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SUT Journal of Mathematics
Vol. 56, No. 2 (2020), 129–145
A necessary test for elliptical symmetry based on
the uniform distribution over the Stiefel manifold
Toshiya Iwashita and Bernhard Klar
(Received April 14, 2020)
Abstract. This paper provides a new procedure for testing the null hypothesis
of multivariate elliptical symmetry. A test for uniformity over the Stiefel mani-
fold based on modified degenerate V-statistics is employed since the test statistic
proposed in this paper consists of independent random matrices, formed by the
scaled residuals (or the Studentized residuals), which are uniformly distributed
over the Stiefel manifold under the null hypothesis. Also, Monte Carlo simula-
tion studies are carried out to evaluate the type I error and power of the test.
Finally, the procedure is illustrated using the Iris data.
AMS 2010 Mathematics Subject Classification. 62H10, 62H15.
Key words and phrases. Elliptical distribution, left-spherical distribution, scaled
residuals, spherical distribution, Stiefel manifold, uniform distribution.
§1. Introduction
The family of elliptical contoured distributions (or elliptical distributions for
short) is a natural generalization of the multivariate normal distribution. The
assumption of elliptical populations is frequently imposed in multivariate anal-
ysis. However, it is indispensable to test whether a sample comes from an el-
liptical population. Therefore, there exists a sizable literature on this subject
(see Fang and Liang [8] for a survey). See also Manzotti et al. [18], Schott [22],
Huffer and Park [14], Batsidis and Zografos [2] and the references therein.
Let X1, . . . , XNbe iid p-dimensional random (column) vectors drawn from
a population with mean vector µand covariance matrix Σ = Σ′>0 (Σ′
means transpose of Σ and Σ >0 indicates that Σ is positive definite). Let
X= [X1, . . . , XN] be the p×Nobservation matrix. Then, the sample mean
129
130 T. IWASHITA AND B. KLAR
vector and covariance matrix can be expressed as
¯
X=1
NX1,
S=1
nXQX′, n =N−1≥p,(1.1)
respectively, where 1is the vector of Nones,
(1.2) Q=IN−1
N11′,
Iddenotes the identity matrix of size dand the prime refers to transpose.
Some of the statistics for testing elliptical symmetry in the above-mentioned
references consist of the so-called scaled residuals (or Studentized residuals)
Wi=S−1/2(Xi−¯
X), i = 1, . . . , N,
where S−1/2indicates the inverse matrix of a symmetric square root of S.
For instance, Manzotti et al. [18] considered the statistic Wi/||Wi||, where
|| · || stands for the Euclidean norm of a vector, which should approximately
possess the uniform distribution over the unit sphere Sp−1on Rpwhen the dis-
tribution of Xihas elliptical symmetry, and they introduced the procedure of
testing elliptical symmetry by using the limiting distribution of the average of
some spherical harmonics over the Wi/||Wi||’s. Huffer and Park [14] provided
Pearson’s χ2-statistic based on ||Wi||2with qc shells, obtained by dividing
Rpinto cspherical shells centered at the origin and qcongruent sectors em-
anating from the origin. They also carried out numerical studies to compare
the power of their test procedure with other tests for elliptical symmetry and
multivariate normality under various alternatives. As pointed out by Fang
and Liang [8] and Batsidis and Zografos [2], however, a downside in using the
scaled residuals is that Wi,i= 1, . . . , N , are no longer independent, and their
distribution is different from the distribution of Σ−1/2(Xi−µ), i= 1, . . . , N.
In a recent paper, Iwashita and Klar [15] considered the (exact) joint distri-
bution of {Wi}N
i=1, that is, the joint distribution of the p×Nrandom matrix
W= [W1, . . . , WN] = S−1/2[X1, . . . , XN]Q=S−1/2XQ,
under elliptical population. Note that, since Qis an N×Nidempotent matrix
with rank(Q) = n(= N−1), there exists an N×nmatrix Ksuch that
(1.3) KK′=Q, K′K=In, K′1=0,
where 0is the column n-vector of zeroes.
The contribution of this paper is to show that U=K′X′(nS)−1/2pos-
sesses the uniform distribution over the Stiefel manifold, and then construct
ELLIPTICALLY SYMMETRY 131
a procedure of a necessary test for elliptical symmetry. Here we note that the
terminology “necessary test” has the same meaning as in Fang et al. [10].
The paper is organized as follows: In Section 2, we show that the proposed
statistic Uhas the uniform distribution over the Stiefel manifold O(n, p) of
orthonormal n-frames in Rpby applying the result in Iwashita and Klar [15].
