Mathematical Methods of Statistics

, Volume 27, Issue 4, pp 294–311 | Cite as

On the Empirical Distribution Function of Residuals in Autoregression with Outliers and Pearson’s Chi-Square Type Tests

  • M. V. BoldinEmail author
  • M. N. Petriev


We consider a stationary linear AR(p) model with observations subject to gross errors (outliers). The distribution of outliers is unknown and arbitrary, their intensity is γn−1/2 with an unknown γ, n is the sample size. The autoregression parameters are unknown, they are estimated by any estimator which is n1/2-consistent uniformly in γ ≤ Γ < ∞. 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., which enables us to construct a test of Pearson’s chi-square type for testing hypotheses about the distribution of innovations. We establish qualitative robustness of this test in terms of uniform equicontinuity of the limiting level with respect to γ in a neighborhood of γ = 0.


autoregression outliers residuals empirical distribution function Pearson’s chi-square test robustness estimators 

2000 Mathematics Subject Classification

primary 62G10 secondary 62M10 62G30 62G35 


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Copyright information

© Allerton Press, Inc. 2018

Authors and Affiliations

  1. 1.Dept. of Mech. and Math.Lomonosov Moscow State Univ.MoscowRussia

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