We propose an outlier robust and distributions-free test for the explosive AR(1) model with intercept based on simplicial depth. In this model, simplicial depth reduces to counting the cases where three residuals have alternating signs. Using this, it is shown that the asymptotic distribution of the test statistic is given by an integrated two-dimensional Gaussian process. Conditions for the consistency of the test are given and the power of the test at finite samples is compared with five alternative tests, using errors with normal distribution, contaminated normal distribution, and Fréchet distribution in a simulation study. The comparisons show that the new test outperforms all other tests in the case of skewed errors and outliers. Although here we deal with the AR(1) model with intercept only, the asymptotic results hold for any simplicial depth which reduces to alternating signs of three residuals.
We simplify simplicial depth for regression and autoregressive growth processes in two directions. At first we show that often simplicial depth reduces to counting the subsets with alternating signs of the residuals. The second simplification is given by not regarding all subsets of residuals. By consideration of only special subsets of residuals, the asymptotic distributions of the simplified simplicial depth notions are normal distributions so that tests and confidence intervals can be derived easily. We propose two simplifications for the general case and a third simplification for the special case where two parameters are unknown. Additionally, we derive conditions for the consistency of the tests. We show that the simplified depth notions can be used for polynomial regression, for several nonlinear regression models, and for several autoregressive growth processes. We compare the efficiency and robustness of the different simplified versions by a simulation study concerning the Michaelis-Menten model and a nonlinear autoregressive process of order one.
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