Wrapping Generalized Laplace and Generalized Discrete Laplace Distributions

Abstract

Generalized Laplace and discrete generalized Laplace models are described and some of their properties are reviewed. For the analysis of directional data, it is natural to consider wrapped versions of these models. A well-known result for the circular and, more recently, for multivariate circular distributions that mixing and wrapping commute, allows us to readily determine the nature of these wrapped models. Some bivariate extensions of these models are discussed, together with some consideration of the feasibility of wrapping such models. Multivariate versions of the models can be envisioned.

1. Introduction

Data which is of a periodic nature requires specialized models in order to make meaningful inferences about the stochastic mechanism that generates the data. A wide variety of circular models can now be found in the literature, including old standbys such as the von Mises density and the cardioid model. Wrapped models have been added to the modeler’s tool box, providing flexible alternatives to more typical models. One feature that is shared by the majority of the popular models is symmetry. SenGupta and Roy [1] have recently introduced the symmetric Linnik family of distributions, of which the wrapped Laplace distribution is a member, with real-life applications and numerical examples. Though symmetry is mathematically and aesthetically attractive, many data sets are definitely asymmetric, and analyzing them using symmetric models is a futile process of the “square peg in a round hole” nature. Clearly, there is a role to be played by asymmetric or skewed models. Such models will be discussed in the present paper. We will furthermore consider wrapped models of certain asymmetric circular models, thus introducing novel asymmetric models for the researcher to utilize. Before we begin the technical part of our paper, it is quite reasonable to respond in advance to some of the reader’s questions regarding the utility of circular models in scientific research. One response to those questions is that circular data occur in all fields of scientific research and that, in the last few decades, the use of circular densities has become commonplace. We mention just nine of the many research areas in which circular data and concepts play a role, in each case illustrated by references to two recent papers for the interested reader to consult.

(1) Archaeology:

Ember, C. R., and Ember, M. [2]. Using circular distributions to model settlement patterns in prehistoric archaeology. Journal of Anthropological Archaeology, 46, 1–15.

Horne, L. [3]. Analysis of burial orientations using circular statistics in ancient cemeteries. Archaeological and Anthropological Sciences, 10(4), 929–942.

(2) Ecology:

Hannon, B. [4]. The Structure of Ecosystems. Journal of Theoretical Biology, 41(3), 535–546.

Wu, H., Li, B-L., Springer, A. and Neill, W.H. [5]. Modelling animal movement as a persistent random walk in two dimensions; expected magnitude of net displacement. Ecological Modelling 2000, 132, 115–124.

(3) Economics:

Bocken, N. M. P., de Pauw, I., Bakker, E., and van der Grinten, B. [6]. Product Design and Business Model Strategies for a Circular Economy. Journal of Industrial and Production Engineering, 33(5), 308–320.

Geissdoerfer, M., Savaget, P., Bocken, N. M. P., and Hultink, E. J. [7]. The Circular Economy—A New Sustainability Paradigm? Journal of Cleaner Production, 143, 757–768.

(4) Engineering:

Liu, X., and Zhang, L. [8]. Application of Von Mises distribution in probabilistic fracture mechanics for fatigue crack growth modeling. Engineering Fracture Mechanics, 201, 181–193.

Zhang, W., and Liao, H. [9]. A modified Von Mises distribution for modeling the yield strength of metallic materials under complex loading conditions. Journal of Materials Science and Technology, 32(6), 577–585.

(5) Forest Science:

Bard, S. M., and Horváth, G. [10]. Modeling directional tree growth with wrapped distributions: Applications in forest science. Forest Ecology and Management, 406, 64–74.

Arado’ttir, A.L., Robertson, A. and Moore, E. [11]. Circular statistical analysis of birch colonization and the directional growth response of birch and black cottonwood in south Iceland. Agricultural and Forest Meteorology 1997, 84, 179–186.

(6) Music:

Cuddy, L. L., and Lerdahl, F. [12]. Circularity and harmonic progressions in Western tonal music: An empirical investigation. Music Perception, 35(1), 69–88.

Noll, M., and Rapp, E. [13]. Analyzing rhythmic structures in non-Western music using circular models. Journal of Music Theory and Analysis, 7(3), 243–261.

(7) Oceanography:

Soukission, T.H. [14]. Probabilistic modeling of directional and linear characteristics of wind and sea waves. Ocean Engineering 2014, 91, 91–110.

Rodríguez, M. L., and Lázaro, J. [15]. A study of wind direction variability in the Mediterranean Sea using wrapped distributions. Continental Shelf Research, 178, 118–128.

(8) Physics:

Chakrabarti, A., and Ghosh, M. [16]. Von Mises distribution for modeling particle orientations in granular systems. Physical Review E, 94(5), 052901.

Paula, M. A., and Oliveira, L. S. [17]. Extension of von Mises distribution for nonlinear elasticity models. International Journal of Solids and Structures, 224, 105–118.

(9) Psychology:

Cremers, J., and Klugkist, I. [18]. One Direction? A Tutorial for Circular Data Analysis Using R With Examples in Cognitive Psychology. Frontiers in Psychology, 9, 2040.

Wiggins, J. S. [19]. A Psychological Taxonomy of Trait-Descriptive Terms: The Interpersonal Domain. Journal of Personality and Social Psychology, 37(3), 395–412.

Finally, we mention the following paper which is cross-disciplinary and could have been placed in either the engineering or ecology categories.

Meyer, P.G., Cherstvy, A.G., Seckler, H., Hering, R., Blaum, N., Jeltsch, F., and Metzler, R. [20]. Directedness, correlations, and daily cycles in springbok motion: From data via stochastic models to movement prediction. Phys. Rev. Research 5, 043129.

This paper is interesting since it considers circular variables in some stochastic process settings.

When we speak of a circular data point, we refer to a real quantity that has been reduced modulo $2\pi$, so that it falls in the interval $[0,2\pi )$. A circular density is a positive function that is defined on the interval $[0,2\pi )$ and integrates to 1 over that interval. A discrete circular density has support on a countable (usually finite) set of points in the interval $[0,2\pi )$. The list of frequently used circular densities includes the von Mises or circular normal model, the cardioid model and, more recently, wrapped models including wrapped Cauchy and wrapped Laplace models. In the present paper, we will develop additional wrapped models. Some will involve asymmetric and generalized asymmetric Laplace distributions, continuous and discrete. The corresponding wrapped versions, as well as mixtures of wrapped models, are also discussed, together with consideration of bivariate models.

