SCIENCE CHINA
Mathematics

. ARTICLES .

January 2015 Vol. 55 No. 1: 1–XX
doi: 10.1007/s11425-000-0000-0

arXiv:1603.01003v2 [] 24 Jul 2016

In memory of 10 years since the passing of Professor Xiru Chen - a great Chinese statistician

A review of 20 years of naive tests of significance for
high-dimensional mean vectors and covariance
matrices
HU Jiang1 & BAI Zhidong1,∗
1Key

Laboratory for Applied Statistics of MOE, Northeast Normal University, Changchun, 130024, P.R.C.
Email: huj156@nenu.edu.cn, baizd@nenu.edu.cn
Received Month 00, 2015; accepted Month 00, 2015

Abstract In this paper, we introduce the so-called naive tests and give a brief review of the new developments.
Naive testing methods are easy to understand and perform robustly, especially when the dimension is large. In
this paper, we focus mainly on reviewing some naive testing methods for the mean vectors and covariance
matrices of high-dimensional populations, and we believe that this naive testing approach can be used widely
in many other testing problems.
Keywords
MSC(2010)

Naive testing methods, Hypothesis testing, High-dimensional data, MANOVA.
62H15, 62E20

Citation: HU J, BAI Z D. A review of 20 years of naive tests of significance for high-dimensional mean vectors and
covariance matrices. Sci China Math, 2015, 55, doi: 10.1007/s11425-000-0000-0

1

Introduction

Since its proposal by Hotelling (1931) [23], the Hotelling T 2 test has served as a good test used in
multivariate analyses for more than eight decades due to its many useful properties: it is uniformly the
most powerful of the affine invariant tests for the hypotheses H0 : µ = 0 for the one-sample problem and
H0 : µ1 = µ2 for the two-sample problem. However, it has a fatal defect in that it is not well defined when
the dimension is larger than the sample size or the degrees of freedom. As a remedy, Dempster (1958) [16]
proposed his non-exact test (NET) to test the hypothesis of the equality of two multivariate population
means, that is, the test of locations in the two-sample problem. In 1996, Bai and Saranadasa [3] further
found that Dempster’s NET not only serves as a replacement for the Hotelling T 2 to test the hypothesis
when the number of degrees of freedom is lower than the dimension but is also more powerful than the
Hotelling T 2 when the dimension is large, but not too large, such that T 2 is well defined. They also
proposed the asymptotic normal test (ANT) to test the same hypothesis and strictly proved that both
the NET and ANT have similar asymptotic power functions that are higher than those of the Hoteling
T 2 test. Thus, their work raised an important question that classical multivariate statistical procedures
need to re-examine when the dimension is high. To call attention to this problem, they entitled their
paper “The Effect of High Dimension”.
That paper was published nearly 20 years ago and has been cited in other studies more than 100 times
to date in Web of Science. It is interesting that more than 95% of the citations were made in the past
∗ Corresponding

author

c Science China Press and Springer-Verlag Berlin Heidelberg 2012

math.scichina.com

www.springerlink.com

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10 years. This pattern reveals that high-dimensional data analysis has attracted much more widespread
attention since the year 2005 than it had received previously. In the theory of hypothesis testing, of course,
the most preferred test is the uniformly most powerful test. However, such a test does not exist unless
the distribution family has the property of a monotone likelihood ratio for which the parameter can only
be univariate. Hence, there is no uniformly most powerful test for multivariate analysis. Therefore, the
optimal procedure can only be considered for smaller domains of significance tests, such as unbiased tests
or invariant tests with respect to specific transformation groups. The Hotelling T 2 was derived based on
the likelihood ratio principle and proved to be the most powerful invariant test with respect to the affine
transformation group (see Page 174 of [1]). A serious point, however, is that the likelihood ratio test
must be derived under the assumption that the likelihood of the data set exists and is known, except for
the unknown parameters. In a real application, it is impossible to verify that the underlying distribution
is multivariate normal or has any other known form of the likelihood function. Thus, we would like to
use another approach to set up a test for some given hypothesis: choose h(θ) as a target function for the
hypotheses such that the null hypothesis can be expressed as h(θ) = 0 and the alternative as h(θ) > 0
and then look for a proper estimator θ̂ of the parameter θ. Then, we reject the hypothesis if h(θ̂) > h0
such that PH0 (h(θ̂) > h0 ) = α. For example, for the Hotelling test of the difference of two sample means,
Σ =S
one can choose h(µ1 , µ2 , Σ ) = (µ1 − µ2 )′Σ −1 (µ1 − µ2 ), the estimators µ̂i = X̄i , i = 1, 2, and Σ̂
for the sample means and sample covariance matrix. Dempster’s NET and Bai and Saranadasa’s ANT
simply use h(µ1 , µ2 ) = kµ1 − µ2 k2 and µ̂i = X̄i , i = 1, 2. That is, the Hotelling test uses the squared
Mahalanobis distance, whereas the NET and ANT use the squared Euclidean distance. We believe that
the reason why the NET and ANT are more powerful for large dimensions than the Hotelling test is
because the target function of the latter involves too many nuisance parameters in Σ , which cannot be
well estimated. Because the new tests focus only on the naive target function instead of the likelihood
ratio, we call them the naive tests, especially the ones that are independent of the nuisance parameters,
which generally ensures higher power.
In 1996, Bai and Saranadasa [3] raised the interesting point that one might prefer adopting a test of
higher power and approximate size rather than a test of exact size but much lower power. The naive
tests have undergone rapid development over the past twenty years, especially over the past 10. In this
paper, we give a brief review of the newly developed naive tests, which are being applied to a wide array
of disciplines, such as genomics, atmospheric sciences, wireless communications, biomedical imaging, and
economics. However, due to the limited length of the paper, we cannot review all of the developments
and applications in all directions, although some of them are excellent and interesting for the field of
high-dimensional data analysis. In this paper, we focus mainly on reviewing some naive testing methods
(NTMs) for the mean vectors and covariance matrices of high-dimensional populations.
Based on the NTMs, many test statistics have been proposed for high-dimensional data analysis.
Throughout this paper, we suppose that there are k populations and that the observations Xi1 , . . . , Xini
are p-variate independent and identically distributed (i.i.d.) random sample vectors from the i-th population, which have the mean vector µi and the covariance matrix Σ i . Moreover, except where noted, we
work with the following model assumptions:
(A1) Xij := (Xij1 , . . . , Xijp )′ = Γi Zij + µi , for i = 1, . . . k, j = 1 . . . , ni , where Γi is a p × m noni
are m-variate i.i.d. random
random matrix for some m > p such that Γi Γ′i = Σi , and {Zij }nj=1
vectors satisfying E(Zij ) = 0 and V ar(Zij ) = Im , the m × m identity matrix;
(A2)

ni
n

→ κi ∈ (0, 1) i = 1, . . . k, as n → ∞, where n =

Denote
X̄i =

Pk

i=1

ni .

ni
ni
1 X
1 X
(i)
Xij and Si =
(Xij − X̄i )(Xij − X̄i )′ = (sij ).
ni j=1
ni − 1 j=1

When k = 1, the subscripts i or 1 are suppressed from ni , n1 , Γi , µi and so on, for brevity.

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3

No. 1
D

Throughout the paper, we denote by → the convergence in probability and by → the convergence in
distribution.
The remainder of the paper is organized as follows: In Section 2, we review the sample location
parameters. In subsection 2.1, we introduce the findings of Bai and Saranadasa [3]. In subsection
2.2, we introduce Chen and Qin [14]’s test based on the unbiased estimator of the target function. In
subsection 2.3, we review Srivastava and Du’s work on the scale invariant NTM, based on the modified
component-wise squared Mahalanobis distance. In subsection 2.4, we introduce Cai et al’s NTM based
on the Kolmogorov distance, i.e., the maximum component of difference. In subsection 2.5, we introduce
some works on the extensions to MANOVA and contrast tests, that is, tests for problems of more than
two samples. In Section 3, we introduce some naive tests of hypotheses on covariances. In subsection
3.1, we introduce the naive test proposed by Ledoit and Wolf [27] on the hypothesis of the one-sample
covariance matrix and the spherical test. In subsection 3.2, we introduce the NTM proposed by Li and
Chen (2012) [28]. In subsection 3.3, we introduce Cai’s NTM on covariances based on the Kolmogorov
distance. We also review the testing of the structure of the covariance matrix in subsection 3.4. In Section
4, we make some general remarks on the development of NTMs.

