# M. A. Reiss - WIAS

## Contact Details

NameM. A. Reiss |
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AffiliationWIAS |
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Location |
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## Pubs By Year |
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## Pub CategoriesStatistics - Theory (28) Mathematics - Statistics (28) Mathematics - Probability (10) Solar and Stellar Astrophysics (6) Physics - Space Physics (3) Mathematics - Numerical Analysis (2) Statistics - Applications (2) Mathematics - Dynamical Systems (1) Mathematics - Optimization and Control (1) Mathematics - Functional Analysis (1) Statistics - Methodology (1) |

## Publications Authored By M. A. Reiss

Given a sample of a Poisson point process with intensity $\lambda_f(x,y) = n \mathbf{1}(f(x) \leq y),$ we study recovery of the boundary function $f$ from a nonparametric Bayes perspective. Because of the irregularity of this model, the analysis is non-standard. We derive contraction rates with respect to the $L^1$-norm for several classes of priors, including Gaussian priors, priors based on (truncated) random series, compound Poisson processes, and subordinators. Read More

**Authors:**C. Möstl, A. Isavnin, P. D. Boakes, E. K. J. Kilpua, J. A. Davies, R. A. Harrison, D. Barnes, V. Krupar, J. P. Eastwood, S. W. Good, R. J. Forsyth, V. Bothmer, M. A. Reiss, T. Amerstorfer, R. M. Winslow, B. J. Anderson, L. C. Philpott, L. Rodriguez, A. P. Rouillard, P. T. Gallagher, T. L. Zhang

We present a major step forward towards accurately predicting the arrivals of coronal mass ejections (CMEs) on the terrestrial planets, including the Earth. For the first time, we are able to assess a CME prediction model using data over almost a full solar cycle of observations with the Heliophysics System Observatory. We validate modeling results on 1337 CMEs observed with the Solar Terrestrial Relations Observatory (STEREO) heliospheric imagers (HI) with data from 8 years of observations by 5 spacecraft in situ in the solar wind, thereby gathering over 600 independent in situ CME detections. Read More

We investigate the statistics of 288 low-latitude coronal holes extracted from SDO/AIA-193 filtergrams over the time range 2011/01/01 to 2013/12/31. We analyse the distribution of characteristic coronal hole properties, such as the areas, mean AIA-193 intensities, and mean magnetic field densities, the local distribution of the SDO/AIA-193 intensity and the magnetic field within the coronal holes, and the distribution of magnetic flux tubes in coronal holes. We find that the mean magnetic field density of all coronal holes under study is 3. Read More

Interplanetary space is characteristically structured mainly by high-speed solar wind streams emanating from coronal holes and transient disturbances such as coronal mass ejections (CMEs). While high-speed solar wind streams pose a continuous outflow, CMEs abruptly disrupt the rather steady structure causing large deviations from the quiet solar wind conditions. For the first time, we give a quantification of the duration of disturbed conditions (preconditioning) for interplanetary space caused by CMEs. Read More

Principal component analysis (PCA) is a standard tool for dimension reduction. In this paper, we analyse the reconstruction error of PCA and prove non-asymptotic upper bounds for the corresponding excess risk. These bounds unify and improve several upper bounds from the literature. Read More

High-speed solar wind streams emanating from coronal holes are frequently impinging on the Earth's magnetosphere causing recurrent, medium-level geomagnetic storm activity. Modeling high-speed solar wind streams is thus an essential element of successful space weather forecasting. Here we evaluate high-speed stream forecasts made by the empirical solar wