# A folded Laplace distribution

- Yucui Liu
^{1}and - Tomasz J. Kozubowski
^{2}Email author

**2**:10

https://doi.org/10.1186/s40488-015-0033-9

© Liu and Kozubowski. 2015

**Received: **28 September 2015

**Accepted: **7 October 2015

**Published: **22 October 2015

## Abstract

We study a class of probability distributions on the positive real line, which arise by folding the classical Laplace distribution around the origin. This is a two-parameter, flexible family with a sharp peak at the mode, very much in the spirit of the classical Laplace distribution. We derive basic properties of the distribution, which include the probability density function, distribution function, quantile function, hazard rate, moments, and several related parameters. Further properties related to mixture representation, Lorenz curve, mean residual life, and entropy are included as well. We also discuss parameter estimation for this new stochastic model and illustrate its potential applications with real data.

## Keywords

## Classification codes

## Introduction

*∞*). The folding is accomplished via the transformation

*X*is a Laplace random variable with PDF (1), so that the PDF of Y becomes

*X*(Cooray 2008):

*μ*=0, this reduces to

*σ*. This is to be expected, as in this case the Laplace distribution is centered about the origin. Thus, the folded Laplace distribution can be thought of as a generalization of exponential distribution, and in fact it shares the form of the density with the latter when the values are larger than

*μ*. Figure 1 provides an illustration of the folded Laplace PDF. As we shall see in the sequel, this is a rather flexible family with a sharp peak at the mode, resembling the Laplace distribution. The latter has gained popularity in recent years in numerous areas of applications see, e.g., Kotz et al. (2001). We hope that this

*positive version*of the Laplace distribution will prove to be another useful stochastic model as well.

Folded distributions are important and popular models that have found many interesting applications. As noted by several authors [see, e.g., Leone et al. (1961), Psarakis and Panaretos (1990), Cooray et al. (2006), Cooray (2008)], they frequently arise when the data are recorded with disregard to their sign. Perhaps the best known such model is the folded normal distribution, developed in the 1960s and 1970s [see, e.g., Elandt (1961), Leone et al. (1961), Johnson (1962, 1963), Rizvi (1971), and Sundberg (1974)]. Since then, this distribution, along with its various extensions, has been re-visited and applied in diverse fields [see, e.g., Nelson (1980), Sinha (1983), Yadavalli and Singh (1995), Naulin (2003), Bohannon et al. (2004), Lin (2004), Lin (2005), Asai and McAleer (2006), Kim (2006), Liao (2010), Jung et al. (2011), Chakraborty and Chatterjee (2013), Tsagris et al. (2014)]. Other families of folded distributions recently studied include folded *t* and generalized *t* distributions [see Psarakis and Panaretos (1990, 2001), Brazauskas and Kleefeld (2011, 2014), Scollnik (2014), and Nadarajah and Bakar (2015)], folded Cauchy distribution [see Johnson et al. (1994, 1995), Cooray (2008), Nadarajah and Bakar (2015)], folded Gumbel distribution [see Nadarajah and Bakar (2015)], folded normal slash distribution [see Gui et al. (2013)], folded beta distribution see [Berenhaut and Bergen (2011)], folded binomial distribution [see Porzio and Ragozini (2009)], folded logistic distribution [see Cooray et al. (2006), Nadarajah and Kotz (2007), Cooray (2008)], folded exponential transformation [see Piepho (2003)], and doubly-folded bivariate normal distribution [see Stracener (1973)]. Let us note that the folded Laplace (FL) distribution, and a more general folded exponential power family, were recently briefly treated in Cooray (2008) and Nadarajah and Bakar (2015), respectively. Our works offers a more comprehensive theory focused on FL distributions, including numerous new results, estimation, and data examples.

We begin our journey in Section 2, where we define the folded Laplace (FL) model and derive its properties. Section 3 is devoted to statistical inference related to the FL model. In particular, we establish the existence and uniqueness of moment estimators of the FL parameters, and derive their asymptotic behavior. This is followed by Section 4, which contains a data example illustrating modeling potential of the FL distribution. Proofs and technical results are collected in the Appendix.