2
2.1

Testing the population locations
Asymptotic powers of T 2 , NET and ANT

In this section, we first consider the simpler one-sample problem by NTM. That is, the null hypothesis
is H0 : µ1 = µ0 . Under the assumption (A1) with k = 1, and testing the hypothesis
H0 : µ = µ 0

v.s. H1 : µ 6= µ0 ,

it is easy to check that EX̄ = µ. Thus, to set up a test of this hypothesis, we need to choose some norms of
the difference µ − µ0 . There are three types of norms to be chosen in the literature: the Euclidean norm,
the Maximum component norm and the Mahalanobis squared norm. Let us begin from the classical one.
The most famous test is the so-called Hotelling T 2 statistic,
T 2 = n(X̄ − µ0 )′ S−1 (X̄ − µ0 )

(2.1)

which was proposed by Hotelling (1931) [23] and is a natural multi-dimensional extension of the squared
univariate Student’s t-statistic. If the Zj s are normally distributed, the Hotelling T 2 statistic is shown to
be the likelihood ratio test for this one-sample problem and to have many optimal properties. Details can
be found in any textbook on multivariate statistical analysis, such as [1, 31]. It is easy to verify that X̄
and S are unbiased, sufficient and complete estimators of the parameters µ and Σ and that, as mentioned
above, the target function is chosen as the Mahalanobis squared distance of the population mean µ from
the hypothesized mean µ0 , which is also the Euclidean norm of Σ−1/2 (µ − µ0 ). Thus, we can see that the
Hotelling T 2 statistic is a type of NTM, and we simply need to obtain its (asymptotic) distribution. It
(n−p) 2
T has an F -distribution with degrees of freedom p
is well known that under the null hypothesis, p(n−1)
and n − p, and when p is fixed, as n tends to infinity, T 2 tends to a chi-squared distribution with degrees
of freedom p. If we assume yn = p/n → y ∈ (0, 1) and Xj are normally distributed, following Bai and
Saranadasa [3], we may easily derive that
s


(1 − yn )3
nyn
nkδk2 D
T2 −
−
→ N (0, 1), as n → ∞,
(2.2)
2nyn
1 − yn
1 − yn
where δ = Σ −1/2 (µ − µ0 ). By (2.2), it is easy to derive that the asymptotic power function of the T 2
test satisfies
s
!
n(1 − y)
2
→ 0.
kδk
βH − Φ −ξα +
2y

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Here and throughout the paper, Φ is used for the distribution function of a standard normal random
variable, and ξα is its upper α quantile. It should be noted that the above asymptotic distribution of
the Hotelling T 2 statistic (2.2) still holds without the normality assumption. The details can be found
in [33].
Next, we derive the asymptotic power for ANT. In this case, the target function is chosen as h(µ) =
P
kµ − µ0 k2 , and the natural estimator of µ is X̄ = n1 ni=1 Xi . It is easy to derive that
EkX̄k2

V ar(kX̄k2 )

= kµk2 +

1
Σ
trΣ
n

(2.3)

2
Σ2 + 4µ′Σ µ
trΣ
n
m
m
X
X
2
1
+ √ EZ13
(γi′ γi )2
µ′ γi (γi′ γi ) + (EZ14 − 3)
n
n
i=1
i=1

=

(2.4)

where γi is the i-th column of the matrix Γ. Under the conditions
(µ − µ0 )′Σ (µ − µ0 ) =
Σ) =
λmax (Σ
we have
2

V ar(kX̄ − µ0 k ) =

1
Σ2 ),
o( trΣ
n
√
Σ2 ),
o( trΣ

m
X
1
2
Σ2 + (EZ14 − 3)
(γi′ γi )2
trΣ
n
n
i=1

(2.5)
(2.6)
!

(1 + o(1)).

Under the conditions (2.5) and (2.6), using the moment method or martingale decomposition method,
one can prove that
kX̄ − µ0 k2 − E(kX̄ − µ0 k2 )
p
→ N (0, 1)
(2.7)
V ar(kX̄ − µ0 k2 )

To perform the test for the hypothesis H0 : µ = µ0 vs. H1 : µ 6= µ0 , it is necessary to construct
ratio-consistent estimators of E(kX̄ − µ0 k2 ) and V ar(kX̄ − µ0 k2 ) under the null hypothesis. It is obvious

Σ can be estimated by n1 tr(S). The variance can be simply estimated by n1 tr(S2 ) − n1 tr2 (S) if
that n1 trΣ
EZ14 = 3. In the general case, it can be estimated by n1 σ̂n2 , where
σ̂n2

=

1
(n)5
−

X

j1 ,··· ,j5
distinct

1
(n)6

tr ((Xj1 − Xj2 )(Xj1 − Xj3 )′ (Xj1 − Xj4 )(Xj1 − Xj5 )′ )

X

j1 ,··· ,j6
distinct

tr ((Xj1 − Xj2 )(Xj1 − Xj3 )′ (Xj6 − Xj4 )(Xj6 − Xj5 )′ ) ,

(2.8)

where the summations above are taken for all possibilities that j1 , · · · , js , s = 5 or 6, distinctly run over
{1, · · · , n}, and (n)l = n(n − 1) · · · (n − l + 1). Using the standard limiting theory approach, one may
prove that σ̂n2 is a ratio-consistent estimator of σn2 , where
Σ2 + (EZ14 − 3)
σn2 = 2trΣ

m
X

(γi′ γi )2 .

i=1

Therefore, the test rejects H0 if
kX̄ − µ0 k2 >

1
1
tr(S) + √ ξα σ̂n .
n
n

From this result, it is easy to derive that under conditions (2.5) and (2.6), the asymptotic power of ANT
is
!
!
√
√
nkµ − µ0 k2
nkµ − µ0 k2
p
p
≃ Φ −ξα +
.
βAN T ≃ Φ −ξα +
σ̂n2
σn2

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Comparing the expressions of the asymptotic powers of Hotelling test and ANT, one sees that the factor
√
1 − y appears in the asymptotic power of Hotelling’s test but not in that of the ANT. This difference
shows that the ANT has higher power than does the T 2 test when y is close to 1.
Moreover, if p, the dimension of the data, is larger than n − 1, the degrees of freedom, then T 2 is not
well defined, and there is no way to perform the significance test using it.
Remark 1. In the real calculation of σ̂n2 , the computation using the expression of (2.8) is very time
consuming. To reduce the computing time, we should rewrite it as
n

σ̂n2

=

1X ′
4
(X Xj )2 −
n j=1 j
(n)2
+

6
(n)3

X

X

j1 ,j2
distinct

X′j1 Xj2 X′j1 Xj3 +

j1 ,j2 ,j3
distinct

1
(n)2

X′j1 Xj1 X′j1 Xj2 −
2
(n)3

X

j1 ,j2 ,j3
distinct

X

X′j1 Xj1 X′j2 Xj2

j1 ,j2
distinct

X′j1 Xj1 X′j2 Xj3 −

4
(n)4

X

X′j1 Xj2 X′j3 Xj4 , (2.9)

j1 ,j2 ,j3 ,j4
distinct

where each summation runs over all possibilities in which the indices involved are distinct.
It is easy to see that to calculate the estimator σ̂n2 using (2.9) is very time consuming: for example, to
calculate the last term, one needs to compute 2pn4 multiplications. To further reduce the computation
time, one may use the inclusion-exclusion principle to change the last five sums into forms that are easier
to calculate. For example, the last sum I6 can be written as
I6 = (X′ X)2 − 2X′ Xa − 4X′ X(2) X + a2 + 2tr(X2(2) ) + 8X′ X(3) − 6b,

(2.10)

where
X =
X(3)

=

n
X

j=1
n
X

Xj ,

a=

n
X

X′j Xj ,

X(2) =

b=

Xj X′j ,

j=1

j=1

X′j Xj Xj ,

n
X

n
X

(X′j Xj )2 .

j=1

j=1

Here, the coefficients of various terms can be found by the following arguments: Let Ω denote the fact that
there are no restrictions between the indices j1 , · · · , j4 , and let Aik denote the restriction that ji = jk ,
i < k 6 4, which is called an equal sign, or an edge between vertices i and j.
The sum I6 in which the indices j1 , · · · , j4 are distinct can be considered the indices running over the
Q
set 16i<k64 (Ω − Aik ). By expanding the product, one may split the sum I6 into a signed sum of several
sums: the first sum runs over Ω, followed by the subtraction of 6 sums with one equal sign; add 15 sums
with two equal signs; subtract 20 sums with three equal signs, and so on, and finally add the sum with all
six equal signs. Now, the first one runs over Ω, that is, there are no restrictions among the four vertices
1, 2, 3, 4, which simply gives the first term in (2.10). The sum with the equal sign A12 is given by
X

X′j1 Xj2 X′j3 Xj4 =

A12

n1
n X
n X
X

X′j Xj X′j3 Xj4 = aX′ X;

j=1 j3 =1 j4 =1

Similarly, the sum under the equal sign A34 is also aX′ X. These two cases give −2X′ Xa in the second
term in (2.10); the other 4 cases with one equal sign give −4X′ X(2) X in the third term. For example,
X
A13

X′j1 Xj2 X′j3 Xj4 =

n1
n X
n X
X

X′j Xj2 X′j Xj4 = X′ X(2) X;

j=1 j2 =1 j4 =1

Under the equal sign A14 , A23 or and A24 , the sum again has the form X′ X(2) X. By similar arguments,
one can show that the sum with the two equal signs A12 and A34 is given by a2 in the fourth term;
the sums with two equal signs A13 and A24 or A14 and A23 are given by 2tr(X2(2) ) in the fifth term;
the sums for the other 12 cases with two equal signs, such as A12 and A23 , are given by 12X′ X(3) ;

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there are 4 cases in which the three equal signs make three indices equal and leave one index free of
the rest (or, equivalently, three edges forming a triangle), which contribute −4X′ X(3) ; and two cases
give a final contribution of 8X′ X(3) in the seventh term of (2.10). There are 16 other cases of three
equal signs that imply all indices j1 , · · · , j4 are equal, giving a sum of b. Additionally, if there are more
than three equal signs, the indices j1 , · · · , j4 are also all identical; thus, we obtain the coefficient for b as


−16 + 64 − 65 + 1 = −6. Therefore, the splitting (2.10) is true.
Similarly, one may show that
I1

= b

I2

= X′ X(3) − b

I3
I4
I5

= a2 − b

= X′ X(2) X − 2X′ X(3) − tr(X2(2) ) + 2b

= X′ Xa − a2 − 2X′ X(3) + 2b.