In the next section, we will review the definitions of asymmetric Laplace and asymmetric discrete Laplace distributions. A minor generalization of these models is presented, for which mixture representations are particularly convenient. Some discussion of parameter estimation for these models is included. What is novel in the current presentation is the coverage of wrapped versions of these models, which are suitable for the analysis of directional data. The derivation of the wrapped models is much simplified by using the well-known fact that the operations of mixing and wrapping commute (see, e.g., SenGupta and Pal [21] for the circular models and, more recently, Greco et al. [22], for multivariate circular distributions). In Section 6, bivariate versions of these models are investigated and multivariate extensions are sketched. In some cases, it is possible to consider wrapped versions of bivariate distributions.

2. Generalized (Asymmetric) Laplace Distributions

There is more than one way to define an asymmetric Laplace distribution, but the simplest construction begins with two independent exponential random variables, $V_1$ and $V_2$ with $V_i\sim exp\left({\lambda }_i\right)$, $i=1,2.$ We then define

$$X=V_1-V_2.$$

(1)

Such a random variable X is said to have an asymmetric Laplace distribution, and to indicate this we will write $X\sim AL({\lambda }_1,{\lambda }_2).$ The characteristic function of X, defined by (1), is readily found to be of the form

$${\varphi }_X\left(t\right)={[1+{\lambda }_1^{-1}{\lambda }_2^{-1}{t}^2-i({\lambda }_1^{-1}-{\lambda }_2^{-1})t]}^{-1},$$

(2)

The case in which ${\lambda }_1={\lambda }_2$ corresponds to the usual symmetric Laplace model. It is readily verified that the asymmetric Laplace distribution, like the symmetric Laplace distribution, is infinitely divisible.

The following mixture representation of (1), the asymmetric Laplace model, was noted by Kozubowski and Podgorski [23].

Suppose that X has a distribution of an exponential (${\lambda }_1$) variable with probability $[{\lambda }_2/({\lambda }_1+{\lambda }_2)]$, and has the negative of an exponential (${\lambda }_2$) variable with probability $[{\lambda }_1/({\lambda }_1+{\lambda }_2)].$ In that case, the random variable has the asymmetric Laplace distribution (1).

An alternative representation of this mixture model is as follows

$$X=I{V}_1+(1-I)(-V_2),$$

(3)

where $V_i\sim exp\left({\lambda }_i\right),i=1,2,$ are independent and I is an independent Bernoulli random variable with $P(I=1)=[{\lambda }_2/({\lambda }_1+{\lambda }_2)]$.

To indicate that a random variable X has such an asymmetric Laplace distribution, we will write $X\sim AL({\lambda }_1,{\lambda }_2)$.

The mixture representation (3) can be modified to yield a more flexible model with an additional parameter. This will be called the generalized asymmetric Laplace model.

$$Y=I{V}_1+(1-I)(-V_2),$$

(4)

where $V_i\sim exp\left({\lambda }_i\right),i=1,2,$ are independent and I is an independent Bernoulli random variable with $P(I=1)=p$

The characteristic function of a generalized asymmetric Laplace (GAL) variable is of the form

$${\varphi }_Y\left(t\right)={\displaystyle \frac{1+it[p{\lambda }_2^{-1}-(1-p){\lambda }_1^{-1}]}{1+{\lambda }_1^{-1}{\lambda }_2^{-1}{t}^2+it({\lambda }_2^{-1}-{\lambda }_1^{-1})}}.$$

(5)

From the mixture representation (4), we obtain

$$E\left(Y\right)=p{\lambda }_1^{-1}-(1-p){\lambda }_2^{-1}$$

and

$$var\left(Y\right)=p(2-p){\lambda }_1^{-2}+(1-p^2){\lambda }_2^{-2}+2p(1-p){\lambda }_1^{-1}{\lambda }_2^{-1}.$$

In the case considered by Kozubowski and Podgorski [23] in which $p=[{\lambda }_2/({\lambda }_1+{\lambda }_2)]$, the moments simplify as follows

$$E\left(X\right)={\lambda }_1^{-1}-{\lambda }_2^{-1}$$

and

$$var\left(X\right)={\lambda }_1^{-2}+{\lambda }_2^{-2}.$$

Note that it is only when $p=[{\lambda }_2/({\lambda }_1+{\lambda }_2)]$ that the GAL density is continuous at 0.

3. Generalized Asymmetric Discrete Laplace Models

Inusah and Kozubowski [24] introduced a discrete Laplace distribution with geometric variables playing the roles of the exponential variables in the continuous Laplace model. Such a discrete Laplace variable X is of the form

$$X=V_1-V_2,$$

(6)

where the $V_i$s are i.i.d. with $V_i\sim Geometric\left(p\right),i=1,2.$

Note that throughout this paper, geometric random variables are defined to have support $\{0,1,2,\dots \},$ and can be viewed as representing the number of failures preceding the first success in series of Bernoulli trials.

An asymmetric version of this discrete Laplace model was also introduced by Inusah and Kozubowski [24] as follows. X has a discrete asymmetric Laplace distribution (DAL) if it can be represented in the form

$$X=V_1-V_2,$$

(7)

where the $V_i$s are independent with $V_i\sim Geometric\left(p_i\right).i=1,2.$

The possible values of a DAL random variable X are $\{\dots ,-3,-2,-1,0,1,2,3,\dots \}$, and its discrete density can be calculated as follows.

For $x\ge 0$, we have

$$\begin{array}{ccc}\hfill P(X=x)& =& \sum _{v_2=0}^{\infty }P(V_1=x+v_2,V_2=v_2)\hfill \\ & & \\ & =& p_1{p}_2{q}_1^x{(1-q_1{q}_2)}^{-1}\hfill \end{array}$$

(8)

Analogously, for $x<0$,

$$\begin{array}{ccc}\hfill P(X=x)& =& \sum _{v_1=0}^{\infty }P(V_1=v_1,V_2=v_1-x)\hfill \\ & & \\ & =& p_1{p}_2{q}_2^{-x}{(1-q_1{q}_2)}^{-1}.\hfill \end{array}$$

(9)

This discrete asymmetric Laplace density (DAL) is obviously a discrete parallel to the asymmetric Laplace model $Y=U_1-U_2$ where the $U_i$s are independent with $U_i\sim exp\left({\lambda }_i\right),i=1,2.$

Trying to continue with the idea of a parallel construction to that of the continuous asymmetric Laplace model, we may first verify that no mixture representation of the DAL model is available. But here, as in the continuous case, there is no reason not to consider a general mixture model, which we will call the generalized discrete asymmetric Laplace model (GDAL). It includes an additional parameter for flexibility. We thus will consider

$$X=I{V}_1+(1-I)(-V_2),$$

(10)

where $V_i\sim geo\left(p_i\right),i=1,2,$ are independent and I is an independent Bernoulli random variable with $P(I=1)=\pi .$

The discrete density of the GDAL distribution can be calculated as follows

$$\begin{array}{ccc}\hfill P(X=x)& =& \pi p_1{q}_1^x,x=1,2,\dots ,\hfill \\ & & \\ & =& \pi p_1+(1-\pi )p_2,x=0,\hfill \\ & & \\ & =& (1-\pi )p_2{q}_2^{-x},x=-1,-2,-3,\dots ,\hfill \end{array}$$

where $p_1,p_2$ and $\pi$ are parameters ranging over the interval $(0,1).$

The characteristic function of the GDAL$(\pi ,p_1,p_2)$ distribution is of the form:

$${\varphi }_X\left(t\right)={\displaystyle \frac{\pi p_1}{1-(1-p_1)e^{it}}}+{\displaystyle \frac{(1-\pi )p_2}{1-(1-p_2)e^{-it}}}.$$

(11)

From the mixture representation of the GDAL distribution, we may calculate its moments. In particular, we have

$$E\left(Y\right)={\displaystyle \frac{\pi (1-p_1)}{p_1}}-{\displaystyle \frac{(1-\pi )(1-p_2)}{p_2}}$$

and

$$var\left(Y\right)=\left\{{\displaystyle \frac{\pi (1-p_1)(2-p_1)}{p_1^2}}+{\displaystyle \frac{(1-\pi )(1-p_2)(2-p_2)}{p_2^2}}\right\}-{\left\{{\displaystyle \frac{\pi (1-p_1)}{p_1}}-{\displaystyle \frac{(1-\pi )(1-p_2)}{p_2}}\right\}}^2.$$

In the case in which $X\sim DAL(p_1,p_2)$, the moments are simpler, of the forms

$$E\left(X\right)=(1-p_1)/p_1-(1-p_2)/p_2$$

and

$$var\left(X\right)=(1-p_1)/p_1^2+(1-p_2)/p_2^2.$$

4. Estimation for the AL, GAL, DAL and GDAL Distributions

The following estimation results are quoted without proofs from Arnold and Arvanitis [25,26].

4.1. Maximum Likelihood for the AL Distribution

Suppose we have a sample $X_1,X_2,\dots ,X_n$ from an asymmetric Laplace distribution with density

$$f_X(x;{\lambda }_1,{\lambda }_2)={\displaystyle \frac{{\lambda }_1{\lambda }_2}{{\lambda }_1+{\lambda }_2}}[e^{-{\lambda }_{1}x}I(x>0)+e^{{\lambda }_{2}x}I(x<0)]$$

Define $U={\sum }_1^n{X}_{i}I(X_i>0)$, $V=-{\sum }_1^n{X}_{i}I(X_i<0)$ and $W={\sum }_1^{n}I(X_i>0)$.

The maximum likelihood estimates of the parameters are given by:

$\widetilde{{\lambda }_1}={\displaystyle \frac{n}{U+\sqrt{UV}}}$ and $\widetilde{{\lambda }_2}={\displaystyle \frac{n}{V+\sqrt{UV}}}$.

4.2. Maximum Likelihood for the DAL

Suppose we have a sample $X_1,X_2,\dots ,X_n$ from an asymmetric discrete Laplace distribution with density (8) and (9).

Define

$$U=\sum _1^n{X}_{i}I(X_i\ge 0)\mathrm{and} V=-\sum _1^n{X}_{i}I(X_i<0).$$

Taking partial derivatives of the log-likelihood and equating to 0, this eventually yields the following maximum likelihood estimates.

$${\widehat{p}}_1={\left\{{\displaystyle \frac{U}{n}}+{\displaystyle \frac{1}{2}}+\sqrt{{\displaystyle \frac{1}{4}}+{\displaystyle \frac{UV}{n^2}}}\right\}}^{-1},$$

(12)

and

$${\widehat{p}}_2={\left\{{\displaystyle \frac{V}{n}}+{\displaystyle \frac{1}{2}}+\sqrt{{\displaystyle \frac{1}{4}}+{\displaystyle \frac{UV}{n^2}}}\right\}}^{-1}.$$

(13)

4.3. Maximum Likelihood for the GAL

Suppose instead we have a sample $X_1,X_2,\dots ,X_n$ from a generalized asymmetric Laplace distribution of the form

$$X=I{Y}_1+(1-I)(-Y_2)$$

where $I\sim Bernoulli\left(p\right)$ and the $Y_i$s are independent with $Y_i\sim exp\left({\lambda }_i\right)$, $i=1,2.$

Again define $U={\sum }_1^n{X}_{i}I(X_i>0)$, $V=-{\sum }_1^n{X}_{i}I(X_i<0)$, and $W={\sum }_1^{n}I(X_i>0)$. Using these statistics, the maximum likelihood estimates are

$$\tilde{p}={\displaystyle \frac{W}{n}},\widetilde{{\lambda }_1}={\displaystyle \frac{W}{U}},\mathrm{and}\widetilde{{\lambda }_2}={\displaystyle \frac{n-W}{V}}.$$

(14)

4.4. Maximum Likelihood for the GDAL

Suppose instead we have a sample $X_1,X_2,\dots ,X_n$ from a generalized asymmetric discrete Laplace distribution of the form

$$X=I{W}_1+(1-I)(-W_2)$$

where $I\sim Bernoulli\left(\pi \right)$ and the $W_i$s are independent with $W_i\sim geo\left(p_i\right)$, $i=1,2.$

In this case, we will define

$$U={\sum }_1^n{X}_{i}I(X_i\ge 0) and V=-{\sum }_1^n{X}_{i}I(X_i<0),$$

and also define $N_0={\sum }_1^{n}I(X_i=0)$, $N_1={\sum }_1^{n}I(X_i>0)$, and $N_2={\sum }_1^{n}I(X_i<0)$

Taking partial derivatives of the log-likelihood and equating to 0, yields the following likelihood equations

$$\begin{array}{ccc}\hfill {\displaystyle \frac{N_1}{\pi }}& =& {\displaystyle \frac{N_2}{1-\pi }}-{\displaystyle \frac{N_0(p_1-p_2)}{\pi p_1+(1-\pi )p_2}}\hfill \\ & & \hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{N_1}{p_1}}& =& {\displaystyle \frac{U}{1-p_1}}-{\displaystyle \frac{N_0\pi }{\pi p_1+(1-\pi )p_2}}\hfill \\ & & \hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{N_2}{p_2}}& -& {\displaystyle \frac{V}{1=p_2}}-{\displaystyle \frac{N_0(1-\pi )}{\pi p_1+(1-\pi )p_2}}\hfill \end{array}$$

(17)

Note these equations are particularly easy to solve if $N_0=0$. In other cases, an iterative solution may be obtained.

