VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA67 10.15559/16-VMSTA67 Research Article Goodness-of-fit test in a multivariate errors-in-variables model AX=B KukushAlexanderalexander_kukush@univ.kiev.ua∗ TsaregorodtsevYaroslav777Tsar777@mail.ru Taras Shevchenko National University of Kyiv, Kyiv, Ukraine Corresponding author. 2016 2012201634287302 10112016 3122016 4122016 © 2016 The Author(s). Published by VTeX2016 Open access article under the CC BY license.

We consider a multivariable functional errors-in-variables model AX≈B , 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 is constructed based on the total least squares estimator. The proposed test is asymptotically chi-squared under null hypothesis. The power of the test under local alternatives is discussed.

 Goodness-of-fit test local alternatives multivariate errors-in-variables model power of test total least squares estimator 62J05 62H15 65F20 Introduction

We study an overdetermined system of linear equations AX≈B , which often occurs in the problems of dynamical system identification [10]. If matrices A and B are observed with additive uncorrelated errors of equal size, then the total least squares (TLS) method is used to solve the system [10].

In papers [3, 7, 9], under various conditions, the consistency of the TLS estimator Xˆ is proven as the number m of rows in the matrix A is increasing, assuming that the true value A0 of the input matrix is nonrandom. The asymptotic normality of the estimator is studied in [3] and [6].

The model AX≈B with random measurement errors corresponds to the vector linear errors-in-variables model (EIVM). In [2], a goodness-of-fit test is constructed for a polynomial EIVM with nonrandom latent variable (i.e., in the functional case); the test can be also used in the structural case, where the latent variable is random with unknown probability distribution. A more powerful test in the polynomial EIVM is elaborated in [4].

In the paper [5], a goodness-of-fit test is constructed for the functional model AX≈B , assuming that the error matrices A˜ and B˜ are independent and the covariance structure of A˜ is known. In the present paper, we construct a goodness-of-fit test in a more common situation, where the total covariance structure of the matrices A˜ and B˜ is known up to a scalar factor. A test statistic is based on the TLS estimator Xˆ . Under the null hypothesis, the asymptotic behavior of the test statistic is studied based on results of [6] and, under local alternatives, based on [9].

The present paper is organized as follows. In Section 2, we describe the observation model, introduce the TLS estimator, and formulate known results on the strong consistency and asymptotic normality of the estimator. In the next section, we construct the goodness-of-fit test and show that the proposed test statistic has an asymptotic chi-squared distribution with the corresponding number of degrees of freedom. The power of the test with respect to the local alternatives is studied in Section 4, and Section 5 concludes. The proofs are given in Appendix.

We use the following notation: ‖C‖=∑i,jcij2 is the Frobenius norm of a matrix C=(cij) , and Ip is the unit matrix of size p. The symbol E denotes the expectation and acts as an operator on the total product of quantities, and cov means the covariance matrix of a random vector. The upper index ⊤ denotes transposition. In the paper, all the vectors are column ones. The bar means averaging over i=1,…,m , for example, a¯:=m−1∑i=1mai , ab⊤‾:=m−1∑i=1maibi⊤ . Convergence with probability one, in probability, and in distribution are denoted as →P1 , →P , and →d , respectively. A sequence of random matrices that converges to zero in probability is denoted as op(1) , and a sequence of stochastically bounded random matrices is denoted as Op(1) . The notation ε=dε1 means that random variables ε and ε1 have the same probability distribution. Positive constants that do not depend on the sample size m are denoted as const , so that equalities like 2·const=const are possible.

 Observation model and total least squares estimator The TLS problem

Consider the observation model A0X0=B0,A=A0+A˜,B=B0+B˜, where A0∈Rm×n , X0∈Rn×d , and B0∈Rm×d . The matrices A and B contain the data, A0 and B0 are unknown nonrandom matrices, and A˜ , B˜ are the matrices of random errors.

