asymmetric type ii compound laplace distributions and its...

CHAPTER 4

Asymmetric Type II Compound Laplace

Distributions and its Properties

4.1 Introduction

Recently there is a growing trend in the literature on parametric families of asym-

metric distributions which are deviated from symmetry as well as from the classical

normality assumptions. Various researchers have developed different methods

to construct asymmetric distributions with heavy tails. In Chapter 2 we have in-

troduced skew slash distributions generated by Cauchy kernal, skew slash t and

asymmetric slash Laplace distributions for modelling microarray data. The form of

the density functions of this family is not convenient. This motivated us to introduce

a convenient distribution which can account asymmetry, peakedness and heavier

tails. In the present chapter we introduce asymmetric type II compound Laplace

Results included in this chapter form the paper Bindu et al. (2012a).

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CHAPTER 4. ASYMMETRIC TYPE II COMPOUND LAPLACE DISTRIBUTIONS AND ITS PROPERTIES

(ACL) density which is the asymmetric version of the type II compound Laplace

distribution and is a generalization of asymmetric Laplace distribution (AL). This

four-parameter probability distribution provides an additional degree of freedom to

capture the characteristic features of the microarray data. We derive the pdf , cdf ,

qf and study various properties of ACL.

4.2 Symmetric Type II Compound Laplace Distribution

The (symmetric) type II compound Laplace distribution (CL) is introduced by Kotz

et al. (2001) which results from compounding a Laplace distribution with a gamma

distribution. Let X follow a classical Laplace distribution given s with density given

by

f(x|s) =s

2e−s|x−θ|, x ∈ R, (4.2.1)

and let s follow a Gamma(α, β) distribution with density

f(s;α, β) =sα−1e−s/β

βαΓ(α), α > 0, β > 0, s > 0. (4.2.2)

Then the unconditional distribution of X is the type II compound Laplace distri-

bution with parameters (θ, α, β), denoted by X ∼ CL(θ, α, β) and the density func-

tion

f(x) =1

2αβ[1 + |x− θ|β]−(α+1), α > 0, β > 0, θ ∈ R, x ∈ R, (4.2.3)

The CL can be represented as a mixture of Laplace distribution as, Y d= θ+σX,

where X has the standard classical Laplace distribution. The mixture on σ = 1/s

of the distribution of X is the type II compound Laplace distribution with parameter

θ, α and β if 1/s or σ has the Gamma(α, β) distribution.

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4.2.1 Distribution, Survival and Quantile Functions

The cumulative distribution function (cdf) of the type II compound Laplace distribu-

tion is given by

F (x) =

1− 1

2[1 + β(x− θ)]−α, for x > θ,

12[1− β(x− θ)]−α, for x ≤ θ.

(4.2.4)

The survival function (sf) of the type II compound Laplace distribution is given

by

S(x) =

12[1 + β(x− θ)]−α, for x > θ,

1− 12[1− β(x− θ)]−α, for x ≤ θ.

(4.2.5)

The qth quantile function (qf ) of CL distribution is,

ξq =

θ + 1

β

[1− 1

[2q]1/α

], for q ∈

(0, 1

2

],

θ + 1β

[1

(2(1−q))1/α − 1], for q ∈

(12, 1).

(4.2.6)

Remark 4.2.1. If X ∼ CL(θ, α, β) then for α→∞, β → 0 such that αβ = s, a constant,

the density f(x) in Eq.(4.2.3) converges to the classical Laplace density, s2e−s|x−θ|, s > 0,

θ ∈ R.

Remark 4.2.2. If X ∼ CL(θ, α, β) then for α = 1 the density f(x) in Eq.(4.2.3) reduce to

the double Lomax density, which is the ratio of two independent standard Laplace densities.

The double Lomax distribution (Bindu (2011c) and Bindu et al. (2013e)) is given by

f(x) =1

2[1 + β|x− θ|]−2, θ ∈ R, x ∈ R. (4.2.7)

Remark 4.2.3. If X ∼ CL(θ, α, β) then the rth moment of X around θ, E(X − θ)r is

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Figure 4.1: Type II compound Laplace density functions (θ = 0, κ = 1) for vari-ous values of α and β.

given as follows.

mr = E(x− θ)r =

αβrB(r + 1, α− r), for r even, 0 < r < α,

0, for r odd, 0 < r < α.

