Two-Sample and ANOVA Tests for High Dimensional Means

33 Pages·2017·0.53 MB·English

TWO-SAMPLE AND ANOVA TESTS FOR HIGH
DIMENSIONAL MEANS
By Song Xi Chen∗ and Jun Li† and Ping-Shou Zhong‡
Peking University∗, Kent State University†, and Michigan State
University‡
Thispaperconsiderstestingtheequalityoftwohighdimensional
means. Two approaches are utilized to formulate L -type tests for
2
betterpowerperformancewhenthetwohighdimensionalmeanvec-
torsdifferonlyinsparselypopulatedcoordinatesandthedifferences
are faint. One is to conduct thresholding to remove the non-signal
bearing dimensions for variance reduction of the test statistics. The
other is to transform the data via the precision matrix for signal
enhancement. It is shown that the thresholding and data transfor-
mation lead to attractive detection boundaries for the tests. Fur-
thermore, we demonstrate explicitly the effects of precision matrix
estimation on the detection boundary for the test with thresholding
and data transformation. Extension to multi-sample ANOVA tests
is also investigated. Numerical studies are performed to confirm the
theoretical findings and demonstrate the practical implementations.
1. Introduction. Modern statistical data in biological and financial
studies are increasingly high dimensional, but with relatively small sample
sizes. This is the so-called “large p, small n” paradigm, where classical mul-
tivariate procedures originally designed for fixed dimension problems may
nolongerbefeasible.Newmethodswhichareadaptivetothe“largep,small
n” paradigm are needed.
An important high dimensional inferential task is to test the equality of
themeanvectorsbetweentwopopulations.LetX ··· ,X beanIIDsam-
i1 ini
ple drawn from a p-dimensional distribution F , for i = 1 and 2 respectively.
i
The dimensionality p can be much larger than the sample sizes n and n so
1 2
that p/n → ∞. Let µ and Σ be the mean and the covariance of F . The
i i i i
primary interest is testing
(1.1) H : µ = µ versus H : µ (cid:54)= µ .
0 1 2 1 1 2
Hotelling’s T2 test (Hotelling, 1931) is a classical test for the above hy-
potheses with fixed dimension p, and is still applicable if p < n +n −2.
1 2
Keywords and phrases:ANOVA,Datatransformation,Largepsmalln,Sparsesignals,
Thresholding, Two-sample tests for means.
1
2
However, Bai and Saranadasa (1996) showed that Hotelling’s test suffers
from a power loss when p/(n +n −2) approaches to 1 from below. When
1 2
p > n +n −2, the test is inapplicable as the sample covariance matrix is
1 2
no longer invertible.
There have been proposals to modify Hotelling’s T2 statistic for high
dimension. Bai and Saranadasa (1996) removed the inverse of the sample
covariancefromtheHotelling’sformulation.ChenandQin(2010)(CQ)con-
sideredalinearcombinationofU-statisticsandshowedthatthecorrespond-
ing test can operate under much relaxed conditions regarding p and sample
sizes without assuming Σ = Σ . Srivastava, Katayama and Kano (2013)
1 2
proposed using the diagonal matrix of the sample covariance to replace the
samplecovarianceunderthenormality.Gregoryetal.(2015)proposedusing
an average of the squared univariate two-sample t-statistics over p compo-
nents as the test statistic. These four tests are basically all targeted on the
L -normoraweightedL -normbetweenµ andµ .Cai,LiuandXia(2014)
2 2 1 2
(CLX)proposedatestbasedonthemax-normofmarginalt-statistics.More
importantly, they implemented a data transformation designed to increase
the signal strength under sparsity as discovered by Hall and Jin (2010) in
the one-sample innovated higher criticism test.
The L -norm based tests are known to be effective in detecting dense
2
signals when the differences between µ and µ are located over a large
1 2
number of components. However, the tests encounter a power loss under the
sparse signal settings. Meanwhile, although Hall and Jin (2010) discovered
that transforming data with a known precision matrix Ω = Σ−1 for the
Gaussian data leads to enhanced signal strength and a lowered detection
boundary, it is uncertain if these results can be maintained with estimated
Ω for the sub-Gaussian data.
