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ESTIMATION FOR STOCHASTIC MODELS DRIVEN BY LAPLACE MOTION KRZYSZTOF PODGÓRSKI AND JÖRG WEGENER Abstract.

Laplace motion is a Lévy process built upon Laplace distributions.

Non-Gaussian stochastic elds that are integrals with respect to this process are considered and methods for their model tting are discussed. The proposed procedures allow for inference about the parameters of the underlying Laplace distributions. A t of dependence structure is also addressed. The importance of a convenient parameterization that admits natural and consistent estimation for this class of models is emphasized. Several parameterizations are introduced and their advantages over one another discussed. The proposed estimation method targets the standard characteristics: mean, variance, skewness and kurtosis. Their sample equivalents are matched in the closest possible way as allowed by natural constraints within this class. A simulation study and an example of potential applications conclude the paper.

This version has been accepted for publication in Methods.

Communications in Statistics: Theory and

A previous version is published as technical report: LUTFMS-5081-2009, 2009:9, Lund

University, Sweden. 1

2

1.

Introduction

There is a growing interest in stochastic models that allow for asymmetry and heavier than Gaussian tails. Their applications range from mathematical nance and economics through time series analysis (Trindade and Zhu, 2007; Jayakumar and Kuttykrishnan, 2007) to engineering (Gurley et al., 1996; Lagaros et al., 2005; Bengtsson et al., 2009; Åberg et al., 2009), environmental (Molz et al., 2006; Palacios and Steel, 2006; Røislien and Omre, 2006) and biological sciences (Julia and Vives-Rego, 2005; Purdom and Holmes, 2005). In order to provide practitioners with proper tools for analysis of such models, ecient methods of statistical inference are needed. One interesting class of non-Gaussian models are the generalized Laplace distributions, also known as the Bessel function distributions. This family features asymmetry, heavy tails and is parameterized by four parameters. Moreover, the number of parameters is in correspondence with the number of well-known standard characteristics of a statistical distribution: mean, variance, skewness and kurtosis. The presented work exploits this parity and targets these characteristics in the tting procedure. The generalized Laplace distributions have been discussed in detail in the monograph (Kotz et al., 2001). In particular, maximum likelihood estimation methods have been reviewed for the special case of shape parameter set to one. At present, explicit expressions for the ML-estimates in the general case are not known. On the other hand, all moments of the generalized Laplace distribution are readily available and the method of moments is a viable alternative. In (Visk, 2009), this method has been discussed for asymmetric multivariate Laplace distributions with shape parameter set to its standard value of one. In this work we focus on one dimensional distributions and extend the method of moment approach by taking the kurtosis parameter into account.

More im-

portantly, we go beyond independent identically distributed samples by discussing dependent data that follow a moving average model with respect to generalized Laplace innovations. This model has been introduced in (Åberg et al., 2010) in the

ESTIMATION FOR STOCHASTIC MODELS DRIVEN BY LAPLACE MOTION

3

context of stochastic elds and engineering applications. The analysis of fatigue damage accumulated by a linear structure subject to non-Gaussian wave load is presented in (Åberg et al., 2009). Stochastic models need to come along with eective statistical inference methods to be fully utilizable in applications - the present paper is a step into this direction. Here, we demonstrate that the standard parameterization for generalized Laplace distributions is not the most fortunate when it comes to estimation: The cases of normal and gamma distributions are not accounted by nite parameter values within this larger class of distributions. To overcome this major deciency, several alternative parameterizations are discussed. For estimation purposes, we promote a parameterization that naturally accounts for both normal and gamma distributions and is consistent with the asymptotically normal behavior of the gamma distribution. For this parameterization, we show how the method of moments performs under the assumption that data stems from a continuous moving average process with a given correlation structure.

2.

Generalized Laplace distributions and their parameterizations

A Lévy process that is built upon a generalized skewed Laplace distribution is here referred to as Laplace motion. It became a popular model in mathematical nance where it is known as the variance-gamma process (Madan et al., 1998; Madan and Seneta, 1991). The monograph (Kotz et al., 2001) is a good reference for an extensive overview of the generalized Laplace distributions as dened next.

Denition 1 (Generalized Laplace Law). The generalized Laplace laws are best described by their characteristic functions iδu

φ(u) = e

−1/ν  σ 2 u2 , 1 − iµu + 2

(1)

where ν > 0, δ, µ ∈ R and σ > 0. The case of ν = 1 is referred to as asymmetric Laplace and µ = 0 corresponds to generalized symmetric Laplace distributions.

4

K. PODGÓRSKI AND J. WEGENER

Although in view of the above denition, the parameterization

(δ, µ, σ, ν)

seems

to be a natural choice for this class, we shall demonstrate that it is not best suited for estimation purposes.

2.1.

Parameterization (δ, µ, σ, ν), δ, µ ∈ R, σ > 0, ν > 0.

There is a rep-

resentation of the generalized Laplace distribution (1) that directly corresponds to this parameterization. Namely, if

Z

is a standard normal variable and

independent gamma variable with shape

1/ν

Γ

is an

(and scale equal to one), i.e. with

density

f (x) =

1 x1/ν−1 e−x , x > 0, Γ(1/ν)

then the characteristic function (1) represents the distribution of the variable

√ Y = σ ΓZ + µΓ + δ . Using the convergence of limiting case of

νΓ

to one when

ν

(2)

converges to zero, we note that the

normal distribution with variance, say, ψ 2

is obtained when

√ σ/ ν

ψ , while ν → 0 and µ = 0. The asymptotic normality under the √ condition σ/ ν → ψ , ν → 0 remains valid even if µ 6= 0 as Γ converges to nor√ mal distribution when ν → 0, given that µ/ ν is convergent. Another important converges to

σ =0

special case is

which corresponds to the gamma distribution with scale

(we allow negative values of

µ)

and shape

1/ν .