(4.2.8)

Put r = 1 in Eq.(4.2.8), we get E(X − θ) = 0. Hence, the mean E(X) = θ, for

α > 1. Therefore E(X − θ)r is the rth central moment µr of the CL distribution.

The expression for variance is given by,

V (X) =2

β2(α− 1)(α− 2), for α > 2.

Hence, the type II compound Laplace distribution has finite mean if α > 1 and has finite

variance if α > 2.

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4.3 Asymmetric Type II Compound Laplace Distribution

Here we introduce asymmetry into the symmetric type II compound Laplace distri-

bution using the method of Fernandez and Steel (1998). The idea is to postulate

inverse scale factors in the positive and negative orthants of the symmetric distri-

bution to convert it into an asymmetric distribution. Thus a symmetric density f

generates the following class of skewed distribution, indexed by κ > 0.

If g(·) is symmetric on <, then for any κ > 0, a skewed density can be obtained

as

f(x) =2κ

1 + κ2

g(xκ), for x > 0,

g(xκ), for x ≤ 0.

In the above expression, when g is the symmetric type II compound Laplace distri-

bution with density Eq.(4.2.3), we get a skewed distribution with the density function

defined as follows.

Definition 4.3.1. A random variable X is said to have an asymmetric type II compound

Laplace distribution (ACL) with parameters (θ, α, β, k), denoted by X ∼ ACL(θ, α, β, κ)

if its probability density function is given by

f(x) =κ α β

1 + κ2

(1 + κβ(x− θ))−(α+1), for x > θ,

(1− βκ(x− θ))−(α+1), for x ≤ θ,

(4.3.1)

and θ ∈ R, α, β, κ > 0.

The parameters (θ, α, β, κ) are the location, shape, scale, and skewness pa-

rameters, respectively.

4.3.1 Distribution, Survival and Quantile Functions

The cdf of the ACL distribution is given by

F (x) =

1− 1

1+κ2[1 + κβ(x− θ)]−α, for x > θ,

κ2

1+κ2[1− β

κ(x− θ)]−α, for x ≤ θ.

(4.3.2)

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When κ = 1 we get the symmetric type II compound Laplace distribution.

The survival function (sf) of the ACL distribution is given by

S(x) =

1

1+κ2[1 + κβ(x− θ)]−α, for x > θ,

1− κ2

1+κ2[1− β


(4.3.3)

The qth quantile function (qf ) of ACL distribution is,

ξq =

θ + κ

β

[1− 1[

q 1+κ2

κ2

]1/α], for q ∈

(0, κ2

1+κ2

],

θ + 1κβ

[1

((1−q)(1+κ2))1/α− 1], for q ∈

(κ2

1+κ2, 1).

(4.3.4)

The cdf and qf can be useful for goodness-of-fit and simulation purposes. For

q = κ2/(1 + κ2), the qth quantile is given by ξq = θ. Hence, for given κ the location

parameter is given by θ = ξ[κ2/(1+κ2)].

Remark 4.3.1. If X ∼ ACL(θ, α, β, κ) then for α → ∞, β → 0 such that αβ = s, a

constant, the density f(x) in Eq.(4.3.1) converges to the AL density of Kotz et al. (2001)

denoted by AL∗(θ, κ,√

2/s).

4.3.2 Properties

Fig. 4.1 shows density plots of symmetric (for various values of α, β) and Fig. 4.2

shows the asymmetric (for various values of κ) type II compound Laplace distribu-

tions. For asymmetric type II compound Laplace distribution both tails are power

tails and the rate of convergence depends on the values of κ. When κ < 1 the

curve moves to the right of the symmetric curve and the left tail moves towards θ

giving heavier right tail, and vice versa when κ > 1. Below we list a few impor-

tant properties of type II compound Laplace distributions. For properties similar to

asymmetric Laplace we refer to Kotz et al. (2001).

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Figure 4.2: Asymmetric type II compound Laplace density functions for variousvalues of κ and for fixed (θ = 0, α = 1.5 , β = 2).

