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The Annals of Statistics
1995, Vol. 23, No. 6, 1960 ] 1974
SHAPE CHANGES IN THE PLANE FOR LANDMARK DATA
BY MICHAEL J. PRENTICE AND KANTI V. MARDIA
Edinburgh University and Leeds University
This paper deals with the statistical analysis of matched pairs of
shapes of configurations of landmarks in the plane. We provide inference
procedures on the complex projective plane for a basic measure of shape
Ž. change in the plane, on observing that shapes of configurations of k q 1
landmarks in the plane may be represented as points on C P
k y 1
and that
complex rotations are the only maps on C S
k y 1
which preserve the usual
Hermitian inner product. Specifically, if u ,... , u are fixed points on
1 n
k y 1 k y 1
Ž. C P represented as C S rU 1 , and v ,... , v are random points on
1 n
k y 1
5
U
5
2
C P such that the distribution of v depends only on v Au for some j j j
unknown complex rotation matrix A , then this paper provides asymptotic
inference procedures for A. It is demonstrated that shape changes of a
kind not detectable as location shifts by standard Euclidean analysis can
be found by this frequency domain method. A numerical example is given.
1. Introduction. There are a variety of practical problems which re-
quire the statistical analysis of matched pairs of shapes of configurations of
landmarks in the plane. For various examples, see Bookstein 1991. We
assume that data are available in the form of a random sample of n
independent and identically distributed matched pairs X , Y ,... , X , Y 1 1 n n
of k q 1 landmarks in the plane, where here each X and each Y is a j j
complex k q 1 vector. The shape of each object is the information on the
landmarks after all ‘‘pose’’ information has been removed, that is, the effect of
location, size and orientation. Thus, as in Mardia and Dryden 1989 , for
example, each X and Y is centred, scaled and multiplied, conventionally by
j j
the Helmert matrix, to give a random sample of matched pairs of complex
U U U
unit k -vectors u , v , where u u s 1 s v v , u denotes transposed com-
j j j j j j j
plex conjugate and each u and v is identified with u exp i u and v exp i f ,
j j j j
respectively, for all real u , f, and where i
2
s y 1. Thus each u and v is a
j j
point on the complex projective hyperplane C P
k y 1
represented as
k y 1
C S rU 1 , in the notation of Kent 1994. Note the important restriction of
invariance under scalar rotations. We propose here to carry out a basic
investigation of the problem of point and regional estimation of shape change
on shape space, rather than on a Euclidean approximation to shape space.
Goodall 1991 addresses this problem using Euclidean approximations, but
here we propose to explore the consequences of investigating shape change by
representing it as a complex rotation and developing estimation and test
procedures on C P
k y 1
Received November 1993; revised March 1995.
AMS 1991 subject classifications. Primary 62H10; secondary 62H11.
Key words and phrases. Shape, unitary matrices, spherical regression, configuration.
1960
SHAPE CHANGES IN THE PLANE 1961
Although it is now conventional to use the coordinate system of Kendall
1984 on shape space, using complex contrasts generated by premultiplica-
tion with the Helmert matrix, there are some advantages in using a Fourier
alternative, described in detail below. After centring and scaling, each X and j
Y is a unit-length complex k q 1 -vector with entries summing to zero. j
Premultiplication by the Helmert matrix constructs k linearly independent
contrasts, but there is no particular reason to choose a coordinate system on
the complex sphere in this way. Indeed, one criticism levelled at the use of
the complex sphere in shape analysis is that the coordinates do not measure
anything biologically meaningful. The following alternative does, however,
give complex spherical coordinates with a geometric meaning. We propose to
replace the Helmert matrix by
y 1 r 2
F s Ž f. s Ž k q 1. exp 2 p irs r Ž k q 1. ,
Ž Ž ..
r s
r s 1,... , k ; s s 1,... , k q 1,
so that the shape vectors u and v are unit norm discrete Fourier transforms j j
of the centred and scaled X and Y , respectively. Note that FF * s I and
j j k
y 1 T
F * F s I y k q 1 11 , as for the Helmert matrix. Thus, for example, if
k q 1
X represents a configuration of landmarks in standard numerical order at j
the vertices of a regular n -sided polygon in the plane, then u s 1, 0,... , 0 j
and, in general, its first element u is a measure of the extent to which X is j 1 j
similar to such a polygon. Similarly u measures the extent to which X is jt j
similar to a configuration with vertices at exp y 2 p ist r k q 1 , s s 1,... ,
k q 1.
