( θ i β+2βπ)] i=1 n [ pc e c(β θ i +2π2βπ) +(1p)c e c( θ i β+2βπ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgUcaRmaala aabaWaaabCaeqaleaacaWGPbGaaGypaiaaigdaaeaacaWGUbaaniab ggHiLdGccaaIOaGaaGymaiabgkHiTiaadchacaaIPaGaamyzamaaCa aaleqabaGaam4yaiaaiIcacqaH4oqCdaWgaaqaaiaadMgaaeqaaiab gkHiTiabek7aIjabgUcaRiaaikdacqaHYoGycqaHapaCcaaIPaaaaO GaaG4waiaaigdacqGHRaWkcaWGJbGaaGikaiabeI7aXnaaBaaaleaa caWGPbaabeaakiabgkHiTiabek7aIjabgUcaRiaaikdacqaHYoGycq aHapaCcaaIPaGaaGyxaaqaamaaqahabeWcbaGaamyAaiaai2dacaaI XaaabaGaamOBaaqdcqGHris5aOWaamWaaeaacaWGWbGaam4yaiaadw gadaahaaWcbeqaaiaadogacaaIOaGaeqOSdiMaeyOeI0IaeqiUde3a aSbaaeaacaWGPbaabeaacqGHRaWkcaaIYaGaeqiWdaNaeyOeI0IaaG Omaiabek7aIjabec8aWjaaiMcaaaGccqGHRaWkcaaIOaGaaGymaiab gkHiTiaadchacaaIPaGaam4yaiaadwgadaahaaWcbeqaaiaadogaca aIOaGaeqiUde3aaSbaaeaacaWGPbaabeaacqGHsislcqaHYoGycqGH RaWkcaaIYaGaeqOSdiMaeqiWdaNaaGykaaaaaOGaay5waiaaw2faaa aaaaa@8DB4@ (17)

logL β = i=1 n c(12π)[ p e c(β θ i +2π2βπ) (1p) e c( θ i β+2βπ) ] i=1 n [ p e c(β θ i +2π2βπ) +(1p) e c( θ i β+2βπ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaey OaIyRaciiBaiaac+gacaGGNbGaamitaaqaaiabgkGi2kabek7aIbaa caaI9aWaaSaaaeaadaaeWbqabSqaaiaadMgacaaI9aGaaGymaaqaai aad6gaa0GaeyyeIuoakiaadogacaaIOaGaaGymaiabgkHiTiaaikda cqaHapaCcaaIPaWaamWaaeaacaWGWbGaamyzamaaCaaaleqabaGaam 4yaiaaiIcacqaHYoGycqGHsislcqaH4oqCdaWgaaqaaiaadMgaaeqa aiabgUcaRiaaikdacqaHapaCcqGHsislcaaIYaGaeqOSdiMaeqiWda NaaGykaaaakiabgkHiTiaaiIcacaaIXaGaeyOeI0IaamiCaiaaiMca caWGLbWaaWbaaSqabeaacaWGJbGaaGikaiabeI7aXnaaBaaabaGaam yAaaqabaGaeyOeI0IaeqOSdiMaey4kaSIaaGOmaiabek7aIjabec8a WjaaiMcaaaaakiaawUfacaGLDbaaaeaadaaeWbqabSqaaiaadMgaca aI9aGaaGymaaqaaiaad6gaa0GaeyyeIuoakmaadmaabaGaamiCaiaa dwgadaahaaWcbeqaaiaadogacaaIOaGaeqOSdiMaeyOeI0IaeqiUde 3aaSbaaeaacaWGPbaabeaacqGHRaWkcaaIYaGaeqiWdaNaeyOeI0Ia aGOmaiabek7aIjabec8aWjaaiMcaaaGccqGHRaWkcaaIOaGaaGymai abgkHiTiaadchacaaIPaGaamyzamaaCaaaleqabaGaam4yaiaaiIca cqaH4oqCdaWgaaqaaiaadMgaaeqaaiabgkHiTiabek7aIjabgUcaRi aaikdacqaHYoGycqaHapaCcaaIPaaaaaGccaGLBbGaayzxaaaaaaaa @9F32@ (18)

Inorder to estimate the parameters,we have to solve the normal equations

logL p =0; logL c =0; logL β =0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaey OaIyRaciiBaiaac+gacaGGNbGaamitaaqaaiabgkGi2kaadchaaaGa aGypaiaaicdacaaI7aGaaGiiaiaaiccacaaIGaWaaSaaaeaacqGHci ITciGGSbGaai4BaiaacEgacaWGmbaabaGaeyOaIyRaam4yaaaacaaI 9aGaaGimaiaaiUdacaaIGaGaaGiiaiaaiccadaWcaaqaaiabgkGi2k GacYgacaGGVbGaai4zaiaadYeaaeaacqGHciITcqaHYoGyaaGaaGyp aiaaicdacaaIUaaaaa@58C4@ (19)

Since (20) cannot be solved analytically, numerical iteration technique is used to get a solution for p,c and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aIbaa@38AF@ . One may use the nlm package in R software to get the maximum likelihood estimators (MLE) for p,c and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aIbaa@38AF@ .

Application in meteorology

The following application shows the effectiveness of Wrapped Holla and Bhattacharya’s Skewed Laplace (WHBSL) distribution, the data wind directions comprising hourly measurements of three days at a site on Black Mountain, ACT, Australia, was reported in Dr. M.A. Cameron and discussed in Fisher.19 The main goal was to provide a regular monitoring of climate change into ACT. Climate change is the greatest threat facing the world today, wind generated electricity is one of a number of ways that we can reduce our reliance on fossil fuel-generated electricity and therefore reduce our greenhouse gas production and limit climate change. The geography and dynamics wind direction this area are important elements of climate system. This study is important given the growing evidence of the ACT climate change and biosphere to global change. The data (n= 22, degree) are as given below (Table 1).