µ

We conclude that if we consider

the generalized Laplace distributions to be closed under weak convergence both the normal and the gamma distributions are their subclasses. rameterization

(µ, σ, ν)

However, the pa-

does not allow to represent the normal distribution using

numerical values of the parameters and is only expressed by their limits. We report the following formulas for the mean, variance, skewness and excess kurtosis

µ + δ, ν √ 2µ2 + 3σ 2 s=µ ν 2 , (µ + σ 2 )3/2 E(Y ) =

µ2 + σ 2 , ν   σ4 ke = 3ν 2 − 2 . (µ + σ 2 )2

V(Y ) =

(3)

ESTIMATION FOR STOCHASTIC MODELS DRIVEN BY LAPLACE MOTION

Let us note that in the asymmetric case (µ

6= 0)

the parameter

δ

5

is not best

suited to represent the location. We see from the equation for the mean that if

µ/ν

dominates

δ,

the latter may have little to do with the location of the distri-

bution and may be hard to interpret from the data. Therefore, it is more natural to consider a centered generalized Laplace distribution given by consider an additional location parameter

λ

Y − µ/ν

and then

that is added to this centered distri-

bution and which thus equals its mean. The representation (2) can be rewritten including the new location parameter

λ

by centering the gamma variable

√ Y = σ ΓZ + µ(Γ − 1/ν) + λ . We note that

λ

(4)

has a straightforward interpretation and can be estimated by the

sample mean. In the following discussion we assume that

λ

value. Obviously, a nonzero

λ = 0,

can be introduced by adding its value to a mean

zero variable. The standard values of the other parameters are case), scale 2.2.

σ = 1,

and shape

ν=1

µ=0

κ

that replaces

µ

(symmetric

(Laplace distribution).

Parameterization: (κ, σ, ν), κ > 0, σ > 0, ν > 0.

parameter

i.e. the standard

Another natural

through

√

2 p ( 2 + µ2 /σ 2 − µ/σ), 2 √ 2 µ= σ(1/κ − κ) 2 κ=

corresponds to the representation of a mean zero generalized Laplace variable, as a dierence of two independent gamma random variables.

Y,

Γ1 and Γ2 are centered

independent gamma random variables with common shape parameter √ √ 2 2 parameters σ/κ and σκ, respectively, i.e. 2 2

√   2σ 1 Y = (νΓ1 − 1) − κ (νΓ2 − 1) . 2 ν κ

1/ν ,

scale

(5)

In this parameterization, the case of centered gamma distribution is obtained by √ 2 taking σ converging to zero and assuming that σ/κ is converging to positive µ 2 √ 2 or σκ converging to −µ in the case of negative µ. Thus, neither the normal 2 nor the gamma case are expressible through nite parameter values. Nonetheless,

6

K. PODGÓRSKI AND J. WEGENER

gamma and normal distributions are important special cases that possibly t well to some data at hand. Thus for inference purposes, it is important to nd a parameterization in which these cases are expressible by concrete parameters values. This is desirable not only for empirical estimation, but also as a measure of the deviation from normal or gamma distributions, expressible in terms of parameter estimators.

Properly dened deviations can in turn be used for goodness-of-t

tests for gamma or normal distributions within the discussed class of distributions.

2.3.

Parameterization: (η, ψ, ν), η ∈ [−1, 1], ψ > 0, ν ≥ 0.

µ∈R

and

κ>0

both control the asymmetry of the distribution (symmetric case

κ = 1, respectively). Another asymmetry parameter √ ρ = 1/κ − κ = 2µ/σ (ρ = 0 denotes symmetry) is a starting point

corresponding to dened by

The parameters

µ=0

and

for a new parameterization. We simply normalize this parameter by considering

ρ/

p 2 + ρ2

to obtain a compact set of values. The new parameters are given by

r

σ 2 + µ2 , ψ > 0, ν ρ µ η=p =p , −1 ≤ η ≤ 1, 2 + ρ2 σ 2 + µ2

ψ=

(6)

and we note the inverting relations

p σ = ψ ν(1 − η 2 ), √ µ = νψη.

(7)

The new parameterization leads to relatively simple formulas for variance, skewness and excess kurtosis

V(Y ) = ψ 2 , s= Consequently,

η ∈ [−1, 1]

√ νη(3 − η 2 ),

ψ>0


ke = 3ν 1 + 2η 2 − η 4 .

is the standard deviation (with

controls skewness (η

=0

ψ=1

(8)

as standard value),

represents the symmetric case, while

corresponds to the integral with respect to gamma motion) and

ν>0

η = ±1

the tails of

the distribution. Moreover, the normal distribution centered at zero with standard deviation

ψ

is obtained by equating

ν = 0.

Whereas the gamma case with shape

ESTIMATION FOR STOCHASTIC MODELS DRIVEN BY LAPLACE MOTION

1/ν

and scale

µ

corresponds to

ψ 2 = µ2 /ν , η = sgn(µ).

7

Thus in contrast to

the previous parameterizations, these two important special cases are naturally expressed by specic values of the new parameters. The standard parameter values are

η = 0, ψ = 1, ν = 0

and correspond

to the standard normal distribution (the standard case in the parameterization

(δ, µ, σ, ν) was the Laplace distribution with ν = 1).

Under the assumption

λ=0

representation (2) reads

p √  1/2 2 1 − η · (νΓ) Z + η · (νΓ − 1)/ ν , Y =ψ

(9)

whereas (5) becomes

! p p 2 − η 2 + η νΓ1 − 1 2 − η 2 − η νΓ2 − 1 · √ · √ . (10) − Y =ψ 2 2 ν ν √ We note that (νΓ−1)/ ν represents a standardized gamma variable that converges in distribution to the standard normal distribution when We also note that

ψ

ν

converges to zero.

can be naturally estimated by the sample standard devia-

tion. Moreover, if a variable has mean zero and variance one, then the parameters

ψ

and

λ

can be introduced by rst multiplying by

from now on, the focus will be on the parameters 2.4.

η

ψ

and then adding

and

λ=0

and

ψ = 1.

Thus

ν.

Parameterization: (ξ, ζ), ξ ∈ [−1, 1], |ξ| ≤ ζ < 1.

we assume

λ.

Throughout this section

There is one important estimation issue with the

parameterizations that have been presented above: They are not stable for very large and very small values of

ν.

For large values of

ν

its estimation may show

considerable variance. Thus it is natural to introduce a corresponding parameter that sits on a bounded set of values. We propose

ζ=

ν , 0 ≤ ζ < 1. 1+ν

A more serious problem is the case of small when

ν

η:

(or, equivalently, small

ζ ).

Namely,

tends to zero the gamma distributions in (9) and (10) tend to normal

distributions irrespectively of of

ν

(11)

Given the case of small

η.

This could lead to inconsistencies in the estimation

ν,

negative sample skewness will be observed nearly

8

K. PODGÓRSKI AND J. WEGENER

half of the time even if

η = 1.