We can rewrite model (2.1) in an implicit way. Introduce three matrices of size m×(n+d) : C0:=[A0B0],C˜:=[A˜B˜],C:=[AB]. Then C=C0+C˜,C0·X0−Id=0.

Let A⊤=[a1…am] , B⊤=[b1…bm] , and we use similar notation for the rows of the matrices C, A0 , B0 , A˜ , B˜ , and C˜ . Rewrite model (2.1) as a multivariate linear one: X0⊤ai0=bi0, bi=bi0+b˜i,ai=ai0+a˜i;i=1,…,m.

Throughout the paper, the following assumption holds about the errors ci˜=[ai˜⊤b˜i⊤]⊤ :

The vectors c˜i , i≥1 , are i.i.d. with zero mean, and, moreover, cov(c˜1)=σ2In+d, with unknown σ>0 0$]]>.

Thus, the total error covariance structure is assumed to be known up to a scalar factor σ2 , and the errors are uncorrelated with equal variances.

For model (2.1), the TLS problem lies in searching such disturbances ΔAˆ and ΔBˆ that minimize the sum of squared corrections min(X∈Rn×d,ΔA,ΔB)(‖ΔA‖2+‖ΔB‖2), provided that (A−ΔA)X=B−ΔB.

The TLS estimator and its consistency

It can happen that for certain random realization, the optimization problem (2.6)–(2.7) has no solution. In the latter case, we set Xˆ=∞ .

The TLS estimator Xˆ of the matrix parameter X0 in the model (2.1) is a Borel-measurable function of the observed matrices A and B such that its values lie in Rn×d∪{∞} and it provides a solution to problem (2.6)–(2.7) in case there exists a solution, and Xˆ=∞ otherwise.

We need the following conditions to provide the consistency of the estimator:

E‖c˜1‖4<∞ .

1mA0⊤A0→VA as m→∞ , where VA is a nonsingular matrix.

The next result on the strong consistency of the estimator follows, for example, from Theorem 4.3 in [9].

Assume conditions (i)–(iii). Then, with probability one, for all m≥m0(ω) , the TLS estimator Xˆ is finite, and, moreover, Xˆ→P1X0 as m→∞ .

Define the loss function Q(X) as follows: q(a,b;X):=(a⊤X−b⊤)(Id+X⊤X)−1(X⊤a−b), Q(X):= ∑i=1mq(ai,bi;X),X∈Rn×d. It is known that the TLS estimator minimizes the loss function (2.9); see formula (24) in [7].

Introduce the following unbiased estimating function related to the elementary loss function (2.8): s(a,b;X)=a(a⊤X−b⊤)−X(Id+X⊤X)−1(X⊤a−b)(a⊤X−b⊤).

Assume conditions (i)–(iii). Then, with probability one, for all m≥m0(ω) , the TLS estimator Xˆ is a solution to the equation ∑i=1ms(ai,bi;X)=0,X∈Rn×d.

In view of Theorem 2, the statement of Lemma 3 follows from Corollary 4(a) in [6].

 Asymptotic normality of the estimator

We need further restrictions on the model. Recall that the augmented errors  c˜i were introduced in Section 2.2, and the vectors ai0 , b˜i , and so on are those from model (2.3)–(2.4).

E‖c˜1‖4+2δ<∞ for some δ>0 0$]]>;

For δ from condition (iv), 1m1+δ/2∑i=1m‖ai0‖2+δ→0 as m→∞ .

Denote by c˜1(p) the pth coordinate of the vector c˜1 .

For all p,q,r=1,…,n+d , we have Ec˜1(p)c˜1(q)c˜1(r)=0 .

Under assumptions (i) and (iv), condition (vi) holds, for example, in two cases: (a) when the random vector c˜1 is symmetrically distributed, or (b) when the components of the vector c˜1 are independent and, moreover, for each p=1,…,n+d , the asymmetry coefficient of the random variable c˜1(p) equals 0.

