Estimation in an implicit multivariate measurement error model with clustering in the regressor
An implicit linear multivariate model DZ ≈ 0 is considered, where the data matrix D is observed with errors, and Z is a parameter matrix. The error matrix is partitioned into two uncorrelated blocks, and the total covariance structure in each block is supposed to be known up to a corresponding scala...
Saved in:
| Date: | 2008 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2008
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/4542 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Estimation in an implicit multivariate measurement error model with clustering in the regressor / M. Polekha // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 117–125. — Бібліогр.: 9 назв.— англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859795672544313344 |
|---|---|
| author | Polekha, M. |
| author_facet | Polekha, M. |
| citation_txt | Estimation in an implicit multivariate measurement error model with clustering in the regressor / M. Polekha // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 117–125. — Бібліогр.: 9 назв.— англ. |
| collection | DSpace DC |
| description | An implicit linear multivariate model DZ ≈ 0 is considered, where the data matrix D is observed with errors, and Z is a parameter matrix. The error matrix is partitioned into two uncorrelated blocks, and the total covariance structure in each block is supposed to be known up to a corresponding scalar factor. Moreover, the row data are clustered into two groups. Based on the method of corrected objective function, the strongly consistent estimators of scalar factors and the kernel of the matrix D are constructed, as the numbers of rows in the clusters tend to infinity.
|
| first_indexed | 2025-12-02T13:21:31Z |
| format | Article |
| fulltext |
Theory of Stochastic Processes
Vol. 14 (30), no. 1, 2008, pp. 117–125
UDC 519.21
MARIA POLEKHA
ESTIMATION IN AN IMPLICIT MULTIVARIATE MEASUREMENT
ERROR MODEL WITH CLUSTERING IN THE REGRESSOR
An implicit linear multivariate model DZ ≈ 0 is considered, where the data matrix D
is observed with errors, and Z is a parameter matrix. The error matrix is partitioned
into two uncorrelated blocks, and the total covariance structure in each block is
supposed to be known up to a corresponding scalar factor. Moreover, the row data
are clustered into two groups. Based on the method of corrected objective function,
the strongly consistent estimators of scalar factors and the kernel of the matrix D
are constructed, as the numbers of rows in the clusters tend to infinity.
1. Introduction
We deal with an implicit multivariate model D̄Z = 0, D = D̄ + D̃, where Z is the
parameter matrix, D is the data matrix, D̄ is the unobserved matrix, and D̃ is the error
matrix. The model is important for the system identification. In fact, our model can be
obtained from an explicit model AX ≈ B, where the matrices A and B are observed with
errors, and X is a matrix parameter. The approximate equality AX ≈ B means that,
for certain unobserved matrices Ā and B̄, the equality ĀX = B̄ holds, and A = Ā+ Ã,
B = B̄ + B̃, where à and B̃ are error matrices. For this model, there are some results
concerning the estimation of the parameter. In [1, 2], the covariance structure of [A B]
was known up to a constant, and the consistency of the total least squares estimator
was proven. In [4], the consistent estimator was constructed for the situation when the
covariance structure of [A B] is unknown. In [3,8,9], the model AX ≈ B was studied
when the covariance structure of A is known up to one scaling factor, while the covariance
structure of B is known up to another scaling factor. For identifiability reasons, it was
assumed in [3,8,9] that the data consist of two independent clusters, and, in [4], the
number of clusters depended of the size of X. The idea of the use of clusters belongs to
A. Wald [6]. He used it for a linear scalar model and proposed a slope estimator based
on the first empirical moments. In [4], similar moments were used, while, in [3,8,9], the
second empirical moments were exploited.
We modify the results of [3]. But our model is more general, although the matrix D
can be considered as a compound matrix [A B], here the input and output signals are
not separated.