In Section 3, we construct the procedure of testing elliptical symmetry by
combining the method of Pycke [21] with the result in the Section 2. In
Section 4, we conduct some numerical experiments to confirm the ability of
our procedure. We also apply our test for the Iris data presented by Fisher [11].
§2. Preliminary
Let Xbe a p-dimensional random vector from an elliptical distribution with
a location parameter µ∈Rpand a scale matrix Λ, a symmetric and positive
definite matrix of order p, having a probability density function (pdf) of the
form
(2.1) f(x) = cp|Λ|−1/2g((x−µ)′Λ−1(x−µ)),
where gis a nonnegative function, and cpis a normalizing constant (see, for
example, Muirhead [19, Section 1.5], Fang and Zhang [9, Section 2.6.5]). Note
that the characteristic function (cf) of Xcan be expressed as
(2.2) Ψ(t) = exp(it′µ)ψ(t′Λt),t∈Rp, i =√−1,
and, if they exist, E[X] = µand Σ = Cov[X] = −2ψ′(0)Λ ≡γΛ>0.
Suppose Xis a k×lrandom matrix and H∈ O(k), where O(k) is the set
of orthogonal matrices of order k. If Xd
=HX for every fixed H, where the
notation “ d
=” denotes equality in distribution, we call the distribution of X
left-spherical. If X′is left-spherical, then Xis right-spherical. When Xis left-
and right-spherical, we call Xspherical (see Dawid [5]).
Throughout this paper, we assume the existence of the covariance matrix Σ
and the pdf as given in (2.1) for nonsingularity of the sample covariance matrix
(1.1) (see Balakrishnan et al. [1], Eaton and Perlman [6] and Okamoto [20]),
and we will write X∼ECp(µ,Λ; ψ) to indicate that Xhas an elliptical distri-
bution whose cf has the form given in (2.2). In a similar way, if a k×lrandom
matrix Xhas a left-spherical distribution with the cf ϕX(T) for k×lmatrix T,
then, we denote X∼LSk×l(ϕX), and X∼SSk×l(ϕX) means Xhas a spherical
distribution with the cf ϕX(T) (see, in some detail, Fang and Zhang [9, Lemma
3.1.1 and Theorem 3.1.4]). Finally, X∼ Uk,l indicates that a k×lrandom
matrix Xis uniformly distributed over the Stiefel manifold O(k, l), i.e., Xis
left-spherical and X′X=Il(see Fang and Zhang [9, Definition 3.1.2]).
132 T. IWASHITA AND B. KLAR
Suppose X1, . . . , XNare independent random copies of X∼ECp(0,Λ; ψ)
and
(2.3) X= [X1, . . . , XN] = [X(1), . . . , X(p)]′.
We define the following subclass of the left-spherical distribution LSp×N(ϕX),
(2.4) Fp×N={X(p×N)∼LSp×N(ϕX); X(1) is spherical}
(appeared in Fang and Zhang [9, p. 123] as F7) and introduce a result related
to Fp×N; similar results appeared in Iwashita and Klar [15].
Lemma 2.1. Let Xbe the observation matrix, defined by (2.3), based on
independent random sample {Xi}N
i=1 from ECp(0,Λ; ψ). Then
(2.5) Y=S−1/2X∼LSp×N(ϕY)
for the respective characteristic functions ϕY, where Sis defined in (1.1).
Proof. Iwashita and Klar [15] showed that if {˜
Xi}N
i=1 is an iid sample from
ECp(0, Ip;ψ), then ˜
Y=˜
S−1/2˜
X∼LSp×N(ϕ˜
Y), where ˜
Xand ˜
Sdenote the
p×Nobservation matrix and p×pcovariance matrix based on {˜
Xi}N
i=1,
respectively. By straightforward manipulation, we have
Y=S−1/2X= (Λ1/2˜
SΛ1/2)−1/2Λ1/2˜
X
= [(Λ1/2˜
SΛ1/2)−1/2Λ1/2˜
S1/2]˜
S−1/2˜
X=HΛ,˜
S˜
Y ,
where HΛ,˜
S= (Λ1/2˜
SΛ1/2)−1/2Λ1/2˜
S1/2∈ O(p) (see Balakrishnan et al. [1]).