Similar results would hold for samples from a normal

distribution. However, since the value of

η > 0 is prohibited in the case of negative

skewness, it would force an estimated value of estimation of the case

η=1

η

would show some inconsistencies. A parameter that

allows for a smooth transition between the case equivalently,

ζ)

to be negative. As a result, the

η =1

and

η = −1,

when

ν

(or,

tends to zero, remedies the deciencies. Thus, we propose a new

asymmetry controlling parameter

ξ = sgn(η)η 2 ζ, −ζ ≤ ξ ≤ ζ

(12)

with the inversion formulas

s

|ξ| ζ , η = sgn(ξ) , 1−ζ ζ s p |ξ| µ = sgn(ξ) = sgn(ξ) |ξ|(1 + ν). 1−ζ ν=

2.5.

Summarizing comments.

For any class of stochastic models, it is impor-

tant to decide on a proper parameterization. When there is a clear choice, the best parameterization should be promoted. From our discussion, three important and dierent parameterizations have emerged:

(δ, µ, σ, ν), (λ, η, ψ, ν),

and

(λ, ξ, ψ, ζ).

For ease of use, Table 1 summarizes those results and shows expressions for the rst two (centered) moments in the respective parameterization. For standardized (zero mean and variance one) distributions, there eectively remain only two free parameters within each parameterization.

Under this as-

sumption, conversion from one set of parameters to the other is shown in Table 2. Unfortunately, there is no universal best amongst them, since the latter one is most convenient for inference purposes while the other two are more explicitly attached to certain theoretical properties of Laplace distributions.

Since there

exist immediate conversions from one to another, we use the parameterization that serves the following discussion best.

ESTIMATION FOR STOCHASTIC MODELS DRIVEN BY LAPLACE MOTION

Parameterization

Section

Mean

µ ν

Variance

µ2 +σ 2 ν

Standardized

√

(− √µ/

√

ν

, õ 2

ν

,√ 2

(δ, µ, σ, ν)

2.1

(λ, η, ψ, ν)

2.3

λ

ψ2

(0, η, 1, ν)

(λ, ξ, ψ, ζ)

2.4

λ

ψ2

(0, ξ, 1, ζ)

Table 1.

+δ

9

σ

1+µ2 /σ

σ

1+µ2 /σ

√

ν

1+µ2 /σ 2

, ν)

Summary of proposed parameterizations  indicated sec-

tions refer to more comprehensive expositions. The last column lists parameters for the standardized version (mean zero, standard deviation one).

Section Parameterization

Section

µ∈R

2.1

2.1



2.3

√ η = µ/ ν

2.3

µ=

√

2.4

νη

ν≥0



2

ξ ∈ [−1, 1]

2.4

|ξ| ≤ ζ < 1 Table 2.

|ξ| 1−ζ

ν = ζ/(1 − ζ) q η = sgn(ξ) |ξ| ζ

ν>0 η ∈ [−1, 1]

q

µ = sgn(ξ)

ν = ζ/(1 − ζ) 2

µ ξ = sgn(µ) 1+ν

η ν ξ = sgn(η) 1+ν

ζ = ν/(1 + ν)

ζ = ν/(1 + ν)



Conversion relations between dierent parameterizations.

The latter are identied by their respective references in the text (section numbers). Here, the underlying distribution is standardized to mean zero and variance one.

3.

Moving averages driven by Laplace motion

The Laplace distributions are innitely divisible and one can consider stochastic measures with values distributed according to generalized Laplace distributions.

10

K. PODGÓRSKI AND J. WEGENER

Here, we summarize the basic facts that lead to the construction of stationary processes based on integration with respect to such measures. For the details we refer to (Åberg et al., 2010). 3.1.

Laplace motion.

ity space

(Ω, F, P).

variables

X

on

Ω

By

Let us consider a measure space

L2 = L2 (Ω, F, P)

for which

E(X 2 )

(X , B, m) and a probabil-

we denote the Hilbert space of random

is nite.

Denition 2 (Stochastic Laplace Measure). A stochastic Laplace measure, Λ, with parameters δ ∈ R, µ ∈ R, σ > 0 and controlled by a measure m, is a

function that maps A ∈ B, m(A) < ∞ into L2 such that Λ(A) has generalized Laplace distribution given by the characteristic function φΛ(A) (u) = e

iuδm(A)

−m(A)  σ 2 u2 . 1 − iµu + 2

For disjoint Ai ∈ B, Λ(Ai ) are independent and with probability one Λ

∞ [

!

Ai

i=1

=

∞ X

Λ(Ai ).

i=1

By standard measure extension arguments, it is obvious that by taking suciently `rich'

(Ω, F, P ), one can dene Λ for an arbitrary measure space (X , B, m).

We note

V(Λ(A)) = (σ 2 + µ2 )m(A). A special and important case of the stochastic Laplace measure can be associated with a Lévy motion corresponding to the Laplace and generalized Laplace distributions. Namely, the symmetric Laplace motion that is dened next can be identied with the Laplace stochastic measure on the Borel sets of the haline

[0, ∞)

with the control measure being the Lebesgue measure.

Denition 3

(Laplace

Motion).

A Laplace motion L(t) with the asymme-

try parameter µ, the space scale parameter σ and the time scale parameter ν , LM(µ, σ, ν), is dened by the following conditions (i)

it starts at the origin, i.e., L(0) = 0;

ESTIMATION FOR STOCHASTIC MODELS DRIVEN BY LAPLACE MOTION

(ii) (iii)

11

it has independent and stationary increments; the increments by the time scale unit L(t + ν) − L(t) have an asymmetric Laplace distribution centered in mean at zero, with parameters µ and σ .

If µ = 0, σ = 1 and ν = 1 the process L(t) is called the standard Laplace motion. In a natural extension of the terminology, a Laplace measure that is controlled by the Lebesgue measure on the real line is also referred to as a Laplace motion. We also have the representation of the random Laplace measures as a dierence of gamma measures.

Proposition 1 (Difference of Random Gamma Measures). Let Γ1 and Γ2 √

be independent gamma measures with parameters

2 2

· σ/κ and

√

2 2

· σκ, respectively

and both controlled by a measure m on an arbitrary measurable space (X , B). Then the Laplace measure Λ with parameters δ = 0, µ =

√σ ( 1 −κ) 2 κ

and σ that is controlled

by m can be represented as Λ(A) = Γ1 (A) − Γ2 (A). 3.2.

Laplace stochastic integrals.

The models discussed in this work are based

on the standard construction of integrals of deterministic functions with respect to random measures. The stochastic integral of the isometry of

1A (x)

L2 (X , B, m)

with the variables

into

Λ(A).

f

L2 (Ω, F, P)

with respect to

Λ is dened through

that relates the indicator functions

This isometry at

f

is denoted by

Z X= which is often shortened to

Proposition 2

R

f (x)dΛ(x) , X

f dΛ.

