We consider the problem of testing hypotheses regarding the covariance matrix of multivariate normal data, if the sample size s and dimension n satisfy lim [n,s→∞] n/s = y. Recently, several tests have been proposed in the case, where the sample size and dimension are of the same order, that is y ∈ (0,∞). In this paper we consider the cases y = 0 and y = ∞. It is demonstrated that standard techniques are not applicable to deal with these cases. A new technique is introduced, which is of its own interest, and is used to derive the asymptotic distribution of the test statistics in the extreme cases y = 0 and y = ∞. --sphericity test,random matrices,Wishart distribution

Birke, Melanie

Dette, Holger

English

Research Papers in Economics

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Birke, Melanie; Dette, Holger
Working Paper
A note on testing the covariance matrix
for large dimension
Technical Report // Universität Dortmund, SFB 475 Komplexitätsreduktion in Multivariaten
Datenstrukturen, No. 2004,02
Provided in cooperation with:
Technische Universität Dortmund
Suggested citation: Birke, Melanie; Dette, Holger (2003) : A note on testing the
covariance matrix for large dimension, Technical Report // Universität Dortmund,
SFB 475 Komplexitätsreduktion in Multivariaten Datenstrukturen, No. 2004,02, http://
hdl.handle.net/10419/49373A note on testing the covariance matrix for large dimension
Melanie Birke
Ruhr-Universit¨ at Bochum
Fakult¨ at f¨ ur Mathematik
44780 Bochum, Germany
e-mail: melanie.birke@ruhr-uni-bochum.de
Holger Dette
Ruhr-Universit¨ at Bochum
Fakult¨ at f¨ ur Mathematik
44780 Bochum, Germany
e-mail: holger.dette@ruhr-uni-bochum.de
F A X :+ 4 923 43 21 45 5 9
December 19, 2003
Abstract
We consider the problem of testing hypotheses regarding the covariance matrix of multi-
variate normal data, if the sample size s and dimension n satisfy lim
n,s→∞
n/s = y. Recently,
several tests have been proposed in the case, where the sample size and dimension are of the
same order, that is y ∈ (0,∞). In this paper we consider the cases y =0a n dy = ∞. It is
demonstrated that standard techniques are not applicable to deal with these cases. A new
technique is introduced, which is of its own interest, and is used to derive the asymptotic
distribution of the test statistics in the extreme cases y =0a n dy = ∞.
AMS Subject Classiﬁcation 62H15; 62E20
Keywords and phrases: sphericity test, random matrices, Wishart distribution
1 Introduction
A common assumption in multivariate statistics is that of an i.i.d. sample of normally distributed
n-dimensional random variables X1,...,X s+1 ∼N(μ,Σ), where μ ∈ Rn denotes the mean vector
and Σ ∈ Rn×n a positive deﬁnite covariance matrix. In this paper we are interested in the problems
of testing the hypothesis of sphericity, that is
H0 : ∃ σ ∈ I R
>0 :Σ = σIn ,H 1 :Σ  = σIn ∀ σ ∈ I R
>0, (1.1)
and in the problem of testing the hypothesis
H
 
0 :Σ=In ,H
 
1 :Σ = In , (1.2)
1if the sample size s and dimension n converge to inﬁnity, where In ∈ Rn×n denotes the identity
matrix. Note that the identity matrix In in (1.2) can be replaced by any other positive deﬁnite
matrix Σ0 ∈ Rn×n by multiplying the data by Σ
−1/2
0 . Classical analysis assumes that the dimension
is ﬁxed, while the sample size converges to inﬁnity. However, recent applications in gene express
experiments require statistical methods, which are applicable in the case, where the dimension is
large compared to the sample size [see Sebastini, Gussoni, Kohane and Ramoni (2003)]. Because
likelihood ratio tests [see e.g. Muirhead (1982) or Anderson (1984)] are not applicable in the case
n ≥ s, Ledoit and Wolf (2002) suggested to use certain alternative tests which were originally
proposed by John (1971) and Nagao (1973) in the case, where the dimension n is ﬁxed and the
sample size converges to inﬁnity. To be precise, let X =( X1,...,X s+1) and deﬁne
S =
1
s
(XX
T − ¯ X ¯ X
T) (1.3)
as the sample covariance matrix, where ¯ X = 1
s+1
s+1
j=1Xj is the mean of the observations. For
testing the hypothesis of sphericity John (1971) proposed to reject H0 for large values of the
statistic
U =
1
n
tr