4.5. Method of Moments for the AL

Suppose we have a sample $X_1,X_2,\dots ,X_n$ from an asymmetric Laplace distribution represented in the form

$$X=V_1-V_2,$$

where the $V_i$s are i.i.d. with $V_i\sim exponential\left({\lambda }_i\right),i=1,2$.

If we equate the first moment of X and the first absolute moment of X to the corresponding sample moments, $M_1$ and $M_2$, we obtain the following MOM estimates

$$\begin{array}{ccc}\hfill {\tilde{\lambda }}_1& =& {\left[{\displaystyle \frac{M_2+M_1+\sqrt{M_2^2-M_1^2}}{2}}\right]}^{-1}\hfill \\ & & \\ \hfill {\tilde{\lambda }}_2& =& {\left[{\displaystyle \frac{M_2-M_1+\sqrt{M_2^2-M_1^2}}{2}}\right]}^{-1}.\hfill \end{array}$$

(18)

4.6. Method of Moments for the DAL

Suppose we have a sample $X_1,X_2,\dots ,X_n$ from an asymmetric discrete Laplace distribution represented in the form

$$X=V_1-V_2,$$

where the $V_i$s are i.i.d. with $V_i\sim geo\left(p_i\right),i=1,2.$

We denote the first two sample moments, based on a sample of size n, by

$M_1=(1/n){\sum }_{i=1}^n{X}_i$ and $M_2=(1/n){\sum }_{i=1}^n{X}_i^2,$ define $S^2=M_2-M_1^2$, and set up the equations $M_1=E\left(X\right)$ and $S^2=var\left(X\right)$ to be solved to obtain method of moments estimates of the $p_i$s. Here, it is convenient to rewrite the equations in terms of the means of the $V_i$s (i.e., ${\mu }_i=(1-p_i)/p_i,ß=1,2).$ The method of moments estimates of the ${\mu }_i$s obtained in this way are of the following form.

$${\tilde{\mu }}_1={\displaystyle \frac{1}{2}}\left[M_1-1+\sqrt{{(M_1-1)}^2+S^2+M_1=M_1^2}\right],$$

(19)

and

$${\tilde{\mu }}_2={\tilde{\mu }}_1-M_1.$$

(20)

4.7. Method of Moments for the GAL

Suppose we have a sample $X_1,X_2,\dots ,X_n$ from a generalized asymmetric Laplace distribution of the form

$$X=I{V}_1+(1-I)(-V_2)$$

where $I\sim Bernoulli\left(p\right)$ and the $V_i$s are independent with $V_i\sim exp\left({\lambda }_i\right)$, $i=1,2.$ In order to estimate the parameters using the method of moments, it is convenient to define ${\delta }_i={\lambda }_i^{-1},i=1,2.$ The first three moments are:

$$\begin{array}{ccc}\hfill E\left(X\right)& =& p{\delta }_1-(1-p){\delta }_2,\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill E\left(X^2\right)& =& 2p{\delta }_1^2+2(1-p){\delta }_2^2,\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill E\left(X^3\right)& =& 6p{\delta }_1^3-6(1-p){\delta }_2^3.\hfill \end{array}$$

(23)

Denote the sample moments by $M_i=(1/n){\sum }_{j=1}^n{X}_j^i,i=1,2,3$.

The moment equations to solve numerically are:

$$\begin{array}{ccc}\hfill M_1& =& p{\delta }_1-(1-p){\delta }_2,\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill M_2& =& 2p{\delta }_1^2+2(1-p){\delta }_2^2,\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill M_3& =& 6p{\delta }_1^3-6(1-p){\delta }_2^3.\hfill \end{array}$$

(26)

4.8. Method of Moments for the GDAL

Suppose we have a sample $X_1,X_2,\dots ,X_n$ from a generalized asymmetric discrete Laplace distribution of the form

$$X=I{W}_1+(1-I)(-W_2)$$

where $I\sim Bernoulli\left(\pi \right)$ and the $W_i$s are independent with $W_i\sim geo\left(p_i\right)$, $i=1,2.$

We wish to estimate the parameters using the method of moments.

The first three moments of X are:

$$\begin{array}{ccc}\hfill E\left(X\right)& =& \pi {\displaystyle \frac{1-p_1}{p_1}}-(1-\pi ){\displaystyle \frac{1-p_2}{p_2}}\hfill \\ & & \\ \hfill E\left(X^2\right)& =& \pi \left\{2{\left({\displaystyle \frac{1-p_1}{p_1}}\right)}^2+{\displaystyle \frac{1-p_1}{p_1}}\right\}+(1-\pi )\left\{2{\left({\displaystyle \frac{1-p_2}{p_2}}\right)}^2+{\displaystyle \frac{1-p_2}{p_2}}\right\}\hfill \\ & & \\ \hfill E\left(X^3\right)& =& \pi \left\{6{\left({\displaystyle \frac{1-p_1}{p_1}}\right)}^3+6{\left({\displaystyle \frac{1-p_1}{p_1}}\right)}^2+\left({\displaystyle \frac{1-p_1}{p_1}}\right)\right\}-(1-\pi )\left\{6{\left({\displaystyle \frac{1-p_2}{p_2}}\right)}^3+6{\left({\displaystyle \frac{1-p_2}{p_2}}\right)}^2+\left({\displaystyle \frac{1-p_2}{p_2}}\right)\right\}\hfill \\ \end{array}$$

If we denote the first three sample moments, based on a sample of size n, by $M_1=(1/n){\sum }_{i=1}^n{X}_i$, $M_2=(1/n){\sum }_{i=1}^n{X}_i^2,$ and $M_3=(1/n){\sum }_{i=1}^n{X}_i^3,$ and set up the equations $M_j=E\left(X^j\right),j=1,2,3$, then our method of moments estimates will be the numerical solution to these three equations.

5. Wrapped Models

In this section, we will consider wrapped versions of the generalized models GAL and GDAL The results for the more basic AL model can, of course, be obtained as a special case of the GAL. The DAL case must be treated separately, since it does not admit a mixture representation.

