generalized laplacian distributions and autoregressive...

CHAPTER 2

Generalized Laplacian Distributions and

Autoregressive Processes

2.1 Introduction

Generalized Laplacian distribution corresponds to the distribution of differences of inde-

pendently and identically distributed (i.i.d.) gamma random variables. Mathai (1993a,b,c),

Mathai et al. (2006) discussed various generalizations of Laplace distribution and their ap-

plications in different contexts such as input-output processes, growth-decay mechanism,

formation of sand dunes, growth of melatonin in human body, formation of solar neutrinos

etc. Another variant namely, the skew Laplace distribution was introduced and studied in

detail by Kozubowski and Podgorski (2008a,b). Skew Laplace distribution arises as a mix-

Some results included in this chapter form part of papers Jose and Manu (2009) and Jose and Manu(2011).

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CHAPTER 2. GENERALIZED LAPLACIAN DISTRIBUTIONS AND AUTOREGRESSIVE PROCESSES

ture of normal distributions with stochastic variance having gamma distribution, and hence

it is also called Variance Gamma Model. The tails of the variance gamma distribution de-

crease more slowly than the normal distribution. Due to the simplicity, flexibility and an

excellent fit to empirical data, the variance gamma model is recently very popular among

financial modelers.

In the present chapter, we consider the generalized Laplacian distribution and a more

general case namely bilateral gamma distribution. We develop first order autoregressive

processes with generalized Laplacian marginals and the generation of the process. The

regression behaviour and simulation of sample paths are discussed. We consider the

distribution of sums and the covariance structure. We introduce the geometric general-

ized Laplacian distribution and study its properties. Geometric generalized Laplacian pro-

cesses are discussed. The kth order autoregressive processes are described. Bilateral

gamma distribution and its estimation of parameters and a numerical illustration is also

carried out.

2.2 Generalized Laplacian distribution and its properties

Mathai (1993a,b,c) introduced a generalized Laplacian distribution with parameters α and

β, which has the characteristic function (c.f.),

φ(t) =1

(1 + β2t2)α, α > 0, β > 0.

Kotz et al. (2001) give a more general form of the generalized Laplacian distribution with

more parameters and we concentrate on this form. The c.f is as follows

φ(t) =eiθt

(1 + 12σ2t2 − iµt)τ

, −∞ < t <∞, σ > 0, −∞ < µ <∞.

26


This c.f. can be factored as

φ(t) = eiθt

(1

1 + i σ√2κt

)τ (1

1− i σ√2κt

)τ

, (2.2.1)

where κ > 0, µ = σ√2

(1κ− κ).

The probability density function (p.d.f.) of generalized Laplace distribution (GL(θ, κ, σ, τ ))

is

h(x) =

√2e√

22σ

( 1κ−κ)(x−θ)

√πστ+ 1

2 Γ(τ)

(√2|x− θ|κ+ 1

κ

)τ− 12

Kτ− 12

(√2

2σ

(κ+

1

κ

)|x− θ|

)for x 6= θ,

(2.2.2)

where Kλ is the modified Bessel function of the third kind with the index λ, see Kotz et

al. (2001). There are some special cases associated with this distribution. A standard GL

density is obtained for θ = 0 and σ = 1. For τ = 1, we have an asymmetric Laplace, and

for κ = 1 and θ = 0, we obtain a symmetric Laplace distribution. The p.d.f. (2.2.2) can

also be written as

f(x) = (αβ)τ exp

(−(α− β)x

2

)(|x|

α + β

)τ−1/2

Kτ−1/2

(α + β

2|x|),

where α and β describe the left and right-tail behavior and τ is a parameter relating to the

peakedness of the pdf (for details see Wu (2008)).

The mean value, variance, coefficients of skewness and kurtosis of generalized Lapla-

cian distribution are obtained as

Mean value = τµ, Variance = τ(µ2 + σ2).