Note that the cf of Ycan be expressed as, for p×Nmatrix T,
ϕY(T) = E[etr(iT ′HΛ,˜
S˜
Y)]
= Eetr(iT ′HΛ,˜
S˜
Y)O(p)
(dH)
= EO(p)
etr(iT ′HΛ,˜
SH˜
Y)(dH)(use ˜
Yd
=H˜
Y , H ∈ O(p)),
where etr(∗) = exp(tr(∗)), (dH) denotes the unit invariant Haar measure
on O(p) (see, e.g., Muirhead [19, p.72]) and i=√−1. By straightforward
ELLIPTICALLY SYMMETRY 133
calculations based on Muirhead [19, Theorem 7.4.1],
O(p)
etr(iT ′HΛ,˜
SH˜
Y)(dH) = O(p)
etr(i˜
Y T ′HΛ,˜
SH)(dH)
=0F1p
2;−1
4˜
Y T ′HΛ,˜
SH′
Λ,˜
ST˜
Y′
=0F1p
2;−1
4˜
Y T ′T˜
Y′
≡O(p)
etr(iT ′H˜
Y)(dH)
= etr(iT ′˜
Y)O(p)
(dH) (use H˜
Yd
=˜
Y)
= etr(iT ′˜
Y),
where pFq(a1, . . . , ap;b1, . . . , bq;X) is the hypergeometric function of matrix
argument (see, for example, Muirhead [19, Definition 7.3.1]). This implies
ϕY(T) = ϕ˜
Y(T), the cf of ˜
Y, and, hence,
Yd
=˜
Y∼LSp×N(ϕ˜
Y),
which completes the proof.
With the help of Lemma 2.1, we are able to obtain the following result.
Theorem 2.2. Suppose {Xi}N
i=1 is an iid sample drawn from ECp(0,Λ; ψ)
and Xis the observation matrix defined in (2.3). Then,
(2.6) Y′∼SSN×p(ϕY′),
where Yis defined in (2.5).
Proof. Let Y= [Y1,Y2, . . . , YN] = Y(1),Y(2), . . . , Y(p)′. Set a= (α′α)−1/2α
for all α∈RN\ {0}. Then, by Theorem 2 in Iwashita and Klar [15],
Yα=S−1/2Xα∼ECp(0,(α′α)Ip;φ),
where φdenotes the cf of Yα, which generally differs from ψ. Hence the
distribution of Yαdepends on αonly through α′α. As Y∼LSp×N(ϕY) by
Lemma 2.1, using Fang and Zhang [9, Theorem 3.6.9], we obtain
Y=S−1/2X∈Fp×N.
As a side note, let P(p) denote the permutation group, a subgroup of O(p);
that is, if a p×pmatrix HP∈ P(p), then H′
PHP=Ip, and the elements of HP
134 T. IWASHITA AND B. KLAR
are either 0 or 1 (see, Fang et al. [7, pp.5–6]). As HPYd
=Y∼LSp×N(ϕY),
we have [Y(i1), . . . , Y(ip)]′d
=Y∈Fp×Nfor any permutation (i1, . . . , ip) of
(1, . . . , p).
Taking into account that rank(Y Y ′) = rank(Y) = p < N, let λ1, λ2, . . . , λp
(λi>0) be the eigenvalues of Y Y ′and Hλbe an orthogonal matrix of the
pertaining eigenvectors such that
HλY Y ′H′
λ= diag(λ1, λ2, . . . , λp)≡Λ, Hλ∈ O(p).
Here we note that Hλis a random matrix on O(p). In a similar way as in the
proof of Lemma 2.1, we obtain for Y∼LSp×N(ϕY) and p×Nmatrix T,
E[etr(iT ′HλY)] = E[etr(iT ′Y)].
This yields
HλYd
=Y∼LSp×N(ϕY),Λ = HλY Y ′H′
λ
d
=Y Y ′∼SSp×p(ϕY Y ′).
Let D= (Y Y ′)1/2be a symmetric square root of Y Y ′, i.e., D2=Y Y ′. Then
Dd
=D′∼SSp×p(ϕD),because the following fact holds for H∈ O(p):
(2.7)
D= (Y Y ′)1/2d
= (HY Y ′H′)1/2= (HD2H′)1/2= (HDH′HDH)1/2=HDH′.
Hence, Λ1/2d
= (Y Y ′)1/2=D ∼ SSp×p(ϕD), where Λ1/2is a symmetric square
root of Λ.
