On the Power of Symmetrized Pearson's Type Test under Local Alternatives in Autoregression with Outliers
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We consider a stationary linear AR(p) model with observations subject to gross errors (outliers). The autoregression parameters are unknown as well as the distribution function G of innovations. 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 H_0 G=G_0 with simmetric G_0. We find the power of the test under local alternatives H_1n(ρ) G=(1-ρ n^-1/2)G_0+ρ n^-1/2H. Our test is the special symmetrized Pearson's type test. Namely, 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 (e.d.f.), which is a counterpart of the (inaccessible) e.d.f. of the autoregression innovations. We obtain a stochastic expansion of this e.d.f. and its symmetrized variant under H_1n(ρ) , which enables us to construct and investigate our symmetrized test of Pearson's type for H_0. 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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