(i) If X ∼ ACL(θ, α, β, κ), then Y = aX + b ∼ ACL(b + aθ, α, β/a, κ) where a ∈ R,

b ∈ R and a 6= 0. Hence, the distribution of a linear combination of a random

variable with ACL(θ, α, β, κ) distribution is also ACL. If X ∼ ACL(θ, α, β, κ),

then Y = (X − θ)/β ∼ ACL(0, α, 1, κ), which can be called as the standard

ACL distribution.

(ii) The mode of the distribution is θ and the value of the density function at θ is

(κ/(1 + κ2))αβ.

(iii) The value of the distribution function at θ is κ2/(1 + κ2) and hence, θ is also

the κ2/(1 + κ2)-quantile of the distribution.

(iv) The rth moment of X around θ, E(X − θ)r, exists for 0 < r < α and is given as

follows.

mr = E(X − θ)r =1 + (−1)rκ2(r+1)

κr(1 + κ2)

α

βrB(r + 1, α− r), (4.3.5)

where B(a, b) is a beta function. It is clear that the moments of order α or

greater do not exist. For the symmetric distribution (κ = 1), all odd moments

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around θ are zero and even moments are given by (α/βr) B(r + 1, α− r).

(v) The type II compound Laplace distributions have heavier tails than classical

Laplace distributions. Note that the tail probability of the type II compound

Laplace density is F ∼ cx−α, as x → ± ∞. The heavy tail characteristic

makes this densities appropriate for modeling network delays, signals and

noise, financial risk or microarray gene expression or interference which are

impulsive in nature.

(vi) The type II compound Laplace distributions are completely monotonic on

(θ,∞) and absolutely monotonic on (−∞, θ). As noted by Dreier(1999), every

symmetric density on (−∞,∞), which is completely monotonic on (0,∞), is a

scale mixture of Laplace distributions.

4.3.3 Stochastic Representation ACL

Here we give two stochastic representations for the ACL distribution based on the

two stochastic representation of the AL distribution. Let X has the ACL(θ, α, β, κ)

and Y has AL(0, 1, κ)

Xd= θ + σY, (4.3.6)

where 1/σ has the Gamma(α, β) distribution (Eq. (4.2.2)) or σ has the Inverse

Gamma distribution with parameters α and β. Then ACL(θ, α, β, κ) can be repre-

sented as normal mixture as follows,

Xd= θ + µW + σ

√WZ, (4.3.7)

where µ = σ(

1κ− κ)/√

2, W is the standard exponential variate, Z follows N(0, 1) in-

dependent of W and σ has the Inverse Gamma(α, β) distribution,. Equation (4.3.7)

says that X can be viewed a continuous mixture of normal random variables whose

scale and mean parameters are dependent and vary according to an exponen-

tial distribution. Then X|W ∼ N(θ + µW, σ2W ), where W is exp(1) and σ has the

Inverse Gamma(α, β) distribution.

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Another representation as the log-ratio of two independent random-variables

with Pareto I distributions is given below.

Xd= θ +

σ√2log

(P1

P2

), (4.3.8)

where σ has the Inverse Gamma(α, β) distribution, P1 ∼ Pareto I(κ, 1) and

P2 ∼ pareto(1/κ, 1).

4.4 Estimation of ACL

In this section we study the problem of estimating four unknown parameters, Θ =

(θ, α, β, κ), of ACL distribution. To estimate the parameter θ we use the quantile

estimation. The quantile estimate of θ is given by θ = ξ[κ2/(1+κ2)]. Given κ, the

quantile estimate of θ is the sample quantile of order κ2/(1 + κ2), which is (for large

n) the ([[nκ2/(1 + κ2)]] + 1)th ordered observation, ([[c]] denoted the integral part of

c). When the data are approximately symmetric the estimate of θ will be close to

median. The method of moments or maximum likelihood estimation method can be

employed to estimate Θ as described below. Let X = (X1, · · · , Xn) be independent

and identically distributed samples from an asymmetric type II compound Laplace

distribution with parameters Θ.