If we set
∆ = diag(±λ1,±λ2, . . . , ±λp),
with an arbitrary choice of the sign in each component, it satisfies (2.7), there-
fore ∆ d
= ∆′∼SSp×p(ϕ∆). Applying Theorem A9.5 in Muirhead [19] to
Y Y ′d
= ∆2, there exists an N×prandom matrix U= [U1,U2, . . . , Up] with
U′U=Ip, such that Yd
= ∆U′.Since
Yd
= ∆U′d
=±λ1U1,±λ2U2, . . . , ±λpUp′
∈Fp×N,
it holds that Y(1)
d
=±√λ1U1=√λ1(±U1)∼ECN(0, IN;φ). Referring to
Corollary and Theorem 2.3 in Fang et al. [7, p.30], we see that ||Y(1)|| d
=√λ1
and Y(1)/||Y1|| d
=U1are independent, and U1∼ U(SN−1), where U(SN−1)
denotes the uniform distribution over the unit sphere in RN.
In the same way, we get, for i= 1, . . . , p,Y(i)
d
=±√λiUi, where √λiand
Ui∼ U(SN−1) are independent. Thus, U∼ UN ,p, independent of ∆, i.e.,
Y′d
=U∆∼SSN×p(ϕY′).
ELLIPTICALLY SYMMETRY 135
According to Lemma 4 and its proof in Dawid [5], if V∼LSk×l(ϕV) and
the fixed q×kmatrix Lsatisfies LL′=Iq, then
(2.8) LV ∼LSq×l(ϕLV ).
Hence, using (1.3), (2.6) and (2.8), we obtain
Z≡K′Y′=K′X′S−1/2∼LSn×p(ϕZ),
and, actually, Z∼SSn×p(ϕZ). Referring to Fang and Zhang [9, p.101], and
noting that (n−1/2Z)′(n−1/2Z) = Ip, we obtain
Z(Z′Z)−1/2=n−1/2Z∼ Un,p.
Summarizing the above yields the following result, which is the key to propose
the test statistic for multivariate elliptical symmetry in the next section.
Corollary 2.3. Let X= [X1,X2, . . . , XN], where {Xi}N
i=1 is an iid sample
drawn from ECp(µ,Λ; ψ), and let Sbe the sample covariance matrix of (1.1).
Then
(2.9) U=K′X′(nS)−1/2∼ Un,p ,
where Kis an N×nmatrix which satisfies the conditions of (1.3).
§3. Test of uniformity over Stiefel manifold O(n, p)
In this section, we propose a new test procedure for uniformity over the Stiefel
manifold O(n, p) as a generalization of tests proposed by Pycke [21]. Therein,
he considered tests for uniformity of circular distributions against multimodal
alternatives by making use of certain degenerate U- and V-statistics. Let
{Θi}m
i=1 be an iid sample drawn from a distribution defined on the interval
[0,2π]. Pycke [21] identified the unit circle S1with the interval [0,2π] in
which the endpoints 0 and 2πare identified, and considered the degenerate U-
and V-statistics
G=−2
m−1
m
i=2
i−1
j=1
log{2−2 cos(Θi−Θj)},
Vq=2
m
m
i=1
m
j=1
cos(Θi−Θj)−q
1−2qcos(Θi−Θj) + q2, q ∈(0,1),(3.1)
as test statistics for uniformity. Pycke [21] determined critical values for vari-
ous significant levels and various sample sizes by Monte Carlo simulation.
136 T. IWASHITA AND B. KLAR
Let S1, . . . , Smbe independent d-dimensional random vectors drawn from
a uniform distribution over the hypersphere Sd−1(d≥2), and let Θij =
arccos(S′
iSj) denote the enclosed angle between Siand Sj. For d= 2, the
relation between Cartesian and polar coordinates yields Θi−Θj= Θij . Here,
we consider
˜
Vℓ,d =1
m2
m
i=1
m
j=1
2 cos (ℓΘij) = 2
m+4
m2
m
i=2
i−1
j=1
cos ℓarccos(S′
iSj),(3.2)
Vq,d =2
m
m
i=1
m
j=1
cos(Θij)−q
1−2qcos(Θij) + q2=2
m
m
i=1
m
j=1
S′
iSj−q
1−2qS′
iSj+q2,(3.3)
where ℓis a natural number and q∈(0,1), as test statistics for uniformity over
the Stiefel manifold. Clearly, (3.3) is a direct generalization of (3.1) for d≥3,
whereas ˜
Vℓ,d uses the individual components of the kernel function pertaining
to Vq(see Pycke [21]). The statistics in (3.2) and (3.3) are V-statistics
V=1
m2
m
i=1
m
j=1
hS′
iSj=1
m2
m
i=1
m
j=1
h(Θij),
with kernels ˜
hl(θ) = 2 cos(lθ), l ≥1,and
hq(θ) = 2
∞
k=1
qk−1cos(kθ) = 2(cos θ−q)
1−2qcos θ+q2, q ∈(0,1),
respectively. The distribution of Θij under the hypotheses of uniformity can
be obtained by direct computations, or one can resort to the distribution of
the correlation coefficient under normality as done in Cai et al. [3, Lemma 12]
(see also Cai and Jiang [4, Lemma 4.1]).