Introduce the following random element in the space of collections of five matrices: Wi=(ai0a˜i⊤,ai0b˜i⊤,a˜ia˜i⊤−σ2In,a˜ib˜i⊤,b˜ib˜i⊤−σ2Id).

The next statement on the asymptotic normality of the estimator follows from the proof of Theorem 8(b) in [6], where, instead of condition (vi), there was a stronger assumption that c˜1 is symmetrically distributed, but the proof of Theorem 8(b) in [6] still works under the weaker condition (vi).

Assume conditions (i) and (iii)–(vi). Then:

1m ∑i=1mWi⟶dΓ=(Γ1,…,Γ5)as m→∞, where Γ is a Gaussian centered random element with matrix components,

m(Xˆ−X0)→dVA−1Γ(X0)as m→∞, Γ(X):=Γ1X−Γ2+Γ3X−Γ4−X(Id+X⊤X)−1(X⊤Γ3X−X⊤Γ4−Γ4⊤X+Γ5), where VA is from condition (iii), and Γi is from condition (2.12).

Under the assumptions of Theorem 4, the components of random element (2.11) are uncorrelated, and therefore, the components of the limit element Γ are uncorrelated as well.

Let f∈Rn×1 . Under the conditions of Theorem 4, the convergence (2.13) implies that m(Xˆ−X0)⊤f→dN(0,S(X0,f)), S(X0,f)=EΓ⊤(X0)VA−1ff⊤VA−1Γ(X0).

Let a consistent estimator fˆ=fˆm of the vector f be given. We want to construct a consistent estimator of matrix (2.16). The matrix S(X0,f) is expressed, for example, via the fourth moments of errors c˜i , and those moments cannot be consistently estimated without additional assumptions on the error probability distribution. Therefore, an explicit expression for the latter matrix does not help to construct the desirable estimator. Nevertheless, we can construct something like the sandwich estimator [1, pp. 368–369].

The next statement on the consistency of the nuisance parameter estimators follows from the proof of Lemma 10 in [6]. Recall that the bar means averaging over the observations; see Section 1.

Assume the conditions of Theorem 4. Define the estimators: σˆ2=1dtr[(bb⊤‾−2Xˆ⊤ab⊤‾+Xˆ⊤aa⊤‾Xˆ)(Id+Xˆ⊤Xˆ)−1], VˆA=aa⊤‾−σˆ2In. Then σˆ2→Pσ2,VˆA→PVA.

The next asymptotic expansion of the TLS estimator is presented in [6], formulas (4.10) and (4.11).

Under the conditions of Theorem 4, we have: m(Xˆ−X0)=−VA−1·1m ∑i=1ms(ai,bi;X0)+op(1).

In view of Lemma 7, introduce the sandwich estimator Sˆ(fˆ) of the matrix (2.16): Sˆ(fˆ)=1m ∑i=1ms⊤(ai,bi;Xˆ)VˆA−1fˆfˆ⊤VˆA−1s(ai,bi;Xˆ), where the estimator VˆA is given in (2.18).

Let f∈Rn×1 , and let fˆ be a consistent estimator of this vector. Under the conditions of Theorem 4, the statistic Sˆ(fˆ) is a consistent estimator of the matrix S(X0,f) , that is, Sˆ(fˆ)→PS(X0,f) .

Appendix contains the proof of this theorem and of all further statements.

 Construction of goodness-of-fit test

For the observation model (2.4), we test the following hypotheses concerning the response b and the latent variable a0 :

H0 There exists such a matrix X∈Rn×d that E(b−X⊤a0)=0,and

H1 For each matrix X∈Rn×d , E(b−X⊤a0)is not identically zero.

In fact, the null hypothesis means that the observation model (1.3)–(1.4) holds. Based on observations ai , bi , i=1,…,m , we want to construct a test statistic to check this hypothesis. Let Tm0:=1m ∑i=1m(bi−Xˆ⊤ai)=b−Xˆ⊤a‾.