Coefficient of skewness =µ(2µ2 + 3σ2)√τ(µ2 + σ2)3/2

.

Coefficient of kurtosis = 3 +3(2µ4 + 4σ2µ2 + 3σ4)

τ(µ2 + σ2)2.

27


Figure 2.1: The histogram and frequency curve corresponding to GL(0, 2, 1, 12)

From the factorization of the c.f. of the generalized Laplacian distribution, it can be noted

that a variable Y following GL(θ, κ, σ, τ) can be represented as Y d= θ+ σ√

2( 1κG1− κG2),

where G1 and G2 are i.i.d. gamma random variables following one parameter gamma

distribution with p.d.f.,

fGi(x) =1

Γ(τ)e−xxτ−1; x > 0, τ > 0, i = 1, 2.

The generalized Laplacian random variable Y also admits the representation of the form

Yd= θ+ µW + σ

√WZ, where Z is a standard normal variate and W is a gamma variate

with parameters (τ ,1). Figure 2.1 gives the histogram and frequency curve corresponding

to GL(0, 2, 1, 12) obtained as the result of a simulation study.

2.3 Generalized Laplacian autoregressive processes

Consider the usual linear, additive first order autoregressive model given by

Xn = aXn−1 + εn, 0 < a < 1, n = 0,±1,±2, . . . , (2.3.1)

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where {εn} is the innovation sequence of i.i.d. random variables. The basic problem is

to find the distribution of {εn} such that {Xn} has the generalized Laplacian distribution

GL(θ, κ, σ, τ) as the stationary marginal distribution. The following theorem summarizes

the results in this context.

Theorem 2.3.1. A necessary and sufficient condition that an AR(1) process of the

form Xn = aXn−1 + εn is stationary with GL(θ, κ, σ, τ) marginals is that the innova-

tions {εn} are i.i.d. as a convolution of the form X + Y1 − Y2 as in (2.3.3) provided

ε0d= X0.

Proof. The necessary part can be proved as follows. In terms of the c.f., we can rewrite

(2.3.1) in the form φXn(t) = φXn−1(at)φεn(t).

Under stationarity this yields φε(t) = φX(t)φX(at)

.

Substituting the c.f. given by (2.2.1) we get,

φε(t) =eiθt

eiθat

(1 + i σ√2κat)τ

(1 + i σ√2κt)τ

(1− i σ√2κat)τ

(1− i σ√2κt)τ

= eiθ(1−a)t

[a+ (1− a)

1

(1 + i σ√2κt)

]τ [a+ (1− a)

1

(1− i σ√2κt)

]τ. (2.3.2)

This implies that ε has a convolution structure of the following form.

εd= X + Y1 − Y2, (2.3.3)

where X is a degenerate random variable taking value θ(1 − a) with probability one and

Y1 and Y2 are τ -fold convolutions of T1 and T2 where,

T1 =

0, with probability a

E1, with probability 1− a

T2 =

0, with probability a

E2, with probability 1− a

29


where E1 and E2 are exponential random variables with means σ√2κ

and σ√2κ respectively.

When τ is integer valued, Y1 and Y2 can be easily generated as gamma convolutions.

The sufficiency part can be proved by the method of induction. We assume that Xn−1

follows GL(θ, κ, σ, τ) with c.f. (2.2.1).

φXn(t) = φXn−1(at)φεn(t)

=

[eiθat

(1 + i σ√2κat)τ (1− i σ√

2κat)τ

][eiθ(1−a)t(1 + i σ√

2κat)τ (1− i σ√

2κat)τ

(1 + i σ√2κt)τ (1− i σ√

2κt)τ

]

= eiθt

(1

1 + i σ√2κt

)τ (1

1− i σ√2κt

)τ

,

which is the same as the generalized Laplacian c.f. This shows that the process {Xn} is

strictly stationary with generalized Laplacian marginals, provided ε0d= X0 which follows

GL(θ, κ, σ, τ ).

