A googness of-fit-test for a multivariate errors-in-variables model
A multivariate errors-in-variables model AX ≈ B is considered, where the data matrices A and B are observed with errors, and a matrix parameter X is to be estimated. A goodness-of-fit test which is based on the moment estimator is constructed. The proposed test is asymptotically chi-squared under nu...
Gespeichert in:
| Datum: | 2006 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2006
|
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/4458 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | A googness of-fit-test for a multivariate errors-in-variables model / A. Kukush, M. Polekha // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 63–74. — Бібліогр.: 6 назв.— англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-4458 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-44582009-11-12T12:00:26Z A googness of-fit-test for a multivariate errors-in-variables model Kukush, A. Polekha, M. A multivariate errors-in-variables model AX ≈ B is considered, where the data matrices A and B are observed with errors, and a matrix parameter X is to be estimated. A goodness-of-fit test which is based on the moment estimator is constructed. The proposed test is asymptotically chi-squared under null hypothesis. The power of the test is discussed. 2006 Article A googness of-fit-test for a multivariate errors-in-variables model / A. Kukush, M. Polekha // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 63–74. — Бібліогр.: 6 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4458 en Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
A multivariate errors-in-variables model AX ≈ B is considered, where the data matrices A and B are observed with errors, and a matrix parameter X is to be estimated. A goodness-of-fit test which is based on the
moment estimator is constructed. The proposed test is asymptotically chi-squared under null hypothesis. The power of the test is discussed. |
| format |
Article |
| author |
Kukush, A. Polekha, M. |
| spellingShingle |
Kukush, A. Polekha, M. A googness of-fit-test for a multivariate errors-in-variables model |
| author_facet |
Kukush, A. Polekha, M. |
| author_sort |
Kukush, A. |
| title |
A googness of-fit-test for a multivariate errors-in-variables model |
| title_short |
A googness of-fit-test for a multivariate errors-in-variables model |
| title_full |
A googness of-fit-test for a multivariate errors-in-variables model |
| title_fullStr |
A googness of-fit-test for a multivariate errors-in-variables model |
| title_full_unstemmed |
A googness of-fit-test for a multivariate errors-in-variables model |
| title_sort |
googness of-fit-test for a multivariate errors-in-variables model |
| publisher |
Інститут математики НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/4458 |
| citation_txt |
A googness of-fit-test for a multivariate errors-in-variables model / A. Kukush, M. Polekha // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 63–74. — Бібліогр.: 6 назв.— англ. |
| work_keys_str_mv |
AT kukusha agoognessoffittestforamultivariateerrorsinvariablesmodel AT polekham agoognessoffittestforamultivariateerrorsinvariablesmodel AT kukusha goognessoffittestforamultivariateerrorsinvariablesmodel AT polekham goognessoffittestforamultivariateerrorsinvariablesmodel |
| first_indexed |
2025-07-02T07:41:54Z |
| last_indexed |
2025-07-02T07:41:54Z |
| _version_ |
1836520167082819584 |
| fulltext |
Theory of Stochastic Processes
Vol. 12 (28), no. 3–4, 2006, pp. 63–74
ALEXANDER KUKUSH AND MARIA POLEKHA
A GOOGNESS OF-FIT-TEST FOR A MULTIVARIATE
ERRORS-IN-VARIABLES MODEL
A multivariate errors-in-variables model AX ≈ B is considered, where
the data matrices A and B are observed with errors, and a matrix para-
meter X is to be estimated. A goodness-of-fit test which is based on the
moment estimator is constructed. The proposed test is asymptotically
chi-squared under null hypothesis. The power of the test is discussed.
1. Introduction
Errors-in-variables (EIV) models are rather important in practical appli-
cations. It is reasonable to develop appropriate goodness–of–fit test for such
models.
Consistent estimators for a multivariate errors-in-variables model under
various conditions are presented in [1 – 3]. A goodness–of–fit test is con-
structed in [4] for a linear structural EIV model, where the distribution of
the latent variable and the error distributions are normal. A polynomial
EIV model is considered in [5], without the normality assumption. Present
paper modifies the results of [5] for a multivariate errors-in-variables model.
We use the following notations: ‖A‖ is Frobenius norm of a matrix A,
Ip is the unit matrix of size p. The symbols E, D, and cov denote the
expectation of a random matrix, the variance of random variable, and the
variance - covariance matrix of a random vector, respectively. Op(1) de-
notes a sequence of stochastically bounded random variables, and op(1) is
a sequence of random variables that converges to 0 in probability. All the
vectors in the paper are column vectors.