Under the conditions of Theorem 4, mTm0=1m ∑i=1m(b˜i−X0⊤a˜i)−m(Xˆ−X0)⊤a0‾+op(1).

We need the following stabilization condition on the latent variable:

1m ∑i=1mai0→μa as m→∞ with μa∈Rn×1 .

Assume conditions (i) and (iii)–(vii). Then mTm0→dN(0,ΣT),ΣT=σ2(1−2μa⊤VA−1μa)(Id+X0⊤X0)+S(X0,μa).

Assume the conditions of Lemma 10. Then:

A strong consistent estimator of the vector μa from condition (vii) is given by the statistic μˆa:=a¯=1m ∑i=1mai.

A consistent estimator of matrix (3.5) is given by the matrix statistic ΣˆT:=σˆ2(1−2μˆa⊤VˆA−1μˆa)(Id+Xˆ⊤Xˆ)+Sˆ(μˆa), where σˆ2 and VˆA are presented in (2.17) and (2.18), respectively, and Sˆ(μˆa) is matrix (2.21) with fˆ=μˆa .

To ensure the nonsingularity of the matrix ΣT , we impose a final restriction on the observation model:

There exists a finite matrix limit Sa:=limm→∞1m ∑i=1m(ai0−μa)(ai0−μa)⊤,

and, moreover, the matrix Sa is nonsingular.

Assume conditions (vii) and (viii). Then 1mA0⊤A0=1m ∑i=1mai0ai0⊤→VA=Sa+μaμa⊤asm→∞, and VA is nonsingular as a sum of positive definite and positive semidefinite matrices. Thus, condition (iii) is a consequence of assumptions (vii) and (viii).

Assume conditions (i) and (iv)–(viii). Then:

Matrix (3.5) is positive definite.

With probability tending to one as m→∞ , the symmetric matrix ΣˆT is positive definite as well.

For m≥1 and ω from the underlying probability space Ω such that ΣˆT is positive definite, we define the test statistic Tm2=m·‖ΣˆT−1/2Tm0‖2.

Lemmas 10 and 11(b) imply the following convergence of the test statistic.

Assume conditions (i) and (iv)–(viii). Then under hypothesis H0 , Tm2→dχd2 as m→∞ .

Given a confidence level α, 0<α<1/2 , let χdα2 be the upper α-quantile of the χd2 probability law, that is, P{χd2>χdα2}=α {\chi _{d\alpha }^{2}}\}=\alpha $]]>. Based on Theorem 14, we construct the following goodness-of-fit test with the asymptotic confidence probability 1−α : IfTm2≤χdα2,then we accept the null hypothesis,and ifTm2>χdα2,then we reject the null hypothesis. {\chi _{d\alpha }^{2}},\hspace{3.0pt}& \text{then we reject the null hypothesis}.\end{array}\]]]>

Power of the test

Consider a sequence of models H1,m:bi=X⊤ai0+g(ai0)m+b˜i,ai=ai0+a˜i,i=1,…,m. Here g:Rn→Rd is a given nonlinear perturbation of the linear regression function.

For arbitrary function f(a0) , denote the limit of averages M(f(a0))=limm→∞f(a0)‾, provided that the limit exists and is finite.

In order to study the behavior of the test statistic under local alternatives H1,m , we impose two restrictions on the perturbation function g:

There exist M(g(a0)) and M(g(a0)a0⊤) .

‖g(a0)‖2‾=o(m) as m→∞ .

Under local alternatives H1,m , we ensure the weak consistency and asymptotic normality of the TLS estimator Xˆ .

Assume conditions (i) and (iv)–(x). Under local alternatives H1,m , we have:

Xˆ→PX0 , σˆ2→Pσ2 .

m(Xˆ−X0)→dVA−1Γ(X0)+VA−1M(a0g⊤(a0)) as m→∞ ,

 where Γ(X) is defined in (2.11), (2.12), and (2.14).