For the case of triangle, k s 2, suppose that the correspondence quoted by
wŽ. x
1 2
Kent 1994 , page 292 between C S and S is modified by relabelling as
T 1
T 2
follows. Given z s z , z g C S , define t s t , t , t g S by 1 2 1 2 3
t s y2 Re z
U
z , t s 2 Im z
U
z , t s z z
U
y z z
U
1 1 2 2 1 2 3 1 1 2 2
Then the Fourier choice of coordinate system above maps into the conven-
tional spherical blackboard of Kendall 1984.
In this paper we propose to begin exploration of the statistical analysis of
shape change on Kendall’s shape space viewed as a normed frequency do-
main. The parallels with spectral analysis of time series are obvious, but
there is the important difference that we usually have a relatively small
number of landmarks length of time series with a reasonably large number
of replicated measurements, as compared with only one realisation of a time
series.
A complex spherical regression model is suggested in Section 2 as a basic
model for shape change in the frequency domain. The main theoretical
results needed are stated in Section 3, and estimation and test procedures are
exemplified in Section 4. In Section 5, these ideas are applied to the data
analysed by Mardia and Walder 1994a, b }a set of landmark coordinates
from lateral cephalograms of male rats from a close bred European strain, for
which x-ray results are available at a number of different times for the same
SHAPE CHANGES IN THE PLANE 1963
for some unknown A in SU k. Large sample statistical procedures for
estimating and testing the unknown parameter A are described here. Since
complex rotations are the only maps on C S
k y 1
which preserve inner prod-
ucts, the parameter A is a fundamental measure of shape change on Kendall’s
shape space.
The development of these procedures is an extension of the work of Chang
1987 on spherical regression, which is the corresponding problem for signed
directions on the real hypersphere S
k y 1
, with the v having a probability
j
density dependent only on v
T
Au. Here, in the complex axial case, A and
j j
A exp i u are distinguishable in U k , but induce the same transformation in
complex projective space, so we shall say that A is ‘‘unique’’ in quotes if
- 1 A s A m A
2
2
is unique without quotes. Note that A g SU k.
2
It is convenient to work with k
2
-vectors
Ž 2.
Ž 2. 2. u s u m u y e
j j j
Ž 2.
and v s v m v y e , where j j j
- 3 e s k
y 1
vec I ,
k
so that u
Ž 2.
is unchanged by scalar rotations of u and has zero expectation if
j j
u is uniformly distributed. It is reasonable to estimate A by the matrix A
j
which maximises
y 1
U
2
Ž 2. 4. r Ž A. s n v Au ,
Ý j j
a complex axial analogue of the ‘‘ vector correlation’’ of Stephens 1979 ; see
also Jupp and Mardia 1980. Since
U
2 y 1 Ž 2.
U Ž 2.
Ž 2. 5. v Au y k s v A u
j j j 2 j
Ž 2. Ž 2.
with A as in 2. 1 and v , u as in 2 .2 , we may equivalently choose to 2 j j
estimate A by the matrix A which maximises
- 6 r A s n
y 1
v
Ž 2.
U
A u
Ž 2.
Ý 2 j 2 j
which is in the form of a vector correlation on a subspace of a k
2
-sphere of
y 1
1 r 2
radius 1 y k , provided
- 7 YY s E v
Ž 2 .U
A u
Ž 2.