The paper is organized as follows. In Section 2 we introduce the model
and construct an estimator. In Section 3 we present a goodness–of–fit test
and show that it is asymptotically chi-squared with p degrees of freedom
under null hypothesis. We introduce a local alternative and investigate the
power of the test in Section 4. Section 5 concludes, and the proofs of the
results are presented in Appendix.
2. The model and the estimator
Consider the model of observations:
(1) A0X = B0, A = A0 + Ã, B = B0 + B̃,
2000 Mathematics Subject Classification. Primary 62H15.
Key words and phrases. Multivariate errors-in-variables model, goodness-of-fit test,
moment estimator, asymptotically chi-squared, power of the test.
63
64 ALEXANDER KUKUSH AND MARIA POLEKHA
where A0 ∈ R
m×n, X ∈ R
n×p, B0 ∈ R
m×p. Here the data matrices A, B
are observed, and A0, B0 are unknown nonrandom matrices, and Ã, B̃ are
matrices of random errors.
Let AT = [a1...am], BT = [b1...bm], and we use similar notations for the
rows of A0, B0, Ã, B̃.
Rewrite the model (1) as a multivariate lineal model:
(2) XT a0
i = b0
i , i = 1, m;
bi = b0
i + b̃i, ai = a0
i + ãi, i = 1, m.
We assume the following conditions:
a) the sequences of errors vectors {ãi, i ≥ 1} and {b̃i, i ≥ 1} are two IID
centered sequences of random errors, independent of each other,
b) for all i, ã
d
= ãi, b̃
d
= b̃i and Eã = 0, Eb̃ = 0;
c) covã =: Sã is known and covb̃ =: Sb̃ is unknown.
The adjusted least squares (ALS) estimator of matrix parameter X is
X̂ := (AT A −EÃT Ã)−1AT B = (
m∑
i=1
aia
T
i − Eãiãi
T )−1
m∑
i=1
aib
T
i =
= (aaT −EããT )−1abT ,
(3) X̂ = H̄−1abT ,
where H := aaT − EããT . Hereafter the bars denote averages, e.g.,
abT =
m∑
i=1
aib
T
i /m.
Lemma 1[6]. Assume that the following conditions are satisfied.
(i) E‖ã‖4 < ∞, E‖b̃‖4 < ∞.
(ii) There exists V := lim
m→∞
a0a0T and V is positive definite.
Then H̄ is nonsingular with probability tending to 1, and
(4) X̂
P−→ X as m → ∞,
(5) Ŝb̃ := bbT − baT X̂
P−→ Sb̃ as m → ∞.
The estimator of X̂ is well-defined for m ≥ m0(ω) a.s., under the condi-
tions of Lemma 1. If the matrix H̄ = H̄(m, ω) is singular, then the estimator
is X̂ = H̄†abT , where H̄† is pseudoinverse matrix.
3. Construction of the test
For the response vector b and the corresponding latent vector a0 we consider
the following hypotheses
H0 : there exists a matrix X ∈ R
n×p, for which the equality holds true:
(6) E(b − XTa0) = 0,
A GOOGNESS OF-FIT-TEST 65
and H1 : for all matrices X ∈ R
n×p,
(7) E(b − XT a0) is not identically equal to 0.
We want to construct a test statistic for the null hypothesis using the ob-
servations ai and bi, i = 1, 2, ..., n.
Let w(a0) be a scalar weight function. Then under null hypothesis we
have equality E[(b−XT a0)w(a0)] = 0. We will construct a vector polynomial
s(a), such that under H0 the following relation is true:
(8) E[(b − XT s(a))w(a)] = 0.
Such a construction is possible if one chooses w(a) as follows: w(a) = eλT a,
a ∈ R
n, λ = (λ1, λ2, ..., λn)
T is fixed, λk �= 0, k = 1, n. We fix such a λ and
assume that the corresponding exponential moment of ã exists and satisfies
the condition:
(iii) E[(1 + ‖ã‖)eλT ã] < ∞.
For the chosen weight function, relation (8) holds if for every a0 one has:
a0 · E(eλT ã) = E(s(a0 + ã)eλT ã).
Then (8) holds for s(a) = a − E(ãeλT ã)
E(eλT ã)
. Denote μ0 = E(eλT ã) and μ1 =
E(ãeλT ã), then s(a) = a − μ1
μ0
.
Define a statistic of the score type
(9) T 0
m =
1
m
m∑
i=1
(bi − X̂Ts(ai))e
λT ai = (b − X̂T s(a))eλT a.
We introduce further assumptions to derive an asymptotic expansion of√
m · T 0
m.
(iv) E[(1 + ‖ã‖2)e2λT ã] < ∞.
This condition is stronger than (iii). For arbitrary function f(a0), we
denote M(f(a0)) = lim
m→∞
f(a0), provided the limit exists and finite; a0
j is
jth component of the vector a0.
(v) ∃M((a0(j))l(a0(k))reλT a0
), for all l, r ≥ 0, l + r ≤ 2, j, k = 1, n.
(vi) ‖a0a0T − V ‖ = o(m−1/4), as m → ∞.
Lemma 2. Assume (i), (ii), and (iv) to (vi). Then
(10)
√
m · T 0
m =
1√
m
m∑
i=1
b̃i(e
λT ai − aT
i f) + XT 1√
m
m∑
i=1
ηi + op(1),
where ηi := (a0
i −s(ai))e
λT ai +(Hi−a0
i a
T
i )f are independent random vectors
with expectation 0, Hi = aia
T
i −Eãiã
T
i , f := V −1M(a0eλT a0
)μ0, and matrix
V comes from (ii).
We need some more assumptions in order to apply the central limit the-
orem in the Lyapunov form to the statistic
√
m · T 0
m.
(vii) ∃δ > 0 : E[(1 + ‖ã‖2+δ)e(2+δ)λT ã] < ∞, and E‖b̃‖2+δ < ∞.
66 ALEXANDER KUKUSH AND MARIA POLEKHA
(viii) There exist M((a0(j))l(a0(k))reλT a0
), for all l, r ≥ 0, l + r ≤ 3, and
M((a0(j))l(a0(k))re2λT a0
), for all l, r ≥ 0, l + r ≤ 2; j, k = 1, n.
(ix) ∃δ > 0 : ‖a0‖4+δ + e(2+δ)λT a0 + ‖a0‖2+δe(2+δ)λT a0 ≤ const.
(x) e2λT a0 · ‖a0‖4 = o(m), as m → ∞.
Condition (vii) absorbs conditions (iii) and (iv), and conditions (viii)
absorbs condition (v). Condition (ix) means that the higher empirical mo-
ments are bounded.
Lemma 3. Assume (ii), and (vii) to (ix). Then
√
m · T 0
m
d−→ N(0, ΣT ),
where ΣT := Sb̃ ·M [E(eλT a−aT f)2]+XT [In, fT ⊗In]·M(U)· [In, fT ⊗In]T X,
M(U) := lim
m→∞
cov(Z(a)), Z(ai) :=
[
(a0
i − s(ai))e
λT ai
vec(Hi) − vec(ai
0ai
T )
]
, i = 1, m,
the symbol ⊗ is Kronecker product, and vector f comes from Lemma 2.
Under the conditions of Lemma 3 and condition (x), a consistent estima-
tor Σ̂ of ΣT is constructed,
Σ̂T := Ŝb̃ · (eλT a − aT f)2+
(11) +X̂T [In, fT ⊗ In] · ĉov
[
(a0 − s(a))eλT a
vec(H − a0aT )
]
· [In, fT ⊗ In]T X̂,
where f̂ , ĉov are approximations described below.
A. Since H̄
P−→ V and s(a)eλT a P−→ M(a0eλT a0
)μ0 as m → ∞, we get
the estimator f̂ = H̄−1s(a)eλT a.
B.
M
(
cov
[
(a0 − s(a))eλT a
vec(H − a0aT )
])
= M
(
Σ11 Σ12
ΣT
12 Σ22
)
.
We want to construct Σ̂ij for M(Σij), i, j = 1, 2, based on observations
ai, i = 1, m. We need the following auxiliary statement.
Lemma 4. Let k ≥ 0, and p(a0) be a polynomial of degree k, and {a0
i , i ≥
1, } be a sequence of nonrandom vectors in R
n, satisfying the condition
(xi) (1 + ‖a0‖2k)e2λT a0 = o(m), as m → ∞.
Let ai = a0
i + ãi, i ≥ 1, and vectors ãi satisfy the conditions a) and b),
and the following condition
(xii) E[(1 + ‖ã‖2k)e2λT ã] < ∞.
Assume also that the limit M(p(a0)eλT a0
) = lim
m→∞
1
m
m∑
i=1
p(a0
i )e
λT a0
i exists
and is finite.
