properties, estimation and application to financial …

Advances and Applications in Statistics

© 2021 Pushpa Publishing House, Prayagraj, India

http://www.pphmj.com

http://dx.doi.org/10.17654/AS071010055

Volume 71, Number 1, 2021, Pages 55-84 ISSN: 0972-3617

Received: March 17, 2021; Revised: April 22, 2021; Accepted: October 10, 2021

2020 Mathematics Subject Classification: 62P05.

Keywords and phrases: modified Bessel function of the third kind, generalized inverse

Gaussian distribution, generalized hyperbolic distribution, EM-algorithm.

∗Corresponding author

Published Online: November 16, 2021

PROPERTIES, ESTIMATION AND APPLICATION

TO FINANCIAL DATA FOR GENERALIZED

HYPERBOLIC DISTRIBUTION WHEN

THE INDEX PARAMETER IS 2

3−

Calvin B. Maina1,*

, Patrick G. O. Weke2, Carolyne A. Ogutu

2 and

Joseph A. M. Ottieno2

1Department of Mathematics and Actuarial Science

Kisii University

P. O. Box 408-40200, Kenya

e-mail: [email protected]

2School of Mathematics

University of Nairobi

Kenya

Abstract

Generalized Hyperbolic Distribution (GHD) arises as a normal

variance-mean mixture with Generalized Inverse Gaussian (GIG) as

the mixing distribution. The GHD nests a number of distributions

obtained as special and limiting cases. In literature, however, Normal

Inverse Gaussian (NIG) and Variance-Gamma (VG) are the most

commonly used special and limiting cases, respectively, in analyzing

C. B. Maina, P. G. O. Weke, C. A. Ogutu and J. A. M. Ottieno 56

financial data. The objective of this paper is to derive another special

case of the GHD, obtain its properties, estimate its parameters and

then apply it to some financial data. The properties are determined by

first expressing them in terms of the corresponding properties of the

mixing distribution. The maximum likelihood estimates are obtained

using the Expectation-Maximization (EM) algorithm which overcomes

numerical difficulties occurring when standard numerical techniques

are used. An application to a dataset concerning the Range Resource

Corporation (RRC) is given. It is shown that the proposed model

captures the skewness and excess kurtosis exhibited by the data. The

maximum likelihood estimates are shown to be obtained easily by the

EM-algorithm.

1. Introduction

A normal distribution has two parameters: the location parameter

representing the mean and the scale parameter representing the variance. For

a continuous mixture, we can fix the mean and vary the variance and vice-

versa. Barndorff-Nielsen [2] introduced a normal mixture where the mean is

a linear function of a varying variance. This is called a normal variance-

mean mixture. When the mixing distribution is Generalized Inverse Gaussian

(GIG), then the mixture is called Generalized Hyperbolic Distribution

(GHD) which nests a number of distributions obtained as special and

limiting cases. Special cases are obtained when the index parameter λ takes

specific values. When ,1=λ we obtain the hyperbolic distribution which

was the first special case used in financial modelling (Eberlein and Keller

[4]). Later on, Barndorff-Nielsen [2] introduced the case when 2

1−=λ

which is the Normal Inverse Gaussian (NIG) distribution. The NIG has been

extensively studied in finance because of its analytical tractability property.

Prause [7] mentioned the case when 2

3−=λ and obtained the mean and

variance and no further work has been done. It is the objective of this

paper to study this special case in detail. The specific objective is to

construct, obtain properties, estimate parameters and apply the model to

Properties, Estimation and Application to Financial Data … 57

Range Resource Corporation financial data. The Maximum Likelihood (ML)

parameter estimates are obtained via the EM-algorithm.

This paper is organized as follows. In Section 2, we have definitions and

properties of modified Bessel function of the third kind. Section 3 deals with

the generalized inverse Gaussian distribution with its special case. Section 4

covers the mixed model and its properties. Parameter estimation based on the

EM-algorithm is covered in Section 5. Data analysis is performed in Section

6 and the conclusion is provided in Section 7.

2. Modified Bessel Function of the Third Kind

The most important mathematical tool used for this work is the modified

Bessel function of the third kind. Its definitions and properties are given in

this section. For more detailed study, see Abramowitz and Stegun [1].

Definition 1 and its properties

Modified Bessel function of the third kind of index λ evaluated at ω

denoted by ( )ωλK is defined by

( ) ∫∞ −λ

λ

+ω−=ω

0

1 .1

2exp

2

1dx

xxxK (1)

An alternative form of Definition 1 is

( ) ∫∞ −λ−

λλ

ω−−

ω=ω

0

21

4exp

22

1dt

tttK (2)

which is obtained by letting .2t

xω=

Some properties are as follows.