Ž. 0 j 2 j
U
2
y 1
or, equivalently, provided E v Au ) k. Note that Y
Y
s 0 if the v are
j j 0 j
uniformly distributed and that r s r y k
y 1
, 0 F r F 1.
2
One iterative scheme to estimate the complex rotation matrix A is to start
at some suitable A
Ž 0.
and choose A
Ž 1.
,... , A
Ž t.
,... so A
Ž t.
maximises
Ž t. y 1 U Ž t y 1. U
tr A n Ý u u A * v v , using complex singular value decompositions
j j j j
at each iteration as in Prentice 1989. We call this A the least squares
estimate of A because A minimises
Ž 2. Ž 2. Ž 2. Ž 2.
U
2
v y A u * v y A u s 2 n y 2 v Au. Ý Ž. Ž. Ý j 2 j j 2 j j j
1964 M. J. PRENTICE AND K. V. MARDIA
In Section 3, we find the asymptotic distribution of A under the assumption
y 1
U
that lim n Ý u u s M , a positive definite symmetric matrix Theorem n ™` j j
1 , and asymptotic confidence regions for A will be based on Theorem 1. For
closed subgroups G 9 : G of U k , we also find the asymptotic distribution of
r G 9 s sup r A and of 2 2
A g G ,
r G y r G 9 , when A g G 9 Theorem 2. 2 2
Note that if the underlying distribution is complex Dimroth]Watson with
probability density
2
Ž 2. 8. b Ž K. exp Ž K v * Au .,
then the procedures of this paper are just maximum likelihood estimation
and likelihood ratio testing. Note also that we assume that u are fixed, or j
else make inferences conditional on the u. Complex axial regression with j
w Ž. errors in variables is not discussed here see Chang 1989 for spherical
x regression with errors in variables , but complex axial regression with highly
concentrated errors is certainly of interest; see Rivest 1989 for the spherical
case and our remarks on F tests in Section 4.
3. Statement of the main asymptotic results. The tangent space at
the identity I of SU k is the collection of skew-Hermitian k = k matrices
p
with zero trace, that is, the matrices H such that H q H * s 0 and tr H s 0.
We denote this tangent space by L SU k and define the exponential map
f: L SU k ™ SU k by
`
y 1
s
Ž 3. 1. f Ž H. s Ž s!. H.
Ý
s s 0
Note that if G is a closed subgroup of SU k and L G is the tangent space at
I of G , then L G is a vector subspace of L SU k and L G is the set of H
p
in L SU k such that f tH is in G for all real t. Note that dim G s
2
2
dim L G and dim U k s k , where dim SU k s k y 1.
The statement and proof of Theorem 1 given below is made easier by
introducing the following notation. If A g G : U k , then define A as in
2
2 .1 and let G be the collection of such A , as A varies over G. Then G is a
2 2 2
2
subgroup of SU k m SU k , which is itself a subgroup of SU k , as A has
2
determinant q 1. The tangent space at I of SU k m SU k is the collection
p
2 2
w Ž .x of k = k matrices Prentice 1989
H
Ž 2.
s Ž I m H. q Ž H m I .,
where H g L SU k. We denote this tangent space by L SU k m SU k
and define f : L SU k ™ L SU k m SU k by
2
- 2 f H s f H
Ž 2.
2
2 2
with f as in 3 .1 , acting on k = k matrices. Note that dim L G s
2
2
dim L G for all G : SU k , so, in particular, dim SU k m SU k s k y 1.
1966 M. J. PRENTICE AND K. V. MARDIA
THEOREM 1. Let G be a closed subgroup of U k , so G is a closed
2
2
U
2
subgroup of T : SU k , and suppose each v has a density g v A u , j j 0 j
where A g G. Suppose furthermore that YY ) 0, with YY as above 3 .3 , and 0 0 0
that n
y 1
Ý u u
U
converges to a Hermitian positive definite matrix M , where
j j
M s N q iN with N real symmetric positive definite and N real skew - sym - 1 2 1 2
metric. Then :
a The least squares estimate of A in G , denoted A G , is consistent
2 2 n 2 2
for A. We may write A G s A G ‘‘ uniquely ,’’ so A G is consis -
0 2 n 2 2 n 2 2 n 2
tent for A.
