estimation of the central moments of a random...

STATISTICS IN TRANSITION new series, March 2017

7


Vol. 18, No. 1, pp. 7–26

ESTIMATION OF THE CENTRAL MOMENTS OF

A RANDOM VECTOR BASED ON THE DEFINITION OF

THE POWER OF A VECTOR

Katarzyna Budny1

ABSTRACT

The moments of a random vector based on the definition of the power of a vector,

proposed by J. Tatar, are scalar and vector characteristics of a multivariate

distribution. Analogously to the univariate case, we distinguish the uncorrected

and the central moments of a random vector. Other characteristics of a

multivariate distribution, i.e. an index of skewness and kurtosis, have been

introduced by using the central moments of a random vector. For the application

of the mentioned quantities for the analysis of multivariate empirical data, it

appears desirable to construct their respective estimators.

This paper presents the consistent estimators of the central moments of a random

vector, for which essential characteristics have been found, such as a mean vector

and a mean squared error. In these formulas, the relevant orders of approximation

have been taken into account.

Key words: central moment of a random vector, estimator, multivariate

distribution, power of a vector.

1. Introduction

One of the fundamental characteristics of the univariate random variable are

the ordinary (raw, uncorrected) and the central moments (e.g. Shao, 2003,

Jakubowski and Sztencel, 2004). Even order moments are measures of dispersion

of the distribution of the random variable, the moments of odd order characterize

their location. In the analysis of multivariate distributions the product moments

(about zero), the central mixed moments or collections thereof, e.g. mean vector,

covariance matrix, are considered as classical generalizations of the above

quantities (e.g. Johnson, Kotz and Kemp, 1992; Fujikoshi, Ulyanov and Shimizu,

2010). The uncorrected and central moments of a random vector are also

considered as the expectations of relevant Kronecker products of a random vector

(e.g. Holmquist, 1988). Thus, from this definition, they are size vector quantities

1 Cracow University of Economics. E-mail: [email protected].

8 K. Budny: Estimation of the central …

(collections of product moments (about zero) and central mixed moments of the

corresponding orders).

On the basis of the definition of the power of a vector, Tatar (1996, 1999)

suggested multivariate generalizations of the uncorrected and the central moments

of a random variable, which are different from the above. Let us recall the basic

definitions.

Definition 1.1. [Tatar 1996, 1999] Let ,,, RH be a Hilbert space with the inner

product , . For any vector Hv and for any number 0 NNr the

r -th power of the vector v is defined as follows:

Rvo 1 and

evenfor,

oddfor

1

1

rvv

rvvv

r

r

r .

Let r

kL be a space of random vectors whose absolute value raised to the

r-th power has finite integral, that is:

dPRLrkr

k XX :: .

In the literature, the measure

dPErr

XX is sometimes called the

moment of the order r of a random vector X and designated as rE X (see

Bilodeau and Brenner, 1999). Tatar (2002), however, by analogy with the

univariate case, defines this expression as the absolute moment of order r of a

random vector X . In this study, we will also regard these values as the absolute

moments of a random vector.

Therefore, let us assume that for the vector kR:X an absolute moment

of order r exists.

Definition 1.2. [Tatar 1996, 1999] The ordinary (raw, uncorrected) moment of

order r of the random vector kR:X is defined as

rkr E X, . 1.1

Let us note that the uncorrected moment of the first order is the mean vector,

that is:

XEmk ,1 .

The definition and basic properties of the central moments of a random vector

based on the definition of the power of a vector will be presented in the next

section.


9

2. The central moments of a random vector

Definition 2.1. [Tatar 1996, 1999] The central moment of order r of a random

vector kR:X , for which the absolute moment of order r exists, is given as

r

kr EE XX , . 1.2

Remark 2.1. According to the concept of the power of the vector, if r is an even

number, then Rkrkr ,, , , which means they are scalar quantities. However,

if r is an odd number, then kkrkr R,, , , so we obtain vectors.

Ordinary and central moments of a random vector based on the definition of

the power of the vector are defined at an arbitrarily fixed inner product in the

kR space. In the following part of the paper we will consider the Hilbert space

kR where we define the Euclidean inner product

k

i

iiwvwv1

, where

),,...,( 1 kvvv kk Rwww ),...,( 1 .