Then there exists a polynomial p1(a) of degree k, a ∈ R
n, such that
(12)
1
m
m∑
i=1
p1(ai)e
λT ai
P−→ M(p(a0)eλT a0
), as m → ∞.
Consider the matrix Σ11 = a0a0T
e2λT a0
Ee2λT ã − Es(a)e2λT aa0T−
−a0Es(a)Te2λT a + Es(a)s(a)T e2λT a =: U1 − U2 − UT
2 + U3. Next,
A GOOGNESS OF-FIT-TEST 67
E(aaT e2λT a) = a0a0T
e2λT a0
m1 + a0e2λT a0
m2 + a0T
e2λT a0
mT
2 + e2λT a0
m3,
E(ae2λT a) = a0e2λT a0
m1 + e2λT a0
m2,
E(aTe2λT a) = a0T
e2λT a0
m1 + e2λT a0
mT
2 , E(e2λT a) = e2λT a0
m1, where
m1 = Ee2λT ã, m2 = Eãe2λT ã, m3 = EããT e2λT ã. Then by Lemma 4, the
estimator of U1 equals
Û1 = aaT e2λT a − ae2λT a · mT
2
m1
− aT e2λT a · m2
m1
− e2λT ã(
m3
m1
− 2mT
2 m2
m2
1
).
Again from the previous expression and the following identity
E(s(a)aT e2λT a) = E(s(a)e2λT a)(a0)T +a0e2λT a0
m2 +e2λT a0
(m3−μ1/μ0 ·mT
2 ),
we get an approximation:
Û2 = s(a)aT e2λT a − ae2λT a · mT
2
m1
− e2λT ã(
m3 − μ1/μ0 · mT
2
m1
− mT
2 m2
m2
1
).
The next approximation is Û3 = s(a)s(a)T e2λT a. Finally,
Σ̂11 =: Û1 − Û2 − ÛT
2 + Û3.
In a similar way one can construct other approximations Σ̂ij and obtain
the approximation (11).
Then the test statistic defined as follows: T 2
m = m · ‖Σ̂−1/2
T T 0
m‖2. Since
Σ̂T is the consistent estimator of ΣT , we obtain by Lemma 3 the following
theorem.
Theorem 1. Suppose that the conditions of lemma 3 and condition (x) are
satisfied. Assume as well that at least one of the following two conditions is
satisfied:
(xiii) M [E(eλT a − aT f)2] > 0, and Sb̃ is positive definite;
(xvi) n ≥ p, rankX = p, and the matrix
M(U) := M
(
cov
[
(a0 − s(a))eλT a
vec(H − a0aT )
])
is nonsingular.
Then T 2
m
d−→ χ2
p, under hypothesis H0.
Let α > 0 and χ2
pα be corresponding quantile of the χ2
p distribution,
i.e., P{χ2
p > χ2
pα} = α. Based on Theorem 1, we construct the following
goodness-of -fit test with asymptotic confidence probability 1 − α. If T 2
m ≤
χ2
pα then we accept the hypothesis H0; if T 2
m > χ2
pα then we reject the null
hypothesis.
4. The power properties of the test
Consider the following sequences of models:
(13) H1,m : bi = XT a0
i +
g(a0
i )√
m
+ b̃i, ai = a0
i + ãi, i = 1, m,
68 ALEXANDER KUKUSH AND MARIA POLEKHA
where g : R
n → R
p is a nonlinear vector function which satisfies the condi-
tions:
(xv) ∃ M(g(a0)eλT a0
) and ∃ M(g(a0)a0T
);
(xvi) ‖g(a0)‖2 · (1 + ‖a0‖2 + e2λT a0) = o(m), as m → ∞.
Then under H1,m we have: 1√
m
m∑
i=1
(bi − X̂T s(ai))e
λT ai
d−→ N(C, ΣT ),
where a vector C is found bellow.
Now, we define a noncentral chi-squared distribution χ2
p(τ) with p degrees
of freedom, and noncentrality parameter τ .
Definition.
For p ≥ 1 and τ ≥ 0, let χ2
p(τ)
d
= ‖N(τe, Ip)‖2, where e ∈ R
p, ‖e‖ = 1, or
equivalently χ2
p(τ)
d
= (γ1 +τ)2 +
p∑
i=2
γ2
i , where {γi} are independent standard
normal variables.
Theorem 2. Suppose that all the conditions of Theorem 1 and conditions
(xv), (xvi) are satisfied. Then, under H1,m, T 2
m
d−→ χ2
p(‖Σ−1/2
T C‖),
(14) where C := μ0 · M(g(a0)eλT a0
) − M(g(a0)a0T
)V −1M(a0eλT a0
).