Assume the conditions of Lemma 15. Then under local alternatives H1,m , we have:

mTm0→dN(CT,ΣT) ,

where ΣT is given by (3.5), and CT:=M(g(a0))−M(g(a0)a0⊤)VA−1μa.

The estimator ΣˆT given in (3.6) tends in probability to the asymptotic covariance matrix ΣT .

Now, we define the noncentral chi-squared distribution χd2(τ) with d degrees of freedom and the noncentrality parameter τ.

For d≥1 and τ≥0 , let χd2(τ)=d‖N(τe,Id)‖2 , where e∈Rd , ‖e‖=1 , or, equivalently, χd2(τ)=d(γ1+τ)2+∑i=2dγi2 , where {γi} are i.i.d. standard normal random variables.

Lemma 16 implies directly the following convergence.

Assume conditions (i) and (iv)–(x). Then under local alternatives H1,m , we have: Tm2→dχd2(τ),τ:=‖ΣT−1/2CT‖, where CT is given in (4.2).

Theorem 18 makes it possible to find the asymptotic power of the test under local alternatives H1,m . It is evident that the asymptotic power is an increasing function of τ=‖ΣT−1/2CT‖ . In other words, the larger τ, the more powerful the test.

 Conclusion

We constructed a goodness-of-fit test for a multivariate linear errors-in-variables model, provided that the errors are uncorrelated with equal (unknown) variances and vanishing third moments. The latter moment assumption makes it possible to estimate consistently the asymptotic covariance matrix ΣT of the statistic Tm0 and construct the test statistic Tm2 , which has the asymptotic χd2 distribution under the null hypothesis. The local alternatives H1,m are presented, under which the test statistic has the noncentral χd2(τ) asymptotic distribution. The larger τ, the larger the asymptotic power of the test.

In future, we will try to construct, like in [5], a more powerful test using within a test statistic the exponential weight function ωλ(a)=eλ⊤a,λ∈Rn×1. To this end, it is necessary to require the independence he terrors b˜i and a˜i and also the existence of exponential moments of the errors a˜i . This is the price for a greater power of the test.

 Appendix

Let r>1 1$]]> be a fixed real number, and {ηk} be an i.i.d. sequence with zero mean and finite moment E|η1|r . Assume also that a sequence {dk} of real numbers satisfies 1mr ∑k=1m|dk|r→0as m→∞. Then dη‾=1m ∑k=1mdkηk→P0.

Without of loss generality, we may and do assume that 1<r<2 . It suffices to check that the following three conditions from Theorem 5 in [8, Chap. VI] (5.1):

∑k=1mP{|dkηk|>m}≤ ∑k=1mE|dkηk|rmr=E|η1|rmr ∑k=1m|dk|r→0asm→∞ m\}\le \displaystyle\sum \limits_{k=1}^{m}\displaystyle\frac{\operatorname{\mathsf{E}}|d_{k}\eta _{k}{|}^{r}}{{m}^{r}}=\displaystyle\frac{\operatorname{\mathsf{E}}|\eta _{1}{|}^{r}}{{m}^{r}}\displaystyle\sum \limits_{k=1}^{m}|d_{k}{|}^{r}\to 0\hspace{2.5pt}\text{as}\hspace{2.5pt}m\to \infty $]]>;

1m2 ∑k=1mD(dkηkI(|dkηk|<m))≤1m2 ∑k=1mE(dk2ηk2I(|dkηk|<m))≤1m2 ∑k=1mE|dkηk|r·m2−r=E|η1|rmr ∑k=1m|dk|r→0asm→∞;

εm:=1m ∑k=1mE(dkηkI(|dkηk|<m))=−1m ∑k=1mE(dkηkI(|dkηk|≥m)),|εm|≤1m ∑k=1mE|dkηk|r·1mr−1=E|η1|rmr ∑k=1m|dk|r→0asm→∞.