Property 2.1 (Symmetry)

( ) ( ).ω=ω λ−λ KK (3)


Property 2.2 (Derivative I)

( ) ( ) ( )[ ].2

111 ω+ω−=ωω∂

∂−λ+λλ KKK (4)

Property 2.3 (Derivative II)

( ) ( ) ( ).1 ω−ωωλ=ωω∂

∂+λλλ KKK (5)

Property 2.4 (Recursive relation)

( ) ( ) ( ).211 ω−ω=ωω

λ−λ+λλ KKK (6)

Definition 2 and its properties

( ) ( )∫∞ ω−−λ

λλ −

+λΓ

Γ

ω=ω

12

12 .1

2

1

2

1

2dtetK

t (7)

This can also be expressed in summation form as given below:

Proposition 1.

(a)

( ) ( )∑∞

=

−ω−λ ω

+λΓ

++λΓ

−+λΓ

+λΓ

ωπ=ω

0

2

2

1

2

1

2

1!

2

1

2i

ii

ii

eK (8)

( )

ω

+λΓ

−+λΓ

++λΓ

+λΓ

+ωπ= ∑

∞

=

−ω−

1

2

2

1

2

1

2

1!

2

1

12

i

ii

ii

e (9)

which can further be expressed as

(b)

( ) [ ( ) ]( )

.8!

1241

21 1

22

ωΓ−−γ+ω

π=ω ∑∏∞

= =

ω−λ

i

n

in

n

ieK (10)


Proof.

( ) ( )∫∞ ω−−λ

λλ −

+λΓ

Γ

ω=ω

12

12 .1

2

1

2

1

2dtetK

t

Let ( ).1−ω= ty Then

( ) ( ) ( )∑∞

=

−ω−λλλ

λλλ

++λΓω

−λω

ω

+λΓ

Γω

ω=ω0

2

12

2

122

1

2

2

2

1

2

1

22

2

i

ii

ieK

( )

ω

−+λΓ

++λΓ

+ωπ= ∑

∞−λ

=

−ω−

1

2

2

1!

2

1

12

i

i

ii

i

e

( )( ) ( )

ω

−+λΓ

++λΓ

+ω−λ+ω

π= ∑∞

=

−ω−

2

2

2

2

1!

2

1

8!1

141

2i

i

ii

i

e

( )( )

( ) ( )( )

ω−λ−λ+ω

−λ+ωπ= ω−

2

222

8!2

9414

8!1

141

2e

( )

ω

−+λΓ

++λΓ

+∑∞

=

−

3

2

2

1!

2

1

i

i

ii

i

( )( )

( )( )( )

ω−λ−λ+

ω−λ+

ωπ= ω−

2

222

8!2

9414

8!1

141

2e

( ) ( ) ( )( )

( )

ω

−+λΓ

++λΓ

+ω

−λ−λ−λ+ ∑∞

=

−

33

222

2

2

1!

2

1

8!3

2549414

i

i

ii

i


( )( )

( ( ) )( )

ω−−λ+

ω−λ+

ωπ= ∏

=

ω−2

12

222

8!2

124

8!1

141

2k

ke

( ( ) )( )

( ( ) )( )∏ ∏

= = ω−−λ+

ω−−λ+

3

1

4

14

22

3

22

8!4

124

8!3

124

k k

kk

( ) ,2

2

1!

2

1

5

ω

−+λΓ

++λΓ

+∑∞

=

−

i

i

ii

i

and therefore

( ) ( ( ) )( )

.8!

1241

21 1

22

ω−−λ+ω

π=ω ∑∏∞

= =

ω−λ

i

i

ki

i

keK

Corollary 2.1. For positive integers ,1,2

1 −=−λ nn

(a)

( ) ( )( ) ( ) .2

!!

!1

212

1

ω−

++ωπ=ω ∑

=

−ω−+

n

i

i

n ini

ineK (11)

(b)

( ) ( )( ) ( ) .2

!1!

!11

2

1

12

1

ω−−

−++ωπ=ω ∑

−

=

−ω−−

n

i

i

n ini

ineK (12)

Corollary 2.2. When ...,,3,2,1=n we have

(a) ( ) ( ) ,2

2

1

2

1ω−

− ωπ=ω=ω eKK (13)

(b) ( ) ( ) ,1

12

2

3

2

3

ω+ωπ=ω=ω ω−

−eKK (14)

(c) ( ) ( ) ,33

12 2

2

5

2

5

ω+ω+ω

π=ω=ω ω−−

eKK (15)


(d) ( ) ( ) ,15156

12 32

2

7

2

7

ω+

ω+ω+ω

π=ω=ω ω−−

eKK (16)

(e) ( ) ( ) ,1051054510

12 432

2

7

2

7

ω+

ω+

ω+ω+ω

π=ω=ω ω−−

eKK (17)

(f) ( ) ( ) .94594542010515

12 5432

2

9

2

9

ω+

ω+

ω+

ω+ω+ω

π=ω=ω ω−−

eKK (18)

