Goodness-ofFit Test in a Structural Errors-in-Variables Model Based on a Score Function

A polynomial structural measurement error model is considered. A goodness-of-fit test is constructed based on the quasi-likelihood estimator, which is asymptotically optimal in a large class of estimators. The power of the test is discussed. The test for the linear model with unknown nuisance parameters is studied in more detail. Similar test can be applied to much more general situation, where the estimator is constructed based on a score function.

In this paper we propose a totally different idea.Our test is rather general, it is based on an estimator generated by an unbiased score function and involves its partial derivatives.The test is applied to a polynomial EIVM.We show how to disturb the initial model in order to construct a local alternative, which makes it possible to measure the power of the test.The test in a linear model is studied in more detail.
The paper is organized as follows.Section 2 presents the test in the most generality, where the model of i.i.d.observations is not specified.Section 3 contains a local alternative in rather general nonlinear EIVM.In Section 4 we introduce the quasi-score-like function for the polynomial EIVM, where all the nuisance parameters are unknown.In Sections 5 and 6 we derive the test statistics for the linear and polynomial EIVM, and Section 7 concludes.The main proofs are given in the Appendix.

General Test
Suppose that a family of d-dimensional distributions {P t | t ∈ Θ} is given.Here Θ is a convex compact set in R d .By the i.i.d.observations z 1 , . . ., z n we want to construct a goodness-of-fit test for the hypothesis Now, we suppose that H 0 holds, with a true value θ.Let q(z, t) be a Borel measurable score function valued in R d .The estimator θ of θ is defined as a measurable solution to the equation We need the following assumptions.
(i) θ is an interior point of Θ.
To construct a goodness-of-fit test, introduce the test vector where I k stands for the unit matrix of size k, and in (3 Theorem 1.Under assumptions (i) to (v), where the matrix Σ is variance-covariance matrix of the vector   vec Thus that a statistic is asymptotically χ 2 -distributed with d 2 degrees of freedom which equals the size of B.
If B is degenerate then we transform T n as follows.Let 1 ≤ r ≤ rank B, and suppose that we can choose exactly r components of the vector f n and form the r-dimensional subvector f (r) n in such a way that B (r) be nonsingular matrix.Here B (r) is the asymptotic covariance matrix of f (r) n .Then Based on this convergence, a goodness-of-fit test is constructed.

Local Alternative in General Errors-in-Variables Model
Suppose that ξ and y are random variables and we know the first and second moments of y given ξ up to unknown vector parameter β: The model ( 5) is called mean-variance model for the couple (ξ, y), see Carroll, Ruppert, Stefanski, and Crainicianu (2006).The ξ is latent variable.Instead of ξ we observe a surrogate random variable x, which is related to ξ by measurement equation where δ ∼ N (0, σ 2 δ ) is independent of y and ξ.The variance σ 2 δ is known and positive.The latent variable ξ has a probability density ρ(ξ, α), which is known up to a vector parameter α.We observe z = (x, y) and want to estimate θ = (α , β ) ∈ R d .This is errors-in-variables model, see Kukush, Malenko, and Schneeweiss (2006) for more details.
Here and hereafter we consider a quadratic-in-y unbiased score function only, which is valued in R d : Let H 0 be the hypothesis that the observations (y i , x i ), i = 1, . . ., n are i.i.d.copies of this model.Suppose that the local alternative to H 0 is Here in (7) ỹi is the observed response variable, and (y i , x i , ξ i ) are i.i.d.copies of the above mentioned model ( 5) and ( 6).We need the following relations to hold P θ -a.s. as n → ∞: and let θ be a measurable solution to the equation Sn (t) = 0. Consider Here consistently with (iv), S ∞ (t, s) = E s q(x, y, t), t, s ∈ Θ.
As a consequence of Theorem 2 we have that under H 1n the statistic Here χ 2 r (C) is the noncentral χ 2 r distribution with noncentrality C, i.e.
The larger C the larger the power of the test.
To estimate θ we use the quasi-score-like function q(x, t) with components This function yields an optimal estimator for a large class of unbiased scores, see Kukush et al. (2006) and (Kukush and Malenko, 2008).

Test in Linear Model
As a particular case consider the linear model, k = 1.Let θ = (β , µ, σ, ϕ) .We have the following result: rank B = 4, for all possible values of the parameters.We are able to choose a 4-dimensional vector f (r) n , r = 4, such that for all possible values of θ, the corresponding matrix B (r) is nonsingular.For example, these are the first, second, third and seventh components of f n .The choice is not unique.
Introduce the reliability ratio K = 1 − σ 2 δ σ −2 , and τ 2 = Kσ 2 δ .Then under the local alternative H 1n the deviation vector f from Theorem 2 is equal n in such a way that the corresponding B (r) is nonsingular, and then the corresponding deviation vector f (r) will not vanish.We will have We mention that the proposed test is very sensitive: even for a linear disturbance function p(ξ), the noncentrality parameter in the limit distribution of the T (r) n will not be equal zero.

Test in Polynomial Model
In the polynomial model of order k ≥ 2, it is not easy to compute rank B, because the matrix Σ is always degenerate.Therefore we propose a modified test vector where h is fixed nonzero vector from R d .Then where and Σ h is variance-covariance matrix for the vector [∂h q(θ)/∂t ; q ] .For known nuisance parameters (i.e. when θ = β) under the condition that the true β k = 0, for any h ∈ R k+1 such that h k = 0 we have that Σ h is nonsingular.This implies that B h is nonsingular as well and Under the local alternative H 1n given in ( 7), the modified test vector has an expansion similar to (10), fh,n = f h,n + fh + o p (1), as n → ∞, where the modified deviation equals Therefore, under H 1n we have

The larger B
−1/2 h fh the larger the power of the test.

Conclusion
We constructed a very general goodness-of-fit test, which works for any unspecified model of observations, where the estimator is generated by an unbiased score function.
In errors-in-variables setup, we proposed a local alternative for the test.In a forthcoming paper we will compare the power of this test and other tests known in the literature.
From ( 12) we have By (iii) and (v) Therefore, we get 2 • .The next step is to prove the asymptotic normality of f n .Consider the (i, j)-th element of the matrix, which forms f n : where = 0, due to the condition S ∞ (t, t) = 0.Here assumptions (iii) and (iv) were used together with the same reasoning as in the first part of the proof.Applying (13) we have Therefore, where A is defined in (3) and the sequence satisfies the Central Limit Theorem.g n d → N (0, Σ) with the matrix Σ given in (4).We have f n d → N (0, AΣA ).

Proof of Theorem 2
Consider the difference between Sn (t) and S n (t).Assumptions ( 8) and ( 9) with j = 0 imply that sup as n → ∞ P θ -a.s.Then Sn (t) → S ∞ (t, θ) uniformly in t ∈ Θ P θ -a.s.(here S ∞ is the same as in assumption (iv)).Then eventually a solution to the equation Sn (t) = 0 exists and the equality Sn ( θ) = 0 holds implying θ → θ as n → ∞ P θ -a.s.We have a.s.
Investigating the fn we repeat the reasoning from the proof of Theorem 1. Denote by sij the (i, j)-th element of the matrix, which forms fn .Then Taking vec operation we obtain the expansion (10) and then the statement of the theorem.