The paper is organized as follows. Section 2 describes the model and introduces main
assumptions. In Section 3, we use the method of corrected objective function to derive
an estimator of the kernel of Z in case of known scalar factors λ0
1 and λ0
2. A preliminary
attempt to derive an objective function for λ0
1 and λ0
2, when they are unknown, is made
in Section 4. In Section 5, we introduce a model with two clusters and the final objective
function for the scalar factors and state the consistency result. In Section 6, the consistent
2000 AMS Mathematics Subject Classification. Primary 65F20, 93E12, 62H30, 62J05, 62H12, 62F12,
65P99.
Key words and phrases. Linear measurement error model, corrected objective function, clusters,
consistent estimator.
117
118 MARIA POLEKHA
estimator for the kernel of Z is proposed, and Section 7 concludes. The proofs of the
results are given in Appendix.
In the paper, we use a standard notation: ‖A‖ is the Frobenius norm of a matrix A;
the symbol tr denotes the trace of a matrix; Ip denotes a unit matrix of size p; and
the symbol E denotes the expectation of a random matrix. All the vectors in the paper
are column vectors. For a symmetric matrix C, μ1(C) ≤ . . . ≤ μp(C) are p smallest
eigenvalues of C.
2. The model and assumptions
Consider the model of observations
(1) DZ ≈ 0,
where Z ∈ R(n+p)×p is the unknown matrix parameter, and the data matrix D ∈
Rm×(n+p) is observed, D = D̄ + D̃. Here, D̄ is the unknown nonrandom matrix, and
D̃ is the matrix of random errors. For the true matrix D̄, we have
D̄Z = 0.
We want to estimate the matrix parameter Z with fixed n and p and increasing m.
We assume that dim KerD̄ = p, and the kernel is not changed with increase in m. Let
Z = [z1, . . . , zp] and rankZ = p. Then
D̄[z1, . . . , zp] = 0, D̄zi = 0, i = 1, ..., p,
and
span(z1, . . . , zp) = KerD̄ =: Vp.
Thus, we have to construct the estimator V̂p of the kernel KerD̄.
Let D̃� = [d̃1, · · · , d̃m]. By concerning the error vectors d̃i, we make some assump-
tions:
(i) Ed̃i = 0, for all i.
(ii) There exists δ > 0 such that supi≥1 E‖d̃i‖4+δ <∞.
(iii) The sequence of random vectors { d̃i, i ≥ 1 } is independent.
(iv) There exists n1, 1 ≤ n1 ≤ n + p − 1, such that D̃ = [D̃1 D̃2], D̃1 ∈ Rm×n1 , and
ED̃�
1 D̃2 = 0.
This means that the error matrix D̃ can be partitioned into two uncorrelated blocks.
(v) ED̃�
k D̃k = λ0
kWk, where Wk are the known positive definite matrices, and λ0
k are
the unknown positive scalars, k = 1, 2.
(vi) 1
m‖D̄‖2 ≤ const, m ≥ 1.
3. The estimator under known scalar factors
Suppose that λ0
1 and λ0
2 are known. We will use the corrected objective function
method [5]. The least squares objective function is
Qls(D̄;Z) := ‖D̄Z‖2, Z ∈ R
(n+p)×p,
or
Qls(D̄;Z) = tr(Z�Ψls(D)Z),
where
Ψls(D̄) := D̄�D̄.
By the corrected objective function method, we have to construct a matrix Qc(D;Z)
such that
E[Qc(D;Z)] = Qls(D̄;Z), for all Z.
ESTIMATION IN AN IMPLICIT MULTIVARIATE MEASUREMENT ERROR MODEL 119
It is possible under the known λ0
j and Wj , j = 1, 2, defined in conditions (iv) - (v). Let
Ψc(D) = D�D −
[
λ0
1W1 0
0 λ0
2W2
]
.
Then
Qc(D;Z) = tr(Z�Ψc(D)Z).
We minimize this objective function under the condition Z�Z = Ip. Consider the ordered
eigenvalues of Ψc(D), μ1 ≤ μ2 ≤ . . . ≤ μn+p, and the corresponding orthonormal eigen-
basis {ϕi, i = 1, 2, . . . , n+ p}. The function Qc(D;Z) is minimized at Z0 = [ϕ1, ..., ϕp].
