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Communication Dans Un Congrès Année : 2013

Goodness-of-fit test for Gaussian regression with block correlated errors

Résumé

Let us consider a multivariate Gaussian regression model of size nJ and assume that its covariance matrix has a block diagonal structure composed of n squared blocks of size J. This assumption means that the n blocks of size J behave independently but that there exists possible correlation among the observations within the same block. The expectation and the covariance matrix of the Gaussian model are known up to a fixed number of real parameters and can depend on some covariates. This framework appears for example when considering n independent observations of a Gaussian multiresponse regression model of size J with heteroscedastic error. Many models of various application fields are also covered by our framework. Let us quote for example mixed effects models with correlated within subject errors and autocorrelated errors models for longitudinal data. Our aim is to test the null hypothesis that the expectation of the Gaussian vector belongs to a linear subspace V of R^{nJ} against the alternative that it does not. This issue belongs to the usually so called goodness-of-fit or lack-of-fit testing procedures. It has been adressed already by several authors under several model assumptions. However to our best knowledge there exists no goodness-of-fit testing procedure adapted to the case of multivariate Gaussian regression model with unknown block diagonal covariance matrix. Therefore we propose a new goodness-of-fit test for testing that the expectation of a Gaussian vector of size nJ with block diagonal covariance matrix composed of n blocks of size J belongs to a specified linear subspace V of R^{nj} . Since we aim at considering a nonparametric alternative, we base our test on multiple testing procedure that is to say a global procedure that involves several tests againts several parametric alternatives. Each of these alternatives is characterized by a linear subspace orthogonal to the linear subspace V . To get an efficient testing procedure we consider an alternative composed of several linear subspaces whose dimension may grow with the number of observations n. We state the existence and the consistency of the estimate of the parameters involved both in the expectation and in the covariance matrix of the model under the null hypothesis and under the alternative hypotheses in order to get asymptotic properties when n goes to infinity. Then we prove that our test as well as its bootstrap version achieve the nominal level and are consistent. We prove also that the test is consistent against local alternative approaching the null hypothesis at the 1/sqrt(n) rate up to a factor sqrt (log log(n)). We illustrate the behaviour of our procedure for finite sample size on the basis of a simulation study.
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Dates et versions

hal-02746445 , version 1 (03-06-2020)

Identifiants

  • HAL Id : hal-02746445 , version 1
  • PRODINRA : 200203

Citer

Sylvie Huet, Estelle Kuhn. Goodness-of-fit test for Gaussian regression with block correlated errors. 29-th European Meeting of Statisticians -EMS, Jul 2013, Budapest, Hungary. pp.352. ⟨hal-02746445⟩

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