Finally, one can calculate the estimator of the variance by
σ̂n2

=

n2 − n + 2
4
n2 − 3n + 4 2 6n − 2 ′
6n − 10
b−
X′ X(3) −
a +
X X(2) X −
tr(X2(2) )
(n − 1)3
(n − 2)2
(n)4
(n)4
(n)4
2n + 2 ′
4
+
X Xa −
(X′ X)2 .
(n)4
(n)4

From the expressions above, the numbers of multiplications to be calculated for the terms above are
n(p + 1), np3 , np + 1, np3 , p2 , p + 1 and p + 1, respectively. Thus, by using this formula, the computation
time will be reduced significantly.
Now, consider the two-sample location problem of multivariate normal distributions, H0 : µ1 = µ2 ,
with a common covariance matrix Σ . The classical test for this hypothesis is the Hotelling T 2 test
n1 n2
(X̄1 − X̄2 )′ S−1 (X̄1 − X̄2 )
T2 =
n1 + n2
Pni
Xij , i = 1, 2 and
where X̄i = n1i j=1


ni
2 X
X
1

S=
(Xij − X̄i )(Xij − X̄i )′  .
n1 + n2 − 2 i=1 j=1

In 1958 and 1960, Dempster published two papers, [16] and [17], in which he argued that if p, the
dimension of the data, is larger than N = n1 + n2 − 2, the degrees of freedom, then T 2 is not well
defined, and there is no way to perform the significance test using T 2 . Therefore, he proposed the socalled NET (non-exact test) as follows: Arrange the data X = (X11 , · · · , X1n1 , X21 , · · · , X2n2 ) as a p × n
matrix, where n = n1 + n2 . Select an n × n orthogonal matrix H and transform the data matrix to
Y = X H = (y1 , · · · , yn ) such that

n
n2
1
y1
∼ N √ µ1 + √ µ2 , Σ
n
n

r n n
1 2
(µ1 − µ2 ), Σ
y2
∼ N
n
y3 , · · · , yn

i.i.d.

∼

Then, he defined his NET by
TD =

N (0, Σ ).

ky2 k2
.
ky3 k2 + · · · + kyn k2

He claimed that as n1 and n2 increase, using the so-called chi-square approximation of kyj k2 for i =
2, 3, · · · , n,
nTD ∼ Fr,nr ,

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where N = n1 + n2 − 2 and r = (trΣ)2 /trΣ2 . Comparing N TD with T 2 , we find that Dempster simply
replaced S by tr(S)Ip to smooth the trouble when S is singular.
Bai and Saranadasa [3] observed that Dempster’s NET is not only a remedy for the T 2 test when it
is not well defined but is also more powerful than T 2 even when it is well defined, provided that the
dimension p is large compared with the degrees of freedom N . Based on Dempster’s NET, Bai and
Saranadasa [3] also proposed the so-called ANT (asymptotic normality test) based on the normalization
of ky2 k2 . They established a CLT (central limit theorem) as follows:
Mn
D
p
→ N (0, 1),
V ar(Mn )

where Mn = kX̄1 − X̄2 k2 − n1nn2 tr(S). To perform the significance test for H0 , Bai and Saranadasa
proposed the ratio-consistent estimator of V ar(Mn ) by


1 2
2(N + 2)(N + 1)N
2
\
tr(S ) − tr S .
V ar(Mn ) =
n21 n22 (N − 1)
N
Bai and Saranadasa [3] proved that when dimension p is large compared with n, both NET and
ANT are more powerful than the T 2 test, by deriving the asymptotic power functions of the three tests.
Under the conditions that p/n → y ∈ (0, 1) and n1 /n → κ ∈ (0, 1), the power function of the T 2 test
asymptotically satisfies
s
!
n(1 − y)
2
→ 0,
κ(1 − κ)kδk
βH (δ) − Φ −ξα +
2y
where δ = Σ −1/2 (µ1 − µ2 ).
Under the assumption A2 and



n
2
Σ )
µ Σµ = o
tr(Σ
n1 n2
√
Σ2 ),
Σ ) = o( trΣ
λmax (Σ
′

(2.11)
(2.12)

where µ = µ1 − µ2 , Bai and Saranadasa [3] also proved that the power function of Dempster’s NET
satisfies


nκ(1 − κ)kµk2
√
→ 0.
βD (µ) − Φ −ξα +
Σ2
2trΣ
Without the normality assumption, under the assumptions A1 and A2 and the conditions (2.11) and
(2.12), Bai and Saranadasa [3] proved that the power function of their ANT has similar asymptotic power
to the NET, that is,


nκ(1 − κ)kµk2
√
βBS (µ) − Φ −ξα +
→ 0.
Σ2
2trΣ
2.2

Chen and Qin’s approach

Chen and Qin (2010) [14] argued that the main term of Bai and Saranadasa’s ANT contains squared
terms of sample vectors that may cause non- robustness of the test statistic against outliers and thus
proposed an unbiased estimator of the target function kµ1 − µ2 k2 , given by
TCQ =

1
n1 (n1 − 1)

X

X′1i X1j +

i6=j

1
n2 (n2 − 1)

X
i6=j

X′2i X2j −

2 X ′
X X2j
n1 n2 i,j 1i

Chen and Qin [14] proved that ETCQ = kµ1 − µ2 k2 , and under the null hypothesis,
V ar(TCQ ) =

2
n1 (n1 − 1)

Σ 21 ) +
tr(Σ

2
n2 (n2 − 1)

Σ22 ) +
tr(Σ

4
Σ1Σ 2 )(1 + o(1)).
tr(Σ
n1 n2

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Similarly to Bai and Saranadasa (1996), under the conditions
n1
→ κ
n

Σ iΣ j Σ lΣ h ) = o tr2 (Σ
Σ 1 + Σ 2 )2 , for i, j, l, h = 1 or 2,
tr(Σ

Σ1 + Σ 2 )2 , for i = 1 or 2,
(µ1 − µ2 )′Σ i (µ1 − µ2 ) = o n−1 tr(Σ

Σ1 + Σ 2 )2 = o (µ1 − µ2 )′Σ i (µ1 − µ2 ) , for i = 1 or 2,
or n−1 tr(Σ

Chen and Qin proved that

(2.13)
(2.14)
(2.15)

TCQ − kµ1 − µ2 k2 D
p
→ N (0, 1).
V ar(TCQ )

To perform the test for H0 : µ1 = µ2 with the target function h(µ1 , µ2 ) = kµ1 − µ2 k2 , they proposed
the estimator for V ar(TCQ ) to be
σ̂n2 =

2
4
2
\
\
\
Σ1Σ 2 ),
Σ21 ) +
Σ 22 ) +
tr(Σ
tr(Σ
tr(Σ
n1 (n1 − 1)
n2 (n2 − 1)
n1 n2

where
\
Σ2i ) =
tr(Σ
\
Σ 1Σ 2 ) =
tr(Σ

1
ni (ni − 1)
1
n1 n2

X
j6=k

n1 X
n2
X

j=1 k=1

X′ik (Xij − X̄i(jk) )X′ij (Xik − X̄i(jk) ),

X′2k (X1j − X̄1(j) )X′1j (X2k − X̄2(k) ),

and X̄i(∗) denotes the sample mean of the i-th sample, excluding the ∗-th vectors, as indicated in the
braces.
Applying the central limit theorem, Chen and Qin derived the asymptotic power functions for two
cases:
 


nκ(1−κ)kµ1 −µ2 k2

if (2.14) holds,
Φ −ξα + √
2
Σ 1 +(1−κ)Σ
2tr(κΣ
 Σ2 )