Remark 2.2. Let us observe that from the basic properties of the power of a

vector and multinomial theorem we get the following formulas:

k

t

l

tt

lll m

lk

i

iiklt

m

EXXEll

lEXXE

1

2

... 11

2

,2

1,..,

and

k

lk

i

iikl XXEXXE ,...,1

1

2

,12

k

t

kk

l

tt

lll m

k

t

l

tt

lll m

EXXEXXEll

lEXXEXXE

ll

lt

m

t

m 1

2

... 11

11

2

... 111

,..,,...,

,..,

where !...!

!

,.., 11 mm ll

l

ll

l

is a multinomial coefficient.

By this inner product, the second order central moment is called the total

variance of the random vector (see Bilodeau and Brenner, 1999) or the variance of

the random vector (see Tatar 1996, 1999). According to these terms, X2D will

denote the central moment of the second order of a random vector.

Let us recall that the variance of a random vector was used to present

multivariate generalization of Chebyshev's inequality (see Osiewalski and Tatar,

1999).

By using the central moments, characteristics of a multivariate distribution

such as index of skewness and kurtosis have also been introduced.


Definition 2.2. [Tatar, 2000] The index of skewness of a random vector kR:X , for which there is an absolute moment of third order, is called

2

32

3

2

3

,2

,3

,1

X

XXX

D

EE

k

k

k

. 2.2

Definition 2.3. [Budny and Tatar, 2009, Budny, 2009] Kurtosis of a random

vector kR:X , for which an absolute moment of fourth order exists, is a quantity

defined as

22

4

2,2

,4

,2

X

XXX

D

EE

k

k

k

. 3.2

Assuming further that the random vector kR:N has a multivariate

normal distribution, we obtain the form

k

ji

ji

k

jiji

jiij

k

i

i

k

1,

22

1,

222

1

4

,2

22

1

N . 4.2

The formula 4.2 was determined (Budny, 2012) using Isserlis theorem

(Isserlis, 1919), setting the algorithm for determination of the central mixed

moments of the normally distributed random vector.

For the application of the central moments for the analysis of multivariate,

empirical data, it appears desirable to construct their respective estimators. The

next section will present their form along with a discussion of basic properties.

3. The multivariate sample central moments

3.1. Construction and basic properties

At the beginning let us recall the form of multivariate sample raw moments

with their basic properties useful in the next part of this paper (Budny, 2014).

Suppose that kR:1

X ,…,kn R:X is a random sample from

multivariate distribution, i.e. a set of n independent, identically distributed (i.i.d.)

random vectors, with a finite thr absolute moment.


11

Definition 3.1.1. The multivariate sample raw moment of order r (the estimator

of the thr raw moment of a random vector) is called:

na

n

i

ri

kr

1

,

X

. 1.1.3

Multivariate sample raw moments are consistent and unbiased estimators, and

their central moments satisfy the condition

ss

krkr nOaE 2

,, , 2.1.3

for all Nsr , .

Let us therefore proceed to formulate the forms of estimators of the central

moments of a random vector.

Definition 3.1.2. The multivariate sample central moment of order r (the

estimator of the thr central moment of a random vector) is defined as

nm

n

i

ri

kr

1,

XX

. 3.1.3

Remark 3.1.1. According to the definition of the power of a vector: if r is an

even number, then krm , is a univariate random variable, and if r is an odd

number, then a random vector is obtained as krm , .

Remark 3.2.1. In the following discussion, while examining the properties of

multivariate sample central moments, we will assume, without loss of generality,

that the mean vector is a zero vector, i.e. 0,1 mk .

We will begin the analysis of the properties of estimators of the central

moments of a random vector by determining the form of their expected values. To

do this, we will first introduce some forms of multivariate sample central

moments, useful in further considerations.

Theorem 3.1.1. Multivariate sample central moments can be represented as

follows:

estimator of central moments of even order:

s

p

pkps

s

p

pkpsksks a

p

spa

p

sam

1

12,122

1

2,22,2,2 ,2 XX

n

l

p

p

sn

i

s

p

lpilpsi

p

l

lplp

1 2

222

0

,21 XXXX

, 4.1.3


estimator of central moments of odd order:

ksm ,12

n

p

sp

ap

sa

n

i

s

p

psipi

s

p

kpsp

ks

1 1

21222

1

,2122

,12

,2 XXXX

X

X

XXXX

ks

n

i

s

p

psip

l

lpillplp

mn

l

p

p

s

,2

1 2

2122

0

2 ,21

.

