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Some Goodness of Fit Tests for Random Sequences

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Abstract

In this paper we had made an attempt to incorporate the results from the theory of square Gaussian random variables in order to construct the goodness of fits test for random sequences (time series). We considered two versions of such tests. The first one was designed for testing the adequacy of the hypotheses on expectation and covariance function of univariate non-centered sequence, the other one was constructed for testing the hypotheses on covariance of the multivariate centered sequence. The simulation results illustrate the behavior of these tests in some particular cases.
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Lithuanian Journal of Statistics 2013, vol. 52, No 1, pp. 5-13
Lietuvos statistikos darbai 2013, 52 t., Nr. 1, 5-13 p.
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www.statisticsjournal.lt
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SOME GOODNESS OF FIT TESTS FOR RANDOM SEQUENCES
Yuriy Kozachenko1, Tetiana Ianevych2
Taras Shevchenko National University of Kyiv, Mechanics and Mathematics Faculty
Address: Volodymyrska str., 64/13, Kyiv, 01601, Ukraine
E-mail: 1ykoz@ukr.net, 2yata452@univ.kiev.ua
Received: August 2013 Revised: September 2013 Published: November 2013
Abstract. In this paper we had made an attempt to incorporate the results from the theory of square Gaussian random variables in
order to construct the goodness of ﬁts test for random sequences (time series). We considered two versions of such tests. The ﬁrst one
was designed for testing the adequacy of the hypotheses on expectation and covariance function of univariate non-centered sequence,
the other one was constructed for testing the hypotheses on covariance of the multivariate centered sequence. The simulation results
illustrate the behavior of these tests in some particular cases.
Keywords: goodness of ﬁt test, multivariate random sequence, time series, square Gaussian random variable.
1. Introduction
The task of investigating the properties of a random sequence is very important for its application. Very often some
phenomena can be observed only at certain points in time. In some cases the values of continuous quantities, such as
temperature and voltage, can be written only at discrete moments of time. And even if the observations can be recorded
continuously we can only use discrete data for computational purposes. That is why we usually deal with random se-
quences or time series in practice. The latter term is used more frequently but we prefer to use the former one designating
the connection to random processes.
There is much literature devoted to this topic, in particular the classic books on the statistical analysis of time series
written by Anderson [1], Box and Jenkins [2], and Brockwell and Davis [4].
To date, many goodness-of-ﬁt tests in time series are residual-based. For example, the classic portmanteau test of
Box and Pierce [3] and its improvement by Ljung and Box [15] are based on the sample autocorrelations of the residuals.
In the context of goodness of ﬁt of nonlinear time series models, the McLeod and Li [16] test is based on the sample
autocorrelations of the squared residuals. Based on a spectral approach of the residuals, Chen and Deo [6] proposed some
new diagnostic tests. More recently, perhaps inﬂuenced by the empirical distribution function approach in the goodness-
of-ﬁt test for independent observations, substantial developments for time series data have taken place in the form of tests
based on empirical processes marked by certain residuals, see for instance Chen and Härdle [7] and Escanciano [8]. For
In the model-based approach to time series analysis, estimated residuals are computed once a ﬁtted model has been
obtained from the data, and these are then tested for “whiteness”, i. e. it is determined whether they behave like white
noise. Tests for residual whiteness generally postulate whiteness of the residuals as the Null Hypothesis, so that signiﬁcant
rejections indicate model inadequacy. These tests require the computation of residuals from the ﬁtted model, which can be
quite tedious when the model does not have a ﬁnite order autoregressive representation. Also, in such cases, the residuals
are not uniquely deﬁned.
In this paper we use another approach gained from the theory of square Gaussian random variables. This theory
was developed in works by Kozachenko et al. [11], [12], [13], for the investigation of stochastic processes. In the book
by Buldygin and Kozachenko [5] the properties of the space of square Gaussian random variables were studied and the
connection with Orlicz spaces of random variables was established. We use this theory for the construction goodness
of ﬁt tests on the expectation and covariance function for the non-centered univariate stationary Gaussian sequence and
covariance function for the centered but multivariate stationary random sequence. Our tests do not require the computation
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6Some Goodness of Fit Tests for Random Sequences
of residuals and can be applied to inﬁnite order representations. This paper is the continuation of the work started in [14],
which was devoted to testing the centered univariate random sequence.