(2.16)
βCQ ∼

nκ(1−κ)kµ1 −µ2 k2

if
(2.15)
holds.
Φ √
2
Σ1 +(1−κ)Σ
Σ2 )
2tr(κΣ

Remark 2. The expression of the asymptotic power under the condition (2.15) ((3.5) in Chen and
Qin [14]) may contain an error in that the denominator
of the quantity inside the function Φ should
q
be σn2 in Chen and Qin’s notation, that is, 2

1
n1 (µ1

− µ2 )′Σ 1 (µ1 − µ2 ) +

1
n2 (µ1

− µ2 )′Σ 2 (µ1 − µ2 ),

instead of σn1 . However, the asymptotic power is 1 under the condition (2.15). Therefore, the typo does
not affect the correctness of the expression of the asymptotic power. This point is shown by the following
facts:
By the condition (2.13), we have
q

Σi )kµ1 − µ2 k2 6 o
Σ2i )kµ1 − µ2 k2 .
(µ1 − µ2 )′Σ i (µ1 − µ2 ) 6 λmax (Σ
tr(Σ

Therefore,

Consequently,


2
σn2
6 o σn1 kµ1 − µ2 k2 .
kµ1 − µ2 k2
σn2
>
>
σn2
o(σn1 )

s

2
σn2
2 >M
σn1

for any fixed constant M , where the last step follows from the condition (2.15). Regarding Chen and
Qin’s expression, one has
kµ1 − µ2 k2
kµ1 − µ2 k2
>
> M.
σn1
σn2

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For the one-sample location problem, Chen and Qin [14] modified TBS and proposed
TCQ =

X
1
Xi′ Xj
n(n − 1)
i6=j

Σ4 = o(tr2Σ 2 ) (which is equivalent to (2.6)), as min{p, n} → ∞,
and showed that under the condition trΣ
q

P

2
n(n−1) tr(

TCQ
i6=j (Xi

−

D

X̄(i,j) )X′i (Xj

− X̄(i,j) )Xj )

→ N (0, 1).

(2.17)

Note that the difference between the statistics in (2.17) and the LHS of (2.7) with the denominator
Σ2 .
replaced by the estimator n1 (trS2 − n1 tr2 S) is in the denominators, that are the estimators of trΣ
Remark 3. It has been noted that the main part TCQ of Chen and Qin’s test is exactly the same as Bai
and Saranadasa’s Mn because they are both unbiased estimators of the target function kµ1 − µ2 k2 and
are functions of the complete and sufficient statistics of the mean vectors and covariance matrices for the
two samples. We believe that Chen and Qin’s idea of an unbiased estimator of the target function helped
them propose a better estimator of the asymptotic variance of the test such that their test performed
better than did Bai and Saranadasa’s ANT in simulation. In addition, there is an improved statistic for
the Chen and Qin test by thresholding methods, which was recently proposed by Chen et al. [13]. Wang
et al [50] also proposed a test for the hypothesis under the elliptically distributed assumption, which can
be viewed as a nonparametric extension of TCQ .
2.3

Srivastava and Du’s approach

While acknowledging the defect of the Hotelling test, as indicated in [3,16], Srivastava and Du (2008) [42]
noted that the NET and ANT are not scale invariant, which may cause lower power when the scales of
different components of the model are very different. Accordingly, they proved the following modification
to the ANT:
TSD,1 = (X̄ − µ0 )′ D−1
S (X̄ − µ0 ),
where DS = Diag(s11 , · · · , spp ) is the diagonal matrix of the sample covariance matrix S. Let R be the
population correlation matrix. Then, under the condition that
trRi
p→∞ p
λ(R)
lim √
p→∞
p

< ∞, for i = 1, 2, 3, 4;

0 < lim

= 0,

where λ(R) is the largest eigenvalue of the correlation matrix R.
Srivastava and Du [42] showed that if n ≍ pη and 12 < η 6 1,
nTSD,1 −
r 
2 trR2 −

(n−1)p
n−3
p2
n−1



cp,n

D

→ N (0, 1),
−1/2

as n → ∞,
−1/2

where R is the sample correlation matrix, i.e., R = DS SDS
and cp,n = 1 + trR2 /p3/2 . They
showed that the asymptotic power of TSD,1 under the local alternative, as n → ∞,


!
(n−1)p
′ −1

 nTSD,1 − n−3
n(µ
−
µ
)
D
(µ
−
µ
)
0
0
Σ
√
,
> ξ−α 
βSD = P 

 → Φ −ξα +
r 
2trR2
p2
2
2 trR − n−1 cp,n

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where DΣ is the diagonal matrix of population covariance matrix Σ . Later, Srivastava [41] modified the
asymptotic results above to cases, where the adjusting term cp,n1 in the last test statistic is replaced by
Pn
1 and the restriction for η is relaxed to 0 < η 6 1. Further, by excluding i=1 (Xi − µ0 )′ D−1
S (Xi − µ0 )
from TSD,1 and modifying DS , Park and Ayyala [34] obtained another NTM test statistic:
TP A =

X
n−5
X′i D−1
S(i,j) Xj
n(n − 1)(n − 3)
i6=j

((i,j))

((i,j))

where DS(i,j) = Diag(s11 , · · · , spp ) is the diagonal matrix of the sample covariance matrix excluding
P
the sample points Xi and Xj , i.e., S(i,j) = (n − 3)−1 k6=i,j (Xk − X̄(i,j) )(Xk − X̄(i,j) )′ and X̄(i,j) =
P
(n − 2)−1 k6=i,j Xk .
Srivastava and Du also considered the two-sample location problem with the common covariance matrix
Σ [42] and proposed the testing statistic
TSD,2 =

n1 n2
(X̄1 − X̄2 )′ D−1
S (X̄1 − X̄2 )
n

Under similar conditions for the CLT of TSD,1 , they proved that
p
TSD,2 − NN−2
D
r 
→ N (0, 1).

2
2 trR2 − pn cn,p

(2.18)

They then further derived the asymptotic power function
−1
κ(1 − κ)µ′ DΣ
µ
√
βSD (µ) ∼ Φ −ξα +
2trR2

!

.

Remark 4. The advantage of this statistic is that the terms Xi , DS(i,j) and Xj are all independent such
that it is easy to obtain the approximation
−1
−1
ETP A = µ′ EDΣ
µ ≃ µ′ DΣ
µ,

which is similar to E(nTS − p(n − 1)/(n − 3)), as given in [42]. This point shows that both TSD,1 and
−1
TP A are NTM tests based on the target function µ′ DΣ
µ. The idea that they use to exclude the bias
p(n − 1)/(n − 3) is similar to TCQ , which removes the bias estimator trS given in TBS . Park and Ayyala
also gave the asymptotic distribution of TP A under the null hypothesis, that is,
p
n(n − 1)TP A
p
→ N (0, 1),
[2
2trR

P
[2 is a ratio-consistent estimator of trR2 , i.e., trR
[2 /trR2 →
where trR
1, and

[2 =
trR

X
1
−1
′
Xi′ D−1
S(i,j) (Xj − X̄(i,j) )Xj DS(i,j) (Xi − X̄(i,j) ).
n(n − 1)
i6=j

They then showed that the asymptotic power of the test TP A is the same as the asymptotic power of TSD .
Recently, Dong et al. [18] gave a shrinkage estimator of the diagonals of the population covariance matrix
DΣ 1 and showed that the shrinkage-based Hotelling test performs better than the unscaled Hotelling test
and the regularized Hotelling test when the dimension is large.
Remark 5. For Σ 1 6= Σ 2 , Srivastava et al. [43] used D = DS1 /n1 + DS2 /n2 instead of DS in TSD,2 . For
the case in which the population covariance matrices are diagonal, Wu et al. [52] constructed a statistic
by summing up the squared component-wise t-statistics for missing data, and Dong et al. [18] proposed
a shrinkage-based diagonalized Hotellings test.