Here χ2
p(‖Σ−1/2
T C‖) is noncentral chi-squared random variable with p degrees
of freedom and noncentrality parameter ‖Σ−1/2
T C‖.
From Theorem 2 we can find the asymptotic power of the test under local
alternative (13). It is easy to see that the asymptotic power of the test is
increasing function of ‖Σ−1/2
T C‖. In other words, the larger ‖Σ−1/2
T C‖, the
more powerful test we will have.
Since in present paper the vector λ is arbitrary chosen and the function
g is unknown, it is reasonable to consider the next two problems.
1) We assume that the weight function w(a) = eλT a is fixed. We discuss
for which g the power is the largest. For simplicity we suppose that {a0
i , i ≥
1} are IID random vectors, independent of {ãi, and b̃i, i ≥ 1}, and a0 d
= a0
i .
Then ‖Σ−1/2
T C‖ = μ0 · ‖Σ−1/2
T [E(g(a0)eλT a0
)−
−E(g(a0)a0T
)E(a0a0T
)−1E(a0eλT a0
)]‖ = μ0‖E(Σ
−1/2
T g(a0)hλ(a
0))‖.
Here hλ is defined from the expansion: eλT a0
= zT a0 + hλ(a
0), z ∈ R
n and
Ehλ(a
0)(vTa0) = 0, for all v ∈ R
n. The ratio ‖Σ−1/2
T C‖2/‖Σ−1/2
T g(a0)‖2
L2
is maximal, if g(a0) = hλ(a
0)w, for certain nonrandom w ∈ R
p, w �= 0.
We have hλ(a
0) = eλT a0 − E(a0a0T
)−1/2E(eλT a0
a0T
)a0E(a0a0T
)−1/2, and its
consistent estimator is
ĥλ(a
0) = eλT a0 − H̄−1/2 · 1
μ0
(eλT as(a)T − eλT a(EãT eλT ã −EãeλT ã))a0H̄−1/2.
The function ĥλ(a
0)w, w �= 0, is asymptotically optimal choice of the func-
tion g for a local alternative (13), when the weight function w is fixed.
A GOOGNESS OF-FIT-TEST 69
2) Now, we consider the second problem. Let the function g be fixed and
we want to choose optimally the weight function w(a0) = eλT a0
. We need
to maximize the function ‖Σ−1/2
T C(λ)‖2 for λi ∈ R
n\{0̄}, i = 1, m. Here
the vector function C = C(λ) is given in (14) provided all corresponding
moments of random vectors {ãi, a0
i , i ≥ 1} are exist. This is a nonlinear
problem, and it can be solved numerically. Of course, one has to incorporate
the approximations for ‖Σ−1/2
T C(λ)‖2 constructed by data.
5. Conclusion
We constructed a goodness-of-fit test for a multivariate errors-in-variables
model if the covariance structure of errors b̃ is unknown, and the exponen-
tial moments and the covariance structure of errors ã are known. Using
an exponential weight function, we obtained an asymptotically chi-squared
statistic under null hypothesis. A local alternative hypothesis is introduced,
under which the test has a noncentral chi-squared asymptotic distribution.
We discussed for what local alternatives the power of the test is the largest.
Appendix
Proof of Lemma 1. First we prove (4).
With probability tending to 1, as m → ∞, we have H̄X̂ = abT . Hence
(15) (a0a0T )−1(a0a0T + ãa0T + a0ãT + ããT − EããT )X̂ =
= (a0a0T )−1(a0a0T X + ãa0T X + a0b̃T + ãb̃T ),
or V −1
m H̄X̂ = V −1
m abT , where Vm := a0a0T is nonsingular for m > m0, and
Vm → V, as m → ∞. We show that
(16) (a0a0T )−1(a0a0T + ãa0T + a0ãT + ããT −EããT )
P−→ In,
(17) and (a0a0T )−1(ãa0T X + a0b̃T + ãb̃T )
P−→ 0.
We deal with each summand in (16) separately.
We have ‖(a0a0T )−1(ãa0T )‖ ≤ ‖V −1
m ‖ · ‖ãa0T‖.