5.1.3

Proof: It is easily seen that from the definition of the power of a vector, we get

above formulas.

The computation leading to explicit forms of the expected value of

multivariate sample central moments is tedious and does not bring any relevant

elements for further consideration. So, the next theorem will present their form

with appropriate order of approximation (for univariate case - see Cramér 1958, p.

336). Prior to the formulation of this result, let us consider the following lemma.

Lemma 3.1.1. Assume that 0\Nr . Let us consider a multivariate distribution

(in population) for which the absolute moment of order r2 exists. Then, for every

1,...1 rt we get

2

,

t

tktr nOaE X , 6.1.3

where the operator "" is defined (Tatar, 2008) as follows:

odd - ,gdy ,

even odd, gdy

odd even, for

even - ,for

lkwv

lkvw

lkwv

lkwv

wv

lk

kl

lk

lk

lk .

Proof: see Appendix.

Property 6.1.3 will play a key role in the study of property of unbiasedness

of multivariate sample central moments. It will be used in the proof of the

theorem, which presents forms of their expected values with the relevant order of

approximation.

Theorem 3.1.2. Under the assumptions of Lemma 3.1.1.

1,,

nOmE krkr . 7.1.3


13

Proof: First, let us consider multivariate sample central moments of even orders.

Towards 4.1.3 we get:

XX ,2 ,12

1

2

,22,2,2 ks

s

p

p

kpsksks asEaEp

saEmE

s

p

pkpsaE

p

sp

2

12,122 ,2 X

n

El

p

p

sn

i

s

p

lpilpsi

p

l

lplp

1 2

222

0

,21 XXXX

. 8.1.3

By the assumption 0,1 mk , an easy computation shows that

n

aEks

ks

,2

,12 ,

X , 9.1.3

n

aEks

ks

,12

,2

X . 10.1.3

Note that for each ni ,...,1 and Nt we obtain the following property

2,

tt

i nOE XX . 11.1.3

Indeed, due to Minkowski's inequality in the tL1 , Schwarz's inequality in

21L and property 2.1.3 , applied to the coordinates of univariate sample mean

(Cramér, 1958, p. 332), we get:

t

k

j

tt

jij

t

ttk

j

jij

ti XXEXXEE

1

1

1

1

,XX

t

k

j

t

t

tij

t

k

j

tt

j

tij

n

AXEXEXE

1

1

2

1

2

1

1

2

122

1

2

21

2

11

2

t

t

k

j

ttij

n

CnAXE

,

for each 0nn where 0, CA .


Let us also note that by Jensen's inequality and Hölder's inequality, after

taking into account properties 2.1.3 and 11.1.3 we have:

lpilpsi

lpilpsi EE XXXXXXXX ,, 2222

2

22,

lp

s

lp

sis

l

ss

ps

si

n

CEEE

XXXX

for each ni ,...,1 , sp ,...,2 , 2,...,0 pl and 0nn where 0C .

Clearly, this leads to the conclusion that

11 2

222

0

,21

nOn

El

p

p

sn

i

s

p

lpilpsi

p

l

lplpXXXX

. 12.1.3

Finally, the use of the properties in the following order: unbiasedness of

multivariate sample raw moments, 6.1.3 , 9.1.3 and 2.1.3 , to the equation

8.1.3 implies

1,2,2

nOmE ksks . 13.1.3

In order to determine the form of the expected value of the multivariate

sample central moments of odd orders, let us note that

s

p

kpsp

ksks aEp

saEmE

1

,2122

,12,12 X

n

Ep

sp

n

i

s

p

psipi

1 1

21222,2 XXXX

X

XXXX

ks

n

i

s

p

p

l

psilp

illplp

mEn

El

p

p

s

,2

1 2

2

0

2122 ,21

.