5.1. Wrapping in General, Continuous and Discrete

Consider a density function $f\left(x\right)$ defined for $x\in (-\infty ,\infty )$. The corresponding wrapped density function $f_w\left(\theta \right)$ (suitable for modeling directional data) is defined by

$$f_w\left(\theta \right)=\sum _{k=-\infty }^{\infty }f(\theta +2k\pi ),\theta \in [0,2\pi ).$$

(27)

Popular choices for the density $f\left(x\right)$ to be wrapped include normal and Cauchy densities. Our interest will be focussed on wrapping generalized asymmetric Laplace densities.

Corresponding wrapped discrete models are suitable for analyzing directional data with N equally spaced directions between 0 and $2\pi$ as possible values.

5.2. The Wrapped GAL Distribution, WGAL

It is well known that the wrapped version of an exponential(${\lambda }_1$) density is a truncated exponential density. Thus, if $f^{\left(1\right)}\left(x\right)={\lambda }_1{e}^{-{\lambda }_{1}x}I(x>0)$, then the corresponding wrapped version of $f^{\left(1\right)}$ is

$$f_w^{\left(1\right)}\left(\theta \right)={\displaystyle \frac{{\lambda }_1{e}^{-{\lambda }_1\theta }I(0<\theta <2\pi )}{1-e^{-{\lambda }_1\left(2\pi \right)}}}.$$

(28)

Less well known, but readily verified is the fact that the wrapped version of a negative exponential variable is a truncated version of the density reflected about the point $\pi$. Thus, if $f^{\left(2\right)}\left(x\right)={\lambda }_2{e}^{{\lambda }_{2}x}I(x<0)$, then the corresponding wrapped version of $f^{\left(2\right)}$ is

$$f_w^{\left(2\right)}\left(\theta \right)={\displaystyle \frac{{\lambda }_2{e}^{-{\lambda }_2(2\pi -\theta )}I(0<\theta <2\pi )}{1-e^{-{\lambda }_2\left(2\pi \right)}}}.$$

(29)

With these results in hand, using the fact that the operations of mixing and wrapping commute, we may readily write down the density of the wrapped version of the generalized asymmetric Laplace density. Recall that the GAL density is of the form

$$f^{\left(GAL\right)}\left(x\right)=p{\lambda }_1{e}^{-{\lambda }_{1}x}I(x>0)+(1-p){\lambda }_2{e}^{{\lambda }_{2}x}I(x<0)$$

where we have used p to denote the mixing parameter $\pi$ to avoid confusion with the $\pi$ that is used when angles are expressed in radians. The corresponding wrapped version of this mixture density is thus the same mixture of the wrapped versions of it components. Thus, we have:

$$f_w^{\left(GAL\right)}\left(\theta \right)=p{\displaystyle \frac{{\lambda }_1{e}^{-{\lambda }_1\theta }}{1-e^{-{\lambda }_1\left(2\pi \right)}}}+(1-p){\displaystyle \frac{{\lambda }_2{e}^{-{\lambda }_2(2\pi -\theta )}}{1-e^{-{\lambda }_2\left(2\pi \right)}}},0\le \theta <2\pi .$$

(30)

These wrapped densities are typically bimodal.

5.3. The Wrapped GDAL Distribution, WGDAL

Discrete directional data typically involves observations that can take on values in a set of N equally spaced directions around the unit circle numbered $0,1,2,\dots ,N-1$. The i-th direction corresponds to the angle of $2\pi i/N$ radians, $i=0,1,\dots ,N-1$.

For modeling such situations, we typically begin with an integer valued random variable with possible values in the set $\dots -4,-3,-2,-1,0,1,2,3\dots$ and rescale it by multiplying it by $2\pi /N.$ The resulting data is then wrapped around the unit circle to yield the desired directional model.

To develop such models we can begin with an integer valued random variable X and then define $Y=X\left(modN\right).$ to be used to model the directional data (after multiplying by $2\pi /N.$ Parallel to the situation in the continuous case, we will call Y the wrapped version of X. In this scenario, the commuting of wrapping and mixing result continues to be available, in the sense that if X has a mixture of distributions then its wrapped version Y will have as its distribution the same mixture of the wrapped versions of the mixture components of X. We can thus develop the form of of the wrapped version of the GDAL distribution in terms of the wrapped versions of the two components in the mixture representation of the GDAL distribution. As is to be expected, there will be a close parallel with the treatment of the GAL distribution, but with geometric components playing the role taken before by exponential components.

Recall that the mixture representation of this generalized discrete asymmetric Laplace model is available in the form

$$X=I{V}_1+(1-I)(-V_2),$$

where $V_i\sim geo\left(p_i\right),i=1,2,$ are independent and I is an independent Bernoulli random variable with $P(I=1)=p_0$ (note that $p_0$ is now playing the role earlier played by $\pi$ to avoid confusion with the value of $\pi$, which will appear when angles are measured in radians. We first consider the wrapped version of a geometric $\left(p^{\left(1\right)}\right)$ distribution. If $X^{\left(1\right)}\sim geo\left(p^{\left(1\right)}\right)$, then we can verify that its wrapped version, $Y^{\left(1\right)}$ has a truncated geometric distribution, thus

$$f_{Y^{\left(1\right)}}\left(y\right)={\displaystyle \frac{p^{\left(1\right)}{(1-p^{\left(1\right)})}^y}{1-{(1-p^{\left(1\right)})}^N}}.y=0,1,\dots ,N-1.$$

(31)

An analogous expression is available for the wrapped version of a negative geometric variable. By a negative geometric variable we mean one with discrete density

$$f_{X^{\left(2\right)}}\left(x\right)=p^{\left(2\right)}{(1-p^{\left(2\right)})}^{-y},y=0,-1,-2,-3,\dots .$$

Its corresponding wrapped version is then of the form

$$f_{Y^{\left(2\right)}}\left(y\right)={\displaystyle \frac{p^{\left(2\right)}{(1-p^{\left(2\right)})}^{(N-1-y)}}{1-{(1-p^{\left(2\right)})}^N}}.y=0,1,\dots ,N-1.$$

With the densities of $Y^{\left(1\right)}$ and $Y^{\left(2\right)}$ at hand, we can immediately write down the discrete density of the wrapped GDAL model, using the representation (31) and the mixing-wrapping result as follows

$$f_Y\left(y\right)=p_0{\displaystyle \frac{p^{\left(1\right)}{(1-p^{\left(1\right)})}^y}{1-{(1-p^{\left(1\right)})}^N}}+(1-p_0){\displaystyle \frac{p^{\left(2\right)}{(1-p^{\left(2\right)})}^{(N-1-y)}}{1-{(1-p^{\left(2\right)})}^N}}.y=0,1,\dots ,N-1.$$

(32)

As in the continuous case, the wrapped density is typically bimodal.