Since Vm is nonsingular matrix, ‖V −1
m ‖ ≤ const · λ−1
min(Vm). By Cauchy-
Schwartz inequality we obtain
E‖ãa0T‖2 = E‖ 1
m
m∑
i=1
ãia
0
i
T‖2 =
1
m2
n∑
j,k=1
E(
m∑
i=1
ãija
0
ik)
2 ≤
≤ 1
m2
n∑
j,k=1
(
m∑
i=1
Eã2
ij ·
m∑
i=1
(a0
ik)
2) ≤ 1
m
· ‖Sã‖ · const,
70 ALEXANDER KUKUSH AND MARIA POLEKHA
therefore ‖ãa0T‖ = Op(1)√
m
. By (ii) we have
(18) ‖(a0a0T )−1 · a0ãT‖ =
Op(1)√
m · λmin(Vm)
.
(19) Similarly we have ‖(a0a0T )−1 · ãa0T‖ =
Op(1)√
m · λmin(Vm)
.
Next, from (i) we get E‖ããT − EããT‖2 = 1
m2
n∑
j,k=1
E(
m∑
i=1
ãij ãik −Eãijãik)
2 =
1
m2
n∑
j,k=1
m∑
i=1
E(ãijãik − Eãijãik)
2 = O(1)
m
. Therefore
(20) ‖(a0a0T )−1 · (ããT −EããT )‖ =
Op(1)√
m · λmin(Vm)
.
By the assumption b) and (ii) we get
E‖a0b̃T‖2 =
1
m2
n∑
j,k=1
E(
m∑
i=1
aij
0b̃ik)
2 =
O(1)
m
.
(21) Thus ‖(a0a0T )−1 · a0b̃T‖ =
Op(1)√
m · λmin(Vm)
.
Similarly we obtain for the last residual:
(22) ‖(a0a0T )−1 · ãb̃T‖ =
Op(1)√
m · λmin(Vm)
Therefore, equalities (18) - (20) yield the convergence (16), and the relations
(19), (21) and (22) yield the convergence (17). Then (15) implies the desired
convergence X̂
P−→ X as m → ∞. Now we prove the convergence (5).
We have
bbT = (XT a0 + b̃)(XT a0 + b̃)T = XT a0a0T X + XT a0b̃T + b̃a0T X + b̃b̃T ,
baT = (XT a0 + b̃)aT = XTa0a0T + XT a0ãT + b̃ãT + b̃a0T ,
and X̂ = X + op(1), then Ŝb = XT a0b̃T + b̃b̃T − XT a0ãT X − b̃ãT X + op(1).
From the proof of the first part of the theorem
XTa0b̃T −XT a0ãT X − b̃ãT X = Op(1)/
√
m. Moreover b̃b̃T P−→ Eb̃b̃T = Sb.
As a result we obtain Ŝb
P−→ Sb, as m → ∞. �
A GOOGNESS OF-FIT-TEST 71
Proof of Lemma 2. We substitute the estimator (3) into statistic (9):
(23) T 0
m = (b − X̂Ts(a))eλT a = (b − (H̄−1abT )T s(a))eλT a =
= (XT a0 + b̃ − XTa0aT · H̄−1s(a) − b̃aT · H̄−1s(a)) · eλT a =
= b̃(eλT a − aT · H̄−1s(a)eλT a)+
+ XT (a0eλT a − a0aT · H̄−1s(a)eλT a) =: F + XT G.
First, we investigate the vector
√
mF.
Since H̄ = aaT −EããT P−→ V, as m → ∞, we denote Λ = H̄ − V,
Λ ≈ 0. The approximate equality ” ≈ ” means equality up to summands,
converging to 0 in probability. Thus H̄−1 = (In+V −1ΛV −1)−1V −1 = V −1−
V −1ΛV −1 + rm, where ‖rm‖ = ‖Λ‖2Op(1). We show that
√
m · ‖Λ‖2 ≈ 0.
From (i), and (ii), and (vi) we have E‖H̄ − V ‖2 =
(24) = E‖ãa0T + a0ãT + ããT − EããT + a0a0T − V ‖2 ≤ O(1)
m
+
o(1)√
m
.
Therefore
√
m · ‖Λ‖2 ≈ 0 and ‖rm‖ = op(1)/
√
m. Moreover
√
m · b̃aT =
√
m(b̃ãT + b̃a0T ) =
√
m · Op(1)√
m
= Op(1),
s(a)eλT a = (a0 + ã − μ1
μ0
)eλT ãeλT a0 P−→ M(a0eλT a0 · μ0),
therefore s(a)eλT a = Op(1). Then we get√
m · b̃aT H̄−1s(a)eλT a ≈ √
m · b̃aT V −1M(a0eλT a0
). This implies
(25)
√
mF ≈ √
m · b̃(eλT a − aT V −1M(a0eλT a0μ0)) =
√
m · b̃(eλT a − aT f),
where f is the vector defined in Lemma 2.