(Characteristic

Function).

Let both

R

f dm and

nite and Λ be a stochastic Laplace measure. Then the integral X =

R R

f 2 dm be f dΛ has

the characteristic function iuδ

φX (u) = e

R

f dm

   Z  σ 2 f 2 (x)u2 dm(x) . exp − log 1 − iµuf (x) + 2 X

(13)

12

K. PODGÓRSKI AND J. WEGENER

X

The random variable

dened by the above integral can be considered as

semi-parametric with three numerical parameters semi-parameters

f

and

m

δ, µ ∈ R

and

σ ≥ 0

and two

that run through innitely dimensional spaces. Such

a semi-parametric distribution, as well as any random variable that has it, will be referred to by

δ = µ = 0, σ = 1

LI(δ, µ, σ; m, f ).

The standard values of the parameters are

and these are default when the parameters are not shown in the

notation. Using the characteristic function from Proposition 2, we obtain a recurrence relation for the moments.

Proposition 3

(Moments)

. Let X =

R

f dΛ, δ = 0, and assume that f N ∈

L2 (X , B, m). Then the following recurrence formula for the moments holds EX

where

N

Z N X E X N −k f k dm Sk−1 , = (N − 1)! (N − k)! k=1

 Pr−1 2 2  sr,k , r odd,  k=0 Sr =  r   µ σ2 2 + P r2 −1 s , r even, k=0 r,k 2

and

 sr,k = µ

r−2k−1

σ2 2

k 

    r−k−1 2 r−k 2 σ + µ . k k

The rst four central moments can be expressed as

Z E X =µ ·

f dm,

E (X − E X) = µ + σ 2

2

2




Z ·

f 2 dm, Z
3 2 2 E (X − E X) =µ 2µ + 3σ · f 3 dm, 2 E (X − E X) =3 µ + σ · f dm + Z
4 2 2 4 + 3 2µ + 4µ σ + σ · f 4 dm 4

2


2 2

Z

2

(14)

ESTIMATION FOR STOCHASTIC MODELS DRIVEN BY LAPLACE MOTION

0.6

13

0.6

0.5

ν=2 µ = 0√ σ= ν

!="

ν=1 µ = 0√ σ= ν

0.5

!=" !=0.5

!=0.5

0.4

0.4

0.3

0.3

!=0

!=0

0.2

0.2

0.1

0.1

0 −5

−4

−3

−2

−1

0

1

2

3

4

0 −5

5

0.6

0.5

−4

−3

−2

−1

0

1

2

3

4

5

0.6

!="

ν=1 √ µ= p·ν ! σ = (1 − p) · ν p = 0.1

ν=2 √ µ= p·ν ! σ = (1 − p) · ν p = 0.1

0.5

!=" !=0.5

!=0.5 0.4

0.4

0.3

0.3

!=0

!=0

0.2

0.2

0.1

0.1

0 −5

−4

−3

−2

Figure 1.

−1

0

1

2

3

From left to right

Symmetric densities (µ with

−4

−3

−2

ν = 1, 2,

= 0). Bottom:

−1

1

2

3

4

5

Top:

respectively.

Asymmetric densities

µ =

ke

s

and excess kurtosis

R

ke

can be written as

f 3 dm
3/2 , (µ2 + σ 2 ) f 2 dm   R 4 f dm σ4 = 3 2− · R
2 . 2 2 2 (µ + σ ) f 2 dm

s =

If we take for

sgn(µ)

2µ2 + 3σ 2

3 2

· R

m the Lebesgue measure in Rd

divided by

tain an extra numerical parameter (with the standard value

LI(µ, σ, ν; f ).

0

p = 0.1.

and thus the skewness coecient

Example 1.

0 −5

5

Densities and their dependence on the kernel parameter

α = 0, 0.5, 1, 2, ∞. √ pν ,

4

The case of

f (x) = 1[0,1] (x)

ν > 0,

we ob-

ν = 1) so we can write

corresponds to the generalized Laplace

distributions as dened in (Kotz et al., 2001). We can also fully parameterize such distributions by taking a family of parameterized kernels. Specically, kernels of the form

f (x) = K exp(−β|x|α ),

where

|x|

is the Euclidean norm in

Rd

and

K

is

a normalization constant, lead to a fully parametrical model. The proportionality constant

K

is chosen such that

V(X) = (µ2 + σ 2 )/ν

for all members of the family.

14

K. PODGÓRSKI AND J. WEGENER

For the one dimensional case we have



Z α

exp(−β|x| ) dx = 2β

−1/α

Γ

R and

f (x) = K(α, β) · e−β|x|

α



, where

K 2 (α, β) =

21/α−1 β 1/α
. Γ α+1 α

Thus by using an explicit form of the integral of

Z

α+1 α

 1/α 2 k f = k

f k:

(2β)1/α
2Γ α+1 α

!k/2−1 ,

we obtain explicit formulas for the moments, skewness and kurtosis in terms of the gamma function. We also observe that for large on

[−1, 1]

and thus the distribution of the integral becomes generalized Laplace

with the parameters of Subsection 2.1 equal to small

√ α, the kernel is converging to 1/ 2

√ √ (0, µ/ 2, σ/ 2, ν/2).

Whereas for

α, the kernel will approximate a constant function on the increasing support.

Thus by virtue of the central limit theorem, the integral converges in distribution to a normal distribution. The shapes of the densities in comparison to this two limiting cases are shown in Figure 1.

Remark 1.

We note that the distribution of

X

is leptokurtic (positive excess kur-

tosis), i.e. it has a more acute peak around the mean and fatter tails than a normally distributed variable (for which the excess kurtosis is zero). For example in the symmetric case, considering by

1/ν :

m

to be the Lebesgue measure on

R ke = 3ν R 2

By varying parameter

ν

f4 f2

R

multiplied

.

we can make the excess kurtosis either very large (very fat

tails) or close to normal (ν converging to zero). In this respect, the distribution of the Laplace integral behaves analogously to the distribution of the underlying Laplace measure.

ESTIMATION FOR STOCHASTIC MODELS DRIVEN BY LAPLACE MOTION

15

Except for a few special cases, there is no explicit formula for the densities of Laplace integrals.

One has to resort to numerical approximations of densities.

Saddlepoint methods (Daniels, 1954) were used in (Galtier, 2009) for computation of level crossing distributions for LMA. Eectiveness of such approximations needs yet to be investigated which is beyond the scope of this paper.

