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Applied Mathematics, 2017, 8, 1918-1938

http://www.scirp.org/journal/am

ISSN Online: 2152-7393

ISSN Print: 2152-7385

DOI:

10.4236/am.2017.812136 Dec. 29, 2017 1918 Applied Mathematics

Some Applications of Higher Moments of the

Linear Gaussian White Noise Process

I. S. Iwueze

, C. O. Arimie

, H. C. Iwu

, E. Onyemachi

Department of Statistics, Federal University of Technology, Owerri, Nigeria

Department of Mathematics and Statistics, University of Portharcourt, Portharcourt, Nigeria

Abstract

The Linear Gaussian white noise process is an independent and identically

distributed (iid) sequence with zero mean and finite variance with distribution

( )

0,N

. Hence, if

, ,,

XX X

is a realization of such an iid sequence,

this paper studies in detail the covariance structure of

,,,,1,2,

dd d

XX Xd=

By this study, it is shown that: 1

) all powers of a Linear Gaussian White Noise

Process are iid but, not normally distributed and 2) the higher moments (v

riance and kurtosis) of

, 2,3,

Xd= 

can be used to distinguish between

the Linear Gaussian white noise process and other processes with similar c

variance structure.

Keywords

Stochastic Process,

Linear Gaussian White Noise Process,

Covariance

Structure, Stationarity, Test for White Noise Process, Test for Normality

1. Introduction

The objective of estimation procedures is to produce residuals (the estimated

noise sequence) with no apparent deviations from stationarity, and in particular

with no dependence among these residuals. If there is no dependence among

these residuals, then we can regard them as observations of independent random

variables; there is no further modeling to be done except to estimate their mean

and variance. If there is significant dependence among the residuals, then we

need to look for the noise sequence that accounts for the dependence [1].

In this paper, we examine the covariance structure of powers of the noise se-

quence when the noise sequence is assumed to be independent and identically

distributed normal (Gaussian) random variates with mean zero and finite va-

How to cite this paper:

Iwueze, I.S., Ari-

mie, C.O.

, Iwu, H.C. and Onyemachi, E

(201

7) Some Applications of Higher Mo-

ments of the Linear Gaussian White Noise

Process

Applied Mathematics

, 1918-1938.

https://doi.org/10.4236/am.2017.812136

Received:

August 28, 2017

Accepted:

December 26, 2017

Published:

December 29, 2017

7 by authors and

Scientific

Research Publishing Inc.

This work is licensed under the Creative

Commons Attribution International

License (CC BY

4.0).

http://creativecommons.org/licenses/by/4.0/

Open Access

I. S. Iwueze et al.

DOI:

10.4236/am.2017.812136 1919 Applied Mathematics

riance,

. Some simple tests for checking the hypothesis that the residuals

and their powers are observed values of independent and identically distributed

random variables are also considered. Also considered are tests for normality of

the residuals and their powers.

The stochastic process

Xt T∈

is said to be strictly stationary if the distri-

bution function is time invariant. That is;

( ) ( )

12 1 2

,,, , ,,

t t t tk tk t k

Fx x x Fx x x

++ +

=

(1.1)

where

( )

12 1 1 2 2

,,, , ,,

m mm

tt t t t t t t t

Fx x x PX x X x X x

=≤≤ ≤

(1.2)

That is, the probability measure for the sequence

is the same as that for

for all

. If a series satisfies the next three equations, it is said to be

weakly or covariance stationary.

( )

( )( )

( )

1. , 1, 2, ,

EX t

EX X

E X X Rt t

µ µσ

µµ



= = ∞





− − = <∞









− −= −









(1.3)

If the process is covariance stationary, all the variances are the same and all

the covariances depend on the difference between

and

. The moments

( )( )

( )

, 0,1, 2, .

t tk

E X X Rk k

µµ



− −= =





(1.4)

are known as the autocovariance function. The autocorrelations which do not

depend on the units of measurements of

are given by

( )

(

)

, 0,1, 2,

= = 

(1.5)

A stochastic process

Xt Z∈

, where

, 1, 0, 1,Z = −



, is called a white

noise if with finite mean and variance all the autocovariances (1.4) are zero ex-

cept at lag zero [

( )

0Rk =

, for

0k ≠

]. In many applications,

Xt Z∈

is as-

sumed to be normally distributed with mean zero and variance,

<∞

, and

the series is called a linear Gaussian white noise process if:

( )

var

0, otherwise

1, 0

0, otherwise





























(1.6)

and

( )

