arXiv subjects

Martin Wahl

Publications and source records attributed to Martin Wahl.

23 records · Page 2Linked to original sources

arXiv2019

Functional estimation and hypothesis testing in nonparametric boundary models

Consider a Poisson point process with unknown support boundary curve $g$, which forms a prototype of an irregular statistical model. We address the problem of estimating non-linear functionals of the form $\int Φ(g(x))\,dx$. Following a nonparametric maximum-likelihood approach, we construct an estimator which is UMVU over Hölder balls and achieves the (local) minimax rate of convergence. These results hold under weak assumptions on $Φ$ which are satisfied for $Φ(u)=|u|^p$, $p\ge 1$. As an application, we consider the problem of estimating the $L^p$-norm and derive the minimax separation rates in the corresponding nonparametric hypothesis testing problem. Structural differences to results for regular nonparametric models are discussed.

math.ST↗

arXiv2018

Perturbation bounds for eigenspaces under a relative gap condition

A basic problem in operator theory is to estimate how a small perturbation effects the eigenspaces of a self-adjoint compact operator. In this paper, we prove upper bounds for the subspace distance, taylored for structured random perturbations. As a main example, we consider the empirical covariance operator, and show that a sharp bound can be achieved under a relative gap condition. The proof is based on a novel contraction phenomenon, contrasting previous spectral perturbation approaches.

math.PR↗

arXiv2015

A theory of nonparametric regression in the presence of complex nuisance components

In this paper, we consider the nonparametric random regression model $Y=f_1(X_1)+f_2(X_2)+ε$ and address the problem of estimating the function $f_1$. The term $f_2(X_2)$ is regarded as a nuisance term which can be considerably more complex than $f_1(X_1)$. Under minimal assumptions, we prove several nonasymptotic $L^2(\mathbb{P}^X)$-risk bounds for our estimators of $f_1$. Our approach is geometric and based on considerations in Hilbert spaces. It shows that the performance of our estimators is closely related to geometric quantities, such as minimal angles and Hilbert-Schmidt norms. Our results establish new conditions under which the estimators of $f_1$ have up to first order the same sharp upper bound as the corresponding estimators of $f_1$ in the model $Y=f_1(X_1)+ε$. As an example we apply the results to an additive model in which the number of components is very large or in which the nuisance components are considerably less smooth than $f_1$. In particular, the results apply to an asymptotic scenario in which the number of components is allowed to increase with the sample size.

math.ST↗

arXiv2015

Variable selection in high-dimensional additive models based on norms of projections

We consider the problem of variable selection in high-dimensional sparse additive models. We focus on the case that the components belong to nonparametric classes of functions. The proposed method is motivated by geometric considerations in Hilbert spaces and consists of comparing the norms of the projections of the data onto various additive subspaces. Under minimal geometric assumptions, we prove concentration inequalities which lead to new conditions under which consistent variable selection is possible. As an application, we establish conditions under which a single component can be estimated with the rate of convergence corresponding to the situation in which the other components are known.

math.ST↗

arXiv2013

On the mod-Gaussian convergence of a sum over primes

We prove mod-Gaussian convergence for a Dirichlet polynomial which approximates $\operatorname{Im}\logζ(1/2+it)$. This Dirichlet polynomial is sufficiently long to deduce Selberg's central limit theorem with an explicit error term. Moreover, assuming the Riemann hypothesis, we apply the theory of the Riemann zeta-function to extend this mod-Gaussian convergence to the complex plane. From this we obtain that $\operatorname{Im}\logζ(1/2+it)$ satisfies a large deviation principle on the critical line. Results about the moments of the Riemann zeta-function follow.

math.NT↗