Sankhyā : The Indian Journal of Statistics
2001, Volume 63, Series A, Pt. 1, pp. 118–127
THE EQUALITY OF LINEAR TRANSFORMS OF THE
ORDINARY LEAST SQUARES ESTIMATOR AND
THE BEST LINEAR UNBIASED ESTIMATOR
By JÜRGEN GROSS & GÖTZ TRENKLER
University of Dortmund, Dortmund, Germany
and
HANS JOACHIM WERNER
University of Bonn, Bonn, Germany
SUMMARY. We consider the equality of linear transforms of the ordinary least squares estimator (OLSE) and the traditional best linear unbiased estimator (BLUE) of Xβ in the Gauss-Markov
linear model L := {y, Xβ, V }, where y is an observable random vector with expectation vector
E(y) = Xβ and dispersion matrix D(y) = V . Of much interest to us are explicit parametric representations of the following three sets: (1) For given X and V , the set of all those matrices C with
C OLSE(Xβ) = C BLUE(Xβ). (2) For given X and C, the set of all those dispersion matrices
V with C OLSE(Xβ) = C BLUE(Xβ). (3) For given X, V and C, the event of all appropriate
(consistent) realizations of y under L on which C OLSE(Xβ) coincides with C BLUE(Xβ). Some
special cases are also considered.
1.
Introduction and Preliminaries
Let IRn , IRn×m , and P n×n denote the set of n-dimensional real column vectors, the set of n × m real matrices, and the set of real n × n symmetric nonnegative definite (nnd) matrices, respectively. Given A ∈ IRn×m , let A0 , R(A),
and N (A) denote the transpose, the range (column space), and the null space,
respectively, of A. In addition, let A− denote an arbitrary generalized inverse
of A satisfying AA− A = A. We mention that equation AA− A = A implies
R(A) ∩ N (A− ) = {0}. Moreover, if A is of full column rank, then A− A = I
also follows from A(I − A− A) = 0. The symbols I and 0, respectively, will stand
for the identity matrix and the zero matrix of whatever size is appropriate to the
context. The orthogonal projector onto the range of the matrix A will be denoted by PA . We note that this orthogonal projector may be defined by PA u = u if
Paper received February 2000.
AMS (1991) subject classification: Primary 62J05.
Key words and phrases. Ordinary least squares estimator (OLSE), best linear unbiased estimator (BLUE), traditional BLUE, wider definition BLUE, BLUE conditions, linearly unbiasedly estimable function, natural restrictions, general Gauss-Markov model, generalized inverse (g-inverse),
orthogonal projector.
equality of linear transforms of olse and blue
119
u ∈ R(A) and PA u = 0 if u ∈ N (A0 ); observe that IRn = R(A) ⊕ N (A0 ), where
⊕ indicates a direct sum. So R(PA ) = R(A) and N (PA ) = N (A0 ). We further
mention that PA = A(A0 A)− A0 , irrespective of the choice of (A0 A)− . The matrix
QA := I − PA is also an orthogonal projector. For QA we have R(QA ) = N (A0 )
and N (QA ) = R(A). We recall that any orthogonal projector P is symmetric and
idempotent, i. e. P 0 = P and P 2 = P , and that conversely every idempotent and
symmetric matrix P is an orthogonal projector, namely P = PP . By A ≥ B we
mean that the matrix A − B is symmetric and nonnegative definite and so A is
greater than or equal to B in the Löwner partial ordering.
Consider the general Gauss-Markov model
L = {y, Xβ, V },
where y is an n × 1 observable random vector with expectation E(y) = Xβ and dispersion (or covariance) matrix D(y) = V ; the vector β is the p×1 vector of unknown
regression coefficients, and X ∈ IRn×p and V ∈ P n×n are known and fixed matrices. There are no rank assumptions on X and V . Situations where V is singular
are not infrequent; e.g., when the observations yj add up to a fixed quantity as in
econometrics with complete expenditure systems, the dispersion matrix V clearly
is singular. Provided our Gauss-Markov model L is the correct data-generating
model, then the “consistency condition”
y ∈ R(X, V )
(1)
holds with probability 1; here (X, V ) denotes the partitioned matrix with V placed
next to X.
The linear transformation γ := Lβ, with L ∈ IRk×p fixed and known, is linearly