## Definition and properties

We start with a formal definition of the folded Laplace (FL) model.

###
**Definition**
**2.1**.

A random variable on *R*
_{+} given by the density function (4), where *μ*≥0 and *σ*>0, is said to have a folded Laplace distribution, denoted by *F*
*L*(*μ*,*σ*).

###
**Remark**
**2.1**.

*F*

*L*(

*μ*,

*σ*) distribution can be written more compactly as

Moreover, it is easy to see that folding Laplace distribution (1) with the mode *μ*>0 or one with the mode −*μ*<0 results in the same distribution, so that we can restrict attention to non-negative values of *μ*. In fact, this is always the case when the folding mechanism (2) is applied to a location family *f*(*x*)=*h*(*x*−*μ*), \(x,\mu \in \mathbb R\).

###
**Remark**
**2.2**.

It is straightforward to see that the FL PDF is unimodal with the mode at *μ*, and becomes more symmetric as the mode *μ* increases, as can be seen in Fig. 1. On the other hand, as *μ* gets towards the origin, the distribution becomes exponential with the PDF given by (5). This figure also shows that the FL PDF becomes flatter as the parameter *σ* increases. In the boundary case *σ*=0, the distribution is understood as a point mass at *μ*, which can be seen by taking the limit of the FL cumulative distribution function (CDF) given below as *σ*→0.

###
**Remark**
**2.3**.

It should be noted that the FL distribution is not a member of exponential family (unless *μ*=0).

###
**Remark**
**2.4**.

Its special cases include the folded Laplace distribution (*p*=1) as well as the folded normal distribution (*p*=2).

###
**Remark**
**2.5**.

*log-Laplace distribution*(see, e.g., Kozubowski and Podgórski 2003a,b). In analogy with the log-normal distribution, this model describes the random variable

*Y*= exp(

*X*), where

*X*is Laplace-distributed with the PDF (1), and is given by the PDF

where *α*=1/*σ*>0 is a tail parameter and *δ*= exp(*μ*) is a scale parameter. However, in contrast with the FL distribution, this distribution has a power-law tails.

###
**Remark**
**2.6**.

The QF above can be used to simulate random variates *Y* from an FL distribution via *Y*=*Q*(*U*), where *U* is a standard uniform variate, available in standard statistical packages. Alternatively, simulation can be accomplished by first generating a Laplace variate *T*=*σ*
*X*+*μ*, where *X* is a standard Laplace, followed by *Y*=|*T*|. The standard Laplace variate can be generated via *X*=*E*
_{1}−*E*
_{2}, where the {*E*
_{
i
}} are independent standard exponential variates [see, e.g., Kotz et al. (2001)].

### 2.1 The hazard rate

The hazard rate (failure rate, mortality rate) of the FL model, defined as the ratio of the PDF to the survival function, is provided in the result below. Its routine derivation, which can be found in Liu (2014), will be omitted.

###
**Proposition**
**2.1**.

*Y*∼

*F*

*L*(

*μ*,

*σ*), then the hazard rate of

*Y*is given by

Moreover, this function is concave up and monotonically increasing from *h*(0)= exp(−*μ*/*σ*)/*σ* to *h*(*μ*)=1/*σ* on the interval [0,*μ*].

### 2.2 The moment generating function and moments

The following result, which was stated in Cooray (2008) and derived in Liu (2014), provides an explicit formula for the moment generating function (MGF) of the FL model.

###
**Proposition**
**2.2**.

*Y*∼

*F*

*L*(

*μ*,

*σ*), then the moment generating function of

*Y*is given by

By taking the derivatives of the MGF at *t*=0, we can recover the moments of the FL distribution. The latter are given in the following result, whose lengthy albeit routine derivation shall be omitted [details can be found in Liu (2014)].