3. Special Case of Generalized Inverse Gaussian Distribution

In this section, we derive the distribution of ( )γδλ ,,GIG and deduce

the special case when .2

3−=λ

Proposition 2. The distribution of ( )γδλ ,,GIG is given by:

( ) ( ) .2

1exp

22

21

γ+δ−δγ

δγ=

λ

−λλz

zK

zzg (19)

Hence, for ,2

3−=λ

( )( )

( ) .2

1expexp

12

22

2

53

γ+δ−δγ

δγ+πδ=

−z

zzzg (20)

Proof. From equation (1),

( ) ∫∞ −λ

λ

+ω−=ω

0

1 .1

2exp

2

1dx

xxxK

Using the parametrization δγ=ω and the transformation ,zx δγ= we

obtain

( ) ,2

1exp

2

1

0

22

1

∫∞ −λ

λλ

γ+δ−

δγ=δγ dzz

zzK


( )∫∞

λ

−λλ

γ+δ−δγ

δγ=

0

221

.2

1exp

21 dzz

zK

z

Thus

( ) ( )

γ+δ−δγ

δγ=

λ

−λλz

zK

zzg

221

2

1exp

2

is a pdf known as the generalized inverse Gaussian distribution.

When ,2

3−=λ using equation (12), we have

( ) ( )

γ+δ−δγ

δγ=

−

−−−z

zK

zzg

22

2

3

12

3

2

3

2

1exp

2

γ+δ−

δγ+δγπ

δγ=

δγ−

−−z

ze

z 222

5

2

3

2

1exp

11

22

( ).

2

1exp

12

22

2

53

γ+δ−

δγ+πδ= δγ−

zz

ez

Proposition 3. (a) The rth moment of the ( )γδλ ,,GIG distribution is

given by

( ) ( )( ) .δγ

δγ

γδ=

λ+λ

K

KZE r

rr (21)

(b) When ,2

3−=λ

( ) ,1

2

δγ+δ=ZE (22)

( )( )

,1

2

3

δγ+γδ=Zvar (23)

( ) ( )( )

,1

3133

362533

3δγ+γ

γδ−γδ−δγ+δ=µ Z (24)


( )( )

.1

31231545

343625

4δγ+γ

δ+γδ+γδ+γδ=µ Z (25)

Proof. For part (b), we have

Mean

( )( )

( )δγ

δγ

γδ=

2

3

2

1

K

K

ZE

.1

2

δγ+δ=

Variance

( ) ( ) ( )2

43

11 δγ+δ−δγ+γ

δ=Zvar

( ).

12

3

δγ+γδ=

Third Central Moment

( )( ) ( )3

6

2

5

3

3

31

2

1

33

δγ+δ+

δγ+γδ−

γδ=µ Z

( )( )

.1

3133

362533

δγ+γγδ−γδ−δγ+δ=

Fourth Central Moment

( )

( ) ( )( ) ( )

( )45

5847325

33425

41

31614

133

δγ+γγδ−δγ+γδ+δγ+γδ−

δγ+δ+γδ+γδ

=µ Z

( ).

1

31231545

343625

δγ+γδ+γδ+γδ+γδ=


4. Proposed Mixed Models

In this section, we construct the normal variance-mean mixture when the

mixing distribution is .,,2

3

γδ−GIG Then we obtain the properties of the

mixed model. The construction can be made through the following:

Proposition 4. If the conditional distribution ( )zzNZX ,~ β+µ and

,,,2

3~

γδ−GIGZ then the mixed model is given by

( ) ( ) ( ) ( ) ( ( ) )( )

,1

expexp

22

2

1

21222

β−αδ+π

αδφφββµ−β−αδδα=−

xKxxxf (26)

where

( ) ( ) .1 22 δµ−+=φ xx (27)

Proof.

( )( )( )

( )∫∞ β+µ−−

π=

0

2

1 2

2

1dzzge

zxf z

zx

( ) ( )

( )δγ+πδγδ=

µ−β

12

exp3 xe

( ) ( )( )∫

∞ −

γ+βδ+µ−+γ+β−×

0 22

22223 .

1

2exp dz

z

xzz

Let

( )( )

.22

22

tx

zγ+β

δ−µ−=


Then

( ) ( ) ( )

( )( )

( ) 22

223

12

exp

δ+µ−γ+β

δγ+πδγδ=

µ−β

x

exf

x

( ) ( )∫

∞ −

+×δ+µ−γ+β×

0

22223 1

2exp dt

tt

xt

( ) ( )

( )( )

( ) 22

223

1

exp

δ+µ−γ+β

δγ+πδγδ=

µ−β

x

ex

( ( ) ( ) ).22222 δ+µ−γ+β× − xK

Write .222 γ+β=α Then

( ) ( ) ( ) ( ) ( ( ) )( )

,1

expexp

22

2

1

21222

β−αδ+π

αδφφββµ−β−αδδα=−

xKxxxf

( ) ( ).1

2

2

δµ−+=φ x

x

One of the attractive features of constructing a distribution by mixing

is that the properties of the mixed model are expressed in terms of the

properties of the mixing distribution. When ( ),~ ZNZX β+µ we have the

following.