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2.4

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11

Cai et al’s idea

Cai et al [10] noted that all of the NTM tests associated with target functions based on the Euclidean
norm or Mahalanobis distance require the condition that
√
nkµk2 / p → ∞,
(2.19)
to distinguish the null and alternative hypotheses with probability tending to 1. This condition does
√
not hold if only a few components of µ have the order O(1/ n) and all others are 0. Therefore, they
proposed using the L∞ norm or, equivalently, the Kolmogorov distance.
Indeed, Cai et al’s work compensates for the case where
√
(2.20)
n max |µi | → ∞.
i6p

Note that neither of the conditions (2.19) and (2.20) implies the other. The condition (2.20) is weaker
than (2.19) only when the bias vector µ − µ0 or µ1 − µ2 is sparse.
Now, we introduce the work of Cai et al. [8], who developed another NTM test based on the Kolmogorov
distance, which performs more powerfully against sparse alternatives in high-dimensional settings.
Supposing that Σ 1 = Σ 2 = Σ and {X1 , X2 } satisfy the sub-Gaussian-type or polynomial-type tails
condition, Cai et al. proposed the test statistic
 2
n1 n2
Xi
TCLX =
,
max
n1 + n2 16i6p ωii
d
d
−1 (X̄ − X̄ ) := (X , . . . , X )′ and Σ
−1 := Ω = (ω )
where Σ
1
2
1
p
ij p×p is the constrained l1 -minimization for the
inverse matrix estimator of Σ −1 . Here, the so-called constrained l1 -minimization for the inverse matrix
estimator is defined by
o
nX
d
−1 = arg min
|ωij |; subject to kSΩ − Ip k∞ 6 γn
Σ
Ω=(ωij )

ij

p
where γn is a tuning parameter, which may generally be chosen as C log p/n for some large constant
C. For more details on the properties of l1 -minimization estimators, the reader is referred to [5]. Under
the null hypothesis H0 and some spectrum of population covariance matrix conditions, for any x ∈ R, as
min{n, p} → ∞,

 x 
1
P(TCLX − 2 log(p) − log log(p) 6 x) → Exp − Exp −
.
π
2

To evaluate the performance of their maximum absolute components test, they also proved the following
result:
Σ) 6 λmax (Σ
Σ) 6 C0 for some constant C0 > 1; kp = O(pr ) for some r 6 1/4;
Suppose that C0−1p
6 λmin (Σ
√
maxi6p |µi |/ σii > 2β log(p)/n with β > 1/ mini (σii ωii ) + ε for some ε > 0. Then, as p → ∞
PH1 (φα (Ω)) → 1,

where kp is the number of non-zero entries of µ.
Remark 6. Note that for the statistic TCLX , one can use any consistent estimator of Σ −1 in the sense
of the L1 -norm and infinity norm with at least a logarithmic rate of convergence.
2.5

MANOVA and Contrasts: more than two samples

In this subsection, we consider the problem of testing the equality of several high-dimensional mean
vectors, which is also called the multivariate analysis of variance (MANOVA) problem. This problem is
to test the hypothesis
H0 : µ 1 = · · · = µ k

vs

H1 : ∃i 6= j, µi 6= µj .

(2.21)

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For samples that are drawn from a normal distribution family, the MANOVA problem in a highdimensional setting has been considered widely in the literature. For example, among others, Tonda
and Fujikoshi [48] obtained the asymptotic null distribution of the likelihood ratio test; Fujikoshi [21]
found the asymptotic null distributions for the Lawley-Hotelling trace and the Pillai trace statistics;
and Fujikoshi et al [22] considered the Dempster trace test, which is based on the ratio of the trace of
the between-class sample covariance matrix to the trace of the within-class sample covariance matrix.
Instead of investigating the ratio of the traces of the two sample matrices, Schott [38] proposed a test
statistic based on the difference between the traces. Next, we introduce three NTM statistics that are
the improvements on TSD , TCQ and TCLX .
Recently, Srivastava and Kubokawa [44] proposed a test statistic for testing the equality of the mean
vectors of several groups with a common unknown non-singular covariance matrix. Denote by 1r =
(1, . . . , 1)′ an r-vector with all entries 1, and define
Y

= (X11 , . . . , X1n1 , . . . , Xk1 , . . . , Xknk ),

L

= (Ik−1 , −1k−1 )(k−1)×k

and



1n1


 0

E= .
 ..

0

0

0

1n2
..
.

0
..
.

0

1nk









.

n×k

Then, Srivastava and Kubokawa proposed the following test statistic:
TSK =

−1
tr(BDS
) − (n − k)p(k − 1)(n − k − 2)−1
p
,
2cp,n (k − 1)(trR2 − (n − k)−1 p2 )

where B = Y′ E(E′ E)−1 L′ [L(E′ E)−1 L′ ]−1 L(E′ E)−1 E′ Y, DS = Diag[(n − k)−1 Y(In − E(E′ E)−1 E′ )Y],
−1/2
−1/2
and cp,n = 1 + tr(R2 )/p3/2 . Note that Diag[A] denotes the
R = DS Y(In − E(E′ E)−1 E′ )YDS
diagonal matrix consisting of the diagonal elements of the matrix A. Under the null hypothesis and the
condition n ≍ pδ with δ > 1/2, TSK is asymptotically distributed as N (0, 1). Thus, as n, p → ∞,
PH0 (TSK > ξα ) → Φ(−ξα ).
Hence, by comparing the results presented in [42] and [41], it is easy to see that cp,n may be removable
under certain conditions.
Motivated by Chen and Qin [14], Hu et al. [24] proposed a test for the MANOVA problem, that is,
THB

=

k
X
i<j

′

(X̄i − X̄j ) (X̄i − X̄j ) − (k − 1)

= (k − 1)

k
X
i=1

1
ni (ni − 1)

X

k1 6=k2

k
X

n−1
i trSi

i=1

X′ik1 Xik2 −

k
X
i<j

2 X ′
Xik1 Xjk2 .
ni nj
k1 ,k2

When k = 2, clearly, THB reduces to Chen and Qin’s test statistic. It is also shown that as p → ∞ and
n → ∞,
P
THB − ki<j kµi − µj k2 D
p
→ N (0, 1).
V ar(THB )

To perform the test, a ratio-consistent estimator of V ar(THB ) for the MANOVA test is proposed in the
paper.

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Cai and Xia [10] also applied their idea to the MANOVA case under the homogeneous covariance
assumption using the following test statistic:
(
)
2
X
Xjli
ni nj
,
TCX = max
16i6p
ni + nj
b̂ii
16j<l6k

′ d
d
−1 (X̄ − X̄ ) := (X
−1 := (ω )
where Σ
j
l
jl1 , . . . , Xjlp ) ; Σ
ij p×p is a consistent estimator, e.g. the constrained
l1 -minimization for the inverse matrix estimate of Σ −1 ; and b̂ii are the diagonal elements of the matrix
B̂, which is defined by

B̂ = P

ni
k X
X
1
b −1 (Xij − X̄i )(Xij − X̄i )′Σ
b −1 .
Σ
ni − k i=1 j=1

To introduce the theory of Cai and Xia’s test, let
′
r
r
n1 n2
nk−1 nk
1
,
(X̄1 − X̄2 )i , . . . ,
(X̄k−1 − X̄k )i
Yi =
σ̂ii
n1 + n2
nk−1 + nk
k(k−1)
×1
2

where σ̂ii is the estimate of the (i, i)-entry of the covariance matrix Σ . Let Σ Y : ̺ × ̺, ̺ = k(k−1)
, be
2
2
the covariance matrix of Yi . Let λY be the largest eigenvalue of Σ Y , and let d be the dimension of the
eigenspace of λ2Y . Let λ2Y,i : 1 6 i 6 ̺ be the eigenvalues of Σ Y arranged in descending order.
Under the null hypothesis H0 and some regularity conditions on the population covariance matrix, for
any x ∈ R, as min{n, p} → ∞,


 

x
d
Σ) exp − 2
,
H(Σ
PH0 (TCX − 2λ2Y log(p) − (d − 2)λ2Y log log(p) 6 x) → exp −Γ−1
2
2λY
Q̺
λ2
where Γ is the gamma function, and H = i=d+1 (1 − λY2,i )−1/2 . Similar to the two-sample location
Y
problem, they also established a theorem to evaluate the consistency of their test:
P
Σ ) 6 λmax (Σ
Σ) 6 C0 for some constant
Suppose that C0−1 6 λmin (Σ
C0 > 1. If kp = maxj<l6k pi=1 I(µj −
p
√
µl 6= 0) = o(pr ), for some r < 1/4 and maxi kδi k2 / σii > 2σ 2 β log p with some β > 1/(mini σii ωii ) + ε
for some constant ε > 0, then, as p → ∞,
PH1 (φα (Ω) = 1) → 1,
where δi = (µ1i − µ2i , · · · , µk−1,i − µk,i )′ .
2.6

Some related work on the tests of high-dimensional locations

Chen et al. [12] proposed another statistic:
TRHT = X̄′ (S + λI)−1 X̄, for λ > 0,
which is called the regularized Hotelling T 2 test. The idea is to employ the technique of ridge regression
to stabilize the inverse of the sample covariance matrix given in (2.1). Assuming that the underlying
distribution is normally distributed, it is proven that under the null hypothesis, for any λ > 0, as
p/n1 → y ∈ (0, ∞)

√ 
1−λm(λ)
p nTRHT /p − 1−p(1−λm(λ))/n
D
→ N (0, 1),
m(λ)−λm′ (λ)
1−λm(λ)
−
λ
(1−p/n+pλm(λ)/n)3
(1−p/n+pλm(λ)/n)4
where m(λ) = p1 tr(S + λI)−1 and m′ (λ) = 1p tr(S + λI)−2 . They also give an asymptotic approximation
method for selecting the tuning parameter λ in the regularization. Recently, based on a supervisedlearning strategy, Shen and Lin [39] proposed a statistic to select an optimal subset of features to maximize
the asymptotic power of the Hotelling T 2 test.