Proposition 3.1. Let d≥2. Then, under the hypotheses of uniformity, Θij,
1≤i < j ≤m, are pairwise iid with the density function
f(θ) = 1
√π
Γ(d/2)
Γ((d−1)/2) ·(sin θ)d−2, θ ∈[0, π].
Using Proposition 3.1, we obtain E[V] = E [h(Θ12)] = µ(say). This can
ELLIPTICALLY SYMMETRY 137
explicitly be computed using, for l, m = 0,1,2, . . . ,
π
0
cos ((2m+ 1)x) (sin x)ldx = 0,
π
0
cos (2mx) (sin x)2ldx =
(−1)m
22l2l
l−mπ, l ≥m,
0, l < m,
π
0
cos (2mx) (sin x)2l+1 dx
=
(−1)m
22l+1
Γ(2l+ 2)
Γ(3/2 + l−m)Γ(3/2 + l+m)π, l ≥m−1,
0, l < m −1,
(see Gradshteyn-Ryzhik [13, Section 3.631]; in some editions, the factor πin
the second formula is missing).
Since Siand Sjare uniformly distributed over the unit sphere, one may
suppose that Proposition 3.1 remains valid if Sjis replaced by a fixed unit
vector. This is indeed the case. To be specific, put Θs
i= arccos (s′Si) with
s∈Rd,∥s∥= 1. Then, Θs
ihas the same distribution as Θij (see Cai and
Jiang [4], p. 31). As a consequence, E [h(s,S1)] = µ, which shows that Vis a
degenerate V-statistic. Putting Φ(s1,s2) = h(s1,s2)−µ, we obtain
E [Φ(S1,S2)] = 0,E [Φ(s,S1)] = 0,
EΦ2(S1,S2)<∞,E [|Φ(S1,S1)|]<∞,
E [Φ(S1,S1)] = 2,if h=˜
hℓ,
2/(1 −q),if h=hq.
Then, the theory of V-statistics yields that m(V−µ) converges in distribution
to a weighted sum of independent chi-squared random variables. In special
cases, the weights can be obtained (see Proposition 1 in Pycke [21] for the
circular case). However, we do not proceed in this direction, since, for small
and medium sample sizes, it is preferable to use finite sample critical values
obtained by simulation.
Let X1, . . . , XmN be iid p-dimensional random vectors drawn from
ECp(µ, Λ; ψ). Partition this sample into mgroups with equal size N, de-
noted by {X(k)
i}N
k=1 for k= 1, . . . , m. Next, based on (2.9), define mrandom
matrices of size n×pby
Uk=U(k)
1, . . . , U(k)
p=K′X′
(k)(nS(k))−1/2, n =N−1≥p,
where
X(k)=X(k)
1, . . . , X(k)
N, nS(k)=X(k)QX′
(k), k = 1,2, . . . , m.
138 T. IWASHITA AND B. KLAR
Here, Qhas appeared in (1.2), and Kis an N×nconstant matrix satis-
fying (1.3). Taking Corollary 2.3 into account, Uk’s are independently and
uniformly distributed over O(n, p). Hence, the pcolumns U(i)
r, r = 1, . . . , p, of
Uiare not independent, but each of U(i)
ris uniformly distributed over the unit
hypersphere Sn−1, independent of U(j)
rof Uj(i=j). Hence we are able to
construct a testing procedure based on {U(k)
r}m
k=1 by making use of (3.2) and
(3.3), which leads to the “necessary test procedure” for elliptical symmetry.
Remark 3.2. The following reasoning explains the phrase “necessary test pro-
cedure”. As a consequence of Corollary 2.3, when the distribution of the Xi’s
enjoys elliptical symmetry, the Uk’s are independent and uniformly distributed
over the Stiefel manifold. Therefore, if uniformity of the Uk’s is not satisfied,
we reject the hypothesis of elliptical symmetry. On the other hand, even if the
Uk’s have the uniformity over O(n, p), this does not imply elliptical symmetry
of the Xi’s – thus we use the terminology “necessary test procedure”.