Therefore, we obtain the estimator V̂p = span(ϕ1, ..., ϕp). It is a linear span of p eigen-
vectors which correspond to p smallest eigenvalues. The next lemma is similar to the
corresponding lemma in [3].
Lemma 1. Assume (i) to (vi). Then∥∥∥∥ 1
m
Ψc(D)− 1
m
Ψls(D̄)
∥∥∥∥→ 0, as m→∞, a.s.
4. The estimator under unknown scalar factors
Let λ0
j , j = 1, 2, be the unknown scalar factors. Then, for any λ1, λ2 ≥ 0, we define
Ψc(D;λ1, λ2) = Ψc(λ1, λ2) := D�D −
[
λ1W1 0
0 λ2W2
]
,
and
Ψls(D̄;λ1, λ2) = Ψls(λ1, λ2) := D̄�D̄ −
[
(λ1 − λ0
1)W1 0
0 (λ2 − λ0
2)W2
]
.
By Lemma 1, ∥∥∥ 1
m
Ψc(λ1, λ2)− 1
m
Ψls(λ1, λ2)
∥∥∥→ 0, as m→∞, a.s.
We will study the properties of the approximating matrix 1
mΨls(λ1, λ2). Assume the
following condition.
(vii) lim infm→∞ μp+1(D̄�D̄/m) > 0.
For large m, we have the approximate equality
1
m
Ψc(λ1, λ2) ≈ 1
m
Ψls(λ1, λ2),
and, for the approximate matrix Ψls(λ0
1, λ
0
2), we have
μi
(
Ψls(λ0
1, λ
0
2)
)
= 0, for all i = 1, . . . , p,
and μp+1
(
Ψls(λ0
1, λ
0
2)
)
> 0. Note that
Vp(λ0
1, λ
0
2) = span(z1, . . . , zp) = Ker(Ψls(λ0
1, λ
0
2)).
Therefore, in order to estimate λ0
1, λ
0
2, it seems natural to use the objective function
Q(λ1, λ2) :=
p∑
i=1
μ2
i
( 1
m
Ψc(λ1, λ2)
)
.
120 MARIA POLEKHA
Unfortunately, its minimization does not yield a consistent estimator of λ0
1, λ
0
2, since, for
the approximating matrix-valued function Ψls(λ1, λ2)/m, there could exist other values
λ∗1 and λ∗2 which are separated from λ0
1 and λ0
2, and such that
μi
(
Ψls(λ∗1, λ
∗
2)
)
= 0, for all 1 ≤ i ≤ p.
Therefore, we cannot estimate the scalars λ0
1 and λ0
2 by this way.
5. Model with two clusters
Consider two copies of model (1):
(2) D(k)Z ≈ 0, k = 1, 2,
where Z ∈ R(n+p)×p is the unknown parameter matrix to be estimated with rankZ = p,
and D(k) ∈ Rmk×(n+p) are observed,
D(k) = D̄(k) + D̃(k),
D̄(k)Z = 0, k = 1, 2.
Here, D̄(k) are the unknown nonrandom matrices, and D̃(k) are the error matrices,
D̃�(k) = [d̃1(k) · · · d̃mk
(k)], k = 1, 2. We assume that dim KerD̄(k) = p, k = 1, 2. We
want to estimate Vp := KerD̄(1) = KerD̄(2) for increasing m1 and m2 under fixed n and
p. For each k = 1, 2 we assume the following.
(a) Ed̃i(k) = 0, i ≥ 1.
(b) There exists δ > 0 such that supi≥1 E‖d̃i(k)‖4+δ <∞.
(c) { d̃i(k), i ≥ 1 } are two mutually independent sequences of random vectors.
(d) There exists n1, 1 ≤ n1 ≤ n + p − 1, such that D̃(k) = [D̃1(k) D̃2(k)], D̃1(k) ∈
Rmk×n1 and ED̃�
1 (k)D̃2(k) = 0.