###
**Proposition**
**2.3**.

*Y*∼

*F*

*L*(

*μ*,

*σ*), then the

*n*th moment of

*Y*is given by

The following result concerning the coefficient of variation connected with the FL model plays an important role in investigating the existence and uniqueness of moment estimators of *μ* and *σ* (see Section 3) and can be a useful aid in FL model validation. Its proof is included in the Appendix.

###
**Proposition**
**2.4**.

*Y*∼

*F*

*L*(

*μ*,

*σ*) with both

*μ*,

*σ*>0, then the coefficient of variation of

*Y*is

where *δ*=*μ*/*σ*∈[0,*∞*). Moreover, we have 0<*C*
_{
Y
}<1.

### 2.3 Skewness and kurtosis

*Y*∼

*F*

*L*(

*μ*,

*σ*) are given by

respectively, where *δ*=*μ*/*σ*∈ [0,*∞*). Since *γ*
_{
Y
}>0, every FL distribution is skewed to the right. When *δ*=0 (which occurs when *μ*=0), then *γ*
_{
Y
}=2 and *κ*
_{
Y
}=9, which are the skewness and the kurtosis of an exponential random variable (to which the FL model reduces in this case).

### 2.4 The mean/median/mode inequality

One common rule of thumb states that for unimodal distributions, the mean, the median, and the mode often occur in either alphabetical or reverse-alphabetical order [see, e.g., Dharmadhikari and Joag-Dev (1988)]. As shown in the following result, which is proved in the Appendix, the FL distribution is not an exception in this regard.

###
**Proposition**
**2.5**.

*Y*∼

*F*

*L*(

*μ*,

*σ*) then

where *M*(*Y*), *m*(*Y*), and \(\mathbb E(Y)\) are the mode, the median, and the mean of *Y*, respectively.

In the special case when *μ*=0, the distribution reduces to the exponential one, and the inequality in (19) turns into 0<*σ* log2<*σ*, which is trivially true.

### 2.5 Truncated distributions and stochastic representation

*μ*, and use the resulting distributions to derive a stochastic representation of the

*FL*model. Straightforward calculations show that the PDF of a folded Laplace variable truncated above at

*μ*,

*X*=

*Y*|

*Y*≤

*μ*, is given by

*μ*,

*W*=

*Y*|

*Y*≥

*μ*, has an exponential distribution with mean

*σ*, shifted by

*μ*units to the right, so its PDF is

We shall skip a routine derivation of the following result, which provides a stochastic representation of an *FL* random variable in terms of the above two truncated variables.

###
**Proposition**
**2.6**.

*Y*∼

*F*

*L*(

*μ*,

*σ*), then

*X*has the PDF (20),

*W*has the PDF (21),

*I*is an indicator variable given by

and all three variables on the right-hand-side of (22) are mutually independent.

### 2.6 The mean residual life

*mean residual life*function,

*Y*, we start with the conditional distribution of

*Y*−

*t*given

*Y*>

*t*, called the

*excess*random variable. In view of Proposition 2.6, it is easy to see that, for

*t*>

*μ*, the excess random variable is simply exponential with mean

*σ*. On the other hand, routine algebra shows that when 0≤

*t*≤

*μ*, the CDF and the PDF of the excess random variable,

*Y*−

*t*|

*Y*>

*t*, are given by

Straightforward calculations lead to the result below, whose proof shall be omitted.

###
**Proposition**
**2.7**.

*X*∼

*F*

*L*(

*μ*,

*σ*) then the mean residual life function (24) is given by

*m*(

*t*)=

*σ*for

*t*≥

*μ*and

for 0≤*t*≤*μ*.

###
**Remark**
**2.7**.

Note that the function *m* given above is continuous on [0,*∞*] with the value of *μ*+*σ* exp(−*μ*/*σ*) for *t*=0, which is to be expected as *m*(0) is just the mean of *Y* itself.