Here we take

a direct approach through fast Fourier transforming the characteristic function. Figure 1 shows several forms of densities as obtained by this method. The normal density and generalized Laplace densities correspond to the limiting cases and

α = ∞, respectively.

the parameter

ν.

α has a similar inuence on the shape as

We observe that

Moreover, large values of

distribution (the densities for

α=2

and

α=0

α

have little eect on the shape of the

α=∞

almost coincide). The densities

have been normalized so that the variances of all presented distributions are equal to one.

3.3.

Moving averages.

Here we review the basic construction of continuous time

moving averages driven by Laplace motion. Let us assume that and the measure i.e.

X = Rn

m is shift invariant on X .

X

is a Hilbert space

Our main focus is on Euclidean spaces,

although most of the properties are valid without this restriction.

Denition 4 (Laplace moving average). For a Laplace measure Λ, controlled by a shift invariant measure m and a kernel f such that

R

f dm and

R

f 2 dm are

nite, the following process will be called a Laplace moving average (LMA): Z f (t − x) dΛ(x).

Xt =

(15)

X From now on we only consider the case of Laplace motion value).

L(t),

λ f)

Such a LMA process will be referred to as

Let

is denoted by

Xt

be

and

Λ

corresponding to a

so the process is centered at zero (in terms of the mean

version obtained by recentering it at

R

X = R

λ f

LMA(λ, µ, σ, ν; f ).

LMA(µ, σ, ν; f ).

Then

R

LMA(µ, σ, ν; f ),

while its

(so that its mean value is equal to

16

K. PODGÓRSKI AND J. WEGENER

a)

the marginal distribution of



−iuµ/ν

φXt (u) = e

b)

R

f

1 · exp − ν

the covariance function

Z

σ 2 + µ2 r(τ ) = ν where

c)

   σ 2 u2 f 2 (x) dx , log 1 − iµuf (x) + 2 −∞

Z

r(τ )

3.3.1.

of

X(t)

∗

and

R(ω)

R(ω) =

of

X(t)

is given by

σ 2 + µ2 Ff (ω)F f˜(ω), ν

Since

it is natural to consider

f

is symmetric,

(Ff (ω))2 .

ν

(λ, η, ψ, ν)

LMA(λ, η, ψ, ν; f ).

(17)

denotes the convolution operator,

Standardized Laplace moving average.

expressed in the

σ 2 + µ2 (f ∗ f˜)(τ ), ν

denotes the Fourier transform. In particular, if

σ 2 +µ2

(16)

is given by

∞

R(ω) =

then

∞

−∞

the spectral density

F

is given by the characteristic function

f (x − τ )f (x) dx =

f˜(x) = f (−x)

where

Xt

Laplace moving averages can be also

parameterization. In such a case, we use the notation

λ

and

λ=0

ψ

and

are merely centering and scaling parameters,

ψ=1

so that the process has mean zero and

variance one. For the latter to hold, a normalization of the kernel is required, such that

R

f 2 = 1.

The process for which the above conditions are satised is called a

standardized Laplace moving average and it is referred to as LMA(η, ν; f ). 4. The kernel

f

Estimation

(or in the special case the parameter

α,

dened in Example 1)

can be used to model the correlation dependence for moving average processes, while the parameters of the Laplace motion characterize distributional features of the data. Consequently, tting the Laplace moving average process is done in two steps. Firstly, the kernel

f

can be estimated from the spectral density function.

In the second step the parameters of the Laplace motion can be tted through the formulas for the moments. Below we present details of this strategy.

ESTIMATION FOR STOCHASTIC MODELS DRIVEN BY LAPLACE MOTION

4.1.

Kernel estimation.

17

If the kernel is from a parametric family one can t its

parameters using the sample correlation function. This approach depends much on the assumed family of kernels.

In a non-parametric approach and under the

f (x) = f (−x) an estimate fb is q 1/2 −1 b b f (x) = (2π) F R(ω) ,

assumption that the kernel satises

where

b R(ω)

mate of

f

given by

is an estimate of the (two-sided) spectral density function. The esti-

is determined up to a non-identiable constant. Thus one can always

pick an estimate such that

R

fb2 (x) dx = 1,

which means that the convolution

fb∗ fb

becomes a correlation function. This is obtained by letting

F fb(x) = (2π)1/2 qR Thus in practice,

f

−1

q b R(ω)

∞ −∞

b R(ω) dω

.

can be estimated from the correlation structure independently

of the marginal distributions of the data. Here, we focus on the estimation of parameters inherited from the generalized Laplace distributions, while the issue of kernel estimation is left for future studies.

4.2. nel

Method of moments estimation. f

is given and normalized so that

R

In this section we assume that the ker-

f 2 = 1.

The rst four moments of the

marginal density of a Laplace integral are used to t the parameters through (14) by replacing theoretical moments with the corresponding sample ones. Fitting the location

λ

and the scale

ψ

is straightforward as we have

Z E(X) = λ

f, V(X) = ψ 2 ,

and one can use sample mean and sample standard deviation as estimators. From now on, we assume that the data is centered around their mean and scaled so their standard deviation is one. This translates to tting a standardized Laplace moving average

LMA(η, ν; f ) process to the so standardized data.

approach, kurtosis and skewness are used to t

η

and

ν.

In the proposed

18

K. PODGÓRSKI AND J. WEGENER

It should be emphasized that the current goal is to propose reasonable and easy to use estimators while the dicult problem of ecient estimation of the multiparameter model is left for some future studies. In particular, it is expected that the maximum likelihood methods lead to more ecient estimation procedures. For example, the maximum likelihood estimators are explicitly available for independent observations from an asymmetric Laplace distribution (the case of have been shown asymptotically ecient (Kotz et al., 2001). lished results it follows that if estimates of

ν=1

and

κ = 1,

From these estab-

the asymptotic variances for the

λ and σ are σ 2 /2 and σ 2 , respectively, (Kotz et al., 2002).

of moments estimators yields

σ2

and

1.25σ 2 ,

ν = 1) and

The method

respectively. This indicates that the

method of moments estimators are not optimal. Unfortunately, the maximum likelihood approach complicates in the complete setup. Firstly, the estimators are not available explicitly when all four parameters are considered even for independent observations (although in this case the likelihood is explicitly given in terms of the Bessel functions).

Secondly, in the dependent case, an explicit likelihood is

not available. Consequently, numerical evaluation of the maximum likelihood estimators would require an advanced approach, whose eciency would be dicult to assess.