S
1/ntr(S)
− In
2
(1.4)
where tr A denotes the trace of the matrix A. For the problem of testing the hypothesis H 
0 that
the covariance matrix is given by the identity Nagao (1973) proposed to reject the hypothesis in
(1.2) for large values of the statistic
V =
1
n
tr(S − I)
2 , (1.5)
Recently, Ledoit and Wolf (2002) investigated the properties of these tests if the sample size s and
the dimension n converge to inﬁnity at the same rate, that is
n
s
→ y ∈ (0,∞). (1.6)
They proved asymptotic normality for an appropriately standardized version of U and V and
showed that in this case the test of sphericity based on the statistic U is still consistent, while
the test for the hypothesis H 
0, which rejects the null hypothesis for large values of the statistic
V, is not necessarily consistent under assumption (1.6). For this reason Ledoit and Wolf (2002)
proposed to reject the null hypothesis H 
0 in (1.2) for large values of the statistic
W =
1
n
tr(S − In)
2 −
n
s

1
n
tr(S)
2
+
n
s
(1.7)
and showed asymptotic normality of a standardized version of W and consistency of the corre-
sponding test. The proof consists of an application of the delta-method to the vector (1
ntrS, 1
ntrS2),
for which the asymptotic normality has been established by Jonsson (1982).
In the present note we investigate the properties of the statistics U,V and W in the extreme cases
where the sample size and dimension are not of the same order, that is
n
s
→ y ∈{ 0,∞}. (1.8)
2We will demonstrate that in these cases appropriately standardized versions of the statistics U,V
and W are still asymptotically normal distributed. However, the method used by Ledoit and Wolf
(2002) to prove these results under assumption (1.6) is not applicable any more. These diﬃculties
are illustrated in Section 2. In Section 3 we use a direct argument to establish asymptotic normality
under the assumption (1.7), which is of its own interest and based on special structures of the
Wishart matrix. This technique avoids the application of the delta-method and may be useful to
investigate other properties of the sample covariance matrix. Finally, some of the more technical
arguments are given in the Appendix [see Section 4].
2 The proof based on the delta-method — revisited
In this section we brieﬂy explain the diﬃculties in the application of the delta-method to establish
asymptotic normality of the statistics U,V and W under assumption (1.8). The variances of 1
n trS
and 1
n trS2 can be calculated by standard methods, that is
Var
1
n
trS

=
2
ns
Var
1
n
trS
2

=
8
ns
+
20n2 +2 0 n
n2s2 +
8n3 +2 0 n2 +2 0 n
n2s3 .
Deﬁne
Fn =

1
n tr(S) − E

1
n tr(S)
	
1
n tr(S2) − E

1
n tr(S2)
	


, (2.1)
then the following result can either be established by similar methods as given in Jonsson (1982)
or by the same arguments as given in the proofs of the results in Section 3.
Lemma 2.1. If assumption (1.8) is satisﬁed with y =0 , then
√
nsFn
D −→ N (0,A)
where the asymptotic covariance matrix A is deﬁned by
A =

24
48


.
If assumption (1.8) is satisﬁed with y = ∞, the variances of both components of the vector Fn are
of diﬀerent order.
It follows from Lemma 2.1 that the delta-method is not applicable in the case y = ∞, because
we cannot standardize both components of Fn simultaneously in order to obtain a nondegenerate
limit distribution. Note that the ﬁrst component tr( 1
nS) requires a standardization with
√
ns in
order to obtain a positive limiting variance. However, such a standardization would yield for the
variance of the second component
Var
√
ns
n
trS
2