### 2.7 The Lorenz curve

*g*is the PDF of the random variable

*Y*, is a standard tool in economics, used to measure the social or wealth inequality [see, e.g., Gastwirth (1971)]. When we substitute the PDF of FL distribution given in (4), we obtain

where \(a =\mu +2\sigma e^{\frac {-\mu }{\sigma }}-\mu e^{\frac {-2\mu }{\sigma }}\) and \(b=\mu +\sigma e^{\frac {-\mu }{\sigma }}\) is the mean of the FL distribution.

### 2.8 Entropy

of an FL random variable *X* with PDF *g*. This is a standard measure of uncertainty, introduced in Shannon (1948). The following result, which is proven in the Appendix, contains relevant details.

###
**Proposition**
**2.8**.

*X*∼

*F*

*L*(

*μ*,

*σ*) then

where *θ*= exp(−*μ*/*σ*).

###
**Remark**
**2.8**.

When *μ*=0, so that *X* reduces to an exponential variable with mean *σ*, we obtain *H*(*X*)= log*σ*+1, which is the entropy in this special case.

## Parameter estimation

Here we consider the problem of estimating the parameters *μ* and *σ* of the folded Laplace distribution. We shall focus on the method of moments, which is computationally straightforward. Maximum likelihood estimation for this case, which is theoretically and computationally much more involved, is currently under investigation and will be reported elsewhere.

*Y*

_{1},

*Y*

_{2},…,

*Y*

_{ n }be independent and identically distributed (IID) random variables that follow the

*F*

*L*(

*μ*,

*σ*) model, and let \(M_{1}=\bar {Y}_{n}=\frac {1}{n}\sum _{i=1}^{n}Y_{i}\) and \(M_{2}=\frac {1}{n}\sum _{i=1}^{n}{Y_{i}^{2}}\) be the first two sample moments. To derive the method of moment estimators (MMEs) of the two parameters we shall use an equivalent alternative parameterization, where

*μ*is replaced by

*M*

_{ i }}, we obtain the following system of two equations in two unknowns:

*σ*and obtain a single equation for the parameter

*δ*:

*δ*with

where *r* is the inverse of the function *h*. The following result summarizes this discussion.

###
**Proposition**
**3.1**.

Let *Y*
_{1},…,*Y*
_{
n
} be a random sample from an *F*
*L*(*μ*,*σ*) distribution, and let *M*
_{1} and *M*
_{2} be the first and the second sample moments based on the {*Y*
_{
i
}}, respectively. Then, there exist unique moment estimators of *δ*=*μ*/*σ* and *σ*, given by (36), whenever the condition (35) is satisfied.

*σ*=0, we obtain \(\hat {\mu }_{n}=M_{1} = \bar {Y}_{n}\) (this special FL distribution assigns the entire mass to a single point \(\hat {\mu }_{n}\)). Further, the right-hand-side inequality in (35) can be stated as

*n*, since the sample coefficient of variation \(S_{n}/\bar {Y}_{n}\) converges to the theoretical one, which is known to be less than 1 (see Proposition 2.4). In the boundary case \(M_{2} = 2{M_{1}^{2}}\) (\(S_{n}/\bar {Y}_{n}=1\)), to be consistent with Eq. (32), we set \(\hat {\delta }_{n}=0\). This should be interpreted as \(\hat {\mu }_{n}=0\), in which case, according to (37), we would set \(\hat {\sigma }=M_{1}\) (so that the resulting FL distribution is exponential). We propose the same interpretation when \(M_{2} > 2{M_{1}^{2}}\). With these practical conventions, the MMEs of

*μ*and

*σ*always exist and are unique.

###
**Remark**
**3.1**.

In view of (30), the MME of *μ* is \(\hat {\mu }_{n}= \hat {\delta }_{n}\hat {\sigma }_{n}\).