The paper consists of 5 sections and 2 annexes. The second section is devoted to the theory of square Gaussian
random variables and contains the main deﬁnitions and results. In particular, we found the estimate for the distribution
of the maximum of the quadratic form of the square Gaussian random variables. Sections 3 and 4 apply the estimator
obtained in section 2 to construct different aggregate tests.
The test in section 3 was constructed for testing the aggregated hypothesis on the expectation and covariance function
of the non-centered stationary Gaussian sequence. It is based on the approach used in stochastic process L2theory. Within
this inference the process identiﬁcation is made on the basis of the two main characteristics: mathematical expectation
and covariance function.
In the section 4 we consider multivariate sequences. The approach analyzing the residuals dominates in the multi-
variate case too. See, for example, papers by Hosking [9], [10], Mahdi and McLeod [17] and the references therein. The
goodness of ﬁt test we have constructed for the centered Gaussian multivariate stationary sequence is based on ﬁtting the
covariance function.
The power properties of our tests were studied through simulations. Section 5 draws some conclusions. Some neces-
sary mathematical calculations are relegated to the annexes at the end.
2. Square Gaussian random variables
Let Ξ={γi,iI}be a family of joint Gaussian random variables for which Eγi=0 for all iI.
Deﬁnition 1. [13] The space SGΞ()is the space of square Gaussian random variables if any element ξSGΞ()can
be presented as
ξ=
γTA
γE
γTA
γ,(1)
where
γT= (γ1,...,γr),γiΞ,i=1,r,Ais a real-valued matrix; or the element ξSGΞ()is the square mean limit of
the sequence {ξn,n1}of the form (1)
ξ=l.i.mnξn.
It was proved by Buldygin & Kozachenko in [5] that SGΞ()is a linear space.
For the square Gaussian random variables the following results hold true.
Theorem 1. [11] Let ~
ξT= (ξ1,ξ2,...,ξd)be a random vector such that ξiSGΞ()and Bbe a symmetric semi-deﬁnite
matrix. Then for all 0<s<1
2the following inequality is true
Ecosh
v
u
u
ts2~
ξTB
~
ξ
E~
ξTB
~
ξ
R(2s),(2)
where R(y) = 1
1yexpy
2,0<y<1.
Theorem 2. Let {ηm,1mM}be a sequence of random variables that can be presented as quadratic form of square
Gaussian random variables (that is, η=~
ξTB
~
ξ, where Bis a symmetric semi-deﬁnite matrix). Then, for any x 0
Pmax
1mM
ηm
Eηm
>xM·W(x),(3)
where W (x) = R2x
1+2x
coshx
1+2xand the function R is deﬁned in Theorem 1.
Proof. For all 0 <s<1/2 and x>0
Pmax
1mM
ηm
Eηm
>xMmax
1mM
Pηm
Eηm
>x
Mmax
1mM
Ecoshqs2ηm
Eηm
coshs2xMR(s2)
cosh(sx).
Yuriy Kozachenko, Tetiana Ianevych 7
Putting s=x
1+2x, which is approximately the minimal point, we obtain (3) and
Pmax
1mM
ηm
Eηm
>xM(12s)1/2exp{−s/2}
cosh(sx)
Me1/2exp n1
2(1+2x)op1+2x
coshx
1+2x=
=O(x1/4ex)as x.
The theorem is proved.
3. Testing a hypotheses on the expectation and covariance function of a univariate sequence
Using the inequality (3) it is possible to test a hypothesis on the expectation and covariance function of the non-
centered univariate stationary Gaussian sequence. Hereinafter we will consider stationarity in a strict sense.
Let us consider the stationary sequence {γ(n),n1}for which Eγ(n) = aand E(γ(n)a)(γ(n+m)a) = B(m),
m0 is its covariance function. We assume that we have N+Mconsecutive observations of this random sequence. Let
us choose the estimators in the following way:
for the expectation
b
am=1
N
N
n=1
γ(n+m),0mM;
for the covariance function
b
B(m) = 1
N
N
n=1
(γ(n)b
a0)(γ(n+m)b
am),0mM.