14.1.3

At the beginning we shall show that for each ni ,...,1 , sp ,...,2 and

2,...,0 pl there is a property

22122 ,

lppsi

lpil nOE XXXX . 15.1.3


15

Indeed, Jensen's and Hölder's inequalities, the properties 2.1.3 and 11.1.3

imply

psi

lpilpsi

lpil EE

42424

22122 ,, XXXXXXXX

12

212

,2224

1212212

2

122 , lps

ps

klps

lps

lp

lpsilps

l

lpsEE XXX

lp

lps

lp

lps

lps

l

lpslps

ps

klpsn

C

n

A

n

A

12

12

212

2

12

112

212

,2224 ,

for each 0nn where 0,, 21 CAA , which obviously is equivalent to 15.1.3 .

This implies, therefore, a property

11 2

2

0

2122 ,21

nOn

El

p

p

sn

i

s

p

p

l

psilp

illplpXXXX

.

16.1.3

Note that the reasoning analogous to the one carried out above leads to

another property expressed as

2

121222,

ppsipi nOE XXXX , 17.1.3

for each ni ,...,1 and sp ,...,1 .

Furthermore, the elementary computation establishes the equality

n

Enssii ,1212

,

XXX , 18.1.3

for every ni ,...,1 .

Owing to the conditions 17.1.3 and .18.1.3 we get the property

11 1

21222,

nOn

Ep

sp

n

i

s

p

psipiXXXX

. 19.1.3


The proof of the theorem will be completed by determining the order of

approximation of a quantity XksmE ,2 , that takes the form

n

El

p

p

s

aEmE

n

i

s

p

llp

ipsip

l

lplp

ksks

1 1

122

0

,2,2

,21 XXXX

XX

Reasoning analogous to that shown in the proof of the property 15.1.3

justifies the expression

2

1

122,

lp

llp

ipsi nOE XXXX 20.1.3

for each ni ,...,1 , sp ,...,1 and pl ,...,0 . Thus, we get the condition

11 1

122

0

,21

nOn

El

p

p

sn

i

s

p

llp

ipsip

l

lplpXXXX

,

which, together with 10.1.3 leads to the property

1,2

nOmE ks X 21.1.3

Finally, after the consecutive application of unbiasedness of multivariate

sample raw moments and the forms 6.1.3 , 19.1.3 , 16.1.3 and 21.1.3 to the

equation 14.1.3 we get

1,12,12

nOmE ksks , 22.1.3

which completes the proof of the theorem.

Corollary 3.1.1 Multivariate sample central moments are asymptotically unbiased

estimators of the central moments of a random vector.

3.2. Consistency of the multivariate sample central moments

Our considerations in this section will focus on finding orders of

approximations of mean squared errors of multivariate sample central moments,

i.e. quantities of the form 2

,, krkrmE .


17

At the beginning we will take into account even-order sample central

moments. Let us note that the expression 2

,2,2 ksksmE from the formula

4.1.3 can be presented in the following form:

2

,2,2

2

,2,2 ksksksks aEmE

n

aEl

p

p

sn

i

s

p

lpilpsi

ksks

p

l

lplp

1 1

22

,2,2

0

,21 XXXX

n

l

p

p

sE

n

ii

s

pp

p

l

p

l z

lpilpsilp

z

z

z

zzzz

zzzz

1, 1, 0 0

2

1

22

21 21

1

1

2

2

,2 XXXX

.

1.2.3

Based on the property 2.1.3 , we have

12

,2,2 nOaE ksks . 2.2.3

Let us also note that

2

1

22

,2,2 ,

lplp

ilpsiksks nOaE XXXX , 3.2.3

for each ni ,...,1 , sp ,...,1 , pl ,...,0 .

In fact, considering Schwarz's inequality and Hölder's inequality we get

lp

ilpsiksksaE XXXX ,22

,2,22

lp

ipsilksks

zEaE2442

,2,2 , XXXX

d

lp

did

ps

did

ld

ksks EEEaE

2

2

22

22

,2,2 ,XXXX ,

where lpsd 2 .

Owing to the properties 2.1.3 , 11.1.3 and the condition 0,1 km , we

get

lp

ilpsiksksaE XXXX ,22

,2,22

lp

d

lp

dd

ps

kd

d

l

d n

C

n

A

n

A

n

A

1

32

,2

2

21 ,


where 0,,, 321 CAAA , which justifies 3.2.3 .