Note that in order for this distribution to be used to model directions on the circle, it is necessary to rescale the random variable Y by multiplying it by $2\pi /N.$

6. Bivariate Models

We will review, and subsequently consider wrapped versions of, two flexible bivariate asymmetric Laplace models introduced in Arnold [27]. Generalized and discrete versions will also be described.

The first of these bivariate asymmetric Laplace models was introduced by Arvanitis [28] and we refer the reader to that source for a detailed discussion of the model. The construction of this model begins with the components used in developing the general bivariate beta model introduced in Arnold and Ng [29]. Thus, we begin with 8 independent gamma variables $U_1,U_2,\dots ,U_8$ with $U_j\sim \Gamma ({\delta }_j,1),j=1,2,\dots ,8.$ and we define $(X,Y)$ by

$$\begin{array}{c}\hfill X={\lambda }_{11}^{-1}(U_1+U_5+U_7)-{\lambda }_{12}^{-1}(U_3+U_6+U_8),\\ \hfill \\ \hfill Y={\lambda }_{21}^{-1}(U_2+U_6+U_7)-{\lambda }_{22}^{-1}(U_4+U_5+U_8),\end{array}$$

(33)

where it is assumed that the constraints, ${\delta }_1+{\delta }_5+{\delta }_7=1$, ${\delta }_3+{\delta }_6+{\delta }_8=1$, ${\delta }_2+{\delta }_6+{\delta }_7=1$, and ${\delta }_4+{\delta }_5+{\delta }_8=1$, have been imposed. This is to ensure that the bivariate distribution has asymmetric Laplace marginals. This model will be called the bivariate asymmetric Laplace model of the first kind, and if $(X,Y)$ is as defined in (33), we will write $(X,Y)\sim BAL\left(I\right)({\lambda }_{11},{\lambda }_{12},{\lambda }_{21},{\lambda }_{22},\underline{\delta })$. Since there were four constraints on the ${\delta }_j$s, this is an eight-parameter model. The marginal distributions depend only on the four $\lambda$ parameters, thus:

$$X\sim AL({\lambda }_{11},{\lambda }_{12}),Y\sim AL({\lambda }_{21},{\lambda }_{22}).$$

(34)

The second bivariate asymmetric Laplace model utilizes the closure under minimization property of the exponential distribution. For this model, we again begin with 8 independent random variables, $V_1,V_2,\dots ,V_8$, but this time we assume that they are exponentially distributed. Thus, we assume that $V_j\sim exp\left({\lambda }_j\right),j=1,2,\dots ,8.$ We then define

$$\begin{array}{c}\hfill X=min\{V_1,V_5,V_7\}-min\{V_3,V_6,V_8\},\\ \hfill \\ \hfill Y=min\{V_2,V_6,V_7\}-min\{V_4,V_5,V_8\}.\end{array}$$

(35)

If $(X,Y)$ has the structure shown in (35), then we will write $(X,Y)\sim BAL\left(II\right)\left(\underline{\lambda }\right)$ and say that it has a bivariate asymmetric Laplace distribution of the second kind with parameter vector $\underline{\lambda }.$ Note that both the first-kind and the second-kind bivariate asymmetric Laplace distributions have an 8-dimensional parameter space. The marginal distributions of the BAL(II) distribution are, by construction, of the asymmetric Laplace form. Thus:

$$\begin{array}{c}\hfill X\sim AL({\lambda }_1+{\lambda }_5+{\lambda }_7,{\lambda }_3+{\lambda }_6+{\lambda }_8),\end{array}$$

(36)

$$\begin{array}{c}\hfill Y\sim AL({\lambda }_2+{\lambda }_6+{\lambda }_7,{\lambda }_4+{\lambda }_5+{\lambda }_8),\end{array}$$

(37)

Bivariate versions of the generalized asymmetric Laplace (GAL) distribution can be constructed by modifying the BAL(I) or the BAL(II) models by the introduction of two additional probability parameters.

The generalized version of the BAL(I) model (GBAL(I)) may be defined as follows

$$\begin{array}{c}\hfill X=I_1{\lambda }_{11}^{-1}(U_1+U_5+U_7)-(1-I_1){\lambda }_{12}^{-1}(U_3+U_6+U_8),\\ \hfill \\ \hfill Y=I_2{\lambda }_{21}^{-1}(U_2+U_6+U_7)-(1-I_2){\lambda }_{22}^{-1}(U_4+U_5+U_8),\end{array}$$

(38)

where it is assumed that the constraints, ${\delta }_1+{\delta }_5+{\delta }_7=1$, ${\delta }_3+{\delta }_6+{\delta }_8=1$, ${\delta }_2+{\delta }_6+{\delta }_7=1$, and ${\delta }_4+{\delta }_5+{\delta }_8=1$, have been imposed, and where the $I_j$s are independent with $I_j\sim Bernoulli\left(p_j\right),j=1,2.$

Moments of the coordinates of this random vector and the covariance, together with other distributional properties, are not difficult to evaluate using the definitions or, in more difficult cases, could be evaluated by straightforward simulation.

The generalized version of the BAL(II) model (GBAL(II)) may be defined as follows

$$\begin{array}{c}\hfill X=I_1[min\left\{(U_1,U_5,U_7\}\right]-(1-I_1)\left[min\{U_3,U_6,U_8\}\right],\\ \hfill \\ \hfill Y=I_2[min\left\{(U_2,U_6,U_7\}\right]-(1-I_2)\left[min\{U_4,U_5,U_8)\right\}],\end{array}$$

(39)

where the ${\delta }_j$s are positive parameters, and where the $I_j$s are independent with $I_j\sim Bernoulli\left(p_j\right),j=1,2.$

For this model as well, moments, etc., are readily computed.

Discrete versions of the BAL(I)–(II) distributions can now be constructed using negative binomial and geometric components instead of gamma and exponential distributed components.