2.3.1 Generation of the process

Irrespective of whether τ is integer-valued or not, the process can be generated using a

compound Poisson representation. In (2.3.2), the expression[a+ (1− a) 1

(1−i σ√2κt)

]τcan

be obtained as the c.f. of a compound Poisson distribution, see Lawrance (1982). In

particular, this is the c.f. of the random sumN∑i=1

aUiVi, where {Vi} are gamma(1, σ√2κ

)

random variables, {Ui} are i.i.d. random variables uniformly distributed on (0,1), N is a

Poisson random variable of mean−τ ln a and all these random variables are independent.

This can be proved as follows.

Let η =N∑i=1

aUiVi, where {Vi} are gamma(1, σ√2κ

) random variables, {Ui} are i.i.d.

random variables uniformly distributed on (0,1), N is a Poisson random variable of mean

30


−τ ln a. Then the c.f. of η can be obtained as

φη(t) =∞∑n=0

[φV (aU t)

]nP [N = n]

=∞∑n=0

[φV (aU t)

]n eτ ln a(−τ ln a)n

n!

= exp(τ ln a

(1− φV (aU t)

))= exp

(τ ln a

(1−

∫ 1

0

1

1− iau σ√2κtdu

))

= exp

(τ ln a

(∫ 1

0

iau σ√2κt

1− iau σ√2κtdu

))

= exp

(τ

(ln

(1− ia σ√

2κt

)− ln

(1− i σ√

2κt

)))=

[a+ (1− a)

1

1− i σ√2κt

]τ.

Similarly we can generate the other gamma process also and taking the difference we can

obtain a generalized Laplacian process.

Remark 2.3.1. If X0 is distributed arbitrary, then also the process is asymptotically

stationary with generalized Laplacian marginal distribution.

Proof. We have

Xn = aXn−1 + εn = anX0 +n−1∑l=0

alεn−l.

φXn(t) = φX0(ant)n−1∏l=0

φε(alt)

= φX0(ant){exp(iθt(1− a)n−1∑l=0

al)}n−1∏l=0

[(1 + i σ√

2κal+1t)

τ(1− i σ√

2κal+1t)

τ

(1 + i σ√2κalt)τ (1− i σ√

2κalt)τ

]

→ eiθt

(1 + i σ√2κt)τ (1− i σ√

2κt)τ

as n→∞.

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2.3.2 Regression behaviour and sample path properties

The forward regression equation for the process is given by

E(Xn|Xn−1 = x) = ax+ (1− a)[θ + τµ].

Similarly the conditional variance is given by

Var(Xn|Xn−1 = x) = (1− a2)τ(µ2 + σ2) where µ =σ√2

(1

κ− κ).

In the backward direction, the conditional distribution of Xn given Xn+1 = x has non-

linear regression and non-constant conditional variance. In this backward case we will

proceed from the joint characteristic function ofXn andXn+1 following the steps described

in Lawrance (1978). Then we have,

φXn,Xn+1(t1, t2) =φX(t1 + at2)φX(t2)

φX(at2).

Differentiating this with respect to t1 and setting t1 = 0, t2 = t;

i E[eitXn+1E(Xn|Xn+1)] =φ′X(at)φX(t)

φX(at)

= φ′X(at)φε(t), (2.3.4)

where φε(t) is as defined in (2.3.2). Lawrance (1978) showed that linear backward regres-

sion is only possible in the Gaussian case and that non-normal autoregressive processes

are not time reversible as a result due to Weiss (1977). In this case, E(Xn|Xn−1) 6=E(Xn|Xn+1) so that the process is not time reversible.