unbiasedly estimable in model L if and only if R(L0 ) ⊆ R(X 0 ). In what follows, let
γ = Lβ be linearly unbiasedly estimable. The linear estimator Gy is then defined
as the “traditional” best linear unbiased estimator (BLUE) of Lβ under our GaussMarkov model L whenever
Gy ∈ argminAy∈CL D(Ay);
here CL := {Ay : AX = L} and minimization is with respect to the Löwner
ordering. Note that all the statistics Ay with AX = L are linear unbiased estimators
for Lβ under the model L. In other words, Gy is a representation of the “traditional”
BLUE of γ = Lβ if and only if GX = L and D(Ay) ≥ D(Gy) for all A such that
AX = L. We observe that if η̂ = By and η̃ = Gy are any two traditional BLUEs
for Lβ in L, then, with probability 1, η̂ = η̃; see, e.g., Corollary 1.2 in Werner and
Yapar (1996). In the light of this observation, we write Gy = BLUE(Lβ) whenever
Gy coincides almost surely with the traditional BLUE for Lβ in the Gauss-Markov
model L.
We would like to emphasize at this point that in this “traditional” BLUE approach we do not make use of the so-called “natural restrictions”; several authors
have noted that if V is singular the classical unbiasedness condition of Gy for Lβ,
that is,
GX = L,
120
j. groß, g. trenkler and h.j. werner
is too strong, see, e.g., Zyskind and Martin (1969), Rao (1972; 1973b), Schönfeld and
Werner (1987), Puntanen and Styan (1989; 1990); as pointed out by Rao (1973b,
p. 280): “We note that the usual condition on unbiasedness employed in all earlier
work on linear estimation . . . is only sufficient”. Since, in addition to (1), we even
have
y − Xβ ∈ R(V ), with probability 1,
(2)
we can be sure that for a given realization y 0 ,
DXβ = Dy 0
holds true for any matrix D such that DV = 0. Therefore, if V is singular, then
(2) implies “natural restrictions” on β that can be utilized to extend the class of
unbiased estimators. The extended estimators enlarge the class of representations of
the traditional BLUE when the associated dispersion matrix is singular; explicit and
convenient representations of the quadratic unbiased and best unbiased estimators
are given for various levels of extension in Schönfeld and Werner (1987).
The following powerful traditional BLUE characterization is well known in the
literature, see, e.g. (18) in Schönfeld and Werner (1987); compare also Theorem 3.1
in Rao (1973b), Schönfeld and Werner (1986) or Rao (1965, 1973a).
Theorem 1. Consider model L = {y, Xβ, V } and let QX be the orthogonal
projector onto N (X 0 ). Then Gy = BLUE(Lβ) if and only if
G(X, V QX ) = (L, 0).
(3)
For L = X, in particular, P y = BLUE(Xβ) if and only if
P (X, V QX ) = (X, 0).
(4)
Two different matrix-based proofs for the second characterization can also be
found in Puntanen, Styan and Werner (2000). The “BLUE-conditions” (3) or (4)
are extremely useful not only in the field of statistics; indeed Werner (1997) has
illustrated that these conditions provide us even with a rather simple technique
which, in very many cases, enables us to show that a conjectured or claimed matrix
inequality in the Löwner sense is correct.
A direct consequence of (3) is also the observation that the BLUE-property is
preserved under linear transformation, that is, if Gy = BLUE(Lβ) then CGy =
BLUE(CLβ) whenever CL is defined. Since Lβ is linearly unbiasedly estimable if
and only if R(L0 ) ⊆ R(X 0 ) or, equivalently, if and only if L = CX for some suitable
matrix C, it suffices to concentrate on BLUE(Xβ); for we have C BLUE(Xβ) =
BLUE(CXβ). An alternative traditional estimator for Xβ in model L is PX y.
This statistic is known as the ordinary least squares estimator of Xβ and is often
denoted by OLSE(Xβ). Since PX X = X, OLSE(Xβ) is an unbiased estimator for
Xβ under model L, and so C OLSE(Xβ) is unbiased for CXβ. For convenience we
therefore write OLSE(CXβ) briefly for C OLSE(Xβ).
We note that cases with OLSE(CXβ) = BLUE(CXβ) might be advantageous,
not only computationally, because for estimating CXβ we then do not need to