Nevertheless, deriving likelihood based estimation methods would be

an important contribution to the statistical analysis of the discussed models. For example, methods developed in (Andrews et al., 2009) and (Davis, Wu 1997) for time series driven by explicit likelihood.

α-stable

distributions tackle similar diculties of a lacking

Such an approach could be possibly adopted to LMA  this

research topic, however, goes beyond the scope of the presented paper. Let

X stand for a generic variable representing any of Xt , t ∈ R of a LMA(λ, η, ψ, ζ; f ).

We consider the rst four moments of

(λ, η, ψ, ζ).

X

In fact, the rst four cumulants

and their relation to the parameters

ki , i = 1, 2, 3, 4,

are more convenient.

Recall the relations between the cumulants and (central) moments:

k2 = E(X − EX)2 , k3 = E(X − EX)3

and

k4 = E(X − EX)4 − 3k22 .

k1 = EX , These are

ESTIMATION FOR STOCHASTIC MODELS DRIVEN BY LAPLACE MOTION

19

related to the mean, variance, skewness and excess kurtosis in the following manner

k3

E(X) = k1 , E(X − EX)2 = k2 , s = Writing the moments adjusted for the kernel the cumulants in terms of the parameters

f

as

3/2 k2

, ke =

k4 . k22

R Kj = kj / f j ,

we can express

λ, µ, σ, ν :

µ (3σ 2 + 2µ2 ), ν 3 = (2µ4 + 4µ2 σ 2 + σ 4 ) . ν

K3 = K4

As mentioned earlier, the rst two (adjusted) moments are used for estimating the location

λ

and

ψ.

From now on we assume

λ = K1 = 0

and

ψ 2 = K2 = 1,

i.e.

the data is standardized to zero mean and standard deviation one. Using

ν

and

η,

we get relatively simple formulas for adjusted sample kurtosis

and skewness

K3 =

√


ν · η 3 − η2 ,

(18)


K4 = 3ν 1 + 2η 2 − η 4 . Since we can always match the sign of

η

with the sign of

(19)

K3 ,

for the sake

of the following argument, we may assume that both are positive. Following the method of moments estimation, we want to replace the right hand sides of the above equations by their sample counterparts and then solve the resulting equations for and

ν.

The solution, say

ηˆ and νˆ,

η

denes the corresponding parameter estimators.

However, the equations do not always have a solution. For example, it may happen that

b4 K

(the sample equivalent of

K4 )

turns out negative, which is equivalent

to a negative value of the sample excess kurtosis. Moreover, from equation (19)

ν=

b4 K , η 2 ∈ [0, 1], 2 4 3 (1 + 2η − η )

and by substituting into expression (18)

b2 3K η 2 (3 − η 2 ) 3 = b4 1 + 2η 2 − η 4 K 2

(20)

20

K. PODGÓRSKI AND J. WEGENER

which has one real solution only if

b4 > 0 K

and

b 2 /K b 4 ≤ 2/3. K 3 In such a case with the sign of

η

matching that of

(21)

b3, K

the solution is explicitly

dened by

  i√ −1 + 2K − K 2 c −1 + 2K − K 2 −K +2+ 3 c+ , η =− + 2 2c 2 c √ b4. b 32 /K c = (−1 + 3K 2 − K 3 + 6K − 21K 2 + 22K 3 − 6K 4 )1/3 and K = K 2

where

Alternatively, the solution can be expressed without using complex numbers as

η 2 = (2 − K)(1 − C(K)), where

     cos 1 arccos 2−3K3 − 1 ; K≤ 3 (1−K) C(K) =     cosh 1 arccosh 2−3K3 − 1 ; K > 3 (1−K)

√ 3− 3 2 √ 3− 3 2

; (22)

.

This argument leads to the following constraint for possible values of kurtosis and skewness for

LI(µ, σ, ν; f )

distributions.

Proposition 4. Let X be a LI(µ, σ, ν; f ) variable and s, ke its skewness and excess kurtosis parameters. Then ke ≥ 0 and if ke 6= 0, then R 2 ( f 3 )2 s2 ≤ R 4 R 2 . ke 3 f · f

To complete the discussion of the estimation procedure, we need to decide about the estimates if the sample kurtosis and skewness do not satisfy the constraints imposed by the properties of the

LI(η, ν)

distributions. From Proposition 4 the

estimators are not dened from equations (18) and (19) only, if

b 4 < 3K b2 . 2K 3

(23)

˜ 4 and skewness K ˜ 3 for which 2K ˜4 = K ˜ 32 and such that (K ˜ 4, K ˜ 3 ) is minimizing a certain distance to (K b4, K b 3 ). One has 3K

In this case, we propose the values of kurtosis

to decide on a type of distance. Because of simplicity and direct interpretation we

ESTIMATION FOR STOCHASTIC MODELS DRIVEN BY LAPLACE MOTION

have opted to minimize the Euclidean distance between and

R = {(a, b); 2b ≥ 3|a|},

21

b 3 )K b 2, K b4) (ˆ a, ˆb) = (sgn(K 3

i.e. the region that corresponds to (21).

There are two mutually exclusive cases when (23) is satised. We express them using

R b4 2kˆe · ( f 3 )2 2K 2ˆb R = = r= . b 32 3|ˆ a| 3ˆ s2 · f 4 3K

One case is when

(24)

r ≤ −4/9 and the closest point from the region R is (a, b) = (0, 0).

This eectively leads to

ν = 0 and thus a normal distribution is chosen as the best

t to the data. The other case is when

−4/9 < r < 1 for which by direct calculation

we obtain

b b b2 ˜ 3) · K ˜ 2 = 6K4 + 4 sgn(K3 ) · K3 sgn(K 3 13 b b b2 ˜ 4 = 9K4 + 6 sgn(K3 ) · K3 . K 13 By choosing values of value for with the

(19),

˜3 K

and

˜4 K

(25)

on the boundary of the region, the corresponding

˜ 3 (which coincides ηb becomes either 1 or −1, depending on the sign of K b 3 ). The value of νb is then obtained by (18) or, equivalently, by sign of K

i.e.

˜ 4 /6 = K ˜ 2 /4. νb = K 3 This case of t (η b

= ±1)

desson, 1992) with scale

Remark 2.

corresponds to a generalized gamma convolution (Bon-

µ b=

√

νb and

The simple condition

shape

1/b ν.

3ˆb ≤ −2|ˆ a| can be further investigated in the con-

text of testing hypothesis for normality within the discussed class of distribution.

As discussed in Subsection 2.4, the parameterization to guarantee consistent estimation for small

ν.

