Title: Affine-equivariant inference for multivariate location under $L_p$ loss functions
Authors: Alexander Duerre - Universite libre de Bruxelles (Belgium)
Davy Paindaveine - Universite libre de Bruxelles (Belgium) [presenting]
Abstract: The fundamental problem of estimating the location of a $d$-variate probability measure under an $L_p$ loss function is considered. The naive estimator, that minimizes the usual empirical $L_p$ risk, has a known asymptotic behaviour but suffers from several deficiencies for $p\neq 2$, the most important one being the lack of equivariance under general affine transformations. We introduce a collection of $L_p$ location estimators that minimize the size of suitable $\ell$-dimensional data-based simplices. For $\ell=1$, these estimators reduce to the naive ones, whereas, for $\ell=d$, they are equivariant under affine transformations. The proposed class contains in particular the celebrated spatial median and Oja median. Under very mild assumptions, we derive an explicit Bahadur representation result for each estimator in the class and establish asymptotic normality. Under a centro-symmetry assumption, we also introduce companion tests for the problem of testing the null hypothesis that the location $\mu$ of the underlying probability measure coincides with a given location $\mu_0$. We compute asymptotic powers of these tests under contiguous local alternatives, which reveals that asymptotic relative efficiencies with respect to traditional parametric Gaussian procedures for hypothesis testing coincide with those obtained for point estimation. Monte Carlo exercises confirm our asymptotic results.