§4. Some numerical experiments
In this section, we carry out Monte Carlo simulations to evaluate the type
I errors and powers for the proposed tests, together with Rayleigh’s test
(Jupp [16]).
To evaluate the type I error, we consider the following three p-dimensional
elliptical distributions as “null distribution”:
(A1) the normal distribution Np(0,Λ) with pdf
fN(x|Λ) = |2πΛ|−1/2exp(−x′(2Λ)−1x),
(A2) the t-distribution with νdegrees of freedom Tp(ν, Λ) with pdf
fT(x|Λ) = Tp|νπΛ|−1/21 + ν−1x′Λ−1x−(p+ν)/2,
where Tp= Γ [(p+ν)/2] /Γ [ν/2]; we set ν= 3,
(A3) the Kotz type distribution Kp(r, s, k, Λ) with pdf
fK(x|Λ) = Kp|πΛ|−1/2(x′Λ−1x)k−1exp −r(x′Λ−1x)s,
where Kp=sΓ(p/2)/Γ((2k+p−2)/2s)r(2k+p−2)/2sand r, s > 0, 2k+p >
2 (see Fang et al. [7, Chapter 3]); we set (r, s, k) = (1/2,1,2).
ELLIPTICALLY SYMMETRY 139
For each of these models, we used the scale matrices Λ = diag(42,32,22)
diag(42,32,22,1) for p= 3,4, respectively, where diag(λ1, . . . , λp) denotes a
diagonal matrix with diagonal elements λ1, . . . , λp.
We also examine four non-elliptical distributions to evaluate the power in
the same manner as Liang et al. [17]:
(B1) the exponential distribution composed of iid univariate exponential dis-
tribution with pdf f(x) = exp(−x),
(B2) the exponential distribution composed of iid univariate exponential dis-
tribution with pdf f(x) = (1/k2) exp −(1/k2)x,k= 1,2, . . . , p,
(B3) the multivariate chi-squared distribution composed of iid univariate χ2
1,
the chi-squared distribution with 1 degree of freedom,
(B4) the skew-normal distribution with pdf
f(x|α,Λ) = 2fN(x|Λ)Φ(α′Λ−1/2x),
where fN(x|Λ) is defined in (A1) and Φ(∗) denotes the standard nor-
mal cumulative distribution, with parameters, α= (2,−3,−1)′,Λ =
diag(42,32,22) for p= 3, and α= (2,−3,−2,−5)′,Λ = diag(42,32,22,1)
for p= 4 (see, for instance, Genton [12, pp. 15, 16]).
For all distributions above, we choose the dimension p= 3,4, the number of
groups mand the sample size of each group Nas (m, N) = (5,10),(10,5), and
set ℓ= 2,3,4, q =2/3 as used by Pycke [21]. By generating 106samples
from (A1), we obtain the critical values for every statistic based on the first
column U(k)
1of Ukfor nominal levels α= 0.10,0.05,0.01; they are summarized
in Tables 1–4. “Rayleigh” means the modified Rayleigh statistic with error of
order m−1(see Jupp [16]) .
We evaluate the type I error rates for (A1)–(A3) and empirical powers for
(B1)–(B4) based on Monte Carlo simulations with 105iterations. Results are
shown in Tables 5 and 6. From these tables, we observe that the type I error
rates for ˜
Vℓ,d and Vq,d are in very good agreement with the nominal rate. The
Rayleigh test, which is the score test of uniformity within the matrix von
Mises-Fisher family, shows no power at all; this is not surprising in view of
the empirical centering in the Studentized residuals which form the basis of
the test procedure. Among the other statistics, ˜
V3,d shows the highest power,
followed by V√2/3,d; generally, the power is rather low due to the small sample
size m.
To conclude this section, we analyze the famous Iris data, which is presented
by Fisher [11], to assess our test procedure. Results are shown in Tables 7–10.
Here we note that in order to avoid the singularity of the sample covariance
140 T. IWASHITA AND B. KLAR
matrices for mgroups of size N= 5, we modified the data set by swapping the
first data set of Iris Setosa (5.1,3.5,1.4,0.2) with the seventh (4.6,3.4,1.4,0.3).
We also give the values of the different statistics calculated using the rth
column U(k)
rof Uk, indicated by the index rin the tables, since the values
of the statistics depend on the order of the elements of the vectors. In the
case p= 4, m = 10, N = 5 at significant level 10%, the maximum for the
values of the statistics V√2/3,4,˜
V2,4indicates deviations to elliptical symmetry.