We denote
Ea:=E(b
ama)2=1
N2
N
n=1
N
k=1
B(nk) = 1
NB(0) + 2
N2
N1
i=1
(Ni)B(i);
EB(m):=Eb
B(m) = B(m)1
N2
N
n=1
N
k=1
B(nkm) =
=11
NB(m)2
N2
N1
i=1
(Ni)B(im),0mM,
and introduce the random variables:
ηa(m):= (b
ama)2Ea,0mM;
and
ηB(m):=b
B(m)EB(m),0mM.
It is easy to prove that these random variables are square Gausssian.
Let us deﬁne~
η(m)T= (ηa(m),ηB(m)), 0 mM. So, for any semi-deﬁnite matrix Bm= (bi j(m))i,j=1,2the random
variable η(m):=~
η(m)TBm~
η(m)is actually the quadratic form of a square Gaussian random variable.
Remark 1.If bi j =(1,i=j;
0,i6=j.(that is, Bis the identity matrix of order 2), then η(m) = η2
a(m)+ η2
B(m)and Eη(m) =
Eη2
a(m) + Eη2
B(m). All the necessary calculations for the terms of Eη(m)are included to the Annex to this section.
Remark 2.If for every mBm=C1(m)is the inverse of matrix C(m), whose components are the covariances between
the vector ~
η(m)items then Eη(m) = const for all m. But, in this case one should be careful since the matrices C(m)have
to be invertible.
Criterion 1. Let the null hypothesis H0state that a and B(m), m 0are the expectation and covariance function of the
non-centered Gaussian stationary sequence {γ(n),n1}and the alternative Haimplies the opposite statement.
8Some Goodness of Fit Tests for Random Sequences
If for signiﬁcance level αand corresponding critical value εαwhich can be found from the equation MW (εα) = α
max
0mM
~
ηT(m)Bm~
η(m)
E(
~
ηT(m)Bm~
η(m)) >εα,
then the hypothesis H0should be rejected and accepted otherwise.
Remark 3.The probability of a type I error for Criterion 1 is less than or equal to α.
Example 1.Let us consider the non-centered Gaussian sequence {γ(n),n1}whose elements can be presented according
to the expression
γ(n) = a+
j=0
β(j)ζnj,n1 (4)
where β(j) = eλj,j0, λ>0 and {ζk,kZ}is a sequence of independent random variables such that for all kEζk=0,
Eζ2
k=1. In this case Eγ(n) = afor all nand
B(m) = E(γ(n)a)(γ(n+m)a) =
j=0
β(j)β(j+m) =
j=0
eλjeλ(j+m)=eλ|m|
1e2λ,mZ.(5)
Using the simulation study we investigated how the Criterion 1 works. We made 10 000 Monte Carlo simulations of
the non-centered Gaussian stationary sequence γ(n), with a=Eγ(n)and covariance function deﬁned by (5) with ﬁxed
parameter λ. For our needs we used the the simulation methods developed in paper [18].
For the symmetric semi-deﬁnite matrix Bmwe choose the identity matrix of order 2 – I2. Then η(m) = ~
ηT(m)Bm~
η(m) =
η2
a(m) + η2
B(m).
1. Let us check the null hypothesis (H0) that states that the stationary Gaussian sequence γ(n)has expectation a=1 and
covariance function deﬁned by the formula (5) with parameter λ=1 versus the alternative hypothesis (Ha) implying that
the stationary Gaussian sequence γ(n)has expectation a=1 and covariance function deﬁned by the formula (5) with
parameter λ=0.5.
We simulated 10 000 realizations of the sequences deﬁned by (4) with parameters a=1 & λ=1 and a=1 & λ=0.5.
Let us deﬁne the necessary constants: the signiﬁcance level α=0.1, M=10, N=1000 (M+N=1010). In this case
the critical value εα=87.82.
For the simulated sequences we obtained an estimate of the probability of a type I error b
α=0 and the estimate for the
probability of a type II error b
β=0.23.