By the mentioned theorem from [8] the presented bounds imply the desired convergence.  □

The next statement is a version of the Lyapunov CLT.

Let {zi} be a sequence of independent centered random vectors in Rp with cov(z)‾=1m∑i=1mcov(zi)→S as m→∞ . Assume also that, for some δ>0 0$]]>, 1m1+δ/2 ∑i=1mE‖zi‖2+δ≤const. Then 1m ∑i=1mzi→dN(0,S).

(a) We have: S(f):=1m ∑i=1ms⊤(ai,bi;X0)VA−1ff⊤VA−1s(ai,bi;X0)=(S(f)−ES(f))+ES(f).

In the proof of Theorem 8(a) in [6], the following expansion of the estimating function is used: s(ai,bi;X0)=Wi1X0−Wi2+Wi3X0−Wi4−X0(Id+X0⊤X0)−1 ×(X0⊤Wi3X0−X0⊤Wi4−Wi4⊤X0+Wi5), where Wij are the components of the matrix collection (2.11).

We show that the term in parentheses on the right-hand side of (5.3) tends to zero in probability. Taking into account expansion (5.5), we write down one of summands of the expression S(f) : Lm:=1m ∑i=1mX0⊤a˜iai0⊤Zai0a˜i⊤X0,Z:=VA−1ff⊤VA−1. Let us explain why Lm−ELm→P0. It suffices to consider the matrix L˜m:=1m ∑i=1ma˜iai0⊤Zai0a˜i⊤. Up to a constant, its entries contain summands of the form 1m ∑i=1ma˜i(j)ai0(p)ai0(q)a˜i(r). Applying Lemma 19 to the expression 1m ∑i=1mai0(p)ai0(q)(a˜i(j)a˜i(r)−Ea˜i(j)a˜i(r)), we have E(a˜i(j)a˜i(r))2≤E‖a˜i‖4<∞ , and for δ from condition (v), we have: 1m1+δ/2 ∑i=1m|ai0(p)ai0(q)|1+δ/2≤1m1+δ/2 ∑i=1m‖ai0‖2+δ→0asm→∞. Thus, by Lemma 19 expression (5.8) tends to zero in probability. Then L˜m−EL˜m→P0, whence we get (5.7).

In a similar way, other summands of S(f) can be studied, and therefore, S(f)−ES(f)→P0. Next, we verify directly the convergence ES(f)→S(X0,f)=EΓ⊤(X0)VA−1ff⊤VA−1Γ(X0)asm→∞. Therefore, S(f)→PS(X0,f) .

(b) Without any problem, in view of Theorem 2 and the consistency of estimators VˆA and fˆ , the following convergences can be shown: S(f)−Sˆ(f)→P0,Sˆ(f):=1m ∑i=1ms⊤(ai,bi;Xˆ)·Z·s(ai,bi;Xˆ);Sˆ(f)−Sˆ(fˆ)→P0. Here Z is the matrix from relations (5.6).

The desired convergence follows from the convergences established in parts (a) and (b) of the proof.  □

For model (2.3)–(2.4), we have: mTm0=1m ∑i=1m(bi0+b˜i−Xˆ⊤ai0−Xˆ⊤a˜i0) =1m ∑i=1m(b˜i−X0⊤a˜i)−m(Xˆ−X0)⊤a0‾+rest,whererest=−(Xˆ−X0)⊤·1m ∑i=1ma˜i=op(1)·Op(1)=op(1).  □

By Theorem 4(b), m(Xˆ−X0)=Op(1). Therefore, expansion (3.4) and condition (vii) imply that mTm0=1m ∑i=1m(b˜i−X0⊤a˜i)+m(Xˆ−X0)⊤μa+op(1). Next, by expansion (2.20) we get: mTm0=1m ∑i=1m(b˜i−X0⊤a˜i+s⊤(ai,bi;X0)VA−1μa)+op(1).