(e) ED̃�
j (k)D̃j(k) = λ0
jWj(k), j = 1, 2, where Wj(k) are the known positive definite
matrices, and λ0
i are the unknown positive scalars.
(f) 1
mk
‖D̄(k)‖2 ≤ const, mk ≥ 1.
(g) lim infmk→∞ μp+1(D̄�(k)D̄(k)/mk) > 0.
(h) lim infm1,m2→∞ σp+1(D̄�(1)D̄(1)/m1 − D̄�(2)D̄(2)/m2) > 0, where σp+1(C) is
the (p+ 1)-th singular value of the symmetric matrix C.
[We mention that, for the matrix C in brackets in (h), σ1(C) = . . . = σp(C) = 0].
Let Z =
[
Z1
Z2
]
, Z1 ∈ Rn1×p, Z2 ∈ Rn2×p, n1 + n2 = n + p. Then D̃(k)Z =
D̃1(k)Z1 + D̃2(k)Z2.
(i) lim infm1→∞ tr(ZT
j
Wj(1)
m1
Zj) > 0, j = 1, 2, for certain fixed Z = [z1 . . . zp] with
span(z1, . . . , zp) = Vp.
(j) Wj(1)
m1
− Wj(2)
m2
→ 0, as m1,m2 →∞, j = 1, 2.
Let, for λ := (λ1, λ2) ∈ [0,∞)× [0,∞),
Ψ(k)
c (λ) := D�(k)D(k)−
[
λ1W1(k) 0
0 λ2W2(k)
]
,
and let μ1k(λ) ≤ μ2k(λ) ≤ . . . ≤ μpk(λ) be p smallest eigenvalues of the matrix Ψ(k)
c (λ)
with the corresponding orthonormal eigenvectors f1k(λ), f2k(λ), . . . , fpk(λ), k = 1, 2.
Note that if the given eigenvalues are multiple, then the eigenvectors are not uniquely
defined. In this case, we define them in such a way that they are Borel measurable vector
functions of D(k) and λ.
ESTIMATION IN AN IMPLICIT MULTIVARIATE MEASUREMENT ERROR MODEL 121
Let Vpk(λ) = span(f1k(λ), f2k(λ), . . . , fpk(λ)). Then an objective function for estimat-
ing λ is
Qc(λ) :=
2∑
k=1
p∑
i=1
μ2
ik(λ) + c‖ sinΘ(λ)‖2,
where c is a fixed positive constant, Θ(λ) is a diagonal matrix of canonical angles between
the subspaces Vp1(λ) and Vp2(λ), and sin Θ(λ) is defined as the diagonal matrix with the
sines of these angles as diagonal elements. Recall the next definition of canonical angles.
Definition 1. [7]. Let �(X) and �(Y ) be two subspaces in Rn with the same dimension.
The given subspaces are spanned by columns of matrices X and Y , respectively. Let the
columns of the matrix X⊥ form an orthogonal basis for (�(X))⊥ which is an orthogonal
complement to �(X). Then nonzero singular values of the matrix XT
⊥Y are the sines of
nonzero canonical angles between the subspaces �(X) and �(Y ).
Let {εt, t ≥ 1} be a fixed positive sequence, and εt → 0 as t→∞. Then the estimator
λ̂ = λ̂(t) is defined as any measurable solution to the inequality
(3) Qc(λ̂) ≤ inf
λ1,λ2≥0
Qc(λ) + εt,
where t := min(m1,m2).
Theorem 1. Suppose that conditions (a) to (j) are satisfied, then
λ̂→ λ0 := (λ0
1, λ
0
2), as m1,m2 →∞, a.s.
