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The stability of the double-layered antiferromagnets and their magnonic properties are investigated by considering two model systems, the linear chain (LC) and more complex chain of railroad trestle (RT) geometry, and a real example of chromium nitride CrN. The spin-paired order ($\cdots{+}{+}{-}{-}\cdots$) in LC requires an alternation of the ferromagnetic and antiferromagnetic (AFM) interactions, while analogous spin-paired order in RT can be stable even for all magnetic exchange interactions being AFM. The rock-salt structure of CrN evokes clear magnetic frustration since Cr atoms in face-centered cubic lattice form links to twelve nearest neighbors all equivalent and AFM. Nonetheless, the magnetostructural transition to an orthorhombically distorted phase below $T_\text{N} = 287~\text{K}$ leads to a diversification of Cr-Cr links, which suppresses the frustration and allows for stable double-layered AFM order of CrN. Based on $\textit{ab initio}$ calculated exchange parameters, the magnon spectra and temperature evolution of ordered magnetic moments are derived., Comment: 7 pages, 8 figures

We prove Fuk-Nagaev and Rosenthal-type inequalities for the sums of independent random matrices, focusing on the situation when the norms of the matrices possess finite moments of only low orders. Our bounds depend on the "intrinsic" dimensional characteristics such as the effective rank, as opposed to the dimension of the ambient space. We illustrate the advantages of such results in several applications, including new moment inequalities for sample covariance matrices and the corresponding eigenvectors of heavy-tailed random vectors. Moreover, we demonstrate that our techniques yield sharpened versions of the moment inequalities for empirical processes.

In two landmark papers, Akaike introduced the AIC and FPE, demonstrating their significant usefulness for prediction. In subsequent seminal works, Shibata developed a notion of asymptotic efficiency and showed that both AIC and FPE are optimal, setting the stage for decades-long developments and research in this area and beyond. Conceptually, the theory of efficiency is universal in the sense that it (formally) only relies on second-order properties of the underlying process $(X_t)_{t\in \mathbb{Z}}$, but, so far, almost all (efficiency) results require the much stronger assumption of a linear process with independent innovations. In this work, we establish sharp oracle inequalities subject only to a very general notion of weak dependence, establishing a universal property of the AIC and FPE. A direct corollary of our inequalities is asymptotic efficiency of these criteria. Our framework contains many prominent dynamical systems such as random walks on the regular group, functionals of iterated random systems, functionals of (augmented) Garch models of any order, functionals of (Banach space valued) linear processes, possibly infinite memory Markov chains, dynamical systems arising from SDEs, and many more., Comment: 89 pages, 3 figures

We present a research outline that aims at investigating group dynamics and peer pressure in the context of industrial robots. Our research plan was motivated by the fact that industrial robots became already an integral part of human-robot co-working. However, industrial robots have been sparsely integrated into research on robot credibility, group dynamics, and potential users' tendency to follow a robot's indication. Therefore, we aim to transfer the classic Asch experiment (see \cite{Asch_51}) into HRI with industrial robots. More precisely, we will test to what extent participants follow a robot's response when confronted with a group (vs. individual) industrial robot arms (vs. human) peers who give a false response. We are interested in highlighting the effects of group size, perceived robot credibility, psychological stress, and peer pressure in the context of industrial robots. With the results of this research, we hope to highlight group dynamics that might underlie HRI in industrial settings in which numerous robots already work closely together with humans in shared environments., Comment: 5 pages, 2 figures

Human-robot interaction (HRI) benefits greatly from advances in the machine learning field as it allows researchers to employ high-performance models for perceptual tasks like detection and recognition. Especially deep learning models, either pre-trained for feature extraction or used for classification, are now established methods to characterize human behaviors in HRI scenarios and to have social robots that understand better those behaviors. As HRI experiments are usually small-scale and constrained to particular lab environments, the questions are how well can deep learning models generalize to specific interaction scenarios, and further, how good is their robustness towards environmental changes? These questions are important to address if the HRI field wishes to put social robotic companions into real environments acting consistently, i.e. changing lighting conditions or moving people should still produce the same recognition results. In this paper, we study the impact of different image conditions on the recognition of arousal and valence from human facial expressions using the FaceChannel framework \cite{Barro20}. Our results show how the interpretation of human affective states can differ greatly in either the positive or negative direction even when changing only slightly the image properties. We conclude the paper with important points to consider when employing deep learning models to ensure sound interpretation of HRI experiments., Comment: Paper contribution to the Social and Cognitive Interactions for Assistive Robotics Workshop (SCIAR) In Conjunction with IROS 2022