To construct the bivariate symmetric discrete Laplace model of the first kind (BDL(I)), we begin with a set of 8 independent random variables $U_1,U_2,\dots ,U_8$ with $U_i\sim neg.bin.({\delta }_j,p),j=1,2,\dots ,8.$ Note that all of the $U_j$s share a common value for p. This results in a construction of a bivariate distribution with symmetric discrete Laplace marginals. It will be seen that it is not possible to use this kind of construction to yield asymmetric marginals. To continue, we now define $(X,Y)$ by

$$\begin{array}{c}\hfill X=(U_1+U_5+U_7)-(U_3+U_6+U_8),\\ \hfill \\ \hfill Y=(U_2+U_6+U_7)-(U_4+U_5+U_8),\end{array}$$

(40)

where it is assumed that the constraints, ${\delta }_1+{\delta }_5+{\delta }_7=1$, ${\delta }_3+{\delta }_6+{\delta }_8=1$, ${\delta }_2+{\delta }_6+{\delta }_7=1$, and ${\delta }_4+{\delta }_5+{\delta }_8=1$, have been imposed. This model will be called the bivariate discrete Laplace model of the first kind and if $(X,Y)$ is as defined in (40) we will write $(X,Y)\sim BDL\left(1\right)\left(\underline{\delta }\right)$. Since there were four constraints on the ${\delta }_j$s, this is a 5 parameter model. The marginal distributions are differences of independent $geometric\left(p\right)$ variables and thus have discrete Laplace densities. Moments are obtainable from the representation (40).

We next consider a bivariate asymmetric discrete Laplace model that utilizes the closure under minimization property of the geometric distribution. For it, we again begin with 8 independent random variables, $V_1,V_2,\dots ,V_8$, but this time we assume that they are geometrically distributed; thus, $V_j\sim geo\left({\tau }_j\right),j=1,2,\dots ,8,$ where ${\tau }_j\in (0,1),$$j=1,2,\dots ,8,$ We then define

$$\begin{array}{c}\hfill X=min\{V_1,V_5,V_7\}-min\{V_3,V_6,V_8\},\\ \hfill \\ \hfill Y=min\{V_2,V_6,V_7\}-min\{V_4,V_5,V_8\},\end{array}$$

(41)

using a construction parallel to that used in the construction of the BAL(II) model earlier described. If $(X,Y)$ has the structure shown in (41), then we will write $(X,Y)\sim BADL\left(II\right)\left(\underline{\tau }\right)$ and say that it has a bivariate asymmetric discrete Laplace distribution of the second kind with parameter vector $\underline{\tau }.$ Note that the second-kind bivariate asymmetric discrete Laplace distributions has an 8-dimensional parameter space. The marginal distributions of the BADL(II) distribution are of the asymmetric discrete Laplace form. Thus:

$$\begin{array}{c}\hfill X\sim ADL(1-(1-{\tau }_1)(1-{\tau }_5)(1-{\tau }_7),1-(1-{\tau }_3)(1-{\tau }_6)(1-{\tau }_8)),\end{array}$$

(42)

$$\begin{array}{c}\hfill Y\sim ADL(1-(1-{\tau }_2)(1-{\tau }_6)(1-{\tau }_7),1-(1-{\tau }_4)(1-{\tau }_5)(1-{\tau }_8)),\end{array}$$

(43)

marginal moments of the BADL(II) distribution are thus readily identified. However, $cov(X,Y)$ is quite complicated and will usually be approximated by simulation. using the definition (41).

Generalized versions of these bivariate discrete Laplace models can be formulated in a manner parallel to that used to generalize the bivariate asymmetric Laplace models.

The generalized version of the BDL(I) model may be defined as follows

$$\begin{array}{c}\hfill X=I_1(U_1+U_5+U_7)-(1-I_1)(U_3+U_6+U_8),\\ \hfill \\ \hfill Y=I_2(U_2+U_6+U_7)-(1-I_2)(U_4+U_5+U_8),\end{array}$$

(44)

where it is assumed that the constraints, ${\delta }_1+{\delta }_5+{\delta }_7=1$, ${\delta }_3+{\delta }_6+{\delta }_8=1$, ${\delta }_2+{\delta }_6+{\delta }_7=1$, and ${\delta }_4+{\delta }_5+{\delta }_8=1$, have been imposed, and where the $I_j$s are independent with $I_j\sim Bernoulli\left({\pi }_j\right),j=1,2,$ and the $U_j$s are independent with $U_j\sim neg.bin.({\delta }_j,p),$$j=1,2,3\dots ,8.$ This model will be called the generalized bivariate discrete Laplace model of the first kind, and if $(X,Y)$ is as defined in (44), we will write $(X,Y)\sim GBDL\left(1\right)({\pi }_1,{\pi }_2,\underline{\delta })$. Since there were four constraints on the ${\delta }_j$s, this is a 7-parameter model.

Means variances and covariance of the coordinates of this random vector are not difficult to evaluate, or could be evaluated by simulation.

The generalized version of the BADL(II) model may be defined as follows

$$\begin{array}{c}\hfill X=I_1[min\left\{(V_1,V_5,V_7\}\right]-(1-I_1)\left[min\{V_3,V_6,V_8\}\right],\\ \hfill \\ \hfill Y=I_2\left[min\{V_2,V_6,V_7\}\right]-(1-I_2)\left[min\{V_4,V_5,V_8)\right\}],\end{array}$$

(45)

where the $I_j$s are independent with $I_j\sim Bernoulli\left({\pi }_j\right),j=1,2.$ and the $V_j$s are independent with $V_j\sim geo\left({\tau }_j\right),j=1,2,\dots ,8.$ This is a 10-parameter model.

Means variances and covariance of the coordinates of this random vector are also not difficult to evaluate, or could be evaluated by simulation.

Remark 1.

Even more general distributions than (44) and (39) can be constructed in which the $I_i$s are dependent-indicator random variables.

6.1. Submodels Involving Less Parameters

Remark 2.

It is not expected that there will be many cases in which the full 8 or more parameter bivariate models discussed in this Section will be needed to “fit” real-world data sets adequately. In practice, lower dimensional sub-models can be expected to suffice. Such models will be obtained by imposing relationships between some parameters of the model, or by setting some, or in some cases many, of the parameters equal to zero, using the following conventions:

(1) 

If $X\sim \Gamma (\alpha ,\beta )$ with $\alpha =0$, then $X=0$ with probability 1.

(2) 

If $X\sim neg.bin(m,p)$ with $m=0$, then $X=0$ with probability 1.

(3) 

If $X\sim exp\left(\lambda \right)$ with $\lambda =0$, then $X=\infty$ with probability 1.

(4) 

If $X\sim geo\left(p\right)$ with $p=0$, then $X=\infty$ with probability 1.

Example 1.