Moreover, the proposed tests based on the 2-variates petal and sepal widths
showed better performance for m= 10, N = 5 as using m= 5, N = 10.
§5. Conclusion
We have constructed a new test procedure for elliptical symmetry by making
use of the uniform distribution over the Stiefel manifold. By simulation, the
proposed test shows good performance of the type I error and power, compared
to Rayleigh’s test. Furthermore, the tests have been applied to the Iris data,
raising doubts that this data set comes from an elliptical distribution.
Acknowledgment
The first author has been partially supported by Grant-in-Aid for Scientific
Research(C), JSPS KAKENHI Grand Numbers 18K11198, 18K03428. The
authors wish to express their thanks to an anonymous reviewer for valuable
comments and suggestions which greatly improved a previous version of this
paper, in particular the original proofs of Lemma 2.1 and Theorem 2.2.
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142 T. IWASHITA AND B. KLAR
Toshiya Iwashita
Department of Liberal Arts, Tokyo University of Science
2641 Yamazaki Noda, Chiba 278–8510, Japan
E-mail:iwashita toshiya@rs.tus.ac.jp
Bernhard Klar
Institut f¨ur Stochastik, Fakult¨at f¨ur Mathematik, Karlsruher Institut f¨ur Technologie
Englerstraße 2, 76131 Karlsruhe 76131, Germany
E-mail:Bernhard.Klar@kit.edu
ELLIPTICALLY SYMMETRY 143
Table 1: Upper tail percentage points for p= 3, N = 5
α m \Statistic Rayleigh V√2/3,3˜
V2,3˜
V3,3˜
V4,3
0.10 5 18.36 12.58 4.66 4.25 4.27
10 18.49 12.74 4.77 4.33 4.38
0.05 5 20.66 13.15 5.33 4.87 4.89
10 20.95 13.21 5.48 5.02 5.05
0.01 5 25.38 14.60 6.84 6.01 6.04
10 26.10 14.24 7.02 6.36 6.37
Table 2: Upper tail percentage points for p= 3, N = 10
α m \Statistic Rayleigh V√2/3,3˜
V2,3˜
V3,3˜
V4,3
0.10 5 36.52 7.36 −3.33 4.07 6.72
10 36.68 2.67 −11.05 4.15 10.39
0.05 5 39.75 7.46 −2.99 4.51 7.16
10 40.02 2.77 −10.72 4.60 10.88
0.01 5 46.24 7.72 −2.23 5.25 7.94
10 46.72 3.00 −9.99 5.37 11.76
Table 3: Upper tail percentage points for p= 4, N = 5
α m \Statistic Rayleigh V√2/3,4˜
V2,4˜
V3,4˜
V4,4
0.10 5 23.32 12.57 4.65 4.25 4.27
10 23.49 12.74 4.78 4.33 4.37
0.05 5 25.93 13.15 5.33 4.87 4.89
10 26.21 13.20 5.48 5.01 5.04
0.01 5 31.27 14.61 6.84 6.02 6.04
10 31.84 14.24 7.02 6.36 6.37
Table 4: Upper tail percentage points for p= 4, N = 10
α m \Statistic Rayleigh V√2/3,4˜
V2,4˜
V3,4˜
V4,4
0.10 5 46.98 7.36 −3.33 4.07 6.72
10 47.12 2.67 −11.06 4.15 10.38
0.05 5 50.61 7.47 −2.99 4.51 7.16
10 50.90 2.77 −10.72 4.60 10.88
0.01 5 57.86 7.72 −2.23 5.26 7.94
10 58.38 3.00 −9.99 5.38 11.74
144 T. IWASHITA AND B. KLAR
Table 5: Monte Carlo type I error rates and powers (p= 3, N = 5, m = 10)
αStatistic Normal tKotz Exp. 1 Exp. 2 χ2Skew Normal
0.10
Rayleigh 0.100 0.101 0.100 0.100 0.100 0.098 0.100