(ξ, ζ)

is most appropriate

We summarize our estimation

procedure using this parameterization in the following result.

Proposition 5. Let (Xt ) be LMA(λ, ξ, ψ, ζ; f ) with f being integrable in any positive integer power and normalized such that

R

f 2 = 1. Further, let S stand for

22

K. PODGÓRSKI AND J. WEGENER

the sample standard deviation, sˆ for the sample skewness and kˆe for the sample excess kurtosis. Assume also that C(·) is given by (22) and r by (24). Dene  s   

ηˆ = sgn(ˆ s)

2 2− 3r

   2 1−C ; 3r

r ≥ 1,

   1; otherwise.  1  ; r ≥ 1,    1 +2ˆ η 2 − ηˆ4   kˆe  9 4 4 νˆ = R 4 1 + sgn(ˆ s) ; − < r < 1, 3 f  26 9r 9    4   0; r≤− . 9

Then the following estimators of (λ, ξ, ψ, ζ): R ˆ = X/ ¯ f, λ ζˆ =

νˆ , 1 + νˆ

ψˆ = S, ξˆ = sgn(ˆ s)ˆ η 2 ζˆ

are consistent. The mean and variance of the tted model match the sample mean and sample variance. The skewness and kurtosis of the tted model match the sample skewness and sample kurtosis, whenever the latter do not violate the constraints given in Proposition 4. Proof.

The moving averages with respect to Lévy measures have the strong mixing

property so they are ergodic (see, for example, (Cambanis, Podgorski, Weron, 1995)). Consequently, the sample versions of the moment estimators converge to their population equivalents, granting consistency. The remaining relations have



been discussed above.

5. 5.1.

Simulation.

Simulation study and application

A simulation study of the proposed estimation methods is pre-

sented in Table 3. We use the parameterization

(λ, ψ, ξ, ζ)

and values of the pa-

rameters have been set so they correspond to the densities illustrated in Figure 1. More precisely, we consider two cases of exponential kernels with

α = 1 and α = 2

ESTIMATION FOR STOCHASTIC MODELS DRIVEN BY LAPLACE MOTION

Values of α

α=1

α=2

α=∞

23

Sample sizes

b λ

b ψ

ξb

ζb

n=100

−0.18, 0.01, 0.15, (0)

0.68, 0.92, 1.23, (1)

−0.09, 0, 0.12, (0)

0, −0.03, 0.41, (0.5)

n=500

−0.08, 0.00, 0.089, (0)

0.88, 0.98, 1.13, (1)

−0.04, 0.00, 0.45, (0)

0.08, 0.39, 0.63, (0.5)

n=100

−0.19, −0.01, 0.18, (0)

0.68, 0.92, 1.23, (1)

−0.01, 0.013, 0.32, (0.05)

0.00, 0.09, 0.45, (0.5)

n=500

−0.08, 0.00, 0.08, (0)

0.86, 0.98, 1.12, (1)

0.00, 0.06, 0.25, (0.05)

0.08, 0.38, 0.62, (0.5)

n=100

−0.17, 0.02, 0.18, (0)

0.66, 0.91, 1.26, (1)

−0.12, 0.00, 0.26, (0)

0.00, 0.20, 0.53, (0.667)

n=500

−0.08, −0.00, 0.07, (0)

0.85, 0.98, 1.14, (1)

−00.07, 0.00, 0.07, (0)

−0.35, 0.57, 0.74, (0.667)

n=100

−0.18, −0.02, 0.16, (0)

0.61, 0.88, 1.25, (1)

−0.02, 0.05, 0.45, (0.067)

0, 0.28, 0.55, (0.667)

n=500

−0.08, 0.00, 0.07, (0)

0.82, 0.97, 1.14, (1)

0.00, 0.083, 0.32, (0.067)

0.29, 0.53, 0.71, (0.667)

n=100

−0.17, −0.01, 0.16, (0)

0.72, 0.94, 1.22, (1)

−0.01, 0.00, 0.12, (0)

0.00, 0.15, 0.53, (0.5)

n=500

−0.08, 0.00, 0.08, (0)

0.88, 0.98, 1.12, (1)

−0.03, 0.00, 0.03, (0)

0.22, 0.43, 0.64, (0.5)

n=100

−0.17, 0.00, 0.16, (0)

0.69, 0.94, 1.21, (1)

−0.00, 0.04, 0.37, (0.05)

0.00, 0.18, 0.53, (0.5)

n=500

−0.08, 0.00, 0.08, (0)

0.87, 0.97, 1.10, (1)

0.00, 0.06, 0.18, (0.05)

0.20, 0.42, 0.62, (0.5)

n=100

−0.19, 0.02, 0.17, (0)

0.65, 0.91, 1.23, (1)

−0.14, 0.00, 0.24, (0)

0.00, 0.31, 0.61, (0.667)

n=500

−0.08, 0.00, 0.08, (0)

0.84, 0.99, 1.15, (1)

−0.07, 0.00, 0.05, (0)

0.42, 0.62, 0.76, (0.667)

n=100

−0.17, 0.01, 0.20, (0)

0.69, 0.92, 1.32, (1)

−0.02, 0.072, 0.52, (0.067)

0, 0.32, 0.61, (0.667)

n=500

−0.08, −0.01, 0.08, (0)

0.83, 0.97, 1.15, (1)

0.00, 0.08, 0.28, (0.067)

0.40, 0.59, 0.75, (0.667)

n=100

−0.12, 0.00, 0.12, (0)

0.86, 0.98, 1.13, (1)

−0.04, 0, 0.04, (0)

0.14, 0.35, 0.57, (0.5)

n=500

−0.06, 0.00, 0.06, (0)

0.93, 1.00, 1.06, (1)

−0.01, 0.00, 0.01, (0)

0.34, 0.45, 0.57, (0.5)

n=100

−0.13, 0.00, 0.12, (0)

0.84, 0.98, 1.13, (1)

0.00, 0.05, 0.20, (0.05)

0.12, 0.33, 0.56, (0.5)

n=500

−0.06, 0.00, 0.06, (0)

0.93, 1.00, 1.07, (1)

0.02, 0.05, 0.11, (0.05)

0.32, 0.44, 0.60, (0.5)

n=100

−0.12, 0.01, 0.13, (0)

0.82, 0.98, 0.1.17, (1)

−0.07, 0.00, 0.06, (0)

0.32, 0.50, 0.70, (0.667)

n=500

−0.06, 0.00, 0.06, (0)

0.92, 1.00, 1.09, (1)

−0.02, 0.00, 0.02, (0)