2.3.3 Observed sample path properties

The sample path properties of the process are studied by generating 100 observations

each from the generalized Laplacian process with various parameter (θ, κ, σ, τ) combina-

32


(a) (b)

Figure 2.2: (a) Sample path of the process for a = 0.8 and (θ, κ, σ, τ) = (0, 1, 1, 2) (b)Sample path of the process for a = 0.3 and (θ, κ, σ, τ) = (0, 1, 1, 2)

(a) (b)

Figure 2.3: (a) Sample path of the process for a = 0.3 and (θ, κ, σ, τ) = (0, 1.3, 1, 2) (b)Sample path of the process for a = 0.3 and (θ, κ, σ, τ) = (0, 1.2, 1.9, 2)

tions. In Figure 2.2(a), we take a = 0.8 and the values of (θ, κ, σ, τ) as (0, 1, 1, 2) and

in Figures 2.2(b), 2.3(a) and 2.3(b) we take a = 0.3 and parameter values as (0, 1, 1,

2), (0, 1.3, 1, 2) and (0, 1.2, 1.9, 2) respectively. The process exhibits both positive and

negative values with upward as well as downward runs as evident from the figures. These

figures point out the rich variety of contexts where the newly developed time series mod-

els can be applied. These observations can be verified by referring to the table showing

P (Xn < Xn−1). These probabilities are obtained by a simulation procedure. Sequences

of 10,000 observations from generalized Laplacian process are generated repeatedly for

ten times, and from each sequence the probability is estimated. Table 2.1 provides the

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(θ, κ, σ, τ) (0,1,1, 2) (0,1.3,1,2) (0,1.2,1.9,2) (0,1.5,2.5,2)

@@

a0.1 0.4990 0.4970 0.4967 0.4933

(0.0022) (0.0035) (0.0032) (0.0032)0.2 0.5004 0.4904 0.4935 0.4881

(0.0026) (0.0027) (0.0029) (0.0043)0.3 0.5011 0.4853 0.4923 0.4783

(0.0027) (0.0038) (0.0032) (0.0017)0.4 0.4989 0.4822 0.4859 0.4743

(0.0034) (0.0033) (0.0024) (0.0043)0.5 0.5000 0.4765 0.4833 0.4661

(0.0041) (0.0031) (0.0039) (0.0030)0.6 0.5001 0.4687 0.4792 0.4584

(0.0027) (0.0032) (0.0023) (0.0025)0.7 0.4996 0.4611 0.4697 0.4446

(0.0027) (0.0032) (0.0032) (0.0034)0.8 0.4992 0.4428 0.4571 0.4196

(0.0031) (0.0027) (0.0026) (0.0041)0.9 0.5012 0.4190 0.4393 0.3908

(0.0036) (0.0043) (0.0034) (0.0028)

Table 2.1: P (Xn < Xn−1) for the generalized Laplacian process.

average of such probabilities from the ten trials along with an estimate of variance for

different values of a and for different combinations of (θ, κ, σ, τ ).

2.3.4 Distribution of sums and joint distribution of (Xn, Xn+1)

When a stationary sequence Xn is used, the distribution of sums Tr = Xn +Xn+1 + · · ·+Xn+r−1 is important. We have

Xn+j = ajXn + aj−1εn+1 + aj−2εn+2 + · · ·+ εn+j.

34


Hence

Tr = Xn +Xn+1 + · · ·+Xn+r−1

=r−1∑j=0

[ajXn + aj−1εn+1 + · · ·+ εn+j]

= Xn

(1− ar

1− a

)+

r−1∑j=1

εn+j

(1− ar−j

1− a

).

The c.f. of Tr is given by

φTr(t) = φXn

(t1− ar

1− a

) r−1∏j=1

φε

(t1− ar−j

1− a

).

φTr(t) =eiθ(t

1−ar1−a )[

1 + i σ√2κ(t1−ar

1−a

)]τ[1− i σ√

2κ

(t1−ar

1−a

)]τ×

r−1∏j=1

eiθt(1−ar−j)

a+ (1− a)1(

1 + i σ√2κt(

1−ar−j1−a

))τ

×

a+ (1− a)1(

1− i σ√2κt(

1−ar−j1−a

))τ .