=8
n2
s2 + o(
n2
s2),
3which converges to inﬁnity. Moreover, even in the case y = 0 the application of the delta-method
with the asymptotic normality in Lemma 2.1 produces substantial diﬃculties. For this we consider
examplarily the statistic U which is obtained as
U = g
1
n
trS,
1
n
trS
2

,
where the function g is deﬁned by g(x,y)=y/x2 − 1. Observing that
E
1
n
trS

=1 ,E
1
n
trS
2

=
n + s +1
s
[see Jonsson (1982)] we obtain that the random variable
√
ns

U − g(1,
n + s +1
s
)

is asymptotically normal distributed with mean 0 and variance

−2
n + s +1
s
,1


24
48


−2n+s+1
s
1


=
n +1
s
2
= o(1).
Consequently, the standardization from the ﬁrst part of Lemma 2.1 yields a degenerate limit
distribution for the statistic
√
ns

U −
n +1
s

in the case where n/s → 0. Moreover, the same diﬃculties appear if this technique is used to
derive the asymptotic distribution of the statistics V and W. In the following section we use an
alternative methodology to derive asymptotic properties of these statistics.
3 The Wishart matrix
Note that under the null hypothesis H0 or H 
0 the estimator S in (1.3) has essentially (up to a
scaling factor) an n-dimensional Wishart distribution with s degrees of freedom. The following
Lemma shows that trS and trS2 have rather simple representations in terms of χ2-distributed
random variables. Throughout this paper the symbol
D = means equality in distribution.
Lemma 3.1. If W denotes an n-dimensional Wishart matrix with s degrees of freedom and
covariance matrix Σ=In,t h e n
tr(W)
D =
1
s
min{n,s} 
i=1

Yi + X|s−n|+i

tr

W
2 D =
1
s2
min{n,s} 
i=1

Yi + X|s−n|+i
2 +
2
s2
min{n,s}−1 
i=1
YiX|s−n|+i+1
4where X  ∼ χ2 ( ) and Yi ∼ χ2 (i) are independent chi-square distributed random variables.
As an application of Lemma 2.1 we can ﬁnd very simple expressions for the statistics U,V,W
in terms of independent random variables with a χ2 distribution. These results can be used to
obtain a direct proof of asymptotic normality based on a central limit theorem for m-dependent
triangular arrays of random variables. We will illustrate the general strategy for the statistic V
in the case n ≥ s, all other cases are treated similary. The following lemma gives an asymptotic
equivalent representation for the statistic V. The proof is straightforward and therefore omitted.
Lemma 3.2. If the null hypothesis (1.2) is valid and lim
n,s→∞
n/s = y ∈ [1,∞] the statistic V
deﬁned in (1.5) satisﬁes

s3
n

V −
n +1
s

D =
s 
i=1
Ys,i + An .
Here An = op(1) and the random variables Ys,i are given by
Ys,i =

1
√
n3 s

V
(n)
i − E

V
(n)
i

− 2

s
n3

Z
(n)
i − E

Z
(n)
i

(3.1)
with Z
(n)
i = Yi + Xn−s+i, V
(n)
i = Z
(n)
i
2
+2YiXn−s+i+1, where Xi,Y i are deﬁned in Lemma 3.1.
Theorem 3.3. If the null hypothesis (1.2) is valid and lim
n,s→∞
n/s = y ∈ [1,∞], then

s3
n

V −
n +1
s

D −→ N

0,
4
y
+8

,
where 1/∞ := 0.
The following theorems contain the corresponding statement for the case
lim
n,s→∞
n/s = y ∈ [0,1)
and for the statistics U and W deﬁned by (1.5) and (1.7), respectively. The proofs are similar and
therefore omitted.
Theorem 3.4. If the null hypothesis (1.2) is valid and lim
n,s→∞n/s = y ∈ [0,1], then
s

V −
n +1
s

D −→ N (0,4+8 y)
Remark 3.5. It was proved by Nagao (1973) that the test, which rejects the hypothesis
H
 