Proposition 5.

( ) ,2

2

+β= µ t

tMetM Zt

X (28)

( ) ( ),ZEXE β+µ= (29)

( ) ( ) ( ),2ZvarZEXvar β+= (30)

( ) ( ) ( ),3 33

3 ZZvarX µβ+β=µ (31)

( ) ( ) ( ) [ ] ( ) [ ].366 223

24

44 ZEZvarZEZZX +β+µβ+µβ=µ (32)


Proof.

Part 1.

( ) ( ) ( )∫ ∫∞ ∞

∞−β+µ=

0,; dzzdxfzzxfetM ZN

txX

( )∫∞µ

+β=

0

2

2exp dzzfz

tte Z

t

.2

2

+β= µ t

tMe Zt

Part 2.

( ) ( )

µ+

+β=

+β

µ

+β

µZ

tt

tZ

tt

tIX eEeZetEetM

22

22

[ ] ( )0IXMXE =

[ ]ZEβ+µ=

which can be obtained using conditional expectation approach given by

[ ] ( )ZXEEXE =

[ ]ZE β+µ=

[ ].ZEβ+µ=

Part 3.

( ) ( ) ( )

+βµ+

+β=

+β

µ

+β

µZ

tt

tZ

tt

tIIX ZetEeZetEetM

22

22

( ) .222

22

+βµ+

µ+

+β

µ

+β

µZ

tt

tZ

tt

tZetEeeEe


Therefore,

( ) [ ] [ ] [ ] [ ]( )2222 2 XEZEZEZEXvar β+µ−µβ+β+µ+=

[ ] [ [ ] ( ) ]222ZEZEZE −β+=

[ ] ( ).2ZvarZE β+=

By the conditional expectation approach, we have

( ) ( ) ( )ZXvarEZXEvarXvar +=

( ) ( )ZvarZE β+µ+=

( ) ( ).2ZvarZE β+=

Part 4.

( ) ( [ ] [ ] ) ( [ ] [ ] [ ] [ ] )3233223 233 ZEZEZEZEZEZEX +−β+−β=µ

( ) ( )ZZvar 333 µβ+β=

( ) ( )

µ+

+βµ=

+β

µ

+β

µZ

tt

tZ

tt

tIIIX ZeEeeZtEetM

2222

22

33

( ) ( )

+βµ+

+β+

+β

µ

+β

µZ

tt

t

tt

tZetEeZeZtEe

2222

22

33

( ) ,23233

22

µ+

+β+

+β

µ

+β

µZ

tt

tZ

tt

teEeeZtEe

( ) [ ] [ ] [ ]ZEZEXEMIIIX µ+µβ+µ== 330 2233

[ ] [ ] [ ]3322 33 ZEZEZE β+βµ+β+


( ) ( ) ( [ ] [ ] [ ]) [ ]( )ZEZEZEZEXEXE β+µµβ+β+µ+= 22222

[ ] [ ] [ ] [ ]22223 2 ZEZEZEZE β+βµ+µβ+µ+µ=

[ ] [ ] [ ] [ ] ,222232

ZEZEZEZE µβ+β+βµ+

[ ] [ ] ( ) [ ] .333322233

ZEZEZEXE β+µβ+βµ+µ=

Therefore,

( ) ( [ ] [ ] ) ( [ ] [ ] [ ] [ ] )3233223 233 ZEZEZEZEZEZEX +−β+−β=µ

( ) ( ).3 33

ZZvar µβ+β=

Part 5.

( ) ( )

+βµ=

+β

µZ

tt

tIVX eZtEetM

2222

2

3

( ) ( )

+β++βµ+

+β

+β

µZ

ttZ

tt

teZteZtEe

22233

22

23

( )

×+βµ+

µ+

+β

µ

+β

µZ

tt

tZ

tt

teZtEeZeEe

2222

22

33

( )

+βµ+

+β

µZ

tt

teZtEe

22

2

( )

++β+

+β

+β

µZ

ttZ

tt

teZeZtEe

22232

22

3


( )

+βµ+

+β

µZ

tt

tZetEe

23

2

3

( )

+×+βµ+

+β

+β

µZ

ttZ

tt

tZeeZtEe

22222

22

3

( )

+βµ+

+β

µZ

tt

teZtEe

233

2

( ) ( )