Given finite i.i.d.~samples in a Hilbert space with zero mean and trace-class covariance operator $\Sigma$, the problem of recovering the spectral projectors of $\Sigma$ naturally arises in many applications. In this paper, we consider the problem of finding distributional approximations of the spectral projectors of the empirical covariance operator $\hat \Sigma$, and offer a dimension-free framework where the complexity is characterized by the so-called relative rank of $\Sigma$. In this setting, novel quantitative limit theorems and bootstrap approximations are presented subject only to mild conditions in terms of moments and spectral decay. In many cases, these even improve upon existing results in a Gaussian setting.

Historical observations of severe weather and simulated severe weather environments (i.e., features) from the Global Ensemble Forecast System v12 (GEFSv12) Reforecast Dataset (GEFS/R) are used in conjunction to train and test random forest (RF) machine learning (ML) models to probabilistically forecast severe weather out to days 4--8. RFs are trained with 9 years of the GEFS/R and severe weather reports to establish statistical relationships. Feature engineering is briefly explored to examine alternative methods for gathering features around observed events, including simplifying features using spatial averaging and increasing the GEFS/R ensemble size with time-lagging. Validated RF models are tested with ~1.5 years of real-time forecast output from the operational GEFSv12 ensemble and are evaluated alongside expert human-generated outlooks from the Storm Prediction Center (SPC). Both RF-based forecasts and SPC outlooks are skillful with respect to climatology at days 4 and 5 with degrading skill thereafter. The RF-based forecasts exhibit tendencies to underforecast severe weather events, but they tend to be well-calibrated at lower probability thresholds. Spatially averaging predictors during RF training allows for prior-day thermodynamic and kinematic environments to generate skillful forecasts, while time-lagging acts to expand the forecast areas, increasing resolution but decreasing objective skill. The results highlight the utility of ML-generated products to aid SPC forecast operations into the medium range.

As robots are expected to get more involved in people's everyday lives, frameworks that enable intuitive user interfaces are in demand. Hand gesture recognition systems provide a natural way of communication and, thus, are an integral part of seamless Human-Robot Interaction (HRI). Recent years have witnessed an immense evolution of computational models powered by deep learning. However, state-of-the-art models fall short in expanding across different gesture domains, such as emblems and co-speech. In this paper, we propose a novel hybrid hand gesture recognition system. Our architecture enables learning both static and dynamic gestures: by capturing a so-called "snapshot" of the gesture performance at its peak, we integrate the hand pose along with the dynamic movement. Moreover, we present a method for analyzing the motion profile of a gesture to uncover its dynamic characteristics and which allows regulating a static channel based on the amount of motion. Our evaluation demonstrates the superiority of our approach on two gesture benchmarks compared to a CNNLSTM baseline. We also provide an analysis on a gesture class basis that unveils the potential of our Snapture architecture for performance improvements. Thanks to its modular implementation, our framework allows the integration of other multimodal data like facial expressions and head tracking, which are important cues in HRI scenarios, into one architecture. Thus, our work contributes both to gesture recognition research and machine learning applications for non-verbal communication with robots., Comment: In Cognitive Computation(Accepted:30/06/2023, Published:17/07/2023),20 pages,20 figures,4 tables;Please find the published version/info to cite: https://doi.org/10.1007/s12559-023-10174-z;Repositories: https://zenodo.org/doi/10.5281/zenodo.10679196, https://zenodo.org/doi/10.5281/zenodo.10693816;This work was co-funded by Horizon Europe project TERAIS under Grant agreement number 101079338

We consider a time-varying first-order autoregressive model with irregular innovations, where we assume that the coefficient function is H\"{o}lder continuous. To estimate this function, we use a quasi-maximum likelihood based approach. A precise control of this method demands a delicate analysis of extremes of certain weakly dependent processes, our main result being a concentration inequality for such quantities. Based on our analysis, upper and matching minimax lower bounds are derived, showing the optimality of our estimators. Unlike the regular case, the information theoretic complexity depends both on the smoothness and an additional shape parameter, characterizing the irregularity of the underlying distribution. The results and ideas for the proofs are very different from classical and more recent methods in connection with statistics and inference for locally stationary processes., Comment: Updated Funding