0
U
b Write A A G s f H for H g L G , with f as in 3 .2 , so
0 2 n 2 2 2 n n 2
U
A A G s f H , with f as in 3. 1. Then H is asymptotically complex
0 n 2 n n
w multivariate normal with mean 0 and probability density with respect to
Ž .x wŽ.
y 1 2
2
.x Lebesgue measure on L G proportional to exp 2 YY YY n tr H N so
2 0 n 1
2
2
2
that y n YY rYY tr H N is asymptotically XX dim G.
0 2 n 1
Note the similarity of parts a and b of our Theorem 1 and Theorem 1 of
2
2
Chang 1986 : N is the real part of M and tr H N ' tr H M.
1 n 1 n
An outline of the proof of Theorem 1 is given in the Appendix.
THEOREM 2. a If A g G , then r G , defined in 2 .6 , has a limiting
0 2
normal distribution with mean YY and variance 0
y 1 y 1 y 1 y 1 y 1
n V Ž G. s n Ž 1 y k. Ž 1 y YY. YY q k Ž 1 y YY. y k YY.
Ž. 0 0 0 2
b If A g H : G , then
0
n YY rYY r G y r H s n YY rYY r G y r H Ž. Ž Ž. Ž.. Ž. Ž Ž. Ž ..
0 2 2 2 0 2
2
has a limiting x dim G y dim H distribution.
c If A g K : H : G , then
0
Ž dim G y dim H. r Ž dim H y dim K.
= r H y r K r r G y r H Ž Ž. Ž.. Ž Ž. Ž ..
2 2 2 2
is asymptotically F dim H y dim K , dim G y dim H.
Again, note some similarity with Theorem 2 of Chang 1986. In a , his
y 1
y 1
2
2
variance was n C q C s n E cos Q y C , whereas here it is
1 2 0
y 1
4
y 1
y 1
2
n E cos Q y YY 1 y k q k , and in b and c the results are
0
identical except for a missing factor of 2 in b.
T
THEOREM 3. Let A A G s f H G for H G g L G. Then
0 n 2 n 2 n 2
1 r 2
1 r 2
n r G y YY and n H G are asymptotically independent.
2 0 n 2
The proofs of Theorems 2 and 3 are available from the authors on request.
If the density g is unknown, then to use Theorem 1 and 2 we need
consistent estimators of YY and YY. Using Theorem 2 a , we can estimate YY
0 2 0
consistently by g s r G if A g G , and a consistent estimator of YY is 0 2 0 2
SHAPE CHANGES IN THE PLANE 1967
w Ž .x
y 1
U
2
g s n k y 1 Ý e 1 y e , where e s v A u and A maximises 2 j j j j G j G
r G. 2
4. Some simple test procedures. Suppose H is a closed subgroup of
SU k , where k ) 2. For now we exclude the case of triangles k q 1 s 3. If
YY and YY , defined in 3 .3 , are known, Theorem 2 b can be used to test, in 0 2
large samples, if the true complex rotation matrix A is in H. If a test of the
simple H : A s A against the general alternative H : A g SU k y H is
0 0 2 0
y 1
required, then from Theorem 2 b , n YY YY r SU k y r A is asymptot-
0 2 2 2 0
2
2
ically distributed as x k y 1 if H is true. Here
0
y 1
U
2 y 1
Ž 4. 1. r Ž SU Ž k .. s n v A u y k
Ý 2 ž j j /
and
y 1
U
2 y 1
Ž 4. 2. r Ž A. s n v A u y k.