+β++β+

+β

+β

µZ

ttZ

tt

teZteZtEe

232244

22

3

( ) .2324

22

+βµ+

µ+

+β

µ

+β

µZ

tt

tZ

tt

tZetEeeEe

Therefore,

[ ] [ ] [ ] [ ] [ ]233222344 12464 ZEZEZEZEXE µβ+µβ+βµ+βµ+µ=

[ ] [ ] [ ] [ ]442322 366 ZEZEZEZE β++β+µ+

since

( ) [ ( )( ) ]44 XEXEX −=µ

[ ] [ ] [ ] [ ] [ ] ( )[ ] ,36442234

XEXEXEXEXEXE −+−=

where

( ) ( ) [ ]( ) ( [ ] [ ] [ ]22233 333 ZEZEZEZEXEXE β+µ+µβ+µβ+µ=

[ ] [ ])3323 ZEZE β+βµ+


[ ] [ ] [ ] [ ]ZEZEZEZE βµ+µβ+µ+βµ+µ= 3222224 3333

[ ] [ ] [ ] [ ] [ ]223333 33 ZEZEZEZEZE µβ+µβ+βµ+µβ+

[ ] [ ] [ ] [ ] [ ],33 3422222ZEZEZEZEZE β+βµ+β+

( ) ( ) [ ] [ ] [ ] [ ]23222422222 ZEZEZEZEXEXE µ+βµ+βµ+µ+µ=

[ ] [ ] [ ] [ ] [ ]ZEZEZEZEZE23323 2222 µβ+βµ+µβ+βµ+

[ ] [ ] [ ]222322224 ZEZEZE βµ+β+βµ+

[ ] [ ] [ ] ,233224

ZEZEZE µβ+β+

[ ] [ ]( ) [ ] [ ]222344464 ZEZEZEXE β+µ+βµ+µ=β+µ=

[ ] [ ] .44433

ZEZE β+µβ+

Therefore,

( ) [ [ ] [ ] [ ] [ ] [ ] [ ] ]4223444 364 ZEZEZEZEZEZEX −+−β=µ

[ [ ] [ ] [ ] [ ] ]3232 236 ZEZEZEZE +−β+

[ ][ [ ] [ ] ] [ ].36 2222ZEZEZEZE +−β+

Finally,

( ) ( ) ( ) [ ] ( ) [ ]223

24

44 366 ZEZvarZEZZX +β+µβ+µβ=µ

which completes the proof.

Corollary 4.1. When ,,,2

3~

γδ−GIGZ we have

(1) ( ) ,1

2

δγ+βδ+µ=XE


(2) ( ) ( )( )

,1

2

22

δγ+γδα+γδ=Xvar

(3) skewness, ( ) ( )

,3

3

2222

3

1

δγα+γ

βδ+δγα+γ

βδ=γ

(4) excess kurtosis,

[ ( ) ( ( ) ( ))]( )

.221213

223

23424

2δα+γδγ

δα+γγ−δγ+αδγ+δγ+α=γ

Proof.

Part 1.

( ) .1

2

δγ+δβ+µ=XE

Part 2.

( ) ( ) ( )2

32

2

11 δγ+γδβ+δγ+

δ=Xvar

( )( )2

2232

1 δγ+γβ+γδ+γδ=

( )( )

.1

2

22

δγ+γδα+γδ=

Part 3.

( )( )

( ( ) )( )33

3625333

2

3

1

31

1

33

δγ+γγδ−γδ−δγ+δβ+

δγ+γβδ=µX

( )33

43333423

1

333

δγ+γγδβ+δβ+γβδ+γβδ=

( )( )

.1

333

2333

δγ+γδα+γγβδ+δβ=


Therefore, denoting skewness by ,1γ we get

( )

( )( )2

3

2

31

X

X

µ

µ=γ

( )( )

( )2

323

32

3

33

43333423 1

1

333

δα+γδ

δγ+γ×δγ+γ

γδβ+δβ+γβδ+γβδ=

( )

( )2

322

332233

δγα+γ

δβ+δγα+γβδ=

( ) ( ).

33

2222

3

δγα+γ

βδ+δγα+γ

βδ=

Part 4. Note that

( )( )45

3436254

41

312315

δγ+γδ+γδ+γδ+γδβ=µ X

( )( ) ( ) ( )δγ+γ

δ+δγ+γ

δβ+δγ+γ

γδ−γδ−δγ+δβ+1

31

61

316

3

3

52

33

3625332

( )( )

( )45

332222322

332334

1

3416

1453

δγ+γγδ+γδ+γδ+δγ+δγβ+

+δγ+γδ+γδδβ

=

( )( )

.1

331345

332243

δγ+γγδ+γδ+δγ+γδ+

Hence, the kurtosis is given by

( )( )( )

( ) ( )( )

δγ+γγδ+γδ+δγ+δγβ++δγ+γδ+γδδβ=

µµ

45

3322322332334

22

4

1

44161453

X

X

( )( )

( )( )

.1

1

3313224

42

45

332243

δα+γδδγ+γ×

δγ+γγδ+γδ+δγ+γδ+


Writing

,222 γ+δ=α

,2 42244 γ+γβ+β=α

we have

( ) .222 73532334624225223 γδ+γδβ+γδβ+γβ+γδβ+δγ=δα+γδγ

Denoting excess kurtosis by ,2γ we obtain

[]