With these as the motivation, this paper considers two modifications to
CQ’s test formulation. First of all, we apply a multi-level thresholding ap-
proach to removing the non-signal bearing dimensions via a multi-layer of
threshold levels to be adaptive to faint signals. The second alteration is to
transform the data by an estimated precision matrix followed by the multi-
level thresholding trying to enlarge the signal strength for more power gain.
The idea of thresholding to remove the non-signal bearing dimensions was
advocated in Donoho and Johnstone (1994) for selecting significant wavelet
coefficients and Fan (1996) for testing the mean with IID Gaussian dis-
tributed components. See also Ji and Jin (2012) for variable selection in
high dimensional regression. In this paper, we show that a two-sample test
based on the multiple thresholding levels (multi-level thresholding test) at-
tains a detection boundary that coincides with the the optimal detection
TWO-SAMPLE AND ANOVA TESTS FOR HIGH DIMENSIONAL MEANS 3
boundary for Gaussian data with identity covariance matrix. Furthermore,
it is found that the detection boundary can be lowered by adding the data
transformation in its formulation. A contribution of the current paper is
that we explicitly establish the effect of precision matrix estimation on the
detection boundary for the test with thresholding and data transformation.
Inadditiontothetwo-sampletests,weextendouranalysistotheANOVA
test for m populations:
(1.2) H∗ : µ = µ = ··· = µ versus H∗ : µ (cid:54)= µ for some i (cid:54)= j,
0 1 2 m 1 i j
where µ is the mean of F for i = 1,··· ,m and m ≥ 2. Multi-level thresh-
i i
olding ANOVA tests with and without data transformation via the preci-
sion matrices are proposed. It is shown that the detection boundaries of the
ANOVA tests resemble those of the two samples outlined above. As far as
weareaware,theresultsregardingthedetectionboundaryforANOVAtests
are the first of this kind in high dimensional testing for the means.
Therestofthepaperisorganizedasfollows.Weanalyzethepowerperfor-
manceoftheCQtestandtheOracletestunderthesparsesettinginSection
2. Thresholding tests without and with data transformation are proposed
in Section 3 for detecting faint signals. Section 4 studies the multi-level
thresholding tests. Extension to the ANOVA tests is provided in Section 5.
Simulation results are presented in Section 6. Section 7 concludes the pa-
per with discussions. Key technical details are reported in the Appendix,
whereas additional proofs and simulation results, and an empirical study to
select differentially expressed gene-sets for a human breast cancer data set
are given in the supplementary material.
2. L -Norm Based Tests under Sparsity. The statistic proposed
2
by Chen and Qin (2010) (herein CQ test) can be written as T = (cid:80)p T
n k=1 nk
where
1 (cid:88)n1 (k) (k) 1 (cid:88)n2 (k) (k)
T = X X + X X
nk n (n −1) 1i 1j n (n −1) 2i 2j
1 1 2 2
i(cid:54)=j i(cid:54)=j
2 (cid:88)n1 (cid:88)n2 (k) (k)
(2.1) − X X ,
n n 1i 2j
1 2
i j
(k)
and X denotes the k-th component of X . It is readily shown that T
ij ij nk
is unbiased to (µ −µ )2, a form of the signal in the k-th dimension.
1k 2k
Tofacilitatesimplernotation,wemodifythestatisticT byrescalingeach
n
T by σ /n +σ /n , the variance of X¯(k)−X¯(k) where σ is the k-
nk 1,kk 1 2,kk 2 1 2 i,kk
th diagonal component of Σ (i = 1,2) and is assumed to be known. If σ
i 1,kk
4
and σ are unknown, we can use σˆ /n +σˆ /n where σˆ and σˆ
2,kk 1,kk 1 2,kk 2 1,kk 2,kk
are the usual sample variances at the k-th dimension. This will make the
CQ test invariant under the scale transformation; see Feng et al. (2015) for
a related investigation. To expedite discussion, we assume σ are known
i,kk
and equal to one without loss of generality. This leads to a modified CQ
statistic
p
(2.2) T˜ = n(cid:88)T , with n = n n /(n +n ).