0.51, 0.62, 0.73, (0.667)

n=100

−0.13, 0.00, 0.13, (0)

0.82, 0.99, 1.18, (1)

0.00, 0.08, 0.31, (0.067)

0.30, 0.52, 0.70, (0.667)

n=500

−0.06, 0.00, 0.06, (0)

0.91, 1.00, 1.09, (1)

0.02, 0.07, 0.14, (0.067)

0.49, 0.60, 0.74, (0.667)

Simulation study of the estimators of λ, ψ, ξ, ζ . 10%, 50% and 90% percentiles based on Monte Carlo samples are reported and actual values of the parameters are given in parentheses. Table 3.

with

β =1

and one Dirac's delta, i.e. assuming independence between observa-

tions. The time step is

dt = 0.3 and we consider two sample sizes:

the parameters of Laplace noise, we took

µ = 0 (ξ = 0)

and

µ=

√

0.1 ∗ ν (ξ = 0.05

ν = 1 (ζ = 0.5) if

ν=1

and

and

100 and 500. For

ν = 2 (ζ = 0.667),

ξ = 0.0667

if

ν = 2).

Four

dierent sets of the Laplace parameters, three examples of kernels and two sample sizes result in the total of

4 × 3 × 2 = 24

location and scale parameters (λ and

ψ)

simulations, as seen in Table 3.

are set to zero and one, respectively.

The

24

K. PODGÓRSKI AND J. WEGENER

The data has been simulated using the approach discussed in (Åberg et al., 2010). Moreover, estimation procedures as described above have been applied for Monte Carlo replicates of the data (500 of them). The kernel has been assumed to be exactly known and its estimation has not been investigated in this study. The eect of dependence in the data on the quality of estimation is apparent from the simulations. Clearly, estimation in the independent case (kernel set to Dirac's delta) is most accurate. The case of long dependence represented by the exponential kernel with parameter

α = 1

is, as expected, the least precise.

observe a bias in the estimation of

ξ

ζ

and

We

that could be reduced by some bias

removal procedures but we leave this topic for some future studies. Generally, we conclude that the proposed estimators perform fairly well however more profound studies about their eciency have to be carried out.

5.2.

Application.

This section illustrates the above estimation procedure and its

feasibility when modeling real-world data. In order to demonstrate its exibility, observations which exhibit skewness and heavy tails serve as a case study. Additionally, the eect of temporal dependencies within observational data is compared to the one from the above simulation study. It is observed that those dependencies, along with limited sample size, vastly inuence the accuracy of parameter estimates. Atmospheric pressure is known to possess distributional properties that vary with location and deviate from standard Gaussian features (Nakamura and Wallace, 1991, and references therein).

Dating back to roughly 1880, sea level

pressure observations are gathered in a single dataset (Schmith et al, 1997). Three to four records per day are collected with typical time intervals of 6 hours. From these records seasonality has been removed and resulting data appear stationary in time. Here, we aim to demonstrate the performance of the estimation procedure for 20 stations in the European Atlantic region. shown in Figure 2 in Figure 2

(right).

(left).

Their geographical distribution is

Sample correlation functions are obtained and shown

We note signicant dierences in long-term dependence at

ESTIMATION FOR STOCHASTIC MODELS DRIVEN BY LAPLACE MOTION

25

75 1

11

70

0.8

12 8

20

10

0.6

6

Corr.

65

19

7 13

60

18

0.4

5

14 16 17 3 4 2

9

55

0.2

0 1

15

50 −40

−30

−20

Figure 2.

−10

0

Left:

10

20

30

40

−0.2 0

10

20

30 40 Lag [~6 hrs]

50

60

70

Air pressure observation sites in the European

North Atlantic region.

Right:

Autocorrelation functions of the pres-

sure data for all stations (particularly long dependence is observed for Stations 8, 7, 10 and 11).

various geographical locations. The sample correlation functions have been used to obtain non-parametric smoothed estimates of the spectra of the signal. These in turn were used for computing moving average kernel estimates as described in Subsection 4.1. A station-wise analysis compares LMA-estimated densities and with their respective sample counterparts. Densities are obtained by Fourier transforming the characteristic function (16) after estimation of parameters as described in Section 4.2. For comparison, sample distributions are estimated in a non-parametric fashion, using a bandwidth of 0.25 in an Epanechnikov formulation (Epanechnikov, 1969). Those estimates are performed over a grid that partitions the range of data into 100 intervals of equal length. Most of the stations have a high quality t as observed for Station 4 in Figure 3. Slightly less accurate ts for Stations 7, 8, 9, 10 and 11 are also presented in this gure. The overall distributional structure is captured with notable precision. Nevertheless, for longterm dependent data (cf. Stations 8, 10 and 11) density estimates may dier. The reason for this dierence becomes evident when comparing Stations 8 and 11 in some detail: correlation functions of both stations show similar longterm behavior, but the density of Station 11 is much poorer assessed than the one for Station 8. Holding the smallest

26

K. PODGÓRSKI AND J. WEGENER

number of records (less than 100000) of the whole dataset, Station 11 has only about 57% of the sample size of Station 8.

The more precise density estimate

in the latter case indicates that the loss of accuracy due to longer temporal dependence can be balanced by an increased number of observations. This result is well in line with the preceding simulation study showing improved accuracy with increasing sample size, especially evident for longterm dependent data. Station 4

Station 7 0.45

0.45 0.4

0.4

0.35

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0 960

970

980

990

1000 1010 1020 1030 1040 1050 1060 Pressure [hPa]

0 940

960

980

Station 8

1000 1020 Pressure [hPa]

1040

1060

1040

1060

Station 9 0.45

0.45

0.4

0.4

0.35

0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15

0.15 0.1

0.1

0.05

0.05

0 940

960

980

1000 1020 Pressure [hPa]

1040

1060

0

960

980

Station 10

1000 1020 Pressure [hPa] Station 11

0.5

0.45

0.45 0.4 0.4 0.35 0.35 0.3 0.3 0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

940

960

Figure 3.

980

1000 1020 Pressure [hPa]

1040

1060

0

960

980

1000 1020 Pressure [hPa]

1040

1060

Density estimates for selected stations. Sample estimates

are given as solid lines, LMA model estimates as dots.

References

27

Acknowledgment The authors wish to acknowledge the support from the EU project SEAMOCS. Research was also supported by the Swedish Research Council Grant 2008-5382.

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Centre for Mathematical Sciences, Mathematical Statistics, Lund University, Box 118, SE-22100 Lund, Sweden

E-mail address :

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