equality of linear transforms of olse and blue
121
know the true dispersion matrix of our model L. The conditions (4) of Theorem 1
immediately tell us that OLSE(Xβ) = BLUE(Xβ) is equivalent to
PX V QX = 0
or, equivalently, to
V N (X 0 ) ⊆ N (X 0 ).
(5)
For further alternative characterizations and an excellent overview of OLSE(Xβ)
= BLUE(Xβ) we refer the reader to Puntanen and Styan (1989; 1990).
If instead of estimating Xβ we are primarily interested in estimating only Lβ =
CXβ (for a given fixed matrix C), it might be expected that OLSE(CXβ) =
BLUE(CXβ) can occur even if OLSE(Xβ) fails to be the BLUE(Xβ). In the
subsequent sections it is therefore our principal aim to give detailed answers to the
following three problems.
Problem 1: Given X ∈ IRn×p and V ∈ P n×n , what is the set C of matrices
C ∈ IRk×n satisfying C OLSE(Xβ) = C BLUE(Xβ)?
Problem 2: Given X ∈ IRn×p and C ∈ IRk×n , what is the set V of matrices
V ∈ P n×n satisfying C OLSE(Xβ) = C BLUE(Xβ)?
Problem 3: Given X ∈ IRn×p , C ∈ IRk×n and V ∈ P n×n , what is the set Y of
vectors y satisfying C OLSE(Xβ) = C BLUE(Xβ)?
We emphasize that throughout this paper we are, as discussed above, only interested in traditional linear estimators. All our proofs will be based on the “BLUE
conditions” (3). Besides we make use only of some elementary results from the
theory of generalized inversion, the following direct sum decomposition
R(X, V ) = R(X, V QX ) = R(X) ⊕ R(V QX ),
cf. e.g., (2.6) in Werner and Yapar (1996), and the following theorem dating back
to Baksalary (1984, Theorem 1).
Theorem 2. The equation AW A0 = B, with fixed given A ∈ IRn×m and B ∈
P n×n , admits a solution in P m×m if and only if R(B) ⊆ R(A). In this case, a
representation of the general solution is W = ZZ 0 , with
Z = A− B 1/2 + (I − A− A)Z,
where B 1/2 ∈ P m×m is the uniquely defined symmetric and nonnegative definite
square root of B, A− is an arbitrary but fixed g-inverse of A and Z is free to vary
in IRm×m .
2.
Solution to Problem 1
We begin this section with the following result characterizing the set C of Problem 1 and giving an explicit parametric representation for this set in terms of X
and V .
122
j. groß, g. trenkler and h.j. werner
Theorem 3. For given X ∈ IRn×p and V ∈ P n×n , consider the model
L = {y, Xβ, V }.
Then the following conditions are equivalent:
(i) C ∈ C, i.e. OLSE(CXβ) = BLUE(CXβ);
(ii) CPX V QX = 0;
(iii) R(C 0 ) ⊆ N (QX V PX );
(iv) R(C 0 ) ⊆ N (X 0 ) ⊕ {u ∈ R(X) : V u ∈ R(X)};
(v) C = Z [I − PX V QX (PX V QX )− ] for some matrix Z ∈ IRk×n .
Proof. Since CPX y = OLSE(CXβ) and PX X = X, it follows that CPX X =
CX. Therefore, according to (3) in Theorem 1, CPX y = BLUE(CXβ) if and only
if
CPX V QX = 0.
(6)
Since orthogonal projectors are symmetric, Eq. (6) turns out to be equivalent
to QX V PX C 0 = 0. We thus have (i) ⇐⇒ (ii) ⇐⇒ (iii). In view of IRn =
R(PX ) ⊕ N (PX ) = R(X) ⊕ N (X 0 ) and N (QX ) = R(X) (see Section 1), it is
also clear that (iii) is equivalent to (iv). For completing the proof it suffices to
show (ii) ⇐⇒ (v). Though this is a direct consequence of a well known theorem
(see, e.g., Rao and Mitra (1971, Theorem 2.3.1)), we add a simple proof. Clearly,
if C = Z [I − PX V QX (PX V QX )− ] for some matrix Z, then CPX V QX = 0 by
means of the defining equation of (PX V QX )− . Conversely, if CPX V QX = 0, then
choosing Z = C results, as claimed, in C = C [I − PX V QX (PX V QX )− ].
This theorem admits some additional implications.
Corollary 4. For given X ∈ IRn×p and V ∈ P n×n , consider the model L =
{y, Xβ, V } and let QX be defined as before. Then C = IRk×n , i. e. OLSE(CXβ) =
BLUE(CXβ) for every C ∈ IRk×n , if and only if V N (X 0 ) ⊆ N (X 0 ) or, equivalently, if and only if OLSE(Xβ) = BLUE(Xβ).
Corollary 5. For given X ∈ IRn×p and V ∈ P n×n , consider the model L =
{y, Xβ, V }. If
R(X) ∩ R(V ) = {0},
(7)
then OLSE(CXβ) = BLUE(CXβ) if and only if C satisfies
R(C 0 ) ⊆ N (X 0 ) ⊕ [R(X) ∩ N (V )] .
Under (7) and