0 :Σ=In
5for large values of the statistic V, is consistent if the dimension n is ﬁxed and the sample size s
converges to inﬁnity. Ledoit and Wolf (2002) demonstrated that this statement is not correct if n
and s converge at the same rate to inﬁnity. Theorem 3.3 shows that the test is still consistent if
lim
n,s→∞
n/s =0 .
The following result shows that the test based on W for the hypothesis H 
0 and the test based on
U for the hypothesis of sphericity is always consistent independently of the value
y = lim
n,s→∞
n/s ∈ [0,∞].
Theorem 3.6. If the null hypothesis (1.2) is valid and lim
n,s→∞
n/s = y ∈ [0,∞] then
s

W −
ns+ s − 2
s2

D →N(0, 4)
Theorem 3.7. If the null hypothesis (1.1) of sphericity is valid and lim
n,s→∞
n/s = y ∈ [0,∞], then
s

U −
n +1
s

D −→ N (0,4)
Acknowledgements. The work of H. Dette was supported by the Deutsche Forschungsgemein-
schaft (SFB 475, Komplexit¨ atsreduktion in multivariaten Datenstrukturen). The authors would
like to thank Isolde Gottschlich who typed numerous versions of the paper with considerable
technical expertise.
4 Appendix: Proofs
Proof of Lemma 3.1. We will only consider the case s ≥ n, the remaining case can be treated by
similar arguments. By its deﬁnition an n-dimensional Wishart matrix with s degrees of freedom
and covariance matrix Σ = In can be represented as
W =
1
s
YY
T (4.1)
where Y =( Yij)
j=1,...,s
i=1,...,n is an (n × s) matrix with independent identically distributed entries, such
that Yij ∼N (0,1). It now follows from Silverstein (1985) that the matrix W is orthogonally
similar to a triangular matrix ˜ A =( ˜ aij)i,j=1,...,n with entries
˜ ai,i =
1
s

 Y
2
n−i+1 +  X
2
s−i+1

,i =1 ,...,n
˜ ai,i+1 =
1
s
 Yn−i  Xs−i+1,i =1 ,...,n− 1
˜ ai+1,i =
1
s
 Yn−i  Xs−i+1,i =1 ,...,n− 1,
6where ˜ Yn =0 , ˜ X2
i ∼ χ2(i); ˜ Y 2
i ∼ χ2(i) are independent χ2 distributed random variables ( ˜ Xi ≥
0; ˜ Yi ≥ 0). Therefore it is easy to see that the matrix W has the same eigenvalues as the triangular
matrix A =( aij)i,j=1,...,n with entries
ai,i =˜ an−i+1,n−i+1 =
1
s

 Y
2
i +  X
2
s−n+i

,i =1 ,...,n
ai,i+1 =˜ an−i,n−i+1 =
1
s
 Yi  Xs−n+i+1,i =1 ,...,n− 1
ai+1,i =˜ an−i+1,n−i =
1
s
 Yi  Xs−n+i+1,i =1 ,...,n− 1
Introducing the notation Xi = ˜ X2
i ,Y i = ˜ Y 2
i the assertion of the Lemma is now obvious.

Proof of Lemma 3.2. By Lemma 3.1 we have

s3
n

V −
n +1
s

D =
s 
i=1

1
√
n3 s

V
(n)
i − E

V
(n)
i

−2

s
n3

Z
(n)
i − E

Z
(n)
i

−
1
√
n3 s

Y
2
s − E

Y
2
s
	
−
2
√
n3 s
(YsXn − E[YsXn])
−
2
√
n3 s
(YsXn+1 − E[YsXn+1])
+2

s
n3 (Ys − E[Ys]).
and a straightforward but tedious calculation now shows that the last four terms are of order op(1)
[see Birke (2003)]. 
Proof of Theorem 3.3. By Lemma 3.2 we obtain the representation

s3
n

V −
n +1
s

D =
s 
i=1
Ys,i + An,
where An = op(1) and the random variables Ys,i are deﬁned in (3.1). Note that the random
variables {Ys,i}i=1,...,s form a triangular array of rowwise m-dependent random variables. Moreover,
E[Ys,i]=0 , and a straightforward but tedious calculation yields
lim
n,s→∞
Var
 s 
i=1
Ys,i