The distribution of Tr can be obtained by inverting the c.f. as the c.f. uniquely determines

the distribution of a random variable. Now the joint distribution of (Xn, Xn+1) can be given

35


in terms of c.f. as

φXn,Xn+1(t1, t2) = E[e(it1Xn+it2Xn+1)

]= E

[e(it1Xn+it2(aXn+εn+1))

]= E

[e(i(t1+at2)Xn+it2εn+1)

]= φXn(t1 + at2)φεn+1(t2)

=eiθ(t1+t2)

[a+ (1− a) 1

(1+i σ√2κt2)

]τ [a+ (1− a) 1

(1−i σ√2κt2)

]τ[1 + i σ√

2κ(t1 + at2)]τ [1− i σ√

2κ(t1 + at2)]τ

.

The above c.f. is not symmetric in t1 and t2, demonstrating once again that the process is

not time reversible.

Remark 2.3.2. The covariance between Xn and Xn−1 is

Cov(Xn, Xn−1) = θ2 + (1− a)τ 2µ2 − θτµ.

The correlation coefficient ρ is

ρ =θ2 + (1− a)τ 2µ2 − θτµ

(1− a2)τ(µ2 + σ2).

When θ = 0,

ρ =τµ2

(1 + a)(µ2 + σ2).

2.4 Geometric generalized Laplacian distribution

Here we introduce a new distribution namely the geometric generalized Laplacian distribu-

tion (GGLD(θ, µ, σ, τ )). Its c.f. is given by

ψ(t) =1

1 + τ ln(1 + 12σ2t2 − iµt)− iθt

. (2.4.1)

36


The above c.f. ψ(t) can be written in the form φ(t) = exp(

1− 1ψ(t)

), where φ(t) is

the c.f. of the generalized Laplacian distribution, which is infinitely divisible. Therefore

the geometric generalized Laplacian distribution is geometrically infinitely divisible (see

Klebanov et al. (1984), Pillai and Sandhya (1990), Seethalekshmi and Jose (2004a,b;

2006)).

Theorem 2.4.1. Let X1, X2, . . . be i.i.d. GGLD(0, µ, σ, τ) random variables and N

be geometric with mean 1p, such that P [N = k] = p(1 − p)k−1, k = 1, 2, . . . , 0 < p < 1.

Then Y = X1 +X2 + · · ·+XNd= GGLD(0, µ, σ, τ/p).

Proof. The c.f. of Y is

φY (t) =∞∑k=1

[φX(t)]kp(1− p)k−1

=p/(1 + τ ln(1 + t2σ2

2− iµt))

1− ((1− p)/(1 + τ ln(1 + t2σ2

2− iµt))

=1

1 + (τ/p) ln(1 + t2σ2

2− iµt)

.

Hence Y d= GGLD(0, µ, σ, τ/p).

Theorem 2.4.2. Geometric generalized Laplacian distribution GGLD(0, µ, σ, τ) is the

limit distribution of geometric sum of random variables following GL(0, µ, σ, τ/n).

Proof. We have [1 + t2σ2

2− iµt]−τ/n = {1 + [1 + t2σ2

2− iµt]τ/n − 1}−1 is the c.f. of a

probability distribution since generalized Laplacian distribution is infinitely divisible. Hence

by Lemma 3.2 of Pillai (1990),

φn(t) = {1 + n[(1 +t2σ2

2− iµt)τ/n − 1]}−1

is the c.f. of geometric sum of i.i.d. generalized Laplacian random variables. Taking limit

37


as n→∞.

φ(t) = limn→∞

φn(t)

= {1 + limn→∞

n[(1 +t2σ2

2− iµt)τ/n − 1]}−1

= [1 + τ ln(1 +t2σ2

2− iµt)]−1.

