On Symmetrized Pearson's Type Test for Normality of Autoregression: Power under Local Alternatives
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We consider a stationary linear AR(p) model with observations subject to gross errors (outliers). The autoregression parameters as well as the distribution function (d.f.) G of innovations are unknown. The distribution of outliers Π is unknown and arbitrary, their intensity is γ n^-1/2 with an unknown γ, n is the sample size. We test the hypothesis for normality of innovations H_Φ G ∈{Φ(x/θ), θ>0}, Φ(x) is the d.f. N(0,1). Our test is the special symmetrized Pearson's type test. We find the power of this test under local alternatives H_1n(ρ) G(x)=A_n(x):=(1-ρ n^-1/2)Φ(x/θ_0)+ρ n^-1/2H(x), ρ≥ 0, θ_0 is the unknown (under H_Φ) variance of innovations. First of all we estimate the autoregression parameters and then using the residuals from the estimated autoregression we construct a kind of empirical distribution function (r.e.d.f.), which is a counterpart of the (inaccessible) e.d.f. of the autoregression innovations. After this we construct the symmetrized variant r.e.d.f. Our test statistic is the functional from symmetrized r.e.d.f. We obtain a stochastic expansion of this symmetrized r.e.d.f. under H_1n(ρ) , which enables us to investigate our test. We establish qualitative robustness of this test in terms of uniform equicontinuity of the limiting power (as functions of γ,ρ and Π) with respect to γ in a neighborhood of γ=0.
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