2. Let us now check the null hypothesis (H0) that states that the stationary Gaussian sequence γ(n)has expectation a=1
and covariance function deﬁned by the formula (5) with parameter λ=1 versus the alternative hypothesis (Ha) implying
that the stationary Gaussian sequence γ(n)has expectation a=0 and covariance function deﬁned by the formula (5) with
parameter λ=1.
We used again the simulated 10 000 realizations of the sequence deﬁned by (4) with parameters a=1 & λ=1 and
another sequence with parameters a=0.25 & λ=1.
The required constants are the same as previously. In this case we obtained an estimate for the probability of a type I
error b
α=0 and the estimate for the probability of a type II error b
β=0.0055.
Remark 4.It is evident that the more observations we have the more sensitive the criterion is. Finding the number Nfor
which the null and alternative hypotheses can be distinguished is the subject of our continuing investigation.
Yuriy Kozachenko, Tetiana Ianevych 9
4. Testing a hypotheses on the covariance function of a centered multivariate sequence
The inequality (3) can also be useful for testing a hypothesis on the covariance function of a centered multivariate
random sequence.
Let us assume that the components of the multivariate random sequence
γ(n),n1 are jointly Gaussian, stationary
(in the strict sense) sequences {γk(n),n1,k=1,K}for which Eγk(n) = 0 and Eγk(n)γl(n+m) = Bkl (m),m0 is the
covariance function of these sequences. It is worth mentioning that for k=l Bkk is the ordinary autocovariance function
of the k-th component and when k6=l Bkl are the joint covariances or sometimes called cross-covariances. Hereinafter we
shell use the term covariance function of the sequence
γ(n).
We suppose that the sequence
γ(n)is observed at points 1,2,...,N+M(N,M>1). As an estimator of the covariance
function Bkl (m)we choose
b
Bkl (m) = 1
N
N
n=1
γk(n)γl(n+m),N1,m=0,M.
The estimator b
Bkl (m)is unbiased:
Eb
Bkl (m) = 1
N
N
n=1
Eγk(n)γl(n+m) = Bkl (m).
The random variables kl (m) = b
Bkl (m)Bkl (m)are square Gaussian since b
Bkl (m)can be presented as
(γk(1),...,γk(N))TA(γl(m+1),...,γl(N+m)), where the matrix
A=
1
N··· 0
.
.
.....
.
.
0··· 1
N
.
Let ~
(m)be a vector with components kl (m).
Criterion 2. Let the null hypothesis H0state that Bkl (m), m 0is the covariance function of the centered Gaussian
stationary sequence
γ(n) = {γk(n),n1}k=1,Kand the alternative Hastate the opposite.
If for signiﬁcance level αand corresponding critical value εα, which can be found from the equation MW (εα) = α,
max
0mM
~
T(m)Bm~
(m)
E(~
T(m)Bm~
(m)) >εα,
then the hypothesis H0should be rejected and accepted otherwise.
Remark 5.The probability of a type I error for the Criterion 2 is less or equal to α.
Remark 6.The simplest way is to choose the matrix Bmto be identical.
If for every mthe matrix Bm=C1(m)is inverse to C(m)which consists of the covariances of the vector ~
(m)
components, then E(~
T(m)B
~
(m)) = const for all m. But in this case we should pay attention to the invertibility of the
matrices C(m).
Let’s illustrate how this criterion works on the example.
Example 2.We consider the K=2 – component stationary centered Gaussian sequence
γ(n) = {γk(n),n1,k=1,2}.
We assume that each component can be presented as moving average
γk(n) =
j=0
βk(j)ξnj,n1,
with coefﬁcients βk(j) = eλkj,λk>0, k=1,2, j0 and the random variables ξjare independent with zero mean and
variance equal to 1.