V√2/3,30.101 0.101 0.101 0.161 0.130 0.228 0.101
˜
V2,30.101 0.100 0.099 0.101 0.102 0.102 0.100
˜
V3,30.100 0.100 0.099 0.226 0.159 0.351 0.099
˜
V4,30.101 0.101 0.100 0.108 0.106 0.137 0.098
0.05
Rayleigh 0.050 0.050 0.049 0.050 0.049 0.048 0.049
V√2/3,30.050 0.050 0.050 0.089 0.069 0.138 0.050
˜
V2,30.051 0.049 0.050 0.050 0.052 0.050 0.050
˜
V3,30.049 0.050 0.049 0.137 0.088 0.241 0.050
˜
V4,30.050 0.050 0.050 0.056 0.054 0.074 0.049
0.01
Rayleigh 0.009 0.009 0.010 0.009 0.010 0.009 0.009
V√2/3,30.010 0.010 0.010 0.023 0.016 0.044 0.010
˜
V2,30.010 0.010 0.009 0.010 0.010 0.011 0.010
˜
V3,30.010 0.009 0.010 0.042 0.022 0.094 0.009
˜
V4,30.009 0.010 0.010 0.011 0.011 0.018 0.009
Table 6: Monte Carlo type I error rates and powers (p= 4, N = 5, m = 10)
αStatistic Normal tKotz Exp. 1 Exp. 2 χ2Skew Normal
0.10
Rayleigh 0.100 0.099 0.101 0.097 0.100 0.100 0.098
V√2/3,40.100 0.102 0.101 0.138 0.110 0.173 0.099
˜
V2,40.099 0.100 0.099 0.099 0.102 0.101 0.098
˜
V3,40.101 0.100 0.100 0.173 0.110 0.238 0.101
˜
V4,40.101 0.100 0.101 0.109 0.103 0.131 0.101
0.05
Rayleigh 0.050 0.049 0.050 0.048 0.050 0.049 0.049
V√2/3,40.050 0.051 0.050 0.075 0.056 0.100 0.049
˜
V2,40.049 0.050 0.049 0.050 0.051 0.051 0.049
˜
V3,40.050 0.049 0.050 0.100 0.057 0.148 0.052
˜
V4,40.049 0.050 0.051 0.056 0.052 0.070 0.052
0.01
Rayleigh 0.009 0.009 0.009 0.010 0.009 0.010 0.009
V√2/3,40.009 0.010 0.010 0.017 0.012 0.028 0.009
˜
V2,40.010 0.009 0.010 0.010 0.009 0.010 0.009
˜
V3,40.010 0.009 0.010 0.026 0.012 0.046 0.010
˜
V4,40.009 0.010 0.009 0.011 0.010 0.016 0.010
ELLIPTICALLY SYMMETRY 145
Table 7: Iris setosa data for p= 4, m = 5, N = 10
Statistic Rayleigh r V√2/3,4˜
V2,4˜
V3,4˜
V4,4
α\Values 29.09
1 7.08 −4.11 3.51 4.17
2 7.00 −3.40 3.61 2.83
3 7.04 −4.03 3.26 4.17
4 7.22 −4.14 0.16 4.05
0.10 46.98 7.36 −3.33 4.07 6.72
0.05 50.61 7.47 −2.99 4.51 7.16
0.01 57.86 7.72 −2.23 5.26 7.94
Table 8: Iris setosa data for p= 4, m = 10, N = 5
Statistic Rayleigh r V√2/3,4˜
V2,4˜
V3,4˜
V4,4
α\Values 23.90
1 11.45 1.78 2.26 2.21
2 10.80 3.15 2.15 1.06
3 11.87 5.30 2.30 −0.65
4 12.91 3.34 −4.34 0.11
0.10 23.49 12.74 4.78 4.33 4.37
0.05 26.21 13.20 5.48 5.01 5.04
0.01 31.84 14.24 7.02 6.36 6.37
Table 9: Iris setosa data with petal and sepal widths for p= 2, m = 5, N = 10
Statistic Rayleigh r V√2/3,2˜
V2,2˜
V3,2˜
V4,2
α\Values 17.53 1 7.05 −2.86 3.10 2.79
2 7.32 −3.97 0.04 4.32
0.10 25.78 7.36 −3.33 4.07 6.72
0.05 28.51 7.47 −2.98 4.51 7.16
0.01 34.05 7.73 −2.23 5.26 7.93
Table 10: Iris setosa data with petal and sepal widths for p= 2, m = 10, N = 5
Statistic Rayleigh r V√2/3,2˜
V2,2˜
V3,2˜
V4,2
α\Values 16.96 1 11.14 3.65 3.53 0.31
2 13.60 2.68 −3.20 4.48
0.10 13.33 12.75 4.77 4.33 4.37
0.05 15.44 13.21 5.48 5.02 5.05
0.01 19.90 14.25 7.01 6.36 6.36