Consider the following two-parameter sub-model of the $BADL\left(II\right)$ model given by restricting its parameter space as follows:

$$\begin{array}{cc}\hfill & {\tau }_2={\tau }_3=\alpha \in (0,1)\hfill \\ \hfill & {\tau }_1={\tau }_4={\tau }_5=\beta \in (0,1)\hfill \\ \hfill & {\tau }_6={\tau }_7={\tau }_8=0.\hfill \end{array}$$

(46)

This simple model, which still exhibits dependence in terms of the $V_i$ variables, is given by

$$\begin{array}{c}\hfill X=min\{V_1,V_5\}-V_3,\\ \hfill \\ \hfill Y=V_2-min\{V_4,V_5\}.\end{array}$$

(47)

Remark 3.

Higher dimensional versions of the BDL(I), BADL(II), GBDL(I), GBADL(II) models are readily described. Since, for example, the completely general trivariate version of the asymmetric discrete Laplace model of Type II will involve 26 independent geometric components each with its own τ parameter, only submodels including just a limited number of components will be tractable and useful for modeling purposes.

6.2. Wrapping Bivariate Densities

Wrapping can be as useful in two dimensions as it is in one dimension. For example, it can be used to develop bivariate circular distributions beginning with convenient real-valued bivariate distributions. In many applications, the basic bivariate distribution will not have an available density. However, this will not be a problem provided that we choose to describe wrapping in terms of reduction modulo $2\pi$.

Thus, in two dimensions, we may begin with an arbitrary two-dimensional random variable $(X,Y)$ and define a two-dimensional wrapped circular variable by

$$({\Theta }_1,{\Theta }_2)=(Xmod\left(2\pi \right),Ymod\left(2\pi \right)).$$

(48)

For example, we might begin with $(X,Y)$ having a bivariate generalized asymmetric Laplace distribution. The bivariate wrapped variable (48) will have wrapped generalized asymmetric Laplace marginals (i.e., of the form (30)). A variety of relatives of the bivariate Laplace models exist, as described in the first part of Section 6, and can be used in this construction. Although the resulting marginal densities are recognizable as having generalized Laplace forms, the joint densities are generally not obtainable in a simple form. Of course, random variables of the form (48) will be easy to simulate provided that realizations of the corresponding random vector $(X,Y)$ can be simulated.

Bivariate axial variables can, of course, be analogously derived by wrapping $mod\left(\pi \right)$ instead of $mod\left(2\pi \right)$.

If we do indeed have a density function $f_0(x,y)$ available to be wrapped, we can immediately write down an expression for the corresponding bivariate wrapped version. Thus,

$$f_{{\Theta }_1,{\Theta }_2}({\theta }_1,{\theta }_2)=\left\{\sum _{j=-\infty }^{\infty }\sum _{k=-\infty }^{\infty }{f}_0({\theta }_1+2j\pi ,{\theta }_2+2k\pi )\right\}I(0<{\theta }_1,{\theta }_2<2\pi )$$

(49)

The series in (49) will always be convergent, although they often will not result in a simple analytic expression for the wrapped density.

7. Applications with Real Examples

For modelling price change, rather than price (which is a positive variable) either in terms of the price difference or a log ratio of consecutive prices, the Laplace distribution defined on the entire real-line R has been a popular choice. In modern-day volatile markets, such a choice stems from the interesting fact that, while it has heavier tails and a higher peak than the normal distribution, the reverse holds with respect to the Cauchy distribution. Furthermore, it plays a pivotal role in the family of Linnik distributions, as does the normal in the family of stable distributions. Anderson and Arnold [30] demonstrate that the Laplace distribution, which has the index parameter $\alpha$ = 2 in the Linnik family, emerges as the best choice in that entire family for modeling the popular Box–Jenkins common stock closing price-change data of IBM discussed by Box et al. [31].

In the same spirit, the wrapped Laplace distribution serves as a good choice for circular data. Recently, SenGupta and Roy [32] obtained data on the number of traffic accidents in the year 2016 at different times of the day in the city of Srinagar from the reports named Accidental Deaths and Suicides published by National Crime Records Bureau, India. The given data were grouped in time intervals of 3 h and were converted from hours to radians. First, the wrapped stable family of distributions, obtained by wrapping the family of stable distributions, was tested for goodness-of-fit, but it did not yield acceptance of any of its member. However, in the wrapped Linnik family, the wrapped Laplace distribution emerged as the best one with acceptable fit, the p-value of the corresponding goodness-of-fit test statistic was 0.1327258.

Wrapped circular distributions have yet another useful and interesting application. Wrapped distributions preserve (see, SenGupta and Roy [32]) the index parameters of their corresponding linear distributions. This property provides elegant and efficient estimators, which are often otherwise difficult to obtain, of the index parameters of the corresponding stable and Linnik families from their wrapped versions. In this context, SenGupta and Roy [32] considered the log-returns data of the Indian gold market. They noted that the data exhibited mild asymmetry, pronounced platykurticity and quite small first-order autocorrelation properties. This motivated them to study the symmetric Linnik family of distributions as an initial approximation of the distribution for the data. The analysis of stock price data is generally carried out on a difference of order 1 in relation to the original series. So, denoting the original stock price data by $x_t$, the transformation $z_t=100(ln\left(x_t\right)-ln\left(x_{t-1}\right))$ was used. The $z_t$ variables were then wrapped by the process described above. This transformation of log returns aimed to achieve symmetry and reduce autocorrelation in the transformed series (for further details, see SenGupta and Roy [32]). SenGupta and Roy [32] then proceeded to estimate the parameters by various methods, assuming symmetry (with the Linnik family) to give an initial approximation. They noted that, based on the method of characteristic function estimation, the wrapped Laplace emerges as the best choice under this approximation.

Thus as exemplified above, data can often exhibit skewed and bimodal behavior, e.g., as with linear data in financial markets, as also with circular data, as in circadian rhythms from bioinformatics. It is only natural to expect that our wide array of generalizations for both linear and circular distributions of Laplace distribution as presented above, covering asymmetry and bimodality, will pave the way to modeling such data as in emerging applied research.

8. Discussions

The exponential distribution and its generalization, the Weibull distribution, are fundamental models, e.g., in Reliability Inference, for data on $R^{+}$, the positive half of the real line. Our proposed models involving generalized Laplace distributions generalize the exponential distribution for data on the entire real-line R. Such generalizations as proposed above do not seem to be yet (e.g., see Lai [33]) available for the Weibull distribution. However, the methods used in the present paper may be applied to also generalize the Weibull distribution in a similar manner, and we intend to address this in our future work.