###
**Remark**
**3.2**.

*δ*is given, then the parameter

*σ*is uniquely estimated as

Alternatively, with a known *σ*, the MME of *δ* is the unique value for which *v*(*δ*)=*δ*+ exp(−*δ*)=*M*
_{1}/*σ*, provided that *M*
_{1}/*σ*≥1. This is easily deduced from the properties of the function *v*, which is continuous and monotonically increasing on the interval [0,*∞*), with *v*(0)=1 and *v*(*δ*)→*∞* as *δ*→*∞*. The condition \(M_{1}/\sigma = \bar {Y}_{n}/\sigma \geq 1\) is expected to be true when *n* is large, since by law of large numbers \(\bar {Y}_{n}\) converges to the mean \(\mathbb E Y\) given by the right-hand-side of the first equation in (31), which is greater than or equal to *σ*. In the boundary case *M*
_{1}/*σ*=1 we have \(\hat {\delta }_{n}=0\) (so that also \(\hat {\mu }_{n}=0\)), indicating an exponential distribution. One can follow the same interpretation when *M*
_{1}/*σ*≤1.

###
**Remark**
**3.3**.

To find the MMEs of *δ* and *σ* in practice one can use standard Newton-Raphson algorithm or utilize a statistical package (such as R) to compute the unique zero of the (well-behaved) function \(h(\delta) - M_{2}/{M_{1}^{2}}\), *δ*∈[0,*∞*).

Standard large sample theory results (see, e.g., Rao 1973) show that the estimators (36) are consistent and asymptotically normal.

###
**Proposition**
**3.2**.

The vector \((\hat {\delta }_{n}, \hat {\sigma }_{n})\) of MMEs given in Proposition 3.1 is

(i) consistent;

###
**Remark**
**3.4**.

*μ*and

*σ*, where \(\hat {\mu }_{n}= \hat {\delta }_{n}\hat {\sigma }_{n}\), are also consistent and asymptotically normal, with \(\sqrt {n}[ (\hat {\mu }_{n}, \hat {\sigma }_{n}) - (\mu,\sigma)]\) converging in distribution to a bivariate normal distribution with the (vector) mean zero and the covariance matrix

The proof of the above is similar to that of Proposition 3.2, and shall be omitted.

## Illustrative data examples

In this section we present an application of the folded Laplace distribution in modeling *West Texas Intermediate* (WTI) and *Brent Oil* historical oil prices. The WTI data are collected from January 3, 1986 to February 15, 2003, with the total of 1416 data points with the WTI Spot Price FOB (dollars per barrel). The data source is US Department of Energy via wikiposit.org (http://wikiposit.org/uid?DOE.RWTC). The Brent Oil prices, taken from the Invest Excel (http://investexcel.net/), cover the period from January 1, 2009 to January 1, 2012, with the total of 778 data points (dollars per barrel).

*Y*

_{ k }=

*S*

_{ k }/

*S*

_{ k−1}, where

*S*

_{ k }represents the oil price on day

*k*. Clearly, the values are positive, with

*n*

_{1}=1415 and

*n*

_{2}=777 daily returns derived from WTI and Brent daily oil prices, respectively. Our goal is to model the oil price returns using the

*F*

*L*(

*μ*,

*σ*) distribution. We apply the method of moments discussed in Section 3 to estimate the parameters

*μ*and

*σ*of the FL model. The results of the estimation are summarized in Table 1, containing the MMEs along with their standard errors. The latter are computed from the asymptotic distribution given by (40–41).

Estimation of the WTI and the Brent oil data. The standard errors (SE) that appear next to the estimates are approximated from the asymptotic distributions of the estimators

Data | Sample | The 1st | The 2nd | ||
---|---|---|---|---|---|

Set | Size n | Moment | Moment | \(\hat {\mu }\) (SE) | \(\hat {\sigma }\) (SE) |

WTI oil prices | 1415 | 1.002 | 1.001 | 1.002 (0.001) | 0.038 (0.001) |

Brent oil prices | 777 | 1.001 | 1.003 | 1.001 (0.001) | 0.015 (0.001) |

## Appendix

Here we collect selected proofs of the results presented above, which are preceded by a technical lemma.