( )223

6254422

322244224

226

82453

δα+γδγγδ+δγ+γ+γδβ+

δγβ+γβ+β+δγβ+γδβ

=γ

[ ( ) ]( )223

42232422424 442413

δα+γδγγδβ+δγβ+δγβ+γδβ+δγ+α=

[ ( ) ( ( ) ( ))]( )

.221213

223

23424

δα+γδγδα+γγ−δγ+αδγ+δγ+α=

Table 1. A summary of the theoretical property of the proposed mixed

model

Item Description Expression

1 ( )XE δγ+

βδ+µ1

2

2 ( )Xvar ( )( )2

22

1 δγ+γδα+γδ

3 Skewness, 1γ

( ) ( )

3

2222

33

δγα+γ

βδ+δγα+γ

βδ

4 Excess kurtosis, 2γ [ ( ) ( ( ) ( ))]

( )223

23424 221213

δα+γδγδα+γγ−δγ+αδγ+δγ+α


5. Parameter Estimation

Parametric methods commonly used in statistical inference are Method

of Moments (MoM) and Maximum Likelihood (ML) method. However,

these methods have some limitations. Equations obtained by these methods

require complex numerical techniques to solve in cases where the parameters

are hard to separate.

Alternative simple methods have been sought. One such method is

the EM-algorithm which was introduced by Dempster et al. [3] for ML

estimation for data containing missing values or data that can be considered

as producing missing values. Karlis [5] pointed out that the mixing operation

can be considered responsible for producing missing data. The statistical

beauty of the EM-algorithm is that it estimates the unobserved values using

the posterior expectations.

It becomes easier if we exploit the normal variance mean structure of the

GHD through this algorithm. Assume that true data are made of observed

part X and unobserved part Z. This then ensures that the log likelihood of the

complete data ( ),, ii zx ni ...,,3,2,1= factorizes into two parts (Kostas

[6]). The EM-algorithm consists of two main steps: the maximization step

which optimizes the log likelihood with respect to the parameters and the

expectation step which estimates the unobserved values using the posterior

expectations. This implies that the joint density of X and Z is given by:

( ) ( ) ( )., zgzxfzxf =

Therefore, the likelihood function for the joint data becomes:

( ) ( ) ( )∏ ∏= =

=µδβαn

i

n

i

iii zfzxfL

1 1

,,,

and the log likelihood

( ) ( ) ( )∑ ∑= =

+=µδβαn

i

n

i

iii zfzxfl

1 1

loglog,,,

( ) ( ).,, 21 γδ+βµ= ll


For these models,

( ) ( ) ( )∑ ∑

= =

β−µ−−−π−=βµn

i

n

ii

ii z

zxz

nl

1 1

2

1 .2

1log

2

12log

2, (33)

M-step

In this step, the log likelihood of the conditional distribution is optimized

with respect to its parameters :, βµ

( ) ( ),,

1

1 ∑=

β−µ−=βµβ∂∂ n

i

ii zxl

( ) ( )∑=

β−µ−=βµµ∂∂ n

ii

ii

z

zxl

1

1 .,

Equating these equations to zero and solving simultaneously, we obtain

∑

∑ ∑

=

= =

−

−=β

n

i i

n

i

n

i ii

i

zzn

zx

z

x

1

1 1

1

1

ˆ

and hence

.ˆ zx β−=µ

Therefore, at the kth iteration of the algorithm, the estimates for β and µ are

given by

( ) ,1

1

ˆ

1

1 11

∑

∑ ∑

=

= =+

−

−=β

n

i i

n

i

n

i ii

i

k

zzn

zx

z

x

(34)

( ) ( ) .ˆ 11zx

kk ++ β−=µ (35)

The case when

γδ− ,,

2

3~Z

The log likelihood of the mixing distribution ( )γδ,2l is also optimized

with respect to δ and γ as:


( ) ( ) ( )∑=

δγ+−−π−δ=γδn

i

i nzn

nl

1

2 1loglog2

52log

2log3, (36)

,2

1

1

22

∑=

γ+δ−δγ+

n

i

ii

zz

n (37)