n nk 1 2 1 2
k=1
Similar to Chen and Qin (2010), by defining
(cid:26)√ √ (cid:27) σ σ
(2.3) ρ = Cov n(X¯(k)−X¯(k)), n(X¯(l)−X¯(l)) = n( 1,kl + 2,kl),
kl 1 2 1 2 n n
1 2
the variance of T˜ under H is σ2 = 2p+2(cid:80) ρ2 , and under H is
n 0 T˜n,0 i(cid:54)=j ij 1
(2.4) σ2 = 2p+2(cid:88)ρ2 +4n (cid:88) δ δ ρ ,
T˜n,1 ij k l kl
i(cid:54)=j k,l∈S
β
where δ = µ − µ and S = {k : δ (cid:54)= 0} is the set of non-zero δ
k 1k 2k β k k
locations.
Under a general multivariate model and some conditions on the covari-
ance, it can be shown (Chen and Qin, 2010) that
T˜ −||µ −µ ||2
n 1 2 d
−→ N(0,1) as p → ∞ and n → ∞.
σ
T˜n,1
So the modified CQ test rejects H if T˜ /σˆ > z where z is the upper
0 n T˜n,0 α α
α quantile of N(0,1) and σˆ is a consistent estimator of σ .
T˜n,0 T˜n,0
To see the performance of the CQ test under the sparse setting, let |S | =
β
p1−β where|·|representsthecardinalityofasetandβ ∈ (0,1)isthesparsity
index. The power of the CQ test is
(2.5) β (||µ −µ ||) = Φ(cid:18)− σT˜n,0z + p1−βnδ¯2(cid:19),
T˜n 1 2 σT˜n,1 α σT˜n,1
where Φ(·) is the cumulative distribution function of N(0,1), and δ¯2 =
(cid:80) δ2/p1−β in (2.5) is the average standardized signal.
k∈Sβ k
Since σ2 ≥ σ2 , the first term within Φ(·) in (2.5) is bounded. Then,
T˜n,1 T˜n,0
the power is largely determined by the second term
p1−βnδ¯2
(2.6) SNR =: ,
T˜n (cid:113)2p+2(cid:80) ρ2 +4n(cid:80) δ δ ρ
i(cid:54)=j ij k,l∈Sβ k l kl
TWO-SAMPLE AND ANOVA TESTS FOR HIGH DIMENSIONAL MEANS 5
which is the signal-to-noise ratio since the numerator is the average signal
strength and the denominator is the standard deviation of the test statistic
underH .Aninspectionon(2.6)revealsthatwhilethenumeratorofSNR
1 T˜n
is contributed only by those signal bearing dimensions, the standard devia-
tion in the denominator is contributed by all T including those with non-
nk
(cid:113)
signals. Specifically, if Σ = Σ = I , SNR = p1−βnδ¯2/ 2p+4p1−βnδ¯2.
1 2 p T˜n
If β > 1/2 and δ¯= o(n−1/2pβ/2−1/4), SNR = o(1), which implies that the
T˜n
test has little power beyond the significant level. A reason for the power loss
is that the variance of T˜ is inflated by those non-signal bearing T .
n nk
To put the above analysis in a broader perspective, we consider an Oracle
who has the knowledge of the signal bearing set S = {k : δ (cid:54)= 0}. The
β k
Oracle test statistic is
(cid:88)
(2.7) O = n T .
n nk
k∈S
β
Similar to the CQ test, the power of the Oracle test is determined by
p1−βnδ¯2
(2.8) SNR =: where
On σ
On
(2.9) σ2 = 2p1−β +2 (cid:88) ρ2 +4n (cid:88) δ δ ρ .