Motivated from option and derivative pricing, this note develops Edgeworth expansions both in the Kolmogorov and Wasserstein metric for many different types of discrete time volatility models and their possible transformations. This includes, among others, H\"{o}lder-type functions of (augmented) Garch processes of any order, iterated random functions or Volterra-processes., Comment: minor extension, some typos removed

Suppose that the (normalised) partial sum of a stationary sequence converges to a standard normal random variable. Given sufficiently moments, when do we have a rate of convergence of $n^{-1/2}$ in the uniform metric, in other words, when do we have the optimal Berry-Esseen bound? We study this question in a quite general framework and find the (almost) sharp dependence conditions. The result applies to many different processes and dynamical systems. As specific, prominent examples, we study functions of the doubling map 2x mod 1 and the left random walk on the general linear group., Comment: Added some additional references

Given a weakly dependent stationary process, we describe the transition between a Berry-Esseen bound and a second order Edgeworth expansion in terms of the Berry-Esseen characteristic. This characteristic is sharp: We show that Edgeworth expansions are valid if and only if the Berry-Esseen characteristic is of a certain magnitude. If this is not the case, we still get an optimal Berry-Esseen bound, thus describing the exact transition. We also obtain (fractional) expansions given $3 < p \leq 4$ moments, where a similar transition occurs. Corresponding results also hold for the Wasserstein metric $W_1$, where a related, integrated characteristic turns out to be optimal. As an application, we establish novel weak Edgeworth expansion and CLTs in $L^p$ and $W_1$. As another application, we show that a large class of high dimensional linear statistics admit Edgeworth expansions without any smoothness constraints, that is, no non-lattice condition or related is necessary. In all results, the necessary weak-dependence assumptions are very mild. In particular, we show that many prominent dynamical systems and models from time series analysis are within our framework, giving rise to many new results in these areas., Comment: Fixed some typos

Whenever we are addressing a specific object or refer to a certain spatial location, we are using referential or deictic gestures usually accompanied by some verbal description. Especially pointing gestures are necessary to dissolve ambiguities in a scene and they are of crucial importance when verbal communication may fail due to environmental conditions or when two persons simply do not speak the same language. With the currently increasing advances of humanoid robots and their future integration in domestic domains, the development of gesture interfaces complementing human-robot interaction scenarios is of substantial interest. The implementation of an intuitive gesture scenario is still challenging because both the pointing intention and the corresponding object have to be correctly recognized in real-time. The demand increases when considering pointing gestures in a cluttered environment, as is the case in households. Also, humans perform pointing in many different ways and those variations have to be captured. Research in this field often proposes a set of geometrical computations which do not scale well with the number of gestures and objects, use specific markers or a predefined set of pointing directions. In this paper, we propose an unsupervised learning approach to model the distribution of pointing gestures using a growing-when-required (GWR) network. We introduce an interaction scenario with a humanoid robot and define so-called ambiguity classes. Our implementation for the hand and object detection is independent of any markers or skeleton models, thus it can be easily reproduced. Our evaluation comparing a baseline computer vision approach with our GWR model shows that the pointing-object association is well learned even in cases of ambiguities resulting from close object proximity.

The volume tensor provides robust estimate of object shape and orientation in space. The tensor is estimated from 3D data set by the Fakir probe, an interactive method using intersections of the objects boundary with a virtual lines. The method thus can be applied to objects that cannot be segmented automatically. Marking the intersections instead of segmenting the whole object reduces the workload required for obtaining sufficiently precise results. We present theoretical results on the variance of estimate of integrals by systematic sampling that enable calculation of the shape estimate precision. To demonstrate the ability of Fakir technique, we measure the changes in shape and orientation of pheasant brain compartments during development., Comment: The paper was submitted to Image Analysis & Stereology