=
4
y
+8 (4.2)
(see the remark at the end of this Appendix). A central limit theorem for m-dependent triangular
arrays was given by Orey (1958), but we found the conditions of a central limit theorem for
7triangular arrays of α-mixing random variables easier to check. To be precise we applied Theorem
2.1 of Liebscher (1996) to the sequence
 s 
i=1
Ys,i

s∈N
.
For this we calculate
E

Y
4
s,i
	
=
1
s2 n6 E

V
(n)
i − E

V
(n)
i
4
−
8
sn 6 E

V
(n)
i − E

V
(n)
i
3 
Z
(n)
i − E

Z
(n)
i

+2 4
1
n6 E

V
(n)
i − E

V
(n)
i
2 
Z
(n)
i − E

Z
(n)
i
2
−
32s
n6 E

V
(n)
i − E

V
(n)
i

Z
(n)
i − E

Z
(n)
i
3
+
16s2
n6 E

Z
(n)
i − E

Z
(n)
i
4
and
E

Y
6
s,i
	
=
1
s3 n9 E

V
(n)
i − E

V
(n)
i
6
−
12
s2 n9 E

V
(n)
i − E

V
(n)
i
5 
Z
(n)
i − E

Z
(n)
i

+
60
sn 9 E

V
(n)
i − E

V
(n)
i
4 
Z
(n)
i − E

Z
(n)
i
2
−
160
n9 E

V
(n)
i − E

V
(n)
i
3 
Z
(n)
i − E

Z
(n)
i
3
+
240s
n9 E

V
(n)
i − E

V
(n)
i
2 
Z
(n)
i − E

Z
(n)
i
4
−
192s2
n9 E

V
(n)
i − E

V
(n)
i

Z
(n)
i − E

Z
(n)
i
5
+
64s3
n9 E

Z
(n)
i − E

Z
(n)
i
6
.
Because the distribution of the random variables V
(n)
i and Z
(n)
i is explicitly known we can evaluate s
i=1 E|Ys,i|k (k =4 ,6) explicitly [see Birke (2003) for more details]. In the following we denote
by pj(s,k)a n dqj(s,k) polynomials of degree k ∈ N and argument s, then we obtain by a tedious
calculation
s 
i=1
E

Y
4
s,i
	
=
p1 (s,6)
sn 6 +
p2 (s,5)
sn 5 +
p3 (s,4)
sn 4 +
p4 (s,3)
sn 3
+
p5 (s,2)
sn 2 +
p6 (s,1)
sn
+
p (s,0)
s
= O

1
s

,
8s 
i=1
E

Y
6
s,i
	
=
q1 (s,9)
s2 n9 +
q2 (s,8)
s2 n8 +
q3 (s,7)
s2 n7 +
q4 (s,6)
s2 n6 +
q5 (s,5)
s2 n5
+
q6 (s,4)
s2 n4 +
q7 (s,3)
s2 n3 +
q8 (s,2)
s2 n2 +
q9 (s,1)
s2 n
+
q10 (s,0)
s2
= o

1
s

,
Consequently it follows from Theorem 2.1 and equation (2.1) in Liebscher (1996)
s 
i=1

1
√
n3 s

V
(n)
i − E

V
(n)
i

− 2

s
n3

Z
(n)
i − E

Z
(n)
i

D →N

0,
4
y
+8

,
which proves the assertion of Theorem 2.1. 
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9

A central limit theorem for m-dependent random variables.

An Introduction to Multivariate Statistical Analysis.

Aspects of Multivariate Statistical Theory.

Central limit theorems for sums of α-mixing random variables.

On some test criteria for covariance matrix.

Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size.

Some limit theorems for the eigenvalues of a sample covariance matrix.

Some optimal multivariate tests.

Statistical challenges in functional genomics.

The distribution of a statistic used for testing sphericity of normal distributions.

The smallest eigenvalue of a large dimensional Wishart matrix.

A note on testing the covariance matrix for large dimension

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A note on testing the covariance matrix for large dimension

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