On ij k l kl
i(cid:54)=j∈S k,l∈S
β β
Since σ2 (cid:28) σ2 , SNR (cid:29) SNR . Specially, if Σ = Σ = I ,
On T˜n,1 On T˜n 1 2 p
p1−βnδ¯2 p(1−β)/2nδ¯2
(2.10) SNR = = √
On (cid:113)2p1−β +4p1−βnδ¯2 2+4nδ¯2
that tends to infinity for β > 1/2 as long as δ¯(cid:16)n−1/2pβ/4−1/4+(cid:15) for any
(cid:15) > 0, which is much smaller than n−1/2pβ/2−1/4 required for the CQ test,
indicating the test is able to detect much fainter signal.
3. Thresholding and Data Transformation. The power of the Or-
acle test is in its exclusion of the non-signal bearing dimensions, whose
locations are unknown in reality. Thresholding can be carried out to exclude
those non-signal bearing dimensions. Based on the large deviation results
(Petrov, 1995), we use a thresholding level λ (s) = 2slogp for s ∈ (0,1) to
n
strike a balance between removing non-signal bearing T and keeping those
nk
with signals. The thresholding statistic is
p (cid:26) (cid:27)
(cid:88)
(3.1) L (s) = nT I nT +1 > λ (s) ,
1 nk nk n
k=1
6
where I(·) is the indicator function. A closely related statistic is
p (cid:26) (cid:27) (cid:26) (cid:27)
(3.2) L (s) = (cid:88) n(X¯(k)−X¯(k))2−1 I n(X¯(k)−X¯(k))2 > λ (s) ,
2 1 2 1 2 n
k=1
where X¯(k) is the sample means of X(k). Here, we use L (s) to refer ei-
i ij n
ther L (s) or L (s) because both L (s) and L (s) have very similar proper-
1 2 1 2
ties. The proposed thresholding statistic can accommodate the column-wise
dependence, which is defined via the α-mixing among the components of
X = (X(1),··· ,X(p))T.
We take a time series view on the dependence among the components of
the high dimensional X. For any integers a < b, let F to be the σ-
X,(a,b)
algebra generated by {X(m) : m ∈ (a,b)} and define the α-mixing coefficient
α (k) = sup |P(A∩B)−P(A)P(B)|,
X
m∈N,A∈F ,B∈F
X,(1,m) X,(m+k,∞)
where N denotes the set of natural numbers. The following conditions are
assumed in our analysis.
(C1): As n → ∞, p/n → ∞, n /(n +n ) → κ and logp = o(n1/3).
1 1 2
(C2): Let X = µ +W . There exists a positive constant H such that
ij i ij
for h ∈ [−H,H]2, E{ehT·[(Wi(jk))2,(Wi(jl))2]} < ∞ for k (cid:54)= l.
(C3): There exists a permutation X˜ of the components of X such that
X˜ is α-mixing satisfying α (k) ≤ Cαk for some α ∈ (0,1) and a positive
X˜
constant C. Moreover, (cid:80)p |σ | < ∞ for i = 1 or 2 such that ρ defined
l=1 i,kl kl
in (2.3) satisfies (cid:80)p |ρ | < ∞ for any k ∈ {1,··· ,p}.
l=1 kl
Condition (C1) specifies the growth rate of p relative to n under which
the large deviation results can be applied. This condition is not required
in Chen and Qin (2010) because it does not involve the thresholding. (C2)
(k) (l)
assumesthat(X ,X )hasabivariatesub-Gaussiandistribution,whichis
ij ij
more general than the Gaussian distribution. Such conditions are commonly
assumed in high dimensional analysis (Bickel and Levina, 2008b; Zhong,
Chen and Xu, 2013; and Cai, Liu and Xia, 2014). Condition (C3) prescribes
weak dependence among the column components of a permuted version X˜
of the original vector X, which implies that X˜ respects an ordering such
thatcomponentsclosertoeachotheraremorestronglycorrelatedthanthose
further apart. As the thresholded L -norm statistics are invariant under any
2
permutation of the data, (C3) only requires that such permutation exists
and there is no need to actually identify the permutation. The exponential
decayfortheα-mixingcoefficientscanberelaxedtopolynomialdecayswith
TWO-SAMPLE AND ANOVA TESTS FOR HIGH DIMENSIONAL MEANS 7
more involved proofs. The last two restrictions in (C3) are for the quantities
of the original data rather than the permuted data due to the permutation
invariance of the L statistics. While the α-mixing is a common approach to
2
accommodate the column-wise dependence, the physical dependence mea-
sure of Wu (2005) may be also used for modeling the weak dependence.