### Lemma 5.1.

*c*≥0 let

###
*Proof*.

*u*= exp(

*x*), the integral on the right-hand-side above becomes

###
*Proof of Proposition 2.4*.

*C*

_{ Y }<1 is equivalent to \(\sqrt {\mathbb Var(Y)}< \mathbb E(Y)\). When we substitute the expressions for the mean and the variance, then, after some algebra, we end up with the inequality

*x*=

*μ*/

*σ*∈(0,

*∞*). Further algebra shows that the above inequality can be formulated as

Note that the function *v* defined in (43) is continuous and *v*
^{′}(*x*)=2(1−*e*
^{−x
})^{2}>0 for *x*>0, showing that *v* is increasing on the interval (0,*∞*). Since *v*(0)=0, the result follows.

###
*Proof of Proposition 2.5*.

*σ*>0 and exp(−

*μ*/

*σ*)>0, we have

*h*(

*t*)=

*t*+

*e*

^{−t }− log(

*e*

^{ t }+

*e*

^{−t }). Observe that

*h*is decreasing on (0,

*∞*). Thus, our inequality will follow if we can show that

However, the function *h* can be written as *h*(*t*)=*e*
^{−t
}− log[1+ exp(−2*t*)], which clearly converges to zero at infinity. Thus, the relation (44) holds and the result follows.

###
*Proof of Proposition 2.8*.

*F*

*L*(

*μ*,

*σ*) random variable

*X*with PDF

*g*given in (4), this results in

*G*is the

*F*

*L*(

*μ*,

*σ*) CDF, so that

*G*(

*μ*)=[1− exp(−2

*μ*/

*σ*)]/2, while

When we compute *B* using Lemma 5.1, and substitute the resulting expression, along with the *A* and *G*(*μ*) given above, into (45), we obtain the result after straightforward algebra. This completes the proof.

###
*Proof of Proposition 3.1*.

*h*defined in (33) is monotonically decreasing on the interval (0,

*∞*) and satisfies the conditions given in (34). Indeed, it is obvious that in this case the MME of

*δ*is as specified in (36), which in turn leads to the MME of

*σ*, obtained by solving either one of the Eq. (31) with

*δ*replaced by \(\hat {\delta }_{n}\). Since the values of

*h*at the boundary of its domain are obtained easily, we consider the issue of the monotonicity. Straightforward calculations show that the derivative of

*h*is

*δ*∈(0,

*∞*) if and only if we have

*v*above is continuous on [0,

*∞*) and differentiable on (0,

*∞*) with

*v*(0)=0, it is enough to prove that the derivative

*v*

^{′}(

*δ*) is strictly positive for each

*δ*∈(0,

*∞*). It is easy to see that latter condition is equivalent to

###
*Proof of Proposition 3.2*.

*r*(·) is the inverse of

*h*. The quantity \( \overline {X}_{n}\) in (47) is the sample mean of \(X_{i}={Y_{i}^{2}}\),

*i*=1,…,

*n*. To prove consistency, apply the law of large numbers to the sequence \(Z_{i}=(Y_{i}, {Y_{i}^{2}})'\) and conclude that the sample mean \(\overline {Z}_{n}\) converges in distribution to the population mean

*H*(

*m*

_{ Z })=(

*δ*,

*σ*). Next, by the classical multivariate central limit theorem, we have the convergence in distribution \(\sqrt {n} (\overline {Z}_{n} - m_{Z}) \stackrel {d}{\rightarrow } \mathrm {N}(0,\Sigma)\), where the right-hand-side denotes the bivariate normal distribution with mean vector zero and covariance matrix

*n*→

*∞*, the variables

*Ω*=

*D*

*Σ*

*D*

^{′}, where

*H*. A rather lengthy calculation yields

## Declarations

### Acknowledgments

The authors thank the associate editor and two anonymous referees for their helpful remarks and information about reference (Cooray 2008), of which we were unaware. This paper was written while the second author was on a sabbatical leave, and the assistance provided by the University of Nevada, Reno is greatly acknowledged. Kozubowski’s research was also partially funded by the European Union’s Seventh Framework Programme for research, technological development and demonstration under grant agreement no 318984 - RARE.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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