Although Deep Neural Networks reached remarkable performance on several benchmarks and even gained scientific publicity, they are not able to address the concept of cognition as a whole. In this paper, we argue that those architectures are potentially interesting for cognitive robots regarding their perceptual representation power for audio and vision data. Then, we identify crucial settings for cognitive robotics where deep neural networks have as yet only contributed little compared to the challenges in cognitive robotics. Finally, we argue that the rather unexplored area of Reservoir Computing qualifies to be an integral part of sequential learning in this context., Comment: Short paper for EUCOG meeting 2017

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., Comment: 15 pages

The goal of this paper is to establish relative perturbation bounds, tailored for empirical covariance operators. Our main results are expansions for empirical eigenvalues and spectral projectors, leading to concentration inequalities and limit theorems. One of the key ingredients is a specific separation measure for population eigenvalues, which we call the relative rank, giving rise to a sharp invariance principle in terms of limit theorems, concentration inequalities and inconsistency results. Our framework is very general, requiring only $p > 4$ moments and allows for a huge variety of dependence structures., Comment: 58 pages

Consider a stationary, linear Hilbert space valued process. We establish Berry-Essen type results with optimal convergence rates under sharp dependence conditions on the underlying coefficient sequence of the linear operators. The case of non-linear Bernoulli-shift sequences is also considered. If the sequence is $m$-dependent, the optimal rate $(n/m)^{1/2}$ is reached. If the sequence is weakly geometrically dependent, the rate $(n/\log n)^{1/2}$ is obtained.

Let $\{{X}_k\}_{k\geq\mathbb{Z}}$ be a stationary sequence. Given $p\in(2,3]$ moments and a mild weak dependence condition, we show a Berry-Esseen theorem with optimal rate $n^{p/2-1}$. For $p\geq4$, we also show a convergence rate of $n^{1/2}$ in $\mathcal{L}^q$-norm, where $q\geq1$. Up to $\log n$ factors, we also obtain nonuniform rates for any $p>2$. This leads to new optimal results for many linear and nonlinear processes from the time series literature, but also includes examples from dynamical system theory. The proofs are based on a hybrid method of characteristic functions, coupling and conditioning arguments and ideal metrics., Comment: Published at http://dx.doi.org/10.1214/15-AOP1017 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org). Minor corrections, results remain unchanged. Special thanks to Florence Merlevede, for pointing out some errors

Consider $d$ dependent change point tests, each based on a CUSUM-statistic. We provide an asymptotic theory that allows us to deal with the maximum over all test statistics as both the sample size $n$ and $d$ tend to infinity. We achieve this either by a consistent bootstrap or an appropriate limit distribution. This allows for the construction of simultaneous confidence bands for dependent change point tests, and explicitly allows us to determine the location of the change both in time and coordinates in high-dimensional time series. If the underlying data has sample size greater or equal $n$ for each test, our conditions explicitly allow for the large $d$ small $n$ situation, that is, where $n/d\to0$. The setup for the high-dimensional time series is based on a general weak dependence concept. The conditions are very flexible and include many popular multivariate linear and nonlinear models from the literature, such as ARMA, GARCH and related models. The construction of the tests is completely nonparametric, difficulties associated with parametric model selection, model fitting and parameter estimation are avoided. Among other things, the limit distribution for $\max_{1\leq h\leq d}\sup_{0\leq t\leq1}\vert \mathcal{W}_{t,h}-t\mathcal{W}_{1,h}\vert$ is established, where $\{\mathcal{W}_{t,h}\}_{1\leq h\leq d}$ denotes a sequence of dependent Brownian motions. As an application, we analyze all S&P 500 companies over a period of one year., Comment: Clarified Assumption 4.3 and Theorems 4.4, 4.6 and 4.8. The results themselves are unchanged

Two samples of the [Ca$_2$CoO$_{3-t}$]$_{0.62}$(CoO$_2$) misfit cobaltate, often denoted as the Ca$_{3}$Co$_{3.93}$O$_{9}$ phase, were prepared from the same ceramic material by the oxygen and argon annealing, resulting in different carrier concentrations in the conducting CoO$_{2}$ layers, n=0.31 and 0.19 hole/Co, respectively. Electrical and thermal transport properties were studied in dependence of magnetic field up to 140 kOe. The magnetothermopower data reveal an extra spin-entropy contribution to Seebeck coefficient that is not expected for carriers of Fermi liquid character. Its magnitude is unprecedentedly large and makes at zero field up to 50$\%$ of the theoretical limit k$_B$/$e$ ln2$ = 59 \mu VK^{-1}$. This spin-entropy contribution is gradually suppressed with increasing magnetic field, and the saturation is even observed when temperatures are low enough. To understand the results, the thermopower is treated in terms of purely thermodynamic Kelvin formula, and so-called Spin liquid model is evoked, providing a reason for the spin-entropy manifestation in the [Ca$_2$CoO$_{3-t}$]$_{0.62}$(CoO$_2$) misfits., Comment: 26 pages including 11 figures

