Summary. The Wasserstein distance is an attractive tool for data analysis but statistical inference is hindered by the lack of distributional limits. To overcome this obstacle, for probability measures supported on finitely many points, we derive the asymptotic distribution of empirical Wasserstein distances as the optimal value of a linear programme with random objective function.This facilitates statistical inference (e.g. confidence intervals for sample-based Wasserstein distances) in large generality. Our proof is based on directional Hadamard differentiability. Failure of the classical bootstrap and alternatives are discussed. The utility of the distributional results is illustrated on two data sets.

Plasmonic nanostructures have been widely incorporated into different semiconductor materials to improve solar energy conversion. An important point is how to manipulate the incident light so that more light can be efficiently scattered and absorbed within the semiconductors. Here, by using a tunable three-dimensional Au pillar/truncated-pyramid (PTP) array as a plasmonic coupler, a superior optical absorption of about 95% within a wide wavelength range is demonstrated from an assembled CdS/Au PTP photoanode. Based on incident photon to current efficiency measurements and the corresponding finite difference time domain simulations, it is concluded that the enhancement is mainly attributed to an appropriate spectral complementation between surface plasmon resonance modes and photonic modes in the Au PTP structure over the operational spectrum. Because both of them are wavelength-dependent, the Au PTP profile and CdS thickness are further adjusted to take full advantage of the complementary effect, and subsequently, an angle-independent photocurrent with an enhancement of about 400% was obtained. The designed plasmonic PTP nanostructure of Au is highly robust, and it could be easily extended to other plasmonic metals equipped with semiconductor thin films for photovoltaic and photoelectrochemical cells.

We derive distributional limits for empirical transport distances between probability measures supported on countable sets. Our approach is based on sensitivity analysis of optimal values of infinite dimensional mathematical programs and a delta method for non-linear derivatives. A careful calibration of the norm on the space of probability measures is needed in order to combine differentiability and weak convergence of the underlying empirical process. Based on this we provide a sufficient and necessary condition for the underlying distribution on the countable metric space for such a distributional limit to hold. We give an explicit form of the limiting distribution for ultra-metric spaces. Finally, we apply our findings to optimal transport based inference in large scale problems. An application to nanoscale microscopy is given.MSC subject classification Primary 60F05, 60B12, 62E20; Secondary 90C08, 90C31, 62G10

The goal of this paper is to give confidence regions for the excursion set of a spatial function above a given threshold from repeated noisy observations on a fine grid of fixed locations. Given an asymptotically Gaussian estimator of the target function, a pair of data-dependent nested excursion sets are constructed that are sub- and super-sets of the true excursion set, respectively, with a desired confidence. Asymptotic coverage probabilities are determined via a multiplier bootstrap method, not requiring Gaussianity of the original data nor stationarity or smoothness of the limiting Gaussian field. The method is used to determine regions in North America where the mean summer and winter temperatures are expected to increase by mid 21st century by more than 2 degrees Celsius.

We generalize the SiZer of Chaudhuri and Marron (J. Amer. Statist. Assoc. 94 (1999) 807-823; Ann. Statist. 28 (2000) 408-428) for the detection of shape parameters of densities on the real line to the case of circular data. It turns out that only the wrapped Gaussian kernel gives a symmetric, strongly Lipschitz semi-group satisfying "circular" causality, that is, not introducing possibly artificial modes with increasing levels of smoothing. Some notable differences between Euclidean and circular scale space theory are highlighted. Based on this, we provide an asymptotic theory to make inference about the persistence of shape features. The resulting circular mode persistence diagram is applied to the analysis of early mechanically-induced differentiation in adult human stem cells from their actin-myosin filament structure. As a consequence, the circular SiZer based on the wrapped Gaussian kernel (WiZer) allows the verification at a controlled error level of the observation reported by Zemel et al. (Nat. Phys. 6 (2010) 468-473): Within early stem cell differentiation, polarizations of stem cells exhibit preferred directions in three different microenvironments.