4.4.1 Method of Moments

To estimate Θ under the method of moments, four first moments, E(Xr), r =

1, 2, 3, 4, are equated to the corresponding sample moments and the resulted sys-

tem of equations are solved for the unknown parameters. These moments can be

obtained from Eq.(4.3.5) but they exist only when α > 4. Hence, the method is not

applicable to the entire parametric space. An alternative method is a maximum

likelihood estimation where the likelihood function is maximized to estimate the

unknown parameters. We describe this alternative method briefly in the following

subsection.

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4.4.2 Maximum Likelihood Estimation

The log-likelihood function of the data X takes the form

logL(Θ;X) = n log κ− n log(1 + κ2) + n logα + n log β − (α + 1)S(θ, β, κ).

Where

S(θ, β, κ) =n∑i=1

Si(θ, β, κ) =n∑i=1

log

[1 + (κβ)(xi − θ)+ +

β

κ(xi − θ)−

],

and (x − θ)+ = (x − θ), if x > θ, and = 0 otherwise, and (x − θ)− = (θ − x), if x ≤ θ,

and = 0 otherwise.

Existence, uniqueness and asymptotic normality of maximum likelihood esti-

mators (MLEs) can be derived on the same lines as described in detail for an AL

distribution in Kotz et al.(2001).

The MLEs of (α, β) for given θ = θ and κ = κ are obtained by solving the

score equations for α and β. This leads to the following equations which are solved

iteratively.

α =n

S(θ, β, κ)

n

β= (α + 1)

n∑i=1

κ(xi − θ)+ + 1κ(xi − θ)−

1 + κβ(xi − θ)+ + βκ(xi − θ)−

.

In our illustrations, the maximization of the likelihood is implemented using the

optim function of the R statistical software, applying the BFGS algorithm (R De-

velopment Core Team (2006)). Estimates of the standard errors were obtained

by inverting the numerically differentiated information matrix at the maximum likeli-

hood estimates. We discuss the performance of our numerical maximization algo-

rithm (programmed in R) using the simulated data sets in the chapter 5. Now we

discuss the stress-strength reliability Pr(X > Y ), when X and Y are two indepen-

93


dent but non-identically distributed random variables belonging to the asymmetric,

heavy-tailed and peaked distribution, ACL.

4.5 Stress-strength Reliability of ACL

In the context of reliability, the stress-strength model describes the life of a compo-

nent which has a random strength X and is subjected to a random stress Y . The

component fails at the instant that the stress applied to it exceeds the strength, and

the component will function satisfactorily wheneverX > Y . Thus, R = Pr(X > Y ) is

a measure of component reliability. The parameter R is referred to as the reliability

parameter. This type of functional can be of practical importance in many applica-

tions. For instance, if X is the response for a control group, and Y refers to a treat-

ment group, Pr(X < Y ) is a measure of the effect of the treatment. R = Pr(X > Y )

can also be useful when estimating heritability of a genetic trait. Bamber (1975)

gives a geometrical interpretation of A(X, Y ) = Pr(X < Y ) + 12Pr(X = Y ) and

demonstrates that A(X, Y ) is a useful measure of the size of the difference be-

tween two populations.

Weerahandi and Johnson (1992) proposed inferential procedures for Pr(X >

Y ) assuming that X and Y are independent normal random variables. Gupta

and Brown (2001) illustrated the application of skew normal distribution to stress-

strength model. Bindu (2011c) introduced the double Lomax distribution, which

is the ratio of two independent and identically distributed Laplace distributions and

presented its application to the IQ score data set from Roberts (1988). The Roberts

IQ data gives the Otis IQ scores for 87 white males and 52 non-white males hired

by a large insurance company in 1971. Where X represent the IQ scores for whites

and Y represent the IQ scores for non-whites and estimated the probability that the

IQ score for a white employee is greater than the IQ score for a non-white em-

ployee.

The functional R = Pr(X > Y ) or λ = Pr(X > Y ) − Pr(X < Y ) is of practi-

cal importance in many situations, including clinical trials, genetics, and reliability.