A parallel development based on the physical dependence is possible. It is
noted that (C3) is not required in Chen and Qin (2010) because their test
formulation does not involve thresholding to remove the non-signal bear-
ing dimensions. The asymptotic normality of the test statistic in Chen and
Qin (2010) is established by the martingale central limit theorem, which
requires some moment conditions and a general multivariate linear innova-
tion model. However, in one aspect, the paper’s assumption is weaker, as we
do not assume the multivariate linear innovation model. This is due to the
thresholding which renders a need for such model.
In addition to the thresholding, we consider enhancing signal strength via
data rotation. Motivated by Hall and Jin (2010)’s study on data rotation
via a banded Cholesky factor for the innovated HC test, and Cai, Liu and
Xia (2014)’s transformation via the CLIME estimator (Cai et al., 2011) in
theirmax-normbasedtest,wewillshowthatthesignalenhancementcanbe
achieved by transforming the data via an estimate of Ω = Σ−1 = (ω )
w ij p×p
where Σ = (1−κ)Σ +κΣ .
w 1 2
Like Hall and Jin (2010), we consider a bandable covariance matrix class
(cid:26)
V((cid:15)0,C,ν) = Σ : 0 < (cid:15)0 ≤ λmin(Σ) ≤ λmax(Σ) ≤ (cid:15)−01,ν > 0,
(cid:27)
|σ | ≤ C(1+|i−j|)−(ν+1) for all i,j : |i−j| ≥ 1 .
ij
This class satisfies both the banding and thresholding conditions of Bickel
and Levina (2008b). We assume the following regarding the covariance ma-
trices.
(C4): Both Σ and Σ belong to the matrix class V((cid:15) ,C,ν).
1 2 0
Although (C3) has assumed the weakly dependence among the compo-
nents of X , imposing (C4) ensures that the data transformed by Ω are
ij
also weakly dependent. Although we assume the off-diagonal decay rates of
Σ (i=1,2) are the same to expedite the technical analysis, all the results
i
can be generalized to the case with different decay rates. In practice, the
precision matrix Ω needs to be estimated. Bickel and Levina (2008a) pro-
posed estimating Ω by banding the Cholesky factor matrices. Cai, Liu and
8
Luo (2011) introduced the CLIME estimator based on the constrained L
1
minimization. As the CLIME estimator has the same rate of convergence
as the estimator of Bickel and Levina (2008a) (Cai et al., 2011) when Ω
belongs to the bandable class, we use the latter to obtain a slightly simpler
banding Cholesky estimator Ωˆ as follows.
√ τ√
Define Y = 1−κX − κX for k = 1,··· ,n and l = 1,··· ,n ,
kl 1k 2l 1 2
whereκ = lim n /(n +n ).ThenVar(Y ) = Σ ≡ (1−κ)Σ +κΣ .LetY
1 1 2 kl w 1 2
n→∞
be an IID copy of Y for any fixed k and l such that Y = (Y(1),··· ,Y(p))T.
kl
Forj = 1,··· ,p,defineYˆ(j) = aTW(j)wherea = {Var(W(j))}−1Cov(Yˆ(j),
j j
W(j)) and W(j) = (Y(1),··· ,Y(j−1))T. Let (cid:15) = Y(j) − Yˆ(j) and d2 =
j j
Var((cid:15) ), and A be the lower triangular matrix with the j-th row being
j
(aT,0 ) and D = diag(d2,··· ,d2) where 0 means a vector of 0 with
j p−j+1 1 p s
length s. Then, the Cholesky decomposition is Ω = (I −A)TD−1(I −A).