the derivative of which with respect to γ is

( ) ( ) ,1

,

1

2 ∑=

γ−δγ+δ−δ=γδγ∂

∂ n

i

izn

nl

and that with respect to δ is

( ) ( ) .1

1

3,

1

2 ∑=

δ−γ+δγ+γ−δ=γδδ∂

∂ n

iiz

nnn

l

Equating the first equation to zero and simplifying, we obtain

,ˆ2

z

z

δ−δ=γ

where ∑ == n

i izn

z1

.1

Substituting for γ in the second equation and

simplifying, we obtain

( ) .01

13

11

1

2

1

1

1

2

∑∑∑

∑∑

==

=−

=

= =δ−

δ

−δ+δγ+

δ

−δ−δ

n

ii

n

i i

n

i i

n

i i

n

i i

zz

znn

z

znn

n

Note that

.1

1

2

∑ =

δ=δγ+n

i iz

n

Hence

.01

1

2

1

2

1 1

24 ∑ ∑∑ ∑= == =

=

+δ+

−δ

n

i

n

i

ii

n

i

n

ii

i zznz

zn


Letting ,2δ=t we obtain the quadratic equation

,01

2

11

2

1 1

2 =

+

+

− ∑∑∑ ∑

=== =

n

i

i

n

i

i

n

i

n

ii

i ztzntz

zn

where

a

acbbt

2

42 −±−=

with

,1

1 1

2 ∑ ∑= =

−=n

i

n

ii

i zzna

,

1

∑=

=n

i

iznb

.

2

1

= ∑

=

n

i

izc

Since ,t=δ at the kth iteration of the algorithm, the estimates for δ and γ

are, respectively,

( ) ,1t

k =δ +

( ) ( ( ) )( ) .

1

211

z

zk

kk

+

++

δ−δ=γ

These estimates involve computation of unknown values for random

variables: ,Z { }niZi ...,,2,1, = and { }....,,2,1,1niZi =− Estimation of

the values of these random variables amounts to performing the E-step.

E-step

The estimation of the random variables: ,Z { }niZi ...,,2,1, = and

{ }niZi ...,,2,1,1 =− is achieved by computing the posterior expectation

for ( )iii xXZE = and ( )iii xXZE =−1 using posterior distribution.


One attractive and useful feature of the GIG distribution in the mixing

mechanism is its conjugate for the normal distribution. That is, given

a conditional distribution ( )zzNZX ,~ β+µ and the mixing/prior

distribution to be ( ),,,~ γδλGIGZ the posterior distribution is

( ) ,,,2

1~

22

αµ−+δ−λ xGIGXZ

where .22 γ+β=α It can easily be shown that the moments around the

origin of the ( )γδλ ,,GIG distribution are given by

( ) ( )( )δγ

δγ

γδ=

λ+λ

K

KZE r

rr

and this formula holds for negative values of r, i.e., for inverse moments

too. When mixing with

γδ− ,,

2

3~ GIGZ posterior distribution becomes

( ( ) ).,,2~22 αµ−+δ− xGIGZ The required posterior expectations can

be computed as:

( ) ( ) ( ( ) )( ( ) )

,22

2

221

22

µ−+δα

µ−+δαα

µ−+δ==xK

xKxxXZE

( )( )

( ( ) )( ( ) )

.22

2

223

22

1

µ−+δα

µ−+δα

µ−+δ

α==−

−−

xK

xK

x

xXZE

These posterior expectations can now be used to compute the parameter

estimates via the EM-algorithm for maximum likelihood estimation

of the parameters for the proposed mixed model distribution. Let is denote

( ( ) )kiii xXZE θ= , and iw denote ( ( ) ),,1 k

iii xXZE θ=− where ( )kθ

denote the kth iteration values of the .,,2

3

γδ−GIG Then


( ) ( )( )( )

( ( ) ( ) ( )( ) )

( ( ) ( ) ( )( ) ),

2

1

2

2

1

12

1

ikkk

ikkk

ki

kk

i

xK

xKxs

φδα

φδααφδ=

( ) ( )( )

( ( ) ( ) ( )( ) )

( ( ) ( ) ( )( ) ),

2

1

2

2

1

3

2

1

ikkk

ikkk

ikk

k

i

xK

xK

x

w

φδα

φδα

φδ

α=

−

−

for ni ...,,2,1= and ( )( ) ( ( ) )( ( ) )

.12

2

k

kk x

xδ

µ−+=φ

Pseudo values calculated at the E-step can now be used to update the

other parameters as follows:

( ) ,ˆ 1t

k =δ +

( ) ( ( ) )( ) ,

ˆ

ˆˆ

1

211

M

Mk

kk

+

++

δ−δ=γ

( ) ,ˆ

1

1 11

∑∑ ∑

=

= =+

−

−=β

n

i i

n

i

n

i iiik

wsn

wxwx

( ) ( ) ,ˆˆ 11sx

kk ++ β−=µ

( ) ( ( ) ) ( ( ) ) .ˆˆˆ 21211 +++ β+γ=α kkk

The EM-algorithm converges to the maximum if the initial values are in

the admissible range.

The likelihood for the model is

( ) ( ) ( )βµ−δγ+δ+α+δγ+−π−=µβδα nnnnnxl loglog21loglog,,,;

( ) ( ( ) )∑ ∑ ∑= = =

αδφ+φ−β+n

i

n

i

n

i

iii xKxx

1 1 1

2

1

2 .log


Denote by ( )kl the log likelihood after k iterations. We stop iterating

when ( ( ) ( ) ) ( ) ,1tollll

kkk <− + where tol is the chosen tolerance level

( ).10.,e.g 6− Clearly, the criteria used for terminating the algorithm were

based on the relative change of the log likelihood.