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We are interested in applying R = Pr(X > Y ) as a measure of the difference

between two populations, in particular where X and Y refer to the log intensity

measurements of red dye (test sample) and the log intensity measurements for

the green dye (control) in cDNA microarray gene expression data. In microarray

gene expression studies the investigators are interested in “is there any significant

difference in expression values for genes”, “what is the estimate of the number

of genes which are differentially expressed”, “what proportion of genes are really

differentially expressed” and so on.

Here we explored the applications of stress-strength analysis in microarray

gene expression studies using the ACL distribution. We used this concept for

checking array normalization in microarray gene expression data. First we derived

the stress-strength reliability R = Pr(X > Y ) for asymmetric type II compound

Laplace. Then we calculated R for microarray datasets for before and after nor-

malization by taking X as the log intensity measurements of red dye (test sample)

and let Y represent the log intensity measurements of green dye (control sample).

We developed R program and Maple program for the computation of Pr(X > Y ).

We also used the stress strength reliability Pr(X > Y ) for comparing test and con-

trol intensity measurements for each gene in replicated microarray experiments

and computed the proportion of differentially expressed genes.

4.5.1 Pr(X > Y ) for the Asymmetric Type II Compound Laplace Distribution

Let X and Y are two continuous and independent random variables. Let f2 denote

the pdf of Y and F1 denote the cdf X. Then Pr(X > Y ) can be given as,

Pr(X > Y ) =

∫ ∞−∞

F2(z)f1(z)dz. (4.5.1)

Now we evaluate the Pr(X > Y ) for two independent ACL distributions. Let

X and Y are continuous and independent variables having ACL distribution with

parameters θi , αi , βi and κi, i = 1, 2 respectively. The pdf and cdf of ACL is given

95


by

f(x) =κ α β

1 + κ2

(1 + κβ(x− θ))−(α+1), for x > θ

(1− βκ(x− θ))−(α+1), for x ≤ θ,

(4.5.2)

and θ ∈ R, α, β, κ > 0.

F (x) =

1− 1

1+κ2[1 + κβ(x− θ)]−α, for x > θ

κ2

1+κ2[1− β


(4.5.3)

From equation (4.3.1) we get the reliability R for the ACL distribution as follows.

For θ1 < θ2, R can be expressed as

R[θ1<θ2] =κ1κ

22α1β1

(1 + κ21)(1 + κ2

2)

{∫ θ1

−∞

[1− β1

κ1

x

]−(α1+1) [1− β2

κ2

y

]−α2

dz +

∫ θ2

θ1

[1 + β1κ1x]−(α1+1)

[1− β2

κ2

y

]−α2

dz

}+

κ1α1β1

(1 + κ21)

∫ ∞θ2

[1 + β1κ1x]−(α1+1)

[1− 1

(1 + κ22)

(1 + β2κ2y)−α2

]dz,

and if θ1 > θ2

R[θ1>θ2] =κ1κ

22α1β1

(1 + κ21)(1 + κ2

2)

∫ θ2

−∞

[1− β1

κ1

x

]−(α1+1) [1− β2

κ2

y

]−α2

dz +

κ1α1β1

(1 + κ21)

{∫ θ1

θ2

[1− β1

κ1

x

]−(α1+1) [1− 1

(1 + κ22)

(1 + β2κ2y)

]−α2

dz +∫ ∞θ1

[1 + β1κ1x]−(α1+1)

[1− 1

(1 + κ22)

(1 + β2κ2y)−α2

]dz

},

where x = (z − θ1) and y = (z − θ2).

Thus, the reliability parameter R can be expressed as

R = R[θ1<θ2]I[θ1<θ2] +R[θ1>θ2]I[θ1>θ2] (4.5.4)

96


where I(.) is the indicator function. The MLE of the R = P (X > Y ) can be

obtained by replacing the parameters θ1, θ2, α1 , α2, β1, β2, κ1 and κ2 in the expres-

sion of R by their MLE’s. Using Maple program we can evaluate the integrals and

compute the maximum likelihood estimator of R.

4.6 Conclusion

In this Chapter we have introduced ACL distribution, which is the heavy tailed

generalization of the AL distribution. We studied various properties of the ACL

and also derived the stress-strength reliability Pr(X > Y ) for ACL.

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