6. Estimator of the kernel
Under the conditions of Theorem 1, one can obtain the estimator λ̂ for the unknown
scalars λ0 and then construct two estimators Vp1(λ̂) and Vp2(λ̂) for a subspace Vp. But
we want to construct a single subspace estimator using both clusters. It is reasonable to
consider a compound data matrix Dc :=
[
D(1)
D(2)
]
and Wcj := Wj(1) +Wj(2), j = 1, 2.
Then Dc = D̄c + D̃c and
H := ED̃T
c D̃c =
[
λ0
1Wc1 0
0 λ0
2Wc2
]
.
Define the matrix
Ĥ = DT
c Dc −
[
λ̂1Wc1 0
0 λ̂2Wc2
]
.
Let Vp(Ĥ) denote the subspace spanned by the first p eigenvectors of Ĥ corresponding
to the smallest eigenvalues, then V̂p = Vp(Ĥ) = span(ẑ1, ẑ2, . . . , ẑp).
Definition 2. Let V1 and V2 be two subspaces in R
n with the same dimension p, sinΘ
be a diagonal matrix of the sines of canonical angles between the subspaces V1 and V2. A
measurement of the proximity of the subspace V1 to the subspace V2 is a norm ‖ sinΘ‖ =√∑
i≥1 sin2 Θii.
Theorem 2. Under the conditions of Theorem 1, V̂p → Vp holds, i.e. ‖ sin Θ̂‖ → 0, a.s.,
as m1,m2 →∞, where Θ̂ is the matrix of canonical angles between the subspaces V̂p and
Vp.
122 MARIA POLEKHA
7. Conclusions
We considered a multivariate errors-in-variables model DZ ≈ 0 for the case where the
total error covariance structure of the data matrix is known up to two scalar factors. The
main assumption was that we observe two independent copies of the model. In practice,
this means that the data can be partitioned in two separated groups, i.e. clusters. Based
on the objective function method and using the second empirical moments, we have
constructed the consistent estimators of the scalar factors and of the kernel of the data
matrix.
Acknowledgement
The author thanks Prof. A. Kukush for posing the problem and fruitful discussions.
Appendix
Proof of Theorem 1.
1. Behavior of Qc(λ0)
We have
Qc(λ0) =
2∑
k=1
p∑
i=1
μ2
ik(λ0) + c‖ sinΘ(λ0)‖2.
By Lemma 1, αk :=
∥∥∥ 1
mk
Ψ(k)
c (λ0)− 1
mk
D̄�(k)D̄(k)
∥∥∥ → 0, as mk → ∞, a.s., for k =
1, 2. We have μi
(
D̄�(k)D̄(k)/mk
)
= 0, 1 ≤ i ≤ p. Let
δN := inf
mk≥N
μp+1
(
1
mk
D̄�(k)D̄(k)
)
.
Then, due to assumption (g), limN→∞ δN > 0. Thus,
lim
m1,m2→∞
2∑
k=1
p∑
i=1
μ2
ik(λ0) = 0,
and ‖ sinΘk(λ0)‖ ≤ αk
δN
, by Wedin’s theorem [7]. Then ‖ sinΘk(λ0)‖ → 0, as mk →
∞, a.s., k = 1, 2. Here, Θk(λ0) is the diagonal matrix of canonical angles between
the subspaces Vp(Ψ
(k)
c (λ0)/mk) and Vp(D̄�(k)D̄(k)/mk), and Vp denotes the span of p
eigenvectors corresponding to p smallest eigenvalues.
Now,
Vp(D̄�(1)D̄(1)/m1) = Vp(D̄�(2)D̄(2)/m2) = span(z1, . . . , zp).
Hence, ‖ sinΘ(λ0)‖ → 0, as m1,m2 →∞, a.s. Thus,
Qc(λ0)→ 0 as m1,m2 →∞, a.s.
By inequality (3), we have
(4) Qc(λ̂)→ 0 as m1,m2 →∞, a.s.
2. λ̂ is eventually bounded
Now we want to construct such a nonrandom L > 0 that, eventually,
(5) ‖λ̂‖ ≤ L.