Property 3.2.3 leads to condition

11 1

22

,2,2

0

,21

nOn

aEl

p

p

sn

i

s

p

lpilpsi

ksks

p

l

lplpXXXX

.

4.2.3

Reasoning analogous to that shown above justifies the expression

2

2

1

222121

,

llpp

z

lpilpsi

nOEzz

zzz

z XXXX , 5.2.3

for each nii ,...,1, 21 , spp ,...,1, 21 , 11 ,...,0 pl , 22 ,...,0 pl .

Therefore,

11, 1, 0 0

2

1

22

21 21

1

1

2

2

,2

nOn

l

p

p

sE

n

ii

s

pp

p

l

p

l z

lpilpsilp

z

z

z

zzzz

zzzz XXXX

.

6.2.3

Taking into account the form 1.2.3 and the properties 2.2.3 , 4.2.3 and

6.2.3 we get

12

,2,2

nOmE ksks . 7.2.3

Now, we will take into consideration the estimators of central moments of odd

order. Let us note that on the basis of 5.1.3 , the expression 2,12,12 ksksm

can be presented as the relevant linear combination of the components of the form

2,12,12 ksksa ,

122,12,12 ,,

psi

lpil

ksksa XXXX ,

122

,12,12 ,,

llp

ipsiksksa XXXX ,

122122 222

22

21

111

11 ,,,

l

lpipsipsi

lpil

XXXXXXXX .

Moving on to determine the order of approximation of the expected values of

these components, we note that 2.1.3 implies


19

12

,12,12

nOaE ksks . 8.2.3

2

,12,12

2

,12,12 ksksksks aEaE .

Carrying out further analysis we, in turn, apply Jensen's, Schwarz's and

Hölder's inequalities and, thanks to them, we obtain

2122

,12,12 ,,psi

lpil

ksksaE XXXX

24242

,12,12 ,psi

lpil

ksksaE XXXX

d

ps

did

lp

did

ld

dd

ksks EEEaE

12

222

21

2

,12,12 ,

XXXX ,

where lpsd 22 .

This estimation, after taking into account the properties 2.1.3 , .11.1.3 and

the fact that 0,1 km leads to

2122

,12,12 ,,psi

lpil

ksksaE XXXX

lpd

ps

kd

d

lp

d

d

l

d

d

d n

C

n

A

n

A

n

A

1

12

,23

2

2

1

1 ,

which is equivalent to the property

2

1122

,12,12 ,,

lppsi

lpil

ksks nOaE XXXX ,

for each ni ,...,1 , sp ,...,1 , pl ,...,0 .

A slight change in the proof above shows that

2

2

122

,12,12 ,,

lp

llp

ipsiksks nOaE XXXX ,

for each ni ,...,1 , sp ,...,1 , pl ,...,0 and


2

2

1221222121

222

22

21

111

11 ,,,

llpp

llp

ipsipsilp

ilnOE XXXXXXXX

for each nii ,...,1, 21 , spp ,...,1, 21 , 11 ,...,0 pl , 22 ,...,0 pl .

The above properties leads, thus, to

12

,12,12

nOmE ksks . 9.2.3

The next theorem will include the summary of the above consideration.

Theorem 3.2.1. Assume that for any natural number 2r , a quantity

2

,, krkrmE exists. Then, the mean squared error of the multivariate sample

central moment of order r is of the form

12

,,,MSE nOmEm krkrkr .

From this theorem and multivariate generalization of Chebyshev's inequality

(Osiewalski and Tatar, 1999) we obtain a very important corollary.

Corollary 3.2.1 The multivariate sample central moments are consistent

estimators of the central moments of a random vector.

4. Numerical illustration

Multivariate sample central moments, and also their respective functions, can

be useful tools for the analysis of multivariate empirical data, for instance for the

multivariate financial items, which have been considered in Budny and Tatar

(2014).

Budny, Szklarska and Tatar (2014) have presented an analysis of socio-

demographic conditions of Poland taking into account the multivariate (four-

dimensional) data from the Central Statistical Office of Poland from 2012, and

administrative division of the country into counties (powiats). The coordinates of

the data are: the total marriage rate, the total divorce rate, the total fertility rate

and the total mortality rate.