The random vectors zi:=b˜i−X0⊤a˜i+s⊤(ai,bi;X0)VA−1μa satisfy condition (5.2) with the number δ from assumptions (iv) and (v). Let us find the variance–covariance matrix Σi of vector (5.13). We have Σi=cov(b˜i−X0⊤a˜i)+cov(s⊤(ai,bi;X0)VA−1μa)+M+M⊤. Here (see (2.11) and (5.5)) M:=Es⊤(ai,bi;X0)VA−1μa(b˜i−a˜iX0)=E(X0⊤a˜iai0⊤−b˜iai0⊤)VA−1μa(b˜i⊤−a˜i⊤X0);M=−X0⊤(Ea˜iai0⊤VA−1μaa˜i⊤)X0−Eb˜iai0⊤VA−1μab˜i⊤ =−ai0⊤VA−1μaσ2(Id+X0⊤X0)=M⊤;cov(b˜i−X0⊤a˜i)=σ2(Id+X0⊤X0);cov(s⊤(ai,bi;X0)VA−1μa)=Es⊤(ai,bi;X0)·Z·s(ai,bi;X0).

Then ΣT:=limm→∞1m(Σ1+⋯+Σm)=σ2(Id+X0⊤X0)+S(X0,μa)−2μa⊤VA−1μaσ2(Id+X0⊤X0), and this coincides with the right-hand side of equality (3.5).

Finally, the desired convergence follows from expansion (5.12) by Lemma 20 and Slutsky’s lemma.  □

The convergence μˆa→P1μa is established by SLLN. The convergence ΣˆT→PΣT follows from Theorem 8 (the role of f and fˆ is played by μa and μˆa , respectively) and the consistency of estimators σˆ2 , μˆa , and VˆA .  □

(a) Hereafter, for symmetric matrices A and B, notation A≥B ( A>B B$]]>) means that the matrix A−B is positive semidefinite (positive definite).

Condition (vi) ensures the independence of the matrix components Γi in relation (2.12). Therefore, S(X0,μa)≥cov((X0⊤a˜i−bˆi)μa⊤VA−1μa)=σ2(μa⊤VA−1μa)2(Id+X0⊤X0).

From equality (3.5) we have ΣT≥σ2(1−μa⊤VA−1μa)2·Id. By condition (viii), VA>μaμa⊤ \mu _{a}{\mu _{a}^{\top }}$]]>. In the case μa=0 , we get ΣT≥σ2Id>0 0$]]>, and in the case μa≠0 , we put z=VA−1μa and obtain: z⊤VAz=μa⊤VA−1μa>(μa⊤z)2=(μa⊤VA−1μa)2; {\big({\mu _{a}^{\top }}z\big)}^{2}={\big({\mu _{a}^{\top }}{V_{A}^{-1}}\mu _{a}\big)}^{2};\]]]> thus, 1>μa⊤VA−1μa {\mu _{a}^{\top }}{V_{A}^{-1}}\mu _{a}\hspace{2.5pt}$]]>, and inequality (5.17) implies ΣT>0 0$]]>.

Statement (b) follows from statement (a) and Lemma 11(b).  □

(a) The local alternative (4.1) is corresponding to the perturbation matrix G0:=g⊤(a10)⋮g⊤(am0). Model (4.1) can be rewritten as a perturbed model (2.1), A0X0=B0,A=A0+A˜,Bper:=B0+1mG0+B˜, or as a perturbed model (2.2), C0=[A0B0],C˜=[A˜B˜],Cper:=[ABper], C0·X0−Id=0.