ESTIMATION IN AN IMPLICIT MULTIVARIATE MEASUREMENT ERROR MODEL 123
(”Eventually” means: for all n ≥ n0(ω), a.s.) From (4) for any ε0 > 0, we have
(6)
2∑
k=1
p∑
i=1
μ2
ik(λ̂) ≤ ε0, eventually.
By Lemma 1,
(7)
∣∣∣∣ 1
m1
Ψ(1)
c (λ̂)− 1
m1
Ψ(1)
ls (λ̂)
∣∣∣∣→ 0, as m1,m2 →∞, a.s.,
where
Ψ(1)
ls (λ) := D̄�(1)D̄(1)−
[
(λ1 − λ0
1)W1(1) 0
0 (λ2 − λ0
2)W2(1)
]
,
λ := (λ1, λ2) ∈ [0,∞)× [0,∞). Let Z satisfy condition (i). Then
tr
(
Z�(Ψ(1)
ls (λ̂)/m1)Z
)
= −tr
(
(λ̂1 − λ0
1)Z
�
1
(
W1(1)/m1
)
Z1 + (λ̂2 − λ0
2)Z
�
2
(
W2(1)/m1
)
Z2
)
.
Suppose that λ̂1 − λ0
1 > L0, where L0 > 0. Then
tr
(
Z�(Ψ(1)
ls (λ̂)/m1Z
)
≤ −L0 tr
(
Z�
1
(
W1(1)/m1
)
Z1
)
+ const ·λ0
2.
But, due to (i) and (j),
lim inf
m1,m2→∞ tr
(
Z�
1 (W1(1)/m1)Z1
)
> 0.
Thus, for L0 large enough and all m1 ≥ m10 , we have
tr
(
Z�(Ψ(1)
ls (λ̂)/m1)Z
)
≤ −1.
So μp+1
(
Ψ(1)
ls (λ̂)/m1
)
is negative and separated from 0. Then we have from (7) that
μp+1
(
Ψ(1)
c (λ̂)/m1
)
is also negative and separated from 0, starting from some random
number. But this contradicts (6). Therefore, for a large enough nonrandom L0, λ̂1−λ0
1 ≤
L0 holds eventually. A similar inequality can be shown for λ̂2 − λ0
2, based on condition
(i) for j = 2. Thus, (5) holds eventually.
3. Consistency of the estimator
Let us have any fixed set Ω0, Pr(Ω0) = 1, such that, for all ω ∈ Ω0 and for m1 ≥
m10(ω), m2 ≥ m20(ω), ‖λ̂(ω)‖ ≤ L holds. Fix ω ∈ Ω0 and consider a bounded sequence
(8) { λ̂(ω;m1,m2) : m1 ≥ m10(ω),m2 ≥ m20(ω)}.
We will show that sequence (8) tends to λ0, as m1,m2 →∞. Let{
λ̂
(
ω;m1(q),m2(q)
)
, q ≥ 1
}
be any convergent subsequence, i.e. for a certain λ∞ ∈ R2
lim
q→∞ λ̂
(
ω;m1(q),m2(q)
)
= λ∞.
We want to prove the convergence of (8) and show that λ∞ = λ0. Let
M (k)(mk) := diag(μ1k, μ2k, . . . , μpk)
and
Y (k)(mk) := [f (k)
1 (mk) . . . f (k)
p (mk)]
124 MARIA POLEKHA
be a matrix, the columns of which are the first p eigenvalues of the matrix, Ψ(k)
c (λ̂)/mk.
These columns form an orthogonal basis for Vpk(λ̂). According to (4), M (k)(mk)→ 0, as
mk →∞. Moreover, sin Θ(λ̂)→ 0, as q →∞. Here, λ̂ = λ̂(m1(q),m2(q)). Therefore, we
can assume ∥∥∥Y (1)(m1)− Y (2)(m2)
∥∥∥→ 0, as m1,m2 →∞,
where mk = mk(q), k = 1, 2. Then
(9)
1
mk
Ψ(k)
c (λ̂)Y (k)(mk) = M (k)(mk)Y (k)(mk).