2.4.1 Geometric generalized Laplacian processes

Here we develop a first order autoregressive process with geometric generalized Laplacian

marginals. Consider an autoregressive structure given by

Xn =

εn, with probability p

Xn−1 + εn, with probability 1− p(2.4.2)

where {Xn} and {εn} are independently distributed and 0 < p < 1. Now we shall con-

struct an AR(1) process with stationary marginal distribution as GGLD(0, µ, σ, τ ).

Theorem 2.4.3. Consider a stationary autoregressive process {Xn} with the structure

as in (2.4.2). Then a necessary and sufficient condition that {Xn} is stationary with

GGLD(0, µ, σ, τ) marginal distribution, is that {εn} follows GGLD(0, µ, σ, pτ).

Proof. In terms of c.f.s, we can write the above model as

φXn(t) = pφεn(t) + (1− p)φXn−1(t)φεn(t).

Assuming stationarity it reduces to the form

φX(t) = pφε(t) + (1− p)φX(t)φε(t).

38


Therefore φε(t) =φX(t)

p+ (1− p)φX(t).

Substituting, φX(t) =1

1 + τ ln(1 + 12σ2t2 − iµt)

.

We get φε(t) =1

1 + pτ ln(1 + 12σ2t2 − iµt)

. (2.4.3)

Hence εnd= GGLD(0, µ, σ, pτ).

Conversely assume that {εn} are identically distributed as given by (2.4.3), so that

φε(t) =1

1 + pτ ln(1 + 12σ2t2 − iµt)

.

Then φX(t) =pφε(t)

1− (1− p)φε(t)

=1

1 + τ ln(1 + t2σ2

2− iµt)

.

Hence it follows that Xn is distributed as GGLD(0, µ, σ, τ).

2.4.2 Extension to higher order processes

Consider a kth order autoregressive process having the following structure,

Xn =

εn, with probability p,

a1Xn−1 + εn, with probability p1,

a2Xn−2 + εn, with probability p2,...

......

akXn−k + εn, with probability pk,

,

where p1 + p2 + · · · + pk = 1 − p, 0 ≤ pi, p ≤ 1, i = 1, 2, . . . , k and {εn} is sequence of

independently and identically distributed random variables.

When a1 = a2 = · · · = ak = a; 0 < a < 1, we can write the above model in terms of

39


characteristic functions as

φXn(t) = pφεn(t) + p1φXn−1(at)φεn(t) + · · ·+ pkφXn−k(at)φεn(t).

If {Xn} is stationary, then

φX(t) = φε(t)[p+ (1− p)φX(at)].

That is

φε(t) =φX(t)

p+ (1− p)φX(at).

This shows that the innovation structure can be obtained as in the first order case.

2.5 Bilateral gamma distribution and bilateral gamma processes

Kuchler and Tappe (2008a,b) introduced and studied bilateral gamma distribution and

its various properties and also developed an associated Levy process namely bilateral

gamma processes which have applications in modelling financial market fluctuations. A

bilateral gamma distribution with parameters α1, λ1, α2, λ2 > 0 (BG(α1, λ1, α2, λ2)) is de-

fined as the distribution of the difference of two independent gamma variables with pa-

rameters (α1, λ1) and (α2, λ2). The characteristic function of a bilateral gamma distribution

is

ϕ(t) =

(λ1

λ1 − it

)α1(

λ2

λ2 + it

)α2

, t ∈ R. (2.5.1)

The p.d.f. is

f(x) =λα1

1 λα22

(λ1 + λ2)α2Γ(α1)Γ(α2)e−λ1x

∫ ∞0

vλ2−1

(x+

v

λ1 + λ2

)α1−1

e−vdv, x > 0.

40


The density function can also be expressed by means of Whittaker function

f(x) =λα1

1 λα22

(λ1 + λ2)12

(α1+α2)Γ(α2)x

12

(α1+α2)−1e−x2

(λ1+λ2)

×W 12

(α1−α2), 12

(α1+α2−1) (x (λ1 + λ2)) , x > 0, (2.5.2)

where

Wλ,µ(z) =zλe

z2

Γ(µ− λ+ 12)

∫ ∞0

tµ−λ−12 e−t

(1 +

t

z

)µ+λ− 12

dt for µ− λ > −1

2.