R(X) ∩ N (V ) = {0}
(8)
we have OLSE(CXβ) = BLUE(CXβ) if and only if C solves CX = 0.
Needless to say, the case CX = 0 and so CXβ = 0 is not of any interest
statistically.
equality of linear transforms of olse and blue
123
Theorem 3 also gives us a detailed answer to the following question. Considering
all γ = Lβ being linearly unbiasedly estimable in model L: for which of these linear
transformations of β we do have OLSE(Lβ) = BLUE(Lβ)?
Corollary 6. For given X ∈ IRn×p and V ∈ P n×n , consider the model L =
{y, Xβ, V }. Then OLSE(Lβ) = BLUE(Lβ) if and only if
R(L0 ) ⊆ X 0 N (QX V PX )
or, equivalently, if and only if
R(L0 ) ⊆ X 0 {u ∈ R(X) : V u ∈ R(X)}.
If (7) and (8) hold simultaneously, then the only matrix L with OLSE(Lβ) =
BLUE(Lβ) is L = 0.
Proof. From Section 1 we know that Lβ is linearly unbiasedly estimable in
model L if and only if R(L0 ) ⊆ R(X 0 ). Since the latter happens if and only if we have
L = CX for some suitable matrix C, our claims are straightforward consequences
of Theorem 3 and Corollary 5.
3.
Solution to Problem 2
In this section, let X ∈ IRn×p and C ∈ IRk×n be given fixed matrices. According
to Problem 2 we now ask for all those matrices V ∈ P n×n for which we have
OLSE(CXβ) = BLUE(CXβ) under model L = {y, Xβ, V }. For solving this
problem, according to Theorem 1, we simply have to inspect the equation
CPX V QX = 0
once more, but now with respect to V ∈ P n×n .
Theorem 7. For given fixed X ∈ IRn×p and C ∈ IRk×n , put M := PX C 0 CPX .
Let PX , QX and PM be defined as described in Section 1, and let P1 := PX − PM .
Then the following conditions are equivalent:
(i) V ∈ V, i.e. OLSE(CXβ) = BLUE(CXβ);
(ii) CPX V QX = 0;
(iii) V N (X 0 ) ⊆ N (X 0 ) ⊕ [R(X) ∩ N (C)];
(iv) PM V QX = 0;
(v) There exist matrices H ∈ P n×n and Z ∈ IRn×n such that
³
´³
´0
V = W 1/2 + P1 Z W 1/2 + P1 Z ,
where W := PM HPM + QX HQX .
(9)
124
j. groß, g. trenkler and h.j. werner
Proof. From Theorem 3 we already know that (i) is equivalent to (ii). In view
of N (CPX ) = N (X 0 ) ⊕ [R(X) ∩ N (C)] and R(QX ) = N (X 0 ), it is also clear that
(ii) is equivalent to (iii). That (ii) ⇐⇒ (iv) is directly seen by means of N (PM ) =
N (M ) = N (CPX ). So (iv) ⇐⇒ (v) remains to be shown. First, let V be of the
form (9) where W = PM HPM + QX HQX . Then V = W + P1 ZW 1/2 + W 1/2 Z 0 P1 +
P1 ZZ 0 P1 and V ∈ P n×n . Since PM QX = 0, PM P1 = 0 and P1 QX = 0, in addition
we get, as desired, PM V QX = 0 for this particular matrix V . To prove the converse,
let V ∈ P n×n be such that PM V QX = 0. Put W = PM V PM + QX V QX . Since
V ≥ 0 and since orthogonal projectors are symmetric, clearly PM V PM ≥ 0 and
QX V QX ≥ 0. Consequently, W ≥ 0. In view of PM V QX = 0, we further have
W = PM V PM +QX V QX = (PM +QX )V (PM +QX ) so that R(W ) ⊆ R(PM +QX ).
Applying Theorem 2 to (PM + QX )V (PM + QX ) = W now results in the desired
representation of V provided we use G = I as the g-inverse of the orthogonal
projector PM + QX and observe that I − PM − QX = PX − PM = P1 . This
completes the proof.
As a corollary we mention the following interesting result.
Corollary 8. Let X ∈ IRn×p and C ∈ IRk×n be given fixed matrices, and let
R(X) ∩ N (C) = {0}.
Then OLSE(CXβ) = BLUE(CXβ) if and only if V of the model L = {y, Xβ, V }
satisfies
V = PX HPX + QX HQX
for some matrix H ≥ 0. Furthermore, in this case we even have V ∈ V if and only
if
V = PX V PX + QX V QX .
(10)
Proof. The first characterization follows directly from Theorem 7, since under
R(X)∩N (C) = {0}, PM = PX and so P1 = 0. Next recall that PX +QX = I. Since
PX V QX = 0 if and only if V = (PX + QX )V (PX + QX ) = PX V PX + QX V QX ,
the second characterization is also established.
4.
Solution to Problem 3
Under the model L = {y, Xβ, V }, we now wish to identify, for given C ∈ IRk×n ,
X ∈ IRn×p and V ∈ P n×n , the event of all those appropriate (i.e. consistent)
realizations of y on which the BLUE of CXβ coincides with the OLSE of CXβ.