We suppose that
(10) Y (k)
(
mk(q)
)→ Y (k)
∞ , q →∞, k = 1, 2,
and Y
(k)
∞ = [y(k)
∞1 . . . y
(k)
∞p ]. (Otherwise we should consider an embedded subsequence of
numbers mk(q′)). The matrix Θ(λ̂) is a matrix of canonical angles between
span(f (1)
1 (m1) . . . f (1)
p (m1)) and span(f (2)
1 (m2) . . . f (2)
p (m2)).
Then, due to (10), we have sinΘ∞ = 0, where Θ∞ is the matrix of canonical angles
between span(y(1)
∞1 . . . y
(1)
∞p) and span(y(2)
∞1 . . . y
(2)
∞p). Thus, these spans coincide. Since
Y (k)(mk)(q) are matrices of orthonormal columns, the matrices Y (1)
∞ and Y
(2)
∞ have the
same property. Then, from the coincidence of the linear spans of the columns of the
matrices, we have Y (1)
∞ = Y
(2)
∞ U, where U is the p× p orthogonal matrix corresponding
to the choose of another basis in the linear span of columns of the matrix Y
(1)
∞ . Due
to (9),
sup
‖λ‖≤L
∥∥∥∥ 1
mk
Ψ(k)
c (λ) − 1
mk
Ψ(k)
ls (λ)
∥∥∥∥→ 0, as q →∞.
Thus
1
mk(q)
Ψ(k)
ls (λ∞)Y (k)
∞ → 0, as q →∞, k = 1, 2.
Let Y∞ = Y
(2)
∞ . Multiplying it by U for k = 1, we obtain
1
mk(q)
Ψ(k)
ls (λ∞)Y∞ → 0,
but this also holds for k = 2.
Next, (
1
m1(q)
Ψ(1)
ls (λ∞)− 1
m2(q)
Ψ(2)
ls (λ∞)
)
Y∞ → 0, as q →∞.
Due to (j), we have(
1
m1(q)
D̄�(1)D̄(1)− 1
m2(q)
D̄�(2)D̄(2)
)
Y∞ → 0, as q →∞.
On the other hand, assumption (h) yields
Y∞ = [y∞1 . . . y∞p ] and span(y∞1 . . . y∞p) = span(z1, . . . , zp). Due to (9),
1
mk(q)
Ψ(k)
ls (λ∞)Z → 0, as q →∞, k = 1, 2.
Therefore, [
(λ∞1 − λ0
1)W1(1)/m1(q) 0
0 (λ∞2 − λ0
2)W2(1)/m2(q)
]
Z → 0,
ESTIMATION IN AN IMPLICIT MULTIVARIATE MEASUREMENT ERROR MODEL 125
(λ∞j − λ0
j )tr
(
Z�
j Wj(1)Zj/mj(q)
)→ 0, as t→∞, j = 1, 2.
Conditions (i) and (j) imply that λ∞j = λ0
j , j = 1, 2. Hence, any convergent subsequence
of the bounded sequence (8) converges to λ0; therefore, the sequence (8) itself converges
to λ0. The convergence holds for all ω ∈ Ω0, Pr(Ω0) = 1. Thus, λ̂→ λ0, as m1,m2 →∞,
a.s. �
Proof of Theorem 2.
By Theorem 1,∥∥∥∥ 1
m1 +m2
(Ĥ − D̄�
c D̄c)
∥∥∥∥→ 0, as m1,m2 →∞, a.s.
Due to condition (g),
μ1
(
1
m1 +m2
D̄�
c D̄c
)
= . . . = μp
(
1
m1 +m2
D̄�
c D̄c
)
= 0,
and, by Lemma 1, μp+1
(
D̄�
c D̄c/m
)
> 0 for large m1,m2. Moreover, the kernel equals
Vp
(
1
m1 +m2
D̄�
c D̄c
)
= span(z1, . . . , zp).