Next, consider
√
mG, where G comes from (23):
√
mG ≈ √
m · (a0eλT a − a0aT V −1s(a)eλT a + a0aT V −1ΛV −1s(a)eλT a).
Since 4
√
m·‖Λ‖ ≈ 0 we have m1/4(a0aT−V ) ≈ 0 and s(a)eλT a = Op(1). Then√
m ·a0aT V −1ΛV −1s(a)eλT a ≈ √
m ·ΛV −1s(a)eλT a =
√
m ·(H̄V −1s(a)eλT a−
s(a)eλT a),
√
mG ≈ √
m((a0 − s(a))eλT a + (H − a0aT )V −1s(a)eλT a). Since√
m · H − a0aT = Op(1) and s(a)eλT a P−→ M(a0eλT a0
), we also have
(26)
√
mG ≈ √
m · (a − s(a))eλT + (H − a0aT )f.
Using (25) and (26), we obtain (10). �
Proof of Lemma 3. From the Lemma 2 we have
(27)
√
m · T 0
m ≈ 1√
m
m∑
i=1
zi,
72 ALEXANDER KUKUSH AND MARIA POLEKHA
where zi =: b̃i(e
λT ai − ai
T f) + XT ((a0
i − s(ai))e
λT ai + (Hi − ai
0ai
T )f) are
independent random vectors and Ezi = 0. Represent the vectors zi as
zi = b̃i(e
λT ai − ai
T f) + XT [In, fT ⊗ In]
[
(a0
i − s(ai))e
λT ai
vec(Hi) − vec(ai
0ai
T )
]
.
Then Eziz
T
i = Sb̃ · E(eλT ai − ai
0f)2+
+XT [In, fT ⊗ In] · cov
[
(a0
i − s(ai))e
λT ai
vec(Hi) − vec(ai
0ai
T )
]
[In, fT ⊗ In]T X,
therefore lim
m→∞
1
m
m∑
i=1
Eziz
T
i = ΣT . The limit exists due to conditions (i), (ii),
(vi), and (viii). Conditions (vii), (ix) guarantee the following boundedness:
∃δ > 0 :
1
m
m∑
i=1
E‖zi‖2+δ ≤ const.
Thus all the conditions of the CLT in Lyapunov form are satisfied, then
1√
m
m∑
i=1
zi
d−→ N(0, ΣT ).
From this and from (27) using Slutsky Lemma we get Lemma 3. �
Proof of Lemma 4. First we prove by induction that there exists a polyno-
mial p1(a), a ∈ R
n, of degree k such that
(28) E(p1(a)eλT a) = p(a0)eλT a0
.
1. Let p(a0) be a polynomial of degree 0. Since EeλT a = eλT a0
EeλT ã =
eλT a0
μ0, there exists a polynomial of degree 0, p1(a) = p(a0)μ−1
0 .
2. Suppose that for arbitrary polynomial of degree less than k, p(a0),
there exists p1(a) such that deg p1(a) < k and the equality (28) is satisfied.
3. Prove the existence of similar polynomial of degree k.
E(p(a)eλT a) = p(a0)eλT a0
μ0 + eλT a0
Ep∗(a0, ã)eλT ã, where p∗ is some poly-
nomial of two variables. The expectation Ep∗(a0, ã)eλT ã can be represented
as p2(a
0)eλT a0
, where deg p2 < deg p = k. Therefore by part 2 of the proof,
for p2(a
0) there exists a polynomial p∗1(a) of degree less than k, such that (28)
is satisfied. Moreover, E(p(a)eλT a/μ0 − p∗1(a)eλT a) = p(a0)eλT a0
. Therefore,
∃ p1(a) := p(a)/μ0 − p∗1(a), degp1 = k.
Now, prove the convergence (12) for the constructed polynomial p1(a). In
fact, we have to prove the next equality
(29)
1
m
m∑
i=1
p1(ai)e
λT ai − 1
m
m∑
i=1
p(a0
i )e
λT a0
i = op(1).
A GOOGNESS OF-FIT-TEST 73
Consider the difference 1
m
m∑
i=1
(p1(ai)e
λT ai − p(a0
i )e
λT a0
i ) =: 1
m
m∑
i=1
zi, Ezi = 0,
D( 1
m
m∑
i=1
zi) = E( 1
m
m∑
i=1
zi)
2 = 1
m2
m∑
i=1
Ez2
i ,
Ez2
i = E(p1(ai)e
λT ai)2 − (p(a0
i )e
λT a0
i )2 ≤ E(p2
1(ai)e
2λT ai), i = 1, m.