That is, we wish to determine the set Y. Since we assume that the model L is
correctly specified, according to the consistency condition (1), we can and so do
require
Y ⊆ R(X) ⊕ R(V QX );
here we have made use of the identity R(X, V ) = R(X) ⊕ R(V QX ). Thus for each
y ∈ Y there uniquely exist y1 ∈ R(X) and y2 ∈ R(V QX ) such that y = y1 + y2 .
equality of linear transforms of olse and blue
125
Now recall that CPX y = OLSE(CXβ) and that Gy = BLUE(CXβ) if and only
if G(X, V QX ) = (CX, 0). Consequently CPX y = CPX (y1 + y2 ) = Cy1 + CPX y2
and Gy = Cy1 , and so CPX y = Gy if and only if CPX y2 = 0 or, equivalently,
y2 ∈ N (CPX ). Since N (CPX ) = N (X 0 ) ⊕ [R(X) ∩ N (C)], the following theorem
is plain.
Theorem 9. Under model L = {y, Xβ, V }, for given fixed matrices X ∈ IRn×p ,
V ∈ P n×n and C ∈ IRk×n , the event Y := { OLSE(CXβ) = BLUE(CXβ)} of all
appropriate realizations with y ∈ R(X, V ) is given by
Y = R(X) ⊕ [R(V QX ) ∩ N (CPX )]
or, equivalently, by
Y = R(X) ⊕ [R(V QX ) ∩ [N (X 0 ) ⊕ [R(X) ∩ N (C)]]] .
(11)
Corollary 10. Let X, V , C and Y be as in the preceding theorem. If R(X) ∩
N (C) = {0}, which in particular happens when C is of full column rank or when C
is chosen as an arbitrary but fixed g-inverse X − of X, then Y reduces to
Y = R(X) ⊕ [V N (X 0 ) ∩ N (X 0 )] .
(12)
Problem 3 has been addressed earlier by Krämer (1980) and by Groß and Trenkler (1997). These authors, however, only considered the full-rank case of the
model L = {y, Xβ, V }, which is characterized by requiring that the model matrices X and V are both of full-column rank. In such models all linear transformations Lβ of β are linearly unbiasedly estimable (compare Section 1), and
so the BLUE of β does in particular exist. Krämer was interested in the event
{ OLSE(β) = BLUE(β)} whereas Groß and Trenkler inspected, for arbitrary but
fixed L, the event { OLSE(Lβ) = BLUE(Lβ)}. Clearly, if X is of full column rank,
then X − X = I (see Section 1) and so β = X − Xβ. With this and Corollary 10
in mind, it is not at all a surprise that our representation (12) coincides with the
representation found earlier by Krämer. It might therefore be also expected that
our representation (11) does coincide with
Y = R(X) ⊕ [N (X 0 ) ∩ [R(V QX ) ⊕ [R(X) ∩ N (C)]]]
which was found by Groß and Trenkler. And this is indeed the case as is not difficult
to see. For observe that for linear subspaces A1 , A2 , A3 , and A4 of IRn , say, with
A4 ⊆ A1 , A2 ∩A4 = {0} and A3 ∩A4 = {0}, we always have A1 ⊕[A2 ∩ [A3 ⊕ A4 ]] =
A1 ⊕ [A3 ∩ [A2 ⊕ A4 ]] .
5.
Example
As an example, where equality of specific linear transforms of OLSE and BLUE
might be of interest, consider the one-way classification model
yij = µ + αi + eij ,
i = 1, · · · , a,
j = 1, · · · , ni ,
126
j. groß, g. trenkler and h.j. werner
where the eij ’s are uncorrelated random variables with means 0 and variances dij σ 2 .
For this example, assume a = 3, n1 = 2, n2 = 2, n3 = 1, and dij = 1 if (i, j) 6= (1, 2).
Moreover, assume that d12 6= 1 but otherwise unknown. Then the error variances
are not homogeneous within groups, and it is straightforward to check that the
inclusion (5) or, equivalently, the inclusion
V R(X) ⊆ R(X)
fails to be satisfied. Therefore (see Section 1) we do not have OLSE(Xβ) =
BLUE(Xβ) and so OLSE(CXβ) = BLUE(CXβ) does not hold for all matrices C.
Theorem 4 and Corollary 7, however, tell us how to determine all matrices C and
L = CX, respectively, for which the equality in question is correct. We obtain that
Lβ = CXβ is a parametric function with OLSE(Lβ) = BLUE(Lβ) if and only if
Lβ consists, elementwise, of any linear combination of µ + α2 and µ + α3 . Particularly, we thus have OLSE(α3 − α2 ) = BLUE(α3 − α2 ) for the contrast α3 − α2 . For
c0 = (0, 0, −1/2, −1/2, 1) we have c0 PX X = c0 X = c0 and c0 Xβ = α3 − α2 . Therefore, c0 PX y = y31 − y21 /2 − y22 /2 = OLSE(α3 − α2 ) = BLUE(α3 − α2 ). We further
mention that for Xβ (choose C = I) the event Y := { OLSE(Xβ) = BLUE(Xβ)}
is, according to (11), given by