By Wedin’s theorem [7], this implies that Θ̂ → 0, as m1,m2 → ∞., where Θ̂ is the
diagonal matrix of the canonical angles between Lp(Ĥ) and span(z1, . . . , zp). �
Bibliography
1. A.Kukush, I.Markovsky, and S.Van Huffel, Consistency of the structured total least squares esti-
mator in a multivariate errors-in-variables model, Journal of Statistical Planning and Inference
133 (2005), no. 2, 315-358.
2. A.Kukush and S.Van Huffel, Consistency of element-wise weighted total least squares estimator
in multivariate errors-in-variables model AX = B, Metrika 59 (2004), no. 1, 75-97.
3. A.Kukush, I.Markovsky, and S.Van Huffel, Estimation in a linear multivariate measurement
error model with clustering in the regressor, Internal Report 05-170. ESAT-SISTA. K.U.Leuven
(Leuven, Belgium) (2005).
4. A.Kukush and M.Polekha, Consistent estimator in multivariate errors-in-variables model under
unknown error covariance structure, Ukrainian Mathematical Journal 59 (2007), no. 8, 1026-
1033.
5. A.Kukush and S.Zwanzing, About the adaptive minimum contrast estimator in a model with
nonlinear functional relations, Ukrainian Mathematical Journal 53 (2001), 1145-1452.
6. A.Wald, The fitting of straight lines if both variables are subject to error, Ann. Math. Stat. 11
(1940), 284-300.
7. G.Stewart and J.Sun, Matrix Perturbation Theory, Academic Press, Boston, 1990.
8. I.Markovsy, A.Kukush, and S.Van Huffel, On errors-in-variables estimation with unknown noise
variance ratio, 14th IFAC Symposium on System Identification. Newcastle. Australia (2006).
9. I.Markovsy, A.Kukush, and S.Van Huffel, Estimation in a linear multivariate measurement
error model with a change point in data, Computational Statistics and Data Analysis 52 (2007),
no. 2, 1167-1182.
����������
��������
��� ������
�� ����� ���%����� &�
% ���
��� $�
%���
��� (�
������
�
�� 9����� % �%�+� &�
% , ##� $���
��
E-mail : poleha@bigmir.net
|
| id | nasplib_isofts_kiev_ua-123456789-4542 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-02T13:21:31Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Polekha, M. 2009-11-25T11:07:39Z 2009-11-25T11:07:39Z 2008 Estimation in an implicit multivariate measurement error model with clustering in the regressor / M. Polekha // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 117–125. — Бібліогр.: 9 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4542 519.21 An implicit linear multivariate model DZ ≈ 0 is considered, where the data matrix D is observed with errors, and Z is a parameter matrix. The error matrix is partitioned into two uncorrelated blocks, and the total covariance structure in each block is supposed to be known up to a corresponding scalar factor. Moreover, the row data are clustered into two groups. Based on the method of corrected objective function, the strongly consistent estimators of scalar factors and the kernel of the matrix D are constructed, as the numbers of rows in the clusters tend to infinity. en Інститут математики НАН України Estimation in an implicit multivariate measurement error model with clustering in the regressor Article published earlier |
| spellingShingle | Estimation in an implicit multivariate measurement error model with clustering in the regressor Polekha, M. |
| title | Estimation in an implicit multivariate measurement error model with clustering in the regressor |
| title_full | Estimation in an implicit multivariate measurement error model with clustering in the regressor |
| title_fullStr | Estimation in an implicit multivariate measurement error model with clustering in the regressor |
| title_full_unstemmed | Estimation in an implicit multivariate measurement error model with clustering in the regressor |
| title_short | Estimation in an implicit multivariate measurement error model with clustering in the regressor |
| title_sort | estimation in an implicit multivariate measurement error model with clustering in the regressor |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4542 |
| work_keys_str_mv | AT polekham estimationinanimplicitmultivariatemeasurementerrormodelwithclusteringintheregressor |