If the components of
γ(n)are not independent then the covariance function of this sequence has a form:
Bkl (m) =
eλlm
1e(λk+λl),m0;
eλk|m|
1e(λk+λl),m<0, k,l=1,2.(6)
10 Some Goodness of Fit Tests for Random Sequences
In the case when k=lwe obtain the covariance function of the k-th component:
Bkk(m) = eλk|m|
1e2λk,mZ,k=1,2.(7)
Let ~
T(m) = (11(m),12 (m),21(m),22(m)) and matrix B=I4is the identity matrix of 4-th order. Then
E(~
T(m)B
~
(m)) =
2
k=1
2
l=1
E2
kl (m),(8)
where
E2
kl (m) = 1
N"1
(1e2λk)(1e2λl)+e2λlm
(1e(λk+λl))2#+
+2
N
m
t=11t
N"e(λk+λl)t
(1e2λk)(1e2λl)+e2λlm
(1e(λk+λl))2#+
+2
N
N1
t=m+11t
N"e(λk+λl)t
(1e2λk)(1e2λl)+e(λk+λl)t(λlλk)m
(1e(λk+λl))2#=
=O1
N,as N.(9)
Using simulations we investigated how the Criterion 2 works. We had made 10 000 Monte Carlo simulations of two
sequences with covariance function deﬁned by (6) and (7). For the simulations we used the methods described in the
paper by Vasylyk et al. [18].
1. Let the null hypothesis (H0) state that the two components of the multivariate sequence
γ(n)are two jointly Gaussian,
centered stationary sequences γ1(n)and γ2(n)with covariance function deﬁned by (6) and (7) with parameters λ1=1 and
λ2=0.1, respectively, and the alternative hypothesis (Ha) state that λ1=0.3 and λ2=3.
We deﬁne the constants as α=0.1, M=10, N=1000 (M+N=1010). Under these deﬁnitions the critical value
εα=87.82.
We simulated 10 000 realizations of the two bivaiate sequences with parameters λ1=1 & λ2=0.1 and λ1=0.1 &
λ2=1. We obtained an estimate for the probability of a type I error b
α=0 and an estimate for the probability of a type II
error b
β=0.9616.
2. Let the null hypothesis (H0) state that the components of the multivariate sequence
γ(n)are two jointly Gaussian,
centered stationary sequences γ1(n)and γ2(n)with covariance function deﬁned by (6) and (7) with parameters λ1=1 and
λ2=0.1, respectively, and the alternative hypothesis (Ha) state that λ1=0.05 and λ2=5.
We used again simulated 10 000 realizations of the two bivariate sequences with parameters λ1=1 & λ2=0.1 and
λ1=0.05 & λ2=5 and deﬁned the constants as previously.
In this case we obtained an estimate for the probability of a type I error b
α=0 and the estimate for the probability of a
type II error b
β=0.0104.
Remark 7.It is evident that the more observations we have the more sensitive the test is.
5. Conclusions
In this paper we estimated the distribution of the maximum of a random sequence which can be presented as a
quadratic form of square Gaussian random variables. This result made it possible to build the criterion for testing a
hypothesis on the expectation and covariance function of a non-centered univariate stationary Gaussian sequence and a
hypothesis on the covariance function of a centered multivariate stationary Gaussian sequence. The simulation studies
were also incorporated.
The inequality obtained in the section 2 can also be useful for testing the similar hypotheses for non-centered mul-
tivariate random sequences. Our test statistics are quite easy to compute and do not require the calculation of residuals
from the ﬁtted model. This is especially advantageous when the ﬁtted model is not a ﬁnite order autoregressive model.
There is of course, a lot of room for improvement of the tests. Comparison with other tests and ﬁnding the number N
for which the null and alternative hypotheses can be distinguishable are also very important issues for further investiga-
tion.
Yuriy Kozachenko, Tetiana Ianevych 11
6. Acknowledgments
We are grateful to two anonymous referees for their insightful comments that have signiﬁcantly improved the paper.
7. Annex to section 3
This Annex includes the requirements for the section 3 calculations for Eη2
a(m)and Eη2
B(m).
