Calculation of simplicial depth estimators for polynomial regression with applications

A fast algorithm for calculating the simplicial depth of a single parameter vector of a polynomial regression model is derived. Additionally, an algorithm for calculating the parameter vectors with maximum simplicial depth within an affine subspace of the parameter space or a polyhedron is presented. Since the maximum simplicial depth estimator is not unique, l1l1 and l2l2 methods are used to make the estimator unique. This estimator is compared with other estimators in examples of linear and quadratic regression. Furthermore, it is shown how the maximum simplicial depth can be used to derive distribution-free asymptotic αα-level tests for testing hypotheses in polynomial regression models. The tests are applied on a problem of shape analysis where it is tested how the relative head length of the fish species Lepomis gibbosus depends on the size of these fishes. It is also tested whether the dependency can be described by the same polynomial regression function within different populations.