For this data, the second, third and fourth multivariate sample central

moments in selected regions of Poland are as follows.

North region (72 counties of the voivodeships: West Pomeranian,

Kuyavian.

Pomernian, Pomeranian, Warmian-Masurian, without city counties):


21

4744,1,2 km ,

36230

01710

12270

00220

,3

.-

.

.-

.-

m k , 5.3820,4 km .

East region (74 counties of the voivodeships: Podlaskie, Lublin,

Subcarpathian.

Lesser Poland, without city counties):

2.8700,2 km ,

34623

05310

29430

32860

,3

.

.-

.

.-

m k , 27.7891,4 km .

West region (97 counties of the voivodeships: Greater Poland, Lubusz,

Lower.

Silesian, Silesian, Opole, without city counties):

1.8058,2 km ,

1.1132

0.0448-

0.1702

0.2345-

,3 km , 9.1024,4 km .

Central region (71 counties of the voivodeships: Masovian,

Świętokrzyskie, Łódź, without city counties):

2.3571,2 km ,

1.4188-

0.0188

0.1529

0.3748-

,3 km , 14.0671,4 km .

City counties (65 counties):

2.9811,2 km ,

0.69907

0.00003

0.0828-

0.0125-

,3 km , 22.4378,4 km .

Dispersion of the distribution of the vector of the socio-demographic situation of Poland is measured by the central moments of the even orders. Note that the highest dispersion of test vector was observed in city counties while the smallest in the counties of the northern region (see Budny, Szklarska, Tatar, 2014). Central moments of odd order of multivariate distribution are parameters of location.


They can also be considered as a measure of the asymmetry (see Tatar, 2000) and as a vector measures indicate the direction of asymmetry. The above considerations can be supplemented (see Budny, Szklarska, Tatar, 2014) using the functions of the central moments of a random vector, e.g. index of skewness (Tatar 2000) and kurtosis (Budny, Tatar 2009, Budny 2009). Let us mention that the index of skewness shows the direction of asymmetry while its square informs also about the size of the asymmetry.

The problem of estimation of this characteristics of multivariate distribution is

left for further study.

5. Conclusions

In this paper we have proposed consistent and asymptotically unbiased

estimators of the central moments of a random vector based on the power of a

vector. Essential characteristics such as mean vectors and mean squared errors

with the relevant orders of approximation have been established for them. The

central moments of even order are parameters of dispersion of the distribution of a

random vector. The moments of odd order characterize its location. These

quantities can be useful tools for the analysis of multivariate data.

Acknowledgement

Publication was financed from the funds granted to the Faculty of Finance and

Law at Cracow University of Economics, within the framework of the subsidy for

the maintenance of research potential.

REFERENCES

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25

APPENDIX

Proof of lemma 3.11.: The property 6.1.3 will be proved by considering the

cases of even and odd powers of random vectors. Reasoning (in each case) will be

based on Schwarz's inequality considered in the relevant Hilbert's space 2

kL or

2

1L .

At the beginning, let sr 2 and 12 pt where sp ,...1 . It is, therefore,

necessary to prove the property

2

1

12,122 ,

pp

kps nOaE X , 1

For the proof we will apply Schwarz's inequality in Hilbert's space 2kL

with the inner product Y,X:Y,X 2 EkL that leads to

242,122

12,122

2 ,

p

kpsp

kps EaEaE XX .

Let us note that 12

,,22

, OaEaDaE krkrkr . Thus, taking into

account the property 2.1.3 we get

1212,122

2 , pp

kps nOaE X ,

which is equivalent to the condition 1 .

Now, we consider the even numbers sr 2 and pt 2 where 1,...1 sp

In view of this, the property

ppkps nOaE

2,22 X . 2

requires the proof.

Note that Schwarz's inequality in the space 21L of the property 2.1.3

justifies the estimations

p

pkps

pkps

n

CEaEaE

2

42,22

2,22

2 XX ,


which obviously leads to 2 .

In turn, for odd natural numbers 12 sr and 12 pt where

1,...1 sp it is necessary to prove the property

2

1

12,22

pp

kps nOaE X . 3

For this proof, let us note that Jensen's inequality, Schwarz's inequality and

the condition 2.1.3 imply a sequence of inequalities, respectively.

estimation of the central moments of a random...

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