Eη2
a(m) = E((b
ama)2Ea)2=E(b
ama)4(Ea)2
Using the Isserlis’ formula for the centered Gaussian random variables e
γ(n):=γ(n)awe obtain
E(b
ama)4=1
N4
N
n=1
N
k=1
N
t=1
N
s=1
Ee
γ(n+m)e
γ(k+m)e
γ(t+m)e
γ(s+m) =
=1
N4
N
n=1
N
k=1
N
t=1
N
s=1
[B(kn)B(st) + B(tn)B(sk) + B(sn)B(tk)] =
=3
N4 N
n=1
N
k=1
B(kn)!2
.(10)
Then
Eη2
a(m) = 3
N4 N
n=1
N
k=1
B(kn)!2
1
N2
N
n=1
N
k=1
B(kn)!2
=2
N4 N
n=1
N
k=1
B(kn)!2
.(11)
Let us make the required calculation for Eη2
B(m).
Eη2
B(m) = E(b
B(m)EB(m))2=E(b
B(m))2(EB(m))2.
E(b
B(m))2=1
N2
N
n=1
N
k=1
E(γ(n)b
a0)(γ(n+m)b
am)(γ(k)b
a0)(γ(k+m)b
am).
Using the Isserlis’ formula we obtain that
Eη2
B(m) = S02S1+S2(EB(m))2,(12)
where
S0=1
N2 N2B2(m) +
N
n=1
N
k=1
[B2(kn) + B(kn+m)B(knm)]!;
S1=1
N2B(m)
N
n=1
N
k=1
B(kn+m) +
+1
N3
N
n=1
N
k=1
B(kn)!2
+ N
k=1
B(kn+m)! N
l=1
B(lnm)!
;
S2=1
N4
N
n=1
N
k=1
B(kn)!2
+2 N
n=1
N
k=1
B(kn+m)!2
.
8. Annex to section 4
In chapter 4 we need to ﬁnd the expectation of 2
kl (m)in order to calculate the value of E(~
T(m)B
~
(m)) (see
formula (8)). Let us do it.
E2
kl (m) = E(b
Bkl (m)Bkl (m))2=E"1
N
N
n=1
(γk(n)γl(n+m)Bkl (m))#2
=
=1
N2E"N
n=1
N
i=1
(γk(n)γl(n+m)Bkl (m))(γk(i)γl(i+m)Bkl (m))#=
=1
N2
N
n=1
N
i=1
Eγk(n)γl(n+m)γk(i)γl(i+m)B2
kl (m).
12 Some Goodness of Fit Tests for Random Sequences
Using again the Isserlis’ formula for the centered Gaussian random variables we obtain
Eγk(n)γl(n+m)γk(i)γl(i+m) = Eγk(n)γl(n+m)Eγk(i)γl(i+m) +
+Eγk(n)γk(i)Eγl(n+m)γl(i+m) +
+Eγk(n)γl(i+m)Eγl(n+m)γk(i) =
=B2
kl (m) + Bkk (in)Bll (in) + Bkl (in+m)Blk (inm).
Then
E2
kl (m) = 1
N2
N
n=1
N
i=1
[Bkk(in)Bl l (in) + Bkl (in+m)Blk (inm)] =
=1
N[Bkk(0)Bl l (0) + B2
kl (m)] +
+2
N
N1
t=11t
N[Bkk(t)Bl l (t) + Bkl(t+m)Bl k(tm)]
Putting the covariance function deﬁned by (6) into the last formula we get (9).
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Yuriy Kozachenko, Tetiana Ianevych 13
KAI KURIŲ ATSITIKTINIŲ SEKŲ SUDERINAMUMO KRITERIJAI
Yuriy Kozachenko, Tetiana Ianevych
Santrauka. Kvadratinių Gauso atsitiktinių dydžių teorijos rezultatai pritaikyti atsitiktinių sekų (laiko eilučių) su-
derinamumo kriterijams sudaryti. Nagrinėjami du tokių kriterijų atvejai. Pirmasis kritetijus skirtas tikrinti hipotezei
apie vienmatės necentruotos sekos vidurkį ir kovariacinę funkcijąs; antrasis kriterijus skirtas tikrinti hipotezei apie dau-
giamatės centruotos sekos kovariacinę funkciją. Modeliavimo rezultatai iliustruoja šių kriterijų elgesį kai kuriais atskirais
atvejais.
Reikšminiai žodžiai: suderinamumo kriterijus, daugiamatė atsitiktinė seka, laiko eilutė, kvadratinis Gauso atsi-
tiktinis dydis.
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