0

15

90

150

182

220

235

240

245

250

255

265

270

280

285

300

315

330

335

340

345

Table 1 Wind direction data

Here we compute the estimates of the unknown parameters with respect to the three distributions, namely WHBSL (c,p,β) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiIcacaWGJb GaaGilaiaadchacaaISaGaeqOSdiMaaGykaaaa@3D5D@ , WvG (μ,λ,α,β,γ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiIcacqaH8o qBcaaISaGaeq4UdWMaaGilaiabeg7aHjaaiYcacqaHYoGycaaISaGa eq4SdCMaaGykaaaa@439C@  and GvM ( μ 1 , μ 2 , κ 1 , κ 2 ,δ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiIcacqaH8o qBdaWgaaWcbaGaaGymaaqabaGccaaISaGaeqiVd02aaSbaaSqaaiaa ikdaaeqaaOGaaGilaiabeQ7aRnaaBaaaleaacaaIXaaabeaakiaaiY cacqaH6oWAdaWgaaWcbaGaaGOmaaqabaGccaaISaGaeqiTdqMaaGyk aaaa@4786@  and obtain the values for loglikelihood, Akaike information criterion (AIC) and Bayesian information criterion (BIC). Table 2 provides the relevant numerical summaries for the three fits with the goodness of fit. Based on these values, we can conclude that the WHBSL (c,p,β) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiIcacaWGJb GaaGilaiaadchacaaISaGaeqOSdiMaaGykaaaa@3D5D@  distribution is comparatively better than the other two distributions in modeling the present data.20

Distribution

 Estimates

 

 AIC

 BIC

WHBSL

 97.0644

-188.1288

-188.1288

(c,p,β) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiIcacaWGJb GaaGilaiaadchacaaISaGaeqOSdiMaaGykaaaa@3D5D@

p ^ =0.4117 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadchagaqcai aai2dacaaIWaGaaGOlaiaaisdacaaIXaGaaGymaiaaiEdaaaa@3D41@

β ^ =7.8209 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqbek7aIzaaja GaaGypaiaaiEdacaaIUaGaaGioaiaaikdacaaIWaGaaGyoaaaa@3DFA@

WVG

μ ^ =4.07 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqbeY7aTzaaja GaaGypaiaaisdacaaIUaGaaGimaiaaiEdaaaa@3C8C@

 -63.40

 136.8

 136.8

(μ,λ,α,β,γ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiIcacqaH8o qBcaaISaGaeq4UdWMaaGilaiabeg7aHjaaiYcacqaHYoGycaaISaGa eq4SdCMaaGykaaaa@439C@

λ ^ =2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4UdWMbaK aacaaI9aGaaGOmaaaa@393D@

α ^ =0.9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqbeg7aHzaaja GaaGypaiaaicdacaaIUaGaaGyoaaaa@3BB9@

 

 

 

β ^ =2.1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqbek7aIzaaja GaaGypaiaaikdacaaIUaGaaGymaaaa@3BB5@

γ ^ =0.50 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqbeo7aNzaaja GaaGypaiaaicdacaaIUaGaaGynaiaaicdaaaa@3C77@

GvM

μ ^ 1 =5.02 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqbeY7aTzaaja WaaSbaaSqaaiaaigdaaeqaaOGaaGypaiaaiwdacaaIUaGaaGimaiaa ikdaaaa@3D79@

-67.20

144.4

141.11

( μ 1 , μ 2 , κ 1 , κ 2 ,δ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiIcacqaH8o qBdaWgaaWcbaGaaGymaaqabaGccaaISaGaeqiVd02aaSbaaSqaaiaa ikdaaeqaaOGaaGilaiabeQ7aRnaaBaaaleaacaaIXaaabeaakiaaiY cacqaH6oWAdaWgaaWcbaGaaGOmaaqabaGccaaISaGaeqiTdqMaaGyk aaaa@4786@

μ ^ 2 =5.70 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqbeY7aTzaaja WaaSbaaSqaaiaaikdaaeqaaOGaaGypaiaaiwdacaaIUaGaaG4naiaa icdaaaa@3D7F@

 

κ ^ 1 =1.04 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqbeQ7aRzaaja WaaSbaaSqaaiaaigdaaeqaaOGaaGypaiaaigdacaaIUaGaaGimaiaa isdaaaa@3D73@

 

 

 

κ ^ 2 =0.0003 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqbeQ7aRzaaja WaaSbaaSqaaiaaikdaaeqaaOGaaGypaiaaicdacaaIUaGaaGimaiaa icdacaaIWaGaaG4maaaa@3EE6@

δ ^ =0.68 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqbes7aKzaaja GaaGypaiaaicdacaaIUaGaaGOnaiaaiIdaaaa@3C7E@

 

 

 

Table 2 Summary of fits for the wind direction data

Conclusion

In this paper, a new wrapped distribution namely Wrapped Holla and Bhattacharya’s Skewed Laplace (WHBSL) distribution is introduced and studied. The pdf and cdf of the distribution are derived and the shapes of the density function for different values of the parameters are obtained by using Mathematica. Expressions for characteristic function and trigonometric moments are derived. The alternative form of the pdf of the same distribution is also obtained using trigonometric moments. Method of maximum likelihood estimation is used for estimating the parameters. For exploring the validity of the model, we apply it to a real data set from Meteorology. The performance of the proposed model is compared with that of wrapped variance gamma distribution and generalised von Mises distribution using log-likelihood, AIC, and BIC. It is concluded that the wrapped HB Skewed Laplace distribution is a better model for the given data set than the other two distributions.

Acknowledgements

None.

Conflict of interest

Author declares that there is no conflict of interest.

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