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The Random Projection was first proposed by Lopes et al. [29] and was further discussed in later
studies [26, 47, 51, 53]. For Gaussian data, the procedure projects high-dimensional data onto random
subspaces of relatively low-dimensional spaces to allow the traditional Hotelling T 2 statistic to work well.
This method can be viewed as a two-step procedure. First, a single random projection is drawn, and it
is then used to map the samples from the high-dimensional space to a low-dimensional space. Second,
the Hotelling T 2 test is applied to a new hypothesis-testing problem in the projected space. A decision
is then returned to the original problem by simply rejecting H0 whenever the Hotelling test rejects it in
the projected spaces.
Some other related work on tests of high-dimensional locations can be found in [4, 11, 19, 20, 25, 30, 49],
which we do not discuss at length in this paper.

3
3.1

NTM on covariance matrices
One-sample scatter test

The standard test for scatters is to test the hypothesis H0 : Σ = Σ 0 vs H1 : Σ 6= Σ 0 . Because Σ 0
−1/2
by the data set and then change the test to the simpler hypothesis
is known, one can multiply Σ 0
H0 : Σ = Ip . The classical test for this hypothesis is the well-known likelihood ratio, which can be found
in any standard textbook, such as Anderson [1]. The likelihood ratio test statistic is given by
TLR = trS − log det(S) − p.
When p is fixed, the test based on TLR has many optimalities, such as unbiasedness, consistency, and
being invariant under affine transformation. However, similar to the Hotelling T 2 test, it has a fatal
defect in that it is not well defined when p is larger than n − 1. When p is large but smaller than n − 1,
the null distribution is not simple to use, even under normality. The popularly used option is the Wilks
theorem. However, when p is large, the Wilks theorem introduces a very serious error to the test because
its size tends to 1 as p tends to infinity. A correction to the likelihood ratio test based on random matrix
theory can be found in [2]. However, when p is large, especially when p/n is close to 1, we believe that
the asymptotic power will be low, much as occurs for the T 2 test. The idea of NTM can also be applied
to this hypothesis. Now, we first introduce the work by Ledoit and Wolf [27].
Ledoit and Wolf considered two hypotheses: H01 : Σ = Ip and H02 : Σ = aIp with a > 0 unknown.
Based on the idea of the Nagao test (see Nagao (1973) [32]), they proposed two test statistics:
!2
1
S
1
2
V = tr(S − Ip ) and U = tr 1
− Ip ,
p
p
p trS
which can be viewed from the perspective of NTM as considering S and 1p trS to be the estimators of
parameters Σ and a in the target functions
2

1
Σ
1
2
Σ
Σ
Σ
(3.1)
− Ip
h(Σ ) = tr(Σ − Ip ) and h(Σ , a) = tr
p
p
a
Σ. They studied the asymptotic properties of
, respectively. Note that under the null hypothesis, a = p1 trΣ
U and V in the high-dimensional setting where p/n → c ∈ (0, ∞) and found that U , for the hypothesis
of sphericity, is robust against large p, even larger than n. However, because V is not consistent against
every alternative, they proposed a new test statistic:

2
p 1
p
1
2
trS + .
W = tr(S − Ip ) −
p
n p
n
Under normality and the assumptions that
1
Σ)
tr(Σ
p

=

α

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No. 1

1
Σ − αI)2 = δ 2
tr(Σ
p
1
Σ j ) → νj < ∞, for j = 3, 4,
tr(Σ
p
They proved the following:
(i) The law of large numbers
1
tr(S)
p
1
tr(S2 )
p

P

→ α
P

→ (1 + c)α2 + δ 2 ;

(ii) The CLT, if δ = 0
n

"

1
p tr(S) − α
n+p+1 2
1
2
α
p tr(S) −
n

D

→ N

#

" # "
2α2
0
c
,

0
4 1 + 1c α3

4

2
c

D

1
c



α3

+ 5 + 2c α4

4 1+

#!

.

Based on these results, they derived that nU − p → N (1, 4). The inconsistency of the test based on V
can be seen from the following facts. When p is fixed, by the law of large numbers, we have S → Σ = Ip ,
P

and hence, V = p1 tr(S − Ip )2 → 0. However, when p/n → c > 0, we have
V

=

2
1
tr(S)2 − tr(S) + 1
p
p

P

→ (1 + c)α2 + δ 2 − 2α + 1 = cα2 + (α − 1)2 + δ 2 .
Σ − Ip )2 = (α − 1)2 + δ 2 , the null hypothesis can be regarded as
Because the target function is p1 tr(Σ
2
2
(α − 1) + δ = 0, and the alternative can be considered as (α − 1)2 + δ 2 > 0. However, the limit of V
has one more term, cα2 , which is positive. Therefore, the test V is not consistent. In fact, it is easy to
construct a counterexample based on this limit: set
cα + (1 − α)2 + δ 2 = c.
1−c
. Accordingly, the limit of V is the same for
When δ = 0, the solution to the equation above is α = 1+c
1−c
the null α = 1 and the alternative α = 1+c .
For W , we have
P
W → cα2 + (α − 1)2 + δ 2 − cα2 + c = c + (α − 1)2 + δ 2 .

When p is fixed, they proved that as n → ∞,

np
P
W → χ2p(p+1)/2
2

or equivalently,
P

nW − p →

2 2
−p
χ
p p(p+1)/2

When p → ∞, the right-hand side of the above tends to N (1, 4), which is the same as the limit when
p/n → c. This behavior shows that the test based on W is robust against p increasing. Chen et al [15]
extended the work to the case without normality assumptions.
Now, the target functions (3.1) can be rewritten as
Σ) =
h1 (Σ

1
2
1
Σ − Ip )2 = trΣ
Σ2 − trΣ
Σ+1
tr(Σ
p
p
p

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and
2

1
Σ
h2 (a, Σ ) = tr
− Ip
=
p
a

1
1
Σ2
Σ 2
p trΣ − ( p trΣ )
.
Σ)2
( p1 trΣ

Then, under the normality assumption, Srivastava [40] gave the unbiased and consistent estimators of
these parameters in the previous target functions, which are as follows:


1[
1
1
1\2
(n − 1)2
2
2
Σ = trS and trΣ
Σ =
trS −
trΣ
(trS) .
p
p
p
p(n − 2)(n + 1)
n−1
Based on these estimators, he proposed the test statistics
TS1 =

1
1
[
\
Σ2
Σ 2
1[
1\2
p trΣ − ( p trΣ )
Σ − 2 trΣ
Σ + 1 and TS2 =
trΣ
,
p
p
1
Σ)2
( p[
trΣ

and proved that under the assumption n ≍ pδ , 0 < δ 6 1, as {n, p} → ∞, we have asymptotically,


1
n
2
Σ − Ip ) ∼ N (0, τ12 )
TS1 − tr(Σ
2
p
and
n
2

TS2 −

1
1
Σ2
Σ 2
p trΣ − ( p trΣ )
Σ)2
( p1 trΣ

2
2
where τ12 = 2n
p (α2 − 2α3 + α4 ) + α2 , τ2 =
null hypothesis, one can easily obtain

!

∼ N (0, τ22 ),

2n(α4 α21 −2α1 α2 α3 +a32 )
pα61

n
D
TS1 → N (0, 1) and
2

+

α22
α41

Σi . Thus, under the
and αi = p1 trΣ

n
D
TS2 → N (0, 1).
2

Later, Srivastava and Yanagihara [45] and Srivastava et al. [46] extended this work to the cases of two or
more population covariance matrices and without normality assumptions, respectively. Furthermore, Cai
and Ma [9] showed that TS1 is rate-optimal over this asymptotic regime, and Zhang et al. [54] proposed
the empirical likelihood ratio test for this problem.
3.2

Li and Chen’s test based on unbiased estimation of target function

Li and Chen (2012) [28] considered the two-sample scatter problem, that is, testing the hypothesis
Σ1 , Σ 2 ) = tr(Σ
Σ1 − Σ 2 )2 . They selected the test
H0 : Σ 1 = Σ 2 . They choose the target function as h(Σ
Σ1 , Σ 2 ) as
statistic by the unbiased estimator of h(Σ
TLC = An1 + An2 − 2Cn1 n2
where
Anh

=

1 X ′
2
(Xhi Xhj )2 −
(nh )2
(nh )3
i6=j

+

Cn1 n2

=

1
(nh )4

X

X

X′hi Xhj X′hj Xhk

i,j,k
distinct

X′hi Xhj X′hk Xhl ,

i,j,k,l
distinct

XX
1
1 X ′
(X1i X2j )2 −
X′1i X2j X′2j X1k
n1 n2 i,j
n2 (n1 )2
j
i6=k

XX
XX
1
1
X′2i X1j X′1j X2k +
−
X′1i X2j X′1k X2l .
n1 (n2 )2
(n
)
(n
)
1
2
2
2
j
i6=k

i6=j k6=l

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17

Under the conditions A1 and A2 and for any i, j, k, l ∈ {1, 2},
Σ iΣ j Σ kΣ l ) = o(tr(Σ
ΣiΣ j )tr(Σ
ΣkΣ l )),
tr(Σ
we have
V ar(TLC )