Ý 2 0 ž j 0 j /
An alternative procedure, involving more computational effort, is provided
by Theorem 1, but not pursued further here. Given a sample estimate A of
n
the unknown complex rotation, suppose A
U
A has spectral decomposi-
n 0
tion X * exp i L X , where X g U k and L is a real diagonal matrix. Then
if H s iX * n X , a test of H : A s A is provided by referring n 0 0
2
2
2
2
y n YY rYY tr H N to x k y 1. 0 2 n 1
It is also possible to investigate the hypothesis H : Au s l u , where here
1 0 0
u is the population mean before-shape and l is a complex number of unit 0
modulus. The hypothesis H asserts that u is an eigenvector of A and hence 1 0
that the mean before-shape u is the same as the mean after-shape v , since 0 0
all shapes are unchanged by scalar rotations. There may be systematic
differences between each before-shape u and its corresponding after-shape j
v , but u s v. Since the subgroup of matrices A in SU k , such that u is j 0 0 0
an eigenvector of A , is isomorphic to SU k y 1 , it follows from Theorem 2 b
that a test of H against H : A g SU k y H is provided by referring 1 2 1
y 1
2
2
2
n Y
Y YY
r SU k y r SU k y 1 to x 2 k y 1 , as k y 1 y k y 1
0 2 2 2
y 1 s 2 k y 1. Here r SU k is as in 4. 1 and r SU k y 1 is found
2 2
most easily by first rotating all u data and all v data so that M s n
y 1
Ý u u
U
j j
is diagonal, M s diag s ,... , s , where s ,... , s and 1 y s are typically
1 k 1 k y 1 k
T
T
T
T
small. After this change of coordinate system, u s z , a , v s w , b ,
j j j j j j
where the z and w are k y 1 vectors and the a and b are real scalars.
j j j j
Then
y 1
U
2 y 1
Ž 4. 3. r Ž SU Ž k y 1 .. s n v A u y k
Ý 2 ž j c j /
and where A is constrained to be of the form block diag A , 1. The iterative
c 1
search procedure for the constrained least squares estimate A is identical to
1
that for A , except that the dimensionality of the problem is reduced by 1.
Of course in practice YY and YY must be estimated by g and g , say, and
0 2 0 2
the validity of the test procedures is then even more approximate. For a
SHAPE CHANGES IN THE PLANE 1969
FIG. 1. Evolution of rat calvarial shape to Basion ] Bregma baseline. Reproduced from Bookstein
wŽ. x 1991 , page 356.
to three decimal places, and the mean shape at age 150 days is
v s 10
y 3
Ž935, y 40 q 98 i , y 71 y 74 i , 68 y 20 i , 5 q 41 i ,
0
T
31 y 106 i , 54 q 286 i. ,
where both shapes have been made unique by requiring that their first
components are real and positive. The unconstrained least squares estimate
of the complex rotation A is such that v and Au coincide to three decimal 0 0
places and A has an eigenvector
w s 10
y 3
Ž820, 22 q 182 i , y 120 q 132 i , y 165 y 98 i ,
T
108 y 1 i , 7 y 65 i , y 240 q 391 i ,
quite close to both u and v. In passing we note that u , v and w are all
0 0 0 0
T
reasonably close to e s 1, 0, 0, 0, 0, 0, 0. The fact that the first components
1
w of u and v are so large confirms what is evident from Figure 2 reproduced
0 0
Ž. x from Bookstein 1991 , Figure 3. 4 .1 , that these cephalograms have land-
marks labelled 1 to 8 in anticlockwise order, and are approximately evenly
spaced. If they were regular octagons, then the shape vectors would have
been e , but of course there is no biological reason to expect this. 1
1970 M. J. PRENTICE AND K. V. MARDIA
wŽ. FIG. 2. Representation of rat calvaria mid - sagittal section. Reproduced from Bookstein 1991 ,
page 68 x.
