Affine-equivariant inference for multivariate location under Lp loss functions

Abstract

We consider the fundamental problem of estimating the location of a d-variate probability measure under an ${\mathit{L}}_{\mathit{p}}$ loss function. The naive estimator, that minimizes the usual empirical ${\mathit{L}}_{\mathit{p}}$ risk, has a known asymptotic behavior but suffers from several deficiencies for $\mathit{p}\ne 2$, the most important one being the lack of equivariance under general affine transformations. In this work, we introduce a collection of ${\mathit{L}}_{\mathit{p}}$ location estimators ${\hat{\mathit{\mu }}}_{\mathit{n}}^{\mathit{p},\ell }$ that minimize the size of suitable ℓ-dimensional data-based simplices. For $\ell =1$, these estimators reduce to the naive ones, whereas, for $\ell =\mathit{d}$, they are equivariant under affine transformations. Irrespective of ℓ, these estimators reduce to the sample mean for $\mathit{p}=2$, whereas for $\mathit{p}=1$, the estimators provide the well-known spatial median and Oja median for $\ell =1$ and $\ell =\mathit{d}$, respectively. Under very mild assumptions, we derive an explicit Bahadur representation result for ${\hat{\mathit{\mu }}}_{\mathit{n}}^{\mathit{p},\ell }$ and establish asymptotic normality. We prove that, quite remarkably, the asymptotic behavior of the estimators does not depend on ℓ under spherical symmetry, so that the affine equivariance for $\ell =\mathit{d}$ is achieved at no cost in terms of efficiency. To allow for large sample size n and/or large dimension d, we introduce a version of our estimators relying on incomplete U-statistics. Under a centro-symmetry assumption, we also define companion tests ${\mathit{\varphi }}_{\mathit{n}}^{\mathit{p},\ell }$ for the problem of testing the null hypothesis that the location μ of the underlying probability measure coincides with a given location ${\mathit{\mu }}_0$. For any p, affine invariance is achieved for $\ell =\mathit{d}$. For any ℓ and p, we derive explicit expressions for the asymptotic power 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. We illustrate finite-sample relevance of our asymptotic results through Monte Carlo exercises and also treat a real data example.