LetY = (cid:112)n /(n +n )X −(cid:112)n /(n +n )X := (Y(1),··· ,Y(p))T.
n,kl 2 1 2 1k 1 1 2 2l n,kl n,kl
Given a τ, regress Y(j) on Y(j) = (Y(j−τ),··· ,Y(j−1))T to obtain the
n,kl n,kl,−τ n,kl n,kl
least square estimate of a = (a ,··· ,a )T:
j,τ j−τ j−1
aˆ = ((cid:88)n1 (cid:88)n2 Y(j) Y(j)T )−1(cid:88)n1 (cid:88)n2 Y(j) Y(j) .
j,τ n,kl,−τ n,kl,−τ n,kl,−τ n,kl
k=1l=1 k=1l=1
Put aˆT = (0T ,aˆT ,0T ) be the j-th row of a lower triangular ma-
j τ−1 j,τ p−j+1
trix Aˆ and Dˆ = diag(d2 ,··· ,d2 ) where d2 = (cid:80)n1 (cid:80)n2 (Y(j) −
τ τ 1,τ p,τ j,τ k=1 l=1 n,kl
aˆT Y(j) )2/(n n ). Thus, an banded estimator of Ω is
j,τ n,kl,−τ 1 2
(3.3) Ωˆ = (I −Aˆ )TDˆ−1(I −Aˆ ).
τ τ τ τ
The consistency of Ωˆ to Ω is established in Bickel and Levina (2008a).
τ
The method of Qiu and Chen (2015) may be used for selecting the suitable
banding width τ, as well as a method outlined in Section 6.
It is noted that the bandable structure assumed in (C4) is not needed
in the max-norm based test of Cai, Liu and Xia (2014). However, they
require other conditions which are not taken by the current paper. More
importantly, the two tests have different detection boundaries by comparing
those established in Theorem 4 of this paper and that given in Section 1.4.1
of Donoho and Jin (2004). In particular, when the sparsity is moderate such
that 1/2 < β < 3/4, the proposed test can attain the Gaussian detection
boundary while the max-norm test can not. These aspects are reflected in
the simulation results and reported in Figures 1-3 of Section 6.
TWO-SAMPLE AND ANOVA TESTS FOR HIGH DIMENSIONAL MEANS 9
Thetransformedthresholdingteststatisticbasedon{Zˆ =: Ωˆ X : 1 ≤
1i τ 1i
i ≤ n } and {Zˆ =: Ωˆ X : 1 ≤ i ≤ n } is
1 2i τ 2i 2
(3.4)Jˆ (s,τ) = (cid:88)p (cid:26)n(Z¯ˆ1(k)−Z¯ˆ2(k))2 −1(cid:27)I(cid:26)n(Z¯ˆ1(k)−Z¯ˆ2(k))2 > λ (s)(cid:27),
n n
ωˆ ωˆ
kk kk
k=1
where Z¯ˆ(k) is the sample mean of Zˆ(k).
i ij
It is worth discussing the sparsity of the transformed signals as it directly
relates to the benefits of data transformation. In general, the sparsity of
µ −µ does not imply that of Ω(µ −µ ). However, as shown in Lemma 6
1 2 1 2
of the supplementary material, under Conditions (C4) and (C5), Ω(τ)(µ −
1
µ ) is sparse if (µ − µ ) is, where Ω(τ) = {ω I(|i − j| ≤ τ)} is a
2 1 2 ij p×p
banded version of Ω. The supplementary material also contains a numerical
confirmation on the sparsity of Ω(τ)(µ −µ ).
1 2
The asymptotic normality of Jˆ (s,τ) is established in the following.
n
Theorem 1. Assume Conditions (C1)-(C4), p = n1/θ for 0 < θ < 1 and
τ (cid:16) (n−1logp)−1/{2(ν+1)}, then for any s ∈ (1−νθ/(ν +1),1),
σ−1 (cid:8)Jˆ (s,τ)−µ (cid:9) −→d N(0,1),
Jn(s,τ) n Jn(s,τ)
where µ and σ are defined by (8.3) and (8.4), respectively.