For these data, 7 r 6 r SU 7 s 0. 9996271 note that 0 F r F 6 r7 , with
2 2
convergence to seven significant figures in only three iterations, and
g SU 7 s 0. 00005325. For a test of H : u s v , 7 r 6 r SU 6 s 2 1 0 0 2
0 .9995555 with convergence to seven significant figures achieved after only
two iterations. The statistic 4. 5 for testing H against the general alterna- 1
2
tive H is T s 17 .79, to be compared with x 13 , significant at about the
2 1
0 .16 level if the asymptotic distributional assumptions are valid. With these
distributional assumptions there is no definite evidence that the population
mean shape at age 150 days is different from the population mean shape at
90 days. Note also that the statistic T , as in 4 .4 , for testing H : A s I 2 0
against the completely general H , is 156. 5. This gives incontrovertible 2
2
evidence that H is false when compared with x 48. Equally, for a test of 0
H against H , the statistic T s T y T is 138.7, also highly significant 0 1 0 2 1
2
when compared with x 35. We have strong evidence of systematic shape
change, A / I , but no real evidence that the mean after-shape differs from
the mean before-shape.
A similar analysis of the shape changes between ages 7 and 14 days leads
to different conclusions. The mean shape at 7 days is
u s 10
y 3
Ž958, 45 q 161 i , y 125 y 54 i , 30 y 34 i ,
0
T
19 q 66 i , 0 y 43 i , 26 q 165 i.
and the mean shape at age 14 days is
v s 10
y 3
954, 33 q 159 i , y 113 y 57 i , 36 y 31 i ,
0
T
13 q 59 i , y 60 i , 65 q 181 i. ,
1972 M. J. PRENTICE AND K. V. MARDIA
wŽ. x coordinate system of Kendall 1984 , Figure 2, page 101. Specifically, con-
sider the artificial data set consisting of matched pairs u , v , where u s j j j
2
1 r 2
a cos j Q, a sin j Q, 1 y a for small positive a , Q s pr 2 m , and v s
j
u. Note that u s u. All shapes u and v are slight perturbations of j q m j j q 4 m j j
an equilateral triangle.
Clearly, if these shapes on S
2
are converted back into triangles standard-
ised to any baseline whatsoever, then the Hotelling’s T
2
statistic calculated
from the changes in location of the remaining landmark will be negligibly
small. Hence a standard matched pairs Euclidean analysis to detect a change
in mean shape will conclude that there is no change. However, a spherical
regression analysis on Kendall’s blackboard leads to a different conclusion.
The sample mean before-shape and sample mean after-shape are both
T
0, 0, 1 , the equilateral triangle. The hypothesis H : A s I of no systematic
0 3
shape change is rejected in favour of H : u s v , but there is no evidence in
1 0 0
favour of the general alternative H : u / v. The conclusion to be drawn is
2 0 0
that there is a systematic shape change, even though there is no evidence
that the mean before-shape is any different from the mean after-shape.
T
Indeed, let A be a rotation through pr2 about 0, 0, 1 , that is,
0 y 1 0
A s 1 0 0.
Then the systematic change in shape is given by v s Au. j j
APPENDIX
PROOF OF THEOREM 1. The major modification of the proofs of Chang
1986 requires the use of vectors that are here quadratic in the original
shape measurements, leading to more complicated terms in Taylor series
expansions. We refer to the corresponding sections in Chang 1986 for ease of
y 1 Ž 2. Ž 2.
U y 1
U
U
T
comparison. Let X s n Ý u v and Z s n Ý u v m u v *. The n j j n j j j j
existence of M implies the existence of M , as in 3 .5 , and of
2
A. 1 M
q
s lim n
y 1
Ý u u
U
m u u
U
T
j j j j
U
Since A G ™ A , for large enough n we can write A A G s f H , n 2 0 0 n 2 n
where H g L G is chosen to have smallest magnitude. We can assume n
U
wŽ. x A s I by replacing v with A v. By analogy with Chang 1986 , page 911 0 j 0 j
pick a specific B g L G and define a real-valued function on L G ,
d
B
g Ž H. s tr f Ž H q tB. Z ,
Ž. n 2 n
dt t s 0
B
B
so that g H s 0. We expand g in a Taylor series about 0: n n n
d
B Ž 2.
g 0 s tr f tB Z s tr B Z. Ž. Ž Ž.. Ž.
n 2 n n
dt t s 0
SHAPE CHANGES IN THE PLANE 1973
Ž 2.