Introduce the symmetric matrix N=C0⊤C0+λmin(A0⊤A0)In+d. Due to condition (iii), as m→∞ , N=mN0+o(m),N0>0. 0.\]]]>

Consider two matrices of size (n+d)×(n+d) : M1=N−1/2C0⊤(Cper−C0)N−1/2, M2=N−1/2((Cper−C0)⊤(Cper−C0)−σ2mIn+d)N−1/2. In view of the proof of Theorem 4.1 in [9], for the convergence Xˆ→PX0, it suffices to show that, as m→∞ , M1→P0,M2→P0, or taking into account (5.21), that M1′:=1mC0⊤C˜+1mC0⊤·1mG0→P0, M2′:=1m((C˜+1m[0G0])⊤(C˜+1m[0G0])−σ2In+d)→P0.

We study the most interesting summands, those that contain G0 (the convergence of other summands was shown in the proof of Theorem 4.1 in [9]). We have M1″:=1m3/2C0⊤G0=1m3/2A0⊤G0X0⊤A0⊤G0, and due to condition (ix), as m→∞ , 1m3/2A0⊤G0=1m3/2 ∑i=1mai0g⊤(ai0)=O(1)m1/2→0,M1″→0. Next, by condition (x), M2″:=1m2G0⊤G0=1m2 ∑i=1mg(ai0)g⊤(ai0), ‖M2″‖≤constm2 ∑i=1m‖g(ai0)‖2→0asm→∞. Finally, M2‴:=1m3/2C˜⊤G0=1m3/2 ∑i=1mc˜ig⊤(ai0), E‖M2‴‖2≤constm3 ∑i=1m‖g(ai0)‖2→0asm→∞,M2‴→P0.

We established the convergence in probability for the summands from (5.26) and (5.27) that contain the perturbation G0 . Therefore, (5.26) and (5.27) are satisfied, relation (5.25) is satisfied as well, and the results of [9] imply convergence (5.24).

The consistency of the estimator σˆ2 under local alternatives H1,m is established by formula (2.17) and boils down to the consistency of σˆ2 under the null hypothesis: the consistency of Xˆ has been proven already, and, moreover, bperbper,⊤‾=1m ∑i=1m(bi+1mg(ai0))(bi+1mg(ai0))⊤=1m ∑i=1mbibi⊤+op(1), abper,⊤‾=ab⊤‾+op(1).

(b) After we established the consistency of Xˆ under alternatives H1,m , we find an expansion similar to (2.20): m(Xˆ−X0)=−VA−11m ∑i=1ms(ai,biper;X0)+op(1) =−VA−11m ∑i=1ms(ai,bi;X0)−VA−11m ∑i=1msiper+op(1). Conditions (ix) and (x) ensure that, for perturbations siper of the estimating function, we have (the main contribution to siper is made by a linear summand ab⊤ from (2.10)): 1m ∑i=1msiper=−1m ∑i=1mai0g⊤(ai0)+op(1).

Lemma 7, Theorem 4, and formulae (5.36) and (5.37) imply the desired convergence of the normalized TLS estimator.  □

(a) Under the local alternatives, we have: mTm0|H1,m=mTm0|H0+M(g(a0))−m(Xˆ|H1,m−Xˆ|H0)⊤μa+op(1). Expansions (2.20), (5.36), and (5.37) imply that m(Xˆ|H1,m−Xˆ|H0)→PVA−1·M(a0g⊤(a0)). Then, by Lemma 10 and Slutsky’s lemma, mTm0|H1,m→dN(CT,ΣT), CT=M(g(a0))−M(g(a0)a0⊤)·VA−1μa.

(b) Under the local alternatives, the estimators σˆ2 , μˆa , VˆA , and Xˆ are still consistent. Moreover, Sˆ(μˆa)=1m ∑i=1ms⊤(ai,biper;Xˆ)VˆA−1μˆaμˆa⊤VˆA−1s(ai,biper;Xˆ) converges in probability to S(X0,μa) because expression (5.40) does not involve terms linear in biper , and the perturbation of the vectors bi does not modify the asymptotic behavior of Sˆ(μˆa) in transition from H0 to the local alternatives.

Thus, estimator (3.6) does converge in probability to matrix (3.5).  □

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