This work develops change-point methods for statistics of high-frequency data. The main interest is in the volatility of an It\^{o} semi-martingale, the latter being discretely observed over a fixed time horizon. We construct a minimax-optimal test to discriminate continuous paths from paths comprising volatility jumps. This is embedded into a more general theory to infer the smoothness of volatilities. In a high-frequency framework we prove weak convergence of the test statistic under the hypothesis to an extreme value distribution. Moreover, we develop methods to infer changes in the Hurst parameter of fractional volatility processes. A simulation study demonstrates the practical value in finite-sample applications.

Let $\bigl\{X_k\bigr\}_{k \in \mathbb{Z}} \in \mathbb{L}^2(\mathcal{T})$ be a stationary process with associated lag operators ${\boldsymbol{\cal C}}_h$. Uniform asymptotic expansions of the corresponding empirical eigenvalues and eigenfunctions are established under almost optimal conditions on the lag operators in terms of the eigenvalues (spectral gap). In addition, the underlying dependence assumptions are optimal, including both short and long memory processes. This allows us to study the relative maximum deviation of the empirical eigenvalues under very general conditions. Among other things, we show convergence to an extreme value distribution, giving rise to the construction of simultaneous confidence sets. We also discuss how the asymptotic expansions transfer to the long-run covariance operator ${\boldsymbol{\cal G}}$ in a general framework., Comment: corrected some typos

For a semi-martingale $X_t$, which forms a stochastic boundary, a rate-optimal estimator for its quadratic variation $\langle X, X \rangle_t$ is constructed based on observations in the vicinity of $X_t$. The problem is embedded in a Poisson point process framework, which reveals an interesting connection to the theory of Brownian excursion areas. We derive $n^{-1/3}$ as optimal convergence rate in a high-frequency framework with $n$ observations (in mean). We discuss a potential application for the estimation of the integrated squared volatility of an efficient price process $X_t$ from intra-day order book quotes., Comment: Extended version including empirical example

The structural and magnetic properties of two mixed-valence cobaltites with formal population of 0.30 Co$^{4+}$ ions per f.u., (Pr$_{1-y}$Y$_{y}$)$_{0.7}$Ca$_{0.3}$CoO$_3$ ($y=0$ and 0.15), have been studied down to very low temperatures by means of the high-resolution neutron diffraction, SQUID magnetometry and heat capacity measurements. The results are interpreted within the scenario of the spin-state crossover from a room-temperature mixture of the intermediate spin Co$^{3+}$ and low spin Co$^{4+}$ (IS/LS) at the to the LS/LS mixture in the sample ground states. In contrast to the yttrium free $y=0$ that retains the metallic-like character and exhibits ferromagnetic ordering below 55 K, the doped system $y=0.15$ undergoes a first-order metal-insulator transition at 132 K, during which not only the crossover to low spin states but also a partial electron transfer from Pr$^{3+}$ 4f to cobalt 3d states take place simultaneously. Taking into account the non-magnetic character of LS Co$^{3+}$, such valence shift electronic transition causes a magnetic dilution, formally to 0.12 LS Co$^{4+}$ or 0.12 $t_{2g}$ hole spins per f.u., which is the reason for an insulating, highly non-uniform magnetic ground state without long-range order. Nevertheless, even in that case there exists a relatively strong molecular field distributed over all the crystal lattice. It is argued that the spontaneous FM order in $y=0$ and the existence of strong FM correlations in $y=0.15$ apparently contradict the single $t_{2g}$ band character of LS/LS phase. The explanation we suggest relies on a model of the defect induced, itinerant hole mediated magnetism, where the defects are identified with the magnetic high-spin Co$^{3+}$ species stabilized near oxygen vacancies.