Jn(s,τ) Jn(s,τ)
That requiring τ (cid:16) (n−1logp)−1/{2(ν+1)} is to allow consistent estimation
ofΩ.Inaddition,thatimposingp = n1/θ istocontroltheaccumulatederror
duetotheincreaseofdimension.Notethatifθ isarbitrarilycloseto0,pwill
grow exponentially fast with n. Moreover, a sufficient rate of convergence of
Ωˆ to Ω requires the thresholding level s being larger than 1−νθ/(ν +1)
τ
which depends on the sparsity parameter ν for the bandable class in (C4).
In practice, θ can be “estimated” by log(n)/log(p). A liberal choice for the
lower bound may be 1−log(n)/log(p) provided ν >> 1. To gain knowledge
ofν,wemaymodelthedecayofσ parametrically,forinstance|σ | = γ|j−i|
ij ij
for a γ ∈ (0,1) or |σ | = (1+|j−i|)−ξ for a ξ > 0, which represent models
ij
with the exponential and the polynomial decay, respectively. Both models
were considered in Hall and Jin (2010) and Qiu and Chen (2015). We can
also use the generalized method of moment estimator advocated in He and
Chen (2016) to estimate the parameters γ and ξ under both models, which
can be translated to estimates of ν.
For the thresholding statistic L (s) without the data transformation, less
n
conditions than those in Theorem 1 are required in establishing its asymp-
totic normality, as shown in the following proposition. In particular, that
p = n1/θ and the lower bound on s are not needed.
10
Proposition 1. Assume Conditions (C1)-(C3). For any s ∈ (0,1),
(3.5) σ−1 (cid:8)L (s)−µ (cid:9) −→d N(0,1),
Ln(s) n Ln(s)
where µ and σ are given by (8.1) and (8.2), respectively.
Ln(s) Ln(s)
The transformed thresholding test rejects H at the level α if
0
(3.6) Jˆ (s,τ) > z σˆ +µˆ ,
n α Jn(s,τ),0 Jn(s,τ),0
where µˆ and σˆ2 are, respectively, consistent estimators of
Jn(s,τ),0 Jn(s,τ),0
(cid:26) 2 (cid:27)
µJn(s,τ),0 = √2π(2slogp)12p1−s {1+o(1)}, and
(cid:26) 2 (cid:27)
σJ2n(s,τ),0 = √2π{(2slogp)32 +(2slogp)12}p1−s {1+o(1)}, satisfying
p
(3.7) µ −µˆ = o{σ } and σˆ /σ → 1.
Jn(s,τ),0 Jn(s,τ),0 Jn(s,τ),0 Jn(s,τ),0 Jn(s,τ),0
Moreover, the asymptotic power of the transformed thresholding test is
(cid:18) z σ µ −µ (cid:19)
β (||µ −µ ||) = Φ − α Jn(s,τ),0 + Jn(s,τ) Jn(s,τ),0 ,
Jˆn(s,τ) 1 2 σ σ
Jn(s,τ) Jn(s,τ)
which is mainly determined by
µ −µ
(3.8) SNR =: Jn(s,τ) Jn(s,τ),0.
Jˆn(s,τ) σ
Jn(s,τ)
A test based on L (s) without the data transformation can be proposed
n
in an analogy to (3.6). It can be shown based on (3.5) that the asymptotic
power of the thresholding only test is
(cid:18) z σ µ −µ (cid:19)
β (||µ −µ ||) = Φ − α Ln(s),0 + Ln(s) Ln(s),0 ,
Ln(s) 1 2 σ σ
Ln(s) Ln(s)
which is determined by its signal-to-noise ratio
µ −µ
(3.9) SNR =: Ln(s) Ln(s),0,
Ln(s) σ
Ln(s)
where µ and σ are the values of µ and σ under H and
Ln(s),0 Ln(s),0 Ln(s) Ln(s) 0
can be obtained by ignoring all the summation terms in (8.1) and (8.2).