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

• Published in: Mathematical methods of statistics Vol. 27; no. 4; pp. 294 - 311
• Main Authors: Boldin, M. V., Petriev, M. N.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.10.2018; Springer Nature B.V
• Subjects: Autoregressive processes; Chi-square test; Distribution functions; Economic models; Mathematics and Statistics; Outliers (statistics); Parameter estimation; Regression analysis; Statistical tests; Statistical Theory and Methods; Statistics; empirical distribution function; robustness; estimators; 62G30; Pearson’s chi-square test; autoregression; outliers; 62G35; primary 62G10; secondary 62M10; residuals
• ISSN: 1066-5307; 1934-8045
• DOI: 10.3103/S1066530718040038

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 n 1/2 -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.