2.5.1 Geometric bilateral gamma distribution

Here, we introduce a new distribution namely the geometric bilateral gamma distribution

(GBG(α1, λ1, α2, λ2)). Its characteristic function is given by

ψ(t) =1

1 + α1 ln(1− itλ1

) + α2 ln(1 + itλ2

). (2.5.3)

The above characteristic function ψ(t) can be written in the form ϕ(t) = exp(

1− 1ψ(t)

),

where ϕ(t) is the characteristic function of the bilateral gamma distribution, which is in-

finitely divisible. Therefore the geometric bilateral gamma distribution is geometrically in-

finitely divisible , see Klebanov et al. (1984).

Theorem 2.5.1. Let X1, X2, . . . are i.i.d. GBG(α1, λ1, α2, λ2) random variables and

N be geometric with mean 1p, such that P [N = k] = p(1−p)k−1, k = 1, 2, . . . , 0 < p < 1.

Then Y = X1 +X2 + · · ·+XNd= GBG(α1/p, λ1, α2/p, λ2).

41


Proof. The characteristic function of Y is

ψY (t) =∞∑k=1

[ψX(t)]kp(1− p)k−1

=p/(1 + α1 ln(1− it

λ1) + α2 ln(1 + it

λ2))

1− ((1− p)/(1 + α1 ln(1− itλ1

) + α2 ln(1 + itλ2

)))

=1

1 + (α1/p) ln(1− itλ1

) + (α2/p) ln(1 + itλ2

).

Hence Y d= GBG(α1/p, λ1, α2/p, λ2).

2.5.2 Geometric bilateral gamma processes

Here, we develop a first-order autoregressive process with geometric bilateral gamma

marginals. Consider an autoregressive structure given by Consider an autoregressive

structure given by

Xn =

εn, with probability p

Xn−1 + εn, with probability 1− p(2.5.4)

where {Xn} and {εn} are independently distributed and 0 < p < 1.

Now we shall construct an AR(1) process with stationary marginal distribution as

GBG (α1, λ1, α2, λ2). The innovation structure can be derived by considering the charac-

teristic functions on both sides of (2.5.4), which will yield

ψXn(t) = pψεn(t) + (1− p)ψXn−1(t)ψεn(t).

Assuming stationarity it reduces to the form

ψX(t) = pψε(t) + (1− p)ψX(t)ψε(t).

42


Therefore ψε(t) =ψX(t)

p+ (1− p)φX(t).

On substitution, we get

ψε(t) =1

1 + pα1 ln(1− itλ1

) + pα2 ln(1 + itλ2

).

Hence εnd=GBG(pα1, λ1, pα2, λ2).

2.5.3 Estimation of parameters

Bilateral gamma distributions are absolutely continuous with respect to the Lebesgue mea-

sure, because they are the convolution of two gamma distributions. In order to perform a

maximum likelihood estimation, the density in terms of Whittaker function given in (2.5.2)

is used. We take a sample of n observations. The logarithm of the likelihood function for

Θ = (α1, α2, λ1, λ2) is given by

lnL(Θ) = n

(α1 lnλ1 + α2 lnλ2 −

α1 + α2

2ln(λ1 + λ2)− ln Γ(α2)

)+

(α1 + α2

2− 1

)( n∑i=1

ln |xi|

)− λ1 − λ2

2

(n∑i=1

xi

)

+n∑i=1

ln(W 1

2(α1−α2), 1

2(α1+α2−1) (|xi| (λ1 + λ2))

)(2.5.5)

The vector Θ0 obtained from the method of moments is taken as starting point for an

algorithm, for example the Hooke-Jeeves algorithm (see Quarteroni et al. (2002)), which

maximizes the logarithmic likelihood function (2.5.5) numerically. This gives a maximum

likelihood estimate Θ of the parameters. For more details see Kuchler and Tappe (2008b).