2 
X
4 2 2
8
2
Σ2
=
2 tr Σ i + n tr(Σ i − Σ 1Σ 2 )
n
i
i
i=1


4
Σ1 − Σ 2 )Γi ◦ Γ′i (Σ
Σ1 − Σ 2 )Γi )
tr(Γ′i (Σ
ni
8
Σ1Σ 2 ),
tr2 (Σ
+
n1 n2

+

where A ◦ B = (aij bij ) denotes the Hadamard product of matrices A and B.
Li and Chen [28] proved that
Σ1 − Σ 2 )2 D
TCL − tr(Σ
p
→ N (0, 1).
V ar(TLC )
p
p
ar(TCL ) := n21 An1 + n22 An2 , which is a ratio-consistent estimator of V ar(TLC )
Li and Chen selected V\
under H0 . Therefore, the test rejects H0 if


2
2
An1 +
An2 .
TLC > ξα
n1
n2
Remark 7. In [28], Li and Chen also considered the test for the covariance between two sub-vectors,
i.e., testing the hypothesis H0 : Σ1,12 = Σ2,12 , where Σi,12 is the off-diagonal blocks of Σi . As the test
statistic is similar, we omit the details here.
3.3

Cai et al’s maximum difference test

Cai et al [6] also applied their maximum elements of the difference of two sample covariance matrices to
test the hypothesis of the equality of the two population covariances. They defined their test statistic as
follows:
(sij1 − sij2 )2
,
Mn = max Mij = max
16i6j6p
16i6j6p θ̂ij1 /n1 + θ̂ij2 /n2
where sijl is the (i, j)-th element of the sample covariance of the l-th sample, and
θ̂ijl

nl

2
1 X
(Xkil − X̄il )(Xkjl − X̄jl ) − sijl
=
nl
k=1

1 6 i 6 j 6 p and l = 1, 2. Here, θ̂ijl can be considered an estimator of the variance of sijl . Then, they
defined the test by
φα = I(Mn > qα + 4 log p − log log p).
where qα is the upper α quantile of the Type I extreme value distribution with the c.d.f.

x 
1
exp − √ exp(− ) ,
2
8π
and therefore

qα = − log(8π) − 2 log log(1 − α)−1 .
Σ2 and certain distributional
Under sparse conditions on the difference of the population covariances Σ 1 −Σ
conditions, they proved that for any t ∈ R



t
1
.
P(M1 − 4 log p + log log p 6 t) → exp − √ exp −
2
8π
As expected, Cai et al’s test is powerful when the difference of the two population covariances is sparse,
and it thus compensates somewhat for Li and Chen’s test.

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Testing the structure of the covariance matrix

In this subsection, we will consider another important test problem, namely, testing the structure of the
covariance matrix. First, we review the test hypothesis that the covariance matrix Σ is banded. That
is, the variables have nonzero correlations only up to a certain lag τ > 1. To elaborate, we denote
Σ = (σij )p×p and consider the following test hypotheses:
H0 : σij = 0, for all |i − j| > τ v.s. H1 : σij 6= 0, for some |i − j| > τ,

(3.2)

or, equivalently,
H0 : Σ = Bτ (Σ) v.s. H1 : Σ 6= Bτ (Σ),
where Bτ (Σ) = (σij I(|i − j| 6 τ )). From the perspective of NTMs, one can also choose the target
functions by the Euclidean distance and the Kolmogorov distance, which are the main concepts of the
tests proposed by Qiu and Chen [36] and Cai and Jiang [7], respectively.
For τ + 1 6 q 6 p − 1 and µ = 0, let
X
X
2
1
2
Xli X(l+q)j Xlk X(l+q)k
X
X
X
X
−
σ[
=
li
lj
(l+q)i
(l+q)j
ll+q
n(n − 1)
n(n − 1)(n − 2) i,j,k
i6=j

+

distinct

1
n(n − 1)(n − 2)(n − 3)

X

Xli X(l+q)j Xlk X(l+q)m .

i,j,k,m
distinct

P
Pp−q [
τ
τ
2
By denoting TQC
:= 2 p−1
q=k+1
l=1 σll+q , one can easily check that TQC is an unbiased estimator of
tr(Σ − Bτ (Σ)). Under the assumptions τ = o(p1/4 ), (A1), (A2) and certain conditions on the eigenvalues
of Σ, Qiu and Chen [36] showed that under the null hypothesis,
τ
nTQC
D
→ N (0, 4),
Vnτ

and the power function asymptotically satisfies
τ
βQC = P(nTQC
/Vnτ > 2ξα |Σ 6= Bk (Σ)) ≃ Φ



2ξα Vnτ
− δnτ
nvnτ



>Φ



ξα Vnτ
− δnτ
tr(Σ2 )



,

Pp c2
Pτ Pp−q [
2
−2 2
2
where Vnτ = l=1 σ
tr (Σ2 )+8n−1 tr(Σ(Σ−Bτ (Σ)))2 +4n−1 ∆tr[(Γ′ (Σ−
ll +2
q=1
l=1 σll+q , vnτ = 4n
′
Bτ (Σ))Γ) ◦ (Γ (Σ − Bτ (Σ))Γ)] and δnτ = tr(Σ − Bτ (Σ)2 /vnτ .
We can also rewrite the test hypothesis (3.2) as
H0 : ρij = 0, for all |i − j| > τ v.s. H1 : ρij 6= 0, for some |i − j| > τ,
where ρij is the population correlation coefficient between two random variables X1i and X1j . Cai and
Jiang [7] proposed a test procedure based on the largest magnitude of the off-diagonal entries of the
sample correlation matrix
τ
TCL
= max |ρ̂ij |,
|i−j|>τ

where ρ̂ij is the sample correlation coefficient. They showed that under the assumptions log p = o(n1/3 ) →
∞ and τ = o(pǫ ) with ǫ > 0, for any t ∈ R,



1
t
τ
2
.
P(n(TCL ) − 4 log p + log log p 6 t) → exp − √ exp −
2
8π
By implication, one can reject the null hypothesis whenever
τ
(TCL
)2 > n−1 [4 log p − log log p − log(8π) − 2 log log(1 − α)−1 ]

with asymptotical size α.

Hu J & Bai Z.

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19

Remark 8. If τ = 1 and under the normal assumption, then the test hypothesis is (3.2), also known as
testing for complete independence, which was first considered by Schott in 2005 [37] for a high-dimensional
random vector and using the Euclidean distance of the sample correlation matrix.
τ
Remark 9. Peng et al. [35] improved the power of the test TQC
by employing the banding estimator
for the covariance matrices. Zhang et al. [54] also gave the empirical likelihood ratio test procedure for
testing whether the population covariance matrix has a banded structure.

4

Conclusions and Comments

All of the NTM procedures show that most classical procedures in multivariate analysis are less powerful
in some parameter settings when the dimension of data is large. Thus, it is necessary to develop new
procedures to improve the classical ones. However, all of the NTM procedures developed to date require
additional conditions on the unknown parameters to guarantee the optimality of the new procedures; e.g.,
all procedures based on asymptotically normal estimations require that the eigenstructure of population
covariance matrix should not be too odd, and all NTMs based on the Kolmogorov distance require the
sparseness of the known parameters. Therefore, there is a strong need to develop data-driven procedures
that are optimal in most cases.
Acknowledgements
The authors would like to thank the referees for their constructive comments, which
led to a substantial improvement of the paper. J. Hu was partially supported by the National Natural Science
Foundation of China (Grant No. 11301063), Science and Technology Development foundation of Jilin (Grant No.
20160520174JH), and Science and Technology Foundation of Jilin during the “13th Five-Year Plan”; and Z. D.
Bai was partially supported by the National Natural Science Foundation of China (Grant No. 11571067).

