2020/04/04 by Michael Boldin, Boldin, Michael, Michael D. Boldin
Mathematics · #62G30 #62G35 #Advanced Statistical Methods and Models #FOS: Mathematics #Primary 62G10 #Statistical Distribution Estimation and Applications #Statistical Methods and Inference #Statistics Theory (math.ST) #secondary 62M10
paper · pdf · doi:10.48550/arxiv.2004.06467
openalex publication_date 2020/04/04 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We consider a stationary linear AR(p) model with observations subject to\ngross errors (outliers). The autoregression parameters as well as the\ndistribution function (d.f.) G of innovations are unknown. The distribution\nof outliers \Π is unknown and arbitrary, their intensity is \γ\nn-1/2 with an unknown \γ, n is the sample size. We test the\nhypothesis for normality of innovations
mathbfH_
Phi
colon G
in\n
Phi(x/
theta),
,
thetagt;0
, \Φ(x) is the d.f. \N(0,1). Our\ntest is the special symmetrized Pearson's type test. We find the power of this\ntest under local alternatives
mathbfH1n(
rho)
colon\nG(x)=An(x):=(1-
rho n-1/2)
Phi(x/
theta0)+
rho n-1/2H(x), \ρ\≥\n0, ,\θ0 is the unknown (under \H_\Φ) variance of innovations.\nFirst of all we estimate the autoregression parameters and then using the\nresiduals from the estimated autoregression we construct a kind of empirical\ndistribution function (r.e.d.f.), which is a counterpart of the (inaccessible)≠.d.f. of the autoregression innovations. After this we construct the\nsymmetrized variant r.e.d.f. Our test statistic is the functional from\nsymmetrized r.e.d.f. We obtain a stochastic expansion of this symmetrized\nr.e.d.f. under \H1n(\ρ) , which enables us to investigate our\ntest. We establish qualitative robustness of this test in terms of uniform\nequicontinuity of the limiting power (as functions of \γ,\ρ and \Π)\nwith respect to \γ in a neighborhood of \γ=0.\n