We have Ez2
i ≤ const · E[(1 + ‖ai‖2k)e2λT ai ] ≤ const · E[(1 + ‖a0
i ‖2k +
‖ãi‖2k)e2λT a0
i e2λT ãi ] ≤ (1 + ‖a0
i ‖2k)e2λT a0
i · const, i = 1, m.
Then by condition (xvi) we get as m → ∞,
1
m2
m∑
i=1
Ez2
i ≤ const
m2
·
m∑
i=1
(1 + ‖a0
i ‖2k)e2λT a0
i =
1
m
(1 + ‖a0‖2k)e2λT a0 → 0,
Thus we obtain (29), and as a result we get the convergence (12). �
Proof of Theorem 1. From the conditions of Theorem 1 we get that ΣT is
positive matrix. Then m · ‖Σ−1/2
T T 0
m‖2 d−→ χ2
p. Since Σ̂T is the consistent
estimator of ΣT , we have T 2
m = m·‖Σ̂−1/2
T T 0
m‖2 d−→ χ2
p under null hypothesis.
�
Proof of Theorem 2. Assume the hypothesis H1,m. Then
(30) X̂ = H̄−1abT + H̄−1 1√
m
· ag(a0)T .
Since H̄−1abT P−→ X under H0, we have H̄−1 = Op(1) and 1
m
a0g(a0)T → 0,
E‖ 1√
m
· ãg(a0)T‖2 → 0, as m → ∞. Therefore from (30) we obtain X̂
P−→ X
under H1,m. However for the statistic T 0
m we have
(31)
√
m · T 0
m|H1,m =
√
m · T 0
m|H0 + (g(a0)eλT a − g(a0)aT H̄−1seλT a),
where T 0
m|H1,m , and T 0
m|H0 are the values of T 0
m under the corresponding
hypotheses H1,m and H0.
Now, consider the last summand in (31). By conditions (xiii) and (xiv)
we have g(a0)eλT a ≈ g(a0)EeλT a = μ0g(a0)eλT a0 → μ0M(g(a0)eλT a0
)
and seλT a P−→ μ0M(a0eλT a0
), H̄−1 P−→ V −1, g(a0)aT ≈ g(a0)EaT =
g(a0)a0T → M(g(a0)a0T
), as m → ∞. Relation (31) yields
(32)
√
m · T 0
m|H1,m
d−→ N(C, ΣT ),
where C is the vector defined in (14).
The conditions of Theorem 1 are satisfied, therefore ΣT > 0. And from
(32) we have the following convergence
(33) m · ‖Σ−1/2
T · T 0
m|H1,m‖2 d−→ χ2
p(‖Σ−1/2
T C‖).
74 ALEXANDER KUKUSH AND MARIA POLEKHA
Further, due to (xii) and (xiii) we have Ŝb̃
P−→ Sb̃, under H1,m, and Σ̂T
P−→
ΣT under H1,m. (we used only the observations ai, i = 1, m, in the construc-
tion of Σ̂T , and they do not change under local alternative H1,m). Thus by
relation (33). We have T 2
m|H1,m
d−→ χ2
p(‖Σ−1/2
T C‖). �
References
1. Kukush, A., and Van Huffel, S., Consistency of element-wise weighted total least
squares estimator in multivariate errors-in-variables model AX = B, Metrika,
59, (2004), no.1, 75-97.
2. Kukush, A., Markovsky, I., and Van Huffel, S., Consistency of the structured
total least squares estimator in a multivariate errors-in-variables model, Journal
of Statistical Planning and Inference, 133, (2005), no.2, 315-358.
3. Kukush A., Markovsky I., and Van Huffel, S., 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. Zhu, L., Cui, H., and K.W.Ng., Testing lack-of-fit for linear errors-in-variables
model. Acta Appl. Math. (to appear).
5. Kukush, A.G., Cheng, C.-L., A goodness-of-fit test for a polynomial errors-in-
variables model, Ukrainian Mathematical Journal, 56, (2004), no.4, 527-543.
6. Cheng, C.-L., and Schneeweiss, H., Polynomial regression with errors in the
variables, J. R. Statist. Soc. B, 60, (1998), 189-199.
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: alexander kukush@univ.kiev.ua
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: poleha@bigmir.net
|