References
1 T. Anderson. An introduction to multivariate statistical analysis. Third Edition. Wiley New York, 2003.
2 Z. D. Bai, D. D. Jiang, J. F. Yao, and S. R. Zheng. Corrections to LRT on large-dimensional covariance matrix by
RMT. The Annals of Statistics, 37(6B):3822–3840, Dec. 2009.
3 Z. D. Bai and H. Saranadasa. Effect of high dimension: by an example of a two sample problem. Statistica Sinica,
6:311–329, 1996.
4 M. Biswas and A. K. Ghosh. A nonparametric two-sample test applicable to high dimensional data. Journal of
Multivariate Analysis, 123:160–171, 2014.
5 T. Cai, W. Liu, and X. Luo. A Constrained l1 Minimization Approach to Sparse Precision Matrix Estimation. Journal
of the American Statistical Association, 106(494):594–607, 2011.
6 T. Cai, W. Liu, and Y. Xia. Two-Sample Covariance Matrix Testing and Support Recovery in High-Dimensional and
Sparse Settings. Journal of the American Statistical Association, 108(501):265–277, 2013.
7 T. T. Cai and T. Jiang. Limiting laws of coherence of random matrices with applications to testing covariance structure
and construction of compressed sensing matrices. Annals of Statistics, 39(3):1496–1525, 2011.
8 T. T. Cai, W. Liu, and Y. Xia. Two-sample test of high dimensional means under dependence. Journal of the Royal
Statistical Society: Series B, 2014.
9 T. T. Cai and Z. Ma. Optimal hypothesis testing for high dimensional covariance matrices. Bernoulli, 19(5B):2359–
2388, 2013.
10 T. T. Cai and Y. Xia. High-dimensional sparse MANOVA. Journal of Multivariate Analysis, 131:174–196, 2014.
11 a. Chakraborty and P. Chaudhuri. A Wilcoxon-Mann-Whitney-type test for infinite-dimensional data. Biometrika,
102(February):239–246, 2015.
12 L. Chen, D. Paul, R. Prentice, and P. Wang. A regularized Hotelling’s T (2) test for pathway analysis in proteomic
studies. Journal of the American Statistical Association, 106(496):1345–1360, 2011.
13 S. X. Chen, J. Li, and P. Zhong. Two-Sample Tests for High Dimensional Means with Thresholding and Data
Transformation. arXiv preprint arXiv:1410.2848, pages 1–44, 2014.
14 S. X. Chen and Y. L. Qin. A two-sample test for high-dimensional data with applications to gene-set testing. The
Annals of Statistics, 38(2):808–835, 2010.
15 S. X. Chen, L. X. Zhang, and P. S. Zhong. Tests for High-Dimensional Covariance Matrices. Journal of The American
Statistical Association, 105(490):810–819, June 2010.

20

Hu J & Bai Z.

Sci China Math

January 2015

Vol. 55

No. 1

16 A. P. Dempster. A high dimensional two sample significance test. The Annals of Mathematical Statistics, 29(1):995–
1010, 1958.
17 A. P. Dempster. A significance test for the separation of two highly multivariate small samples. Biometrics, 16(1):41,
Mar. 1960.
18 K. Dong, H. Pang, T. Tong, and M. G. Genton. Shrinkage-based diagonal Hotelling T2s tests for high-dimensional
small sample size data. Journal of Multivariate Analysis, 143:127–142, 2016.
19 L. Feng. Scalar-Invariant Test for High-Dimensional Regression Coefficients. (2005):1–19.
20 L. Feng and F. Sun. A note on high-dimensional two-sample test. Statistics & Probability Letters, 105:29–36, 2015.
21 Y. Fujikoshi. Multivariate analysis for the case when the dimension is large compared to the sample size. Journal of
the Korean Statistical Society, 33(1):1–24, 2004.
22 Y. Fujikoshi, T. Himeno, and H. Wakaki. Asymptotic results of a high dimensional MANOVA test and power
comparison when the dimension is large compared to the sample size. Journal of the Japan Statistical Society,
34(1):19–26, 2004.
23 H. Hotelling. The generalization of student’s ratio. The Annals of Mathematical Statistics, 2(3):360–378, 1931.
24 J. Hu, Z. Bai, C. Wang, and W. Wang. On testing the equality of high dimensional mean vectors with unequal
covariance matrices. Annals of the Institute of Statistical Mathematics, pages 1–20, 2014.
25 M. Hyodo and T. Nishiyama. A one-sample location test based on weighted averaging of two test statistics in
high-dimensional data. 2014.
26 L. Jacob, P. Neuvial, and S. Dudoit. DEGraph : differential expression testing for gene networks. 2012.
27 O. Ledoit and M. Wolf. Some hypothesis tests for the covariance matrix when the dimension is large compared to the
sample size. The Annals of Statistics, 30(4):1081–1102, 2002.
28 J. Li and S. X. Chen. Two sample tests for high-dimensional covariance matrices. The Annals of Statistics, 40(2):908–
940, Apr. 2012.
29 M. E. Lopes, L. Jacob, and M. J. Wainwright. A More Powerful Two-Sample Test in High Dimensions using Random
Projection. Advances in Neural Information Processing Systems, 1(2):1206–1214, 2011.
30 P. K. Mondal, M. Biswas, and A. K. Ghosh. On high dimensional two-sample tests based on nearest neighbors.
Journal of Multivariate Analysis, 141:168–178, 2015.
31 R. J. Muirhead. Aspects of multivariate statistical theory, volume 42. Wiley, 1982.
32 H. Nagao. On some test criteria for covariance matrix. The Annals of Statistics, 1(4):700–709, 1973.
33 G. Pan and W. Zhou. Central limit theorem for Hotelling’s T 2 statistic under large dimension. The Annals of Applied
Probability, 21(5):1860–1910, 2011.
34 J. Park and D. N. Ayyala. A test for the mean vector in large dimension and small samples. Journal of Statistical
Planning and Inference, 143(5):929–943, 2013.
35 L. Peng, S. X. Chen, and W. Zhou. More powerful tests for sparse high-dimensional covariances matrices. Journal of
Multivariate Analysis, 149:124–143, 2016.
36 Y. Qiu and S. X. Chen. Test for bandedness of high-dimensional covariance matrices and bandwidth estimation.
Annals of Statistics, 40(3):1285–1314, 2012.
37 J. R. Schott. Testing for complete independence in high dimensions. Biometrika, 92(4):951–956, 2005.
38 J. R. Schott. Some high-dimensional tests for a one-way MANOVA. Journal of Multivariate Analysis, 98(9):1825–1839,
Oct. 2007.
39 Y. Shen and Z. Lin. An adaptive test for the mean vector in large-p-small-n problems. Computational Statistics &
Data Analysis, 89:25–38, 2015.
40 M. S. Srivastava. Some Tests Concerning the Covariance Matrix in High Dimensional Data. Journal of the Japan
Statistical Society, 35(2):251–272, 2005.
41 M. S. Srivastava. A test for the mean vector with fewer observations than the dimension under non-normality. Journal
of Multivariate Analysis, 100(3):518–532, Mar. 2009.
42 M. S. Srivastava and M. Du. A test for the mean vector with fewer observations than the dimension. Journal of
Multivariate Analysis, 99(3):386–402, Mar. 2008.
43 M. S. Srivastava, S. Katayama, and Y. Kano. A two sample test in high dimensional data. Journal of Multivariate
Analysis, 114:349–358, Feb. 2013.
44 M. S. Srivastava and T. Kubokawa. Tests for multivariate analysis of variance in high dimension under non-normality.
Journal of Multivariate Analysis, 115:204–216, 2013.
45 M. S. Srivastava and H. Yanagihara. Testing the equality of several covariance matrices with fewer observations than
the dimension. Journal of Multivariate Analysis, 101(6):1319–1329, July 2010.
46 M. S. Srivastava, H. Yanagihara, and T. Kubokawa. Tests for covariance matrices in high dimension with less sample
size. Journal of Multivariate Analysis, 130:289–309, 2014.
47 M. Thulin. A high-dimensional two-sample test for the mean using random subspaces. Computational Statistics &
Data Analysis, 74:26–38, 2014.
48 T. Tonda and Y. Fujikoshi. Asymptotic Expansion of the Null Distribution of LR Statistic for Multivariate Linear

Hu J & Bai Z.

49
50
51
52
53
54

Sci China Math

January 2015

Vol. 55

No. 1

21

Hypothesis when the Dimension is Large. Communications in Statistics - Theory and Methods, 33(5):1205–1220, Jan.
2004.
A. Touloumis, S. Tavaré, and J. C. Marioni. Testing the mean matrix in high-dimensional transposable data. Biometrics, 71(1):157–166, 2015.
L. Wang, B. Peng, and R. Li. A High-Dimensional Nonparametric Multivariate Test for Mean Vector. Journal of the
American Statistical Association, 1459(June 2015):00–00, 2015.
S. Wei, C. Lee, L. Wichers, G. Li, and J. Marron. Direction-projection-permutation for high dimensional hypothesis
tests. arXiv preprint arXiv:1304.0796, pages 1–29, 2013.
Y. Wu, M. G. Genton, and L. a. Stefanski. A multivariate two-sample mean test for small sample size and missing
data. Biometrics, 62(3):877–885, 2006.
J. Zhang and M. Pan. A high-dimension two-sample test for the mean using cluster. Computational Statistics and
Data Analysis, 97:87–97, 2016.
R. Zhang, L. Peng, and R. Wang. Tests for covariance matrix with fixed or divergent dimension. The Annals of
Statistics, 41(4):2075–2096, 2013.

