Simplified simplicial depth for regression and autoregressive growth processes

Abstract We simplify simplicial depth in two directions for regression and autoregressive growth processes. At first we show that simplicial tangent depth often reduces to counting the subsets with alternating signs of the residuals if the regressors are ordered. 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 and provide an application on crack growth.