Now we consider the estimation of parameters. We use the conditional least square

approach to estimate the four parameters of bilateral gamma distribution. The conditional

least square estimators are strongly consistent under certain conditions, see Klimko and

Nelson (1978). Let x0, x1, ..., xk be a given set of observations of an AR(1) process {Xn}

43


with bilateral gamma marginals. Then the conditional estimators of the parameters can be

obtained by minimizing

E =k∑

n=1

[xn − E(Xn|Xn−1 = xn−1)]2 =k∑

n=1

[xn − axn−1 − (1− a)

(α1

λ1

− α2

λ2

)]2

.

We get the estimates of the parameters by solving the following equations recursively.

a =

∑kn=1

[xn −

(α1

λ1− α2

λ2

)] [xn−1 −

(α1

λ1− α2

λ2

)]∑k

n=1

[xn−1 −

(α1

λ1− α2

λ2

)]2

α1 =λ1

(1− a)

1

k

[k∑

n=1

xn − ak∑

n=1

xn−1

]+λ1

λ2

α2

α2 =λ2

(1− a)

1

k

[a

k∑n=1

xn−1 −k∑

n=1

xn

]+λ2

λ1

α1

λ1 =

{1

(1− a)α1

1

k

[k∑

n=1

xn − ak∑

n=1

xn−1

]+α2

α1

1

λ2

}−1

λ2 =

{1

(1− a)α2

1

k

[k∑

n=1

xn − ak∑

n=1

xn−1

]+α1

α2

1

λ1

}−1

.

These equations can be easily solved using standard software packages like Matlab,

Maple etc.

Tjetjep and Seneta (2006) discussed the generalized method of moments to estimate

the parameters in situations where the method of moments is not easily tractable. This

resembles the method of minimum chi-squared estimation but applied to moments. This

is based on the quantity:

M = minparameters

∑i

M (i),

where

M (i) =

(MP (i)−ME(i)

ME(i)

)2

,

44


(a) (b)

Figure 2.4: (a) The histogram of the log transformed data (b) The Q-Q plot for the dataand bilateral gamma random variables

where MP (i) is the moment expressed in terms of parameters and ME(i) is the sample

moment. When the value of M = 0 it is clear that the totality of estimating equations for

the model are consistent with each other and that there is a solution for the parameter

estimates.

2.5.4 Application to a financial data

Now we apply the model to a data on daily exchange rates of Pound to Indian rupee during

the period 2004 to 2008. The data is available in ‘Hand Book of Statistics on the Indian

Economy’, published by Reserve Bank of India.

From the data, the summary statistics obtained are as follows.

Mean value = −1.8543× 10−5, Variance = 5.3628× 10−6,

Coefficient of skewness = −0.1139, Coefficient of kurtosis = 4.0465.

45


The parameter values are obtained by the generalized method of moments as

α1 = 0.003927, α2 = 0.007339, λ1 = 84.9434, λ2 = 101.0374.

Here these estimated values make M equal to zero, showing that the model is consistent

with the data. The histogram of the log transformed data is given in Figure 2.4(a) and the

Q-Q plot is given in Figure 2.4(b). The Q-Q plot confirms that the distribution is a good fit

for the data.

2.6 Conclusion

In this chapter the generalized Laplacian distribution is considered and its properties are

studied. We developed first order autoregressive processes with generalized Laplacian

marginals and the generation of the process. The regression behaviour and simulation

of sample paths are discussed. We consider the distribution of sums and the covari-

ance structure. We introduced the geometric generalized Laplacian distribution and its

properties are studied. Geometric generalized Laplacian processes are discussed. The

kth order autoregressive processes are described. A more general case namely bilateral

gamma distribution is also considered. Its parameters are estimated using conditional least

squares and generalized method of moments and a numerical illustration is also carried

out.

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