Robustesse de tests de comparaisons de moyennes vis à vis de la non-indépendance des observations

Abstract

This is a study of the robustness of the characteristics of usual tests for the comparison of a mean value with a given value or the comparision of two means, if the successive GAUSSian observations are not independent. The counter models (X t ), t∈N+, for non-independence are characterized by their authocorrelations and are such that EX t =μ, Var X t =σ2. Furthermore, M1 is a stationary MARKOV process; M2 is a stationary model so that the first m authocorrelation coeffficients are different from zero; M3 is the model of the symmetry condition: ↭k ∈N+: Corr (X t X t+k ) = ρ = const,. It seems that the choice of these models covers a vast set of situations. – At first, conditions for a class M′ of models (including M1 and M2) are established which, for instance, imply the quadratic mean convergence of the usual mean estimators (and variance estimators) to μ (and σ2) as well as the existence of a certain bias parameter. The bias equations for the samples, and for the test-power function, are established for each model. In the model M3 the study is non-asymptotic, while in model M′ it is asumptotic. In all cases a bias parameter is obtained that is independent of the parameters being tested. Its value determines the sign and the amplitude of the bias. The bias is large also for the dependence levels in some cases. However, the amplitudes depend on the counter model; the tests become less robust when the autocorrelations increase in absolute value maintaining the same sign.