The orthoperovskites TbCoO$_3$ and DyCoO$_3$ with Co$^{3+}$ in a non-magnetic low-spin state have been investigated by neutron diffraction down to 0.25 K. Magnetic ordering is evidenced below $T_N=3.3$ K and 3.6 K, respectively, and the ordered arrangements are of canted type, A$_x$G$_y$ for TbCoO$_3$ and G$_x$A$_y$ for DyCoO$_3$ in Bertaut's notation. The experiments are confronted with the first-principle calculations of the crystal field and magnetism of Tb$^{3+}$ and Dy$^{3+}$ ions, located in the $Pbnm$ structure on sites of $C_s$ point symmetry. Both these ions exhibit an Ising behavior, which originates in the lowest energy levels, in particular in accidental doublet of non-Kramers Tb$^{3+}$ ($4f^8$ configuration) and in ground Kramers doublet of Dy$^{3+}$ ($4f^9$) and it is the actual reason for the non-collinear AFM structures. Very good agreement between the experiment and theory is found. For comparison, calculations of the crystal field and magnetism for other systems with Kramers ions, NdCoO$_3$ and SmCoO$_3$, are also included.

Crystal and magnetic structures of the $x=0.2$ member of La$_{\rm 0.8-x}$Tb$_{\rm x}$Ca$_{0.2}$CoO$_3$ perovskite series have been determined from the powder neutron diffraction. Enhancement of the diffraction peaks due to ferromagnetic or cluster glass ordering is observed below $T_C=55$ K. The moments evolve at first on Co sites, and ordering of Ising-type Tb$^{3+}$ moments is induced at lower temperatures by a molecular field due to Co ions. The final magnetic configuration is collinear F$_x$ for cobalt subsystem, while it is canted F$_x$C$_y$ for terbium ions. The rare-earth moments align along local Ising axes within \textit{ab}-plane of the orthorhombic $Pbnm$ structure. The behavior in external fields up to $70-90$ kOe has been probed by the magnetization and heat capacity measurements. The dilute terbium ions contribute to significant coercivity and remanence that both steeply increase with decreasing temperature. A remarkable manifestation of the Tb$^{3+}$ Ising character is the observation of a low-temperature region of anomalously large linear term of heat capacity and its field dependence. Similar behaviours are detected also for other terbium dopings $x=0.1$ and 0.3., Comment: 8 pages, 11 figures

Fifteen parameters characterizing the crystal field of rare-earth ions in the RMO$_3$ perovskites (R = Pr, Nd, M = Ga, Co) are calculated by expanding the local Hamiltonian expressed in the basis of Wannier functions into a series of spherical tensor operators. The method contains a single adjustable parameter that characterizes the hybridization of R($4f$) states with the states of oxygen ligands. Subsequently the energy levels and magnetic moments of trivalent R ion are determined by diagonalization of an effective Hamiltonian which, besides the crystal field, contains the $4f$ electron-electron repulsion, spin-orbit coupling and interaction with magnetic field. In the Ga compounds the energy levels of the ground multiplet agree within few meV with those determined experimentally by other authors. For all four compounds in question the temperature dependence of magnetic susceptibility is measured on polycrystalline samples and compared with the results of calculation. For NdGaO$_3$ theory is also compared with the magnetic measurements on a single crystal presented by Luis {\it et al.} Phys. Rev. B {\bf 58}, 798 (1998). A good agreement between the experiment and theory is found.

We consider the model of nonregular nonparametric regression where smoothness constraints are imposed on the regression function $f$ and the regression errors are assumed to decay with some sharpness level at their endpoints. The aim of this paper is to construct an adaptive estimator for the regression function $f$. In contrast to the standard model where local averaging is fruitful, the nonregular conditions require a substantial different treatment based on local extreme values. We study this model under the realistic setting in which both the smoothness degree $\beta>0$ and the sharpness degree $\mathfrak {a}\in(0,\infty)$ are unknown in advance. We construct adaptation procedures applying a nested version of Lepski's method and the negative Hill estimator which show no loss in the convergence rates with respect to the general $L_q$-risk and a logarithmic loss with respect to the pointwise risk. Optimality of these rates is proved for $\mathfrak{a}\in(0,\infty)$. Some numerical simulations and an application to real data are provided., Comment: Published in at http://dx.doi.org/10.1214/14-AOS1248 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)