Ž 2.
Ž 2.
T
However, tr B X s tr B Z since tr B e v * m v s 0. If H g L G , n n
d d
B
g 9 Ž 0.? H s trŽ f Ž sH q tB. Z. s tr Ž C Ž B , H. Z ., Ž. n 2 n n
ds dt s s 0 t s 0
1 Ž 2. T T
Ž. w Ž .x Ž. Ž. where C B , H s HB q BH q H m B * q B m H *. Thus 0 s 2
B
Ž 2.
g H s tr B X q tr C B , H Z q R , where R is suitably bounded and n n n n n
Ž 2.
' Ž. Ž. Ž. so, using Lemma 1, it follows that if a B s ytr B n X , then a B s n n n
q T
' ' Ž Ž.. Ž. Ž Ž. Ž.. YY tr n C B , H M q 1 y YY tr n C B , H vec M e q R , where R 0 n 0 n n n
q
is suitably bounded. It remains to simplify tr C B , H M and
T
q
tr C B , H vec M e. With M as in A.1 ,
1 q q q
tr Ž C Ž B , H. M. s tr Ž Ž BH m I. M. q tr Ž Ž HB m I. M.
2
qtr I m BH *
T
M
q
q tr I m HB *
T
M
q
Ž Ž.. Ž Ž.. 4 Ž. Ž.
s tr HMB q tr BMH s 2 tr HN B.
1
T
Similarly, 2 tr C B , H vec M e simplifies eventually to 0.
Thus
Ž 2.
' '
a Ž B. s ytr B n X s 2 YY tr n Ž H N B .q R ,
Ž. Ž.
n n 0 n 1 n
where R is suitably bounded. n
wŽ. x As in Chang 1986 , Lemma 4 , we have the following lemma:
LEMMA 1. a has covariance quadratic form
n
YY Q B , B s y 4 YY tr B N B for B , B g L G.
2 1 2 2 1 1 2 1 2
PROOF.
E B
Ž 2.
u
Ž 2.
Ž 2.
y YY u
Ž 2.
v
Ž 2.
y YY u
Ž 2.
* B
Ž 2.
u
Ž 2.
Ž. Ž. Ž. ž j j 0 j j 0 j j /
A. 2
U
Ž 2. Ž 2. Ž 2. Ž 2.
s u * B * Ž Y. W Y Ž. B u ,
j j j j
2 2
where Y s u , U * and W is the variance matrix of the canonical random
j j j
variable z
Ž 2.
T
Now Y u s e and Y Bu s f say , where f s 0, f ,... , f and f may be
j j 1 j j 2 p
assumed real, as arbitrary scalar rotations may be applied to the columns of
T
U. Thus A. 2 reduces to g Wg , where g s f m e q e m f and the entries in
j 1 1
g are nonzero only if its double suffix includes the value 1 exactly once.
Hence A. 2 reduces to
k k
U
4 v f f s y 4 YY tr Bu u B Ž. Ý Ý 1 q , 1 m q m 2 j j
q s 2 m s 2
T
U U
since v s d YY with YY as in 3 .3 , and f f s tr u B Bu. We
1 q , 1 m q m 2 2 j j
obtain the covariance quadratic form of a B , which is real, by averag-
n
ing over j. Thus YY Q B , B s y 4 YY Re tr B N B. Since a B s
2 1 2 2 1 1 2 n
2 YY Re tr H N B q o 1 , it follows by the same argument as that of
0 n 1 p
wŽ. x Chang 1986 , Lemma 4 that H is asymptotically complex normal with
n