12 1

1, 0

, , ,,

0, otherwise

kk t tk t t tk

corr X X X X X

+ + + +−



= =







(1.7)

where

is known as the partial autocorrelation function. For large

, the

I. S. Iwueze et al.

DOI:

10.4236/am.2017.812136 1920 Applied Mathematics

sample autocorrelations:

(

)

(

)

(

)

( )

t tk

X XX X

−

−−

−

∑

(1.8)

of an iid sequence

, ,,

XX X

with finite variance are approximately distri-

buted as

0,N







[1] [2] [3]. We can use this to do significance tests for the

autocorrelation coefficients by constructing a confidence interval. Here

, ,,

XX X

is a realization of such an iid sequence, about

(

)

100 1 %

−

the sample autocorrelations should fall between the bounds:

−

(1.9)

where

−

is the

−

quartile of the normal distribution. If the null and al-

ternative hypothesis are:

( ) ( )

: 0 0 and : 0 for some 0

Hk k Hk k

ρρ

= ∀≠ ≠ ≠

(1.10)

where

( )

are autocorrelations at lag

computed for

, ,,

XX X

We can also test the joint hypothesis that all

of the

( )

correlation

coefficients are simultaneously equal to zero. The null and alternative hypothesis

are:

( ) ( ) ( ) ( )

H : 1 2 0 and H : 0 for 1,2, ,

XX X X

m iim

ρρ ρ ρ

==== ≠=

(1.11)

The most popular test for (1.11) is the [4] portmanteau test which admits the

following form

( ) ( )

BP X

Qmn k





∑

(1.12)

where

is the so-called lag truncation number [5] and (typically) assumed to be

fixed [6]. Under the assumption that

, ,,

XX X

is an iid sequence,

( )

is asymptotically a chi-squared random variable with m degree of

freedom. [7] modified the

( )

statistic to increase the power of the test in fi-

nite samples as

( )

() 2

Q m nn









= +



−



∑

(1.13)

Several values of

are often used and simulation studies suggest that the

choice of

( )

lnmn≈

provides better power performance [8].

Another Portmanteau test formulated by [9] can be used as a further test for

iid hypothesis, since if the data are iid, then the squared data are also iid. It is

based on the same statistic used for the Ljung-Box test as

I. S. Iwueze et al.

DOI:

10.4236/am.2017.812136 1921 Applied Mathematics

(

)

( )

(

)

Q m nn









= +



−





∑

(1.14)

where the sample autocorrelations of the data are replaced by the sample auto-

correlations of the squared data,

( )

According to [6], the methodology for testing for white noise can be roughly

divided into two categories: time domain tests and frequency domain tests. Oth-

er time domain tests include the turning point test, the difference-sign test, the

rank test [1]. Another time domain test is to fit an autoregressive model to the

data and choosing the order which minimizes the AICC statistic. A selected or-

der equal to zero suggests that the data is white noise [1].

Let

(

) (

)

[ ]

e, π

2π

ωρω

∞

= ∈−

∑

(1.15)

be the normalized spectral density of

Xt Z∈

. The normalized spectral density

function for the linear Gaussian white noise process is

( )

[ ]

π,π

2π

ωω

= ∈−

(1.16)

The equivalent frequency domain expressions to H

and H

are

( )

[ ]

, π,π

2π

ωω

= ∈−

and H

( )

[ ]

, π,π

2π

ωω

≠ ∈−

(1.17)

In the frequency domain, [10] proposed test statistics based on the famous

and

processes [6], and a rigorous theoretical treatment of their limiting

distributions was provided by [11]. Some contributions to the frequency domain

tests can be found in [12] and [13], among others. This study will concentrate on

the time domain approach only.

A stochastic process

Xt Z

∈

may have the covariance structure (1.6) even

when it is not the linear Gaussian white noise process. Examples are found in the

study of bilinear time series processes [14] [15]. Researchers are often con-

fronted with the choice of the linear Gaussian white noise process for use in

constructing time series models or generating other stationary processes in si-

mulation experiments. The question now is, “How do we distinguish between

the linear Gaussian white noise process from other processes with similar cova-

riance structure”? Additional properties of the linear Gaussian white noise

process are needed for proper identification and characterization of the process

from other processes with similar covariance structure. Therefore, the ultimate

aim of this study is on the use of higher moments for the acceptability of the li-

near Gaussian white noise process. The first moment (mean) and second or

higher moments (variance, covariances, skewness and kurtosis) of powers of the

linear Gaussian white noise process was established in Section 2. The methodol-

ogy was discussed in Section 3, the results are contained in Section 4 while Sec-

tion 5 is the conclusion.

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