Remark. Since the method of moment estimates is difficult to obtain

directly, we suggest to use the moment estimates for NIG as initial values as

presented by Karlis [5] formulation. Where iteration stops unexpectedly, a

slight modification of the parameters will suffice.

For NIG, the moments are:

,ˆ9

ˆˆ4ˆ9

ˆˆ

3,

3

ˆˆ,ˆˆ

ˆˆ,ˆ

ˆˆˆ

2

4221

2

2

21

22

32

αγγ+α

γδ=γγγ=β

γ+βγ=δ

γδβ−=µ sss

x

where x is the sample mean, 2s is the sample variance, while

23231 µµ=γ

and ,32242 −µµ=γ with ( )∑ =

− −=µ n

i

kik xxn

11 , i.e., the sample

skewness and kurtosis, respectively.

Therefore, it can be easily shown that:

( ).

53

3ˆ

212 γ−γ

=γs

The other parameters can be obtained sequentially by substituting the

value of .γ̂

6. Application

For data analysis, we consider Range Resource Corporation (RRC). The

historical weekly prices considered are from 3/01/2000 to 1/07/2013. RRC is

one of the top gainers in the energy sector. It is an independent oil and gas

exploration and production company based in Fort Worth, Texas. Range

is best known for its pioneering of the Devonian-aged Marcellus Shale in


Pennsylvania, which is now the most productive natural gas field in the

United States. Range has over 1 billion USD invested in Southwestern

Pennsylvania, while it also has operations in the Southwestern United States.

Founded in 1976, the current President and Chief Executive Officer is

Jeffrey L. Ventura.

Return for the dataset. Let ( )tP denote the price process of a security

at time t, in particular of a stock. In order to allow comparison of

investments in different securities, we investigate the rates of return defined

by

.loglog 1−−= ttt PPX

Figure 1. Histogram of the weekly log-returns of RRC, for the period

2000-2013 (702 observations).

The reason for this is that the return over n periods, for example n days,

is then just the sum

.loglog 11121 −−+−+++ −=++++ tntntttt PPXXXX ⋯


Figure 2. Fitting proposed model to RRC weekly returns.

We observe that

.784721.2,1890753.0,824736.2,2333151.0 21 =γ−=γ== sx

Therefore, the moment estimates for the NIG distribution are:

,950864.2ˆ,02456226.0ˆ,3714399.0ˆ =δ−=β=γ

.4284473.0ˆ,3722511.0ˆ =µ=α

Proposed model. The table shows Starting values and EM parameter

estimates for the Proposed model at different tolerance levels. The log

likelihood value and the number of iterations are also given. The

monotonicity property of the EM-algorithm can be seen from the table.


Table 2. Parameter estimates at different tolerance level

Parameter Starting values ( )510−=tolEM ( )610−=tolEM ( )810−=tolEM

α̂ 0.3722511 0.277191 0.2777899 0.2778586

β̂ –0.02456226 –0.0323062 –0.03234023 –0.03234413

δ̂ 2.950864 4.095578 4.098373 4.098694

µ̂ 0.4284473 0.4880231 0.4882531 0.4882795

Log likelihood –1695.205 –1695.459 –1695.488

No. of iteration 104 147 235

AIC 3398.41 3398.918 3398.976

7. Conclusion

This special case of the GHD fits the Range Resource Corporation

weekly returns quite well. It was able to capture the skewness and

excess kurtosis inherent in the data. The algorithm proposed was easily

programmable and straightforward. This special case is thus a competitive

model in statistical modelling of financial data.

Figure 3. Q-Q plot for proposed model.


Acknowledgment

The authors gratefully acknowledge the financial support from Kisii

University.

Also, the authors thank the referees deeply for every careful reading of

the manuscript and valuable suggestions that improved the paper.

References

[1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Function, Dover,

New York, 1972.

[2] O. Barndorff-Nielsen, Exponentially decreasing distributions for the logarithm of

particle size, Proc. R. Soc. Lond. A 353 (1977), 409-419.

[3] A. P. Dempster, N. M. Laird and D. Rubin, Maximum likelihood from incomplete

data using the EM algorithm, J. Roy. Statist. Soc. Ser. B 39 (1977), 1-38.

[4] E. Eberlein and U. Keller, Hyberbolic distributions in finance, Bernoulli

1(3) (1995), 281-299.

[5] D. Karlis, An EM type algorithm for maximum likelihood estimation of the

normal-inverse Gaussian distribution, Statist. Probab. Lett. 57 (2002), 43-52.

[6] F. Kostas, Tests of fit for symmetric variance gamma distributions, 15th European

Young Statisticians Meeting, 2007.

[7] K. Prause, Modelling financial data using generalized hyperbolic distributions,

FDM Preprint 48, University of Freiburg, 1999.

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