γ






 γ 



Y =  δ + µ  : γ, δ, ², µ ∈ IR .








 δ−µ

²
Next, in virtue of Corollary 8, the set V of matrices V with OLSE(Xβ) = BLUE(Xβ)
is
V = {PX HPX + QX HQX : H ≥ 0}
where

PX
1
1

1
= 0
2
0
0
1
1
0
0
0
0
0
1
1
0
0
0
1
1
0

0
0

0,

0
2

QX = I − PX
1
−1

1
=  0
2
0
0
−1
1
0
0
0
0
0
1
−1
0

0 0
0 0

−1 0  .

1 0
0 0
We conclude with mentioning that Eq. (10) can also be used here to check whether
a particular given dispersion matrix belongs to V. Needless to say, for d 6= 1
the matrix V = diag(1, d, 1, 1, 1) does fail. This is in accordance with our earlier
observation above.
Acknowledgements. The research of the first and the second author were supported by the Deutsche Forschungsgemeinschaft under grants Tr 253/2-3 and SFB
475. Financial support of the third author by the Deutsche Forschungsgemeinschaft,
Sonderforschungsbereich 303, at the University of Bonn, is gratefully acknowledged.
equality of linear transforms of olse and blue
127
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J. Gross & G. Trenkler
Department of Statistics
University of Dortmund
Vogelpothsweg 87
D-44221 Dortmund, Germany
E-mail: [email protected]
[email protected]
H. J. Werner (corresponding author)
Inst. for Econometrics & Operations Research
Econometrics Unit
University of Bonn, Adenauerallee 24-42
D-53113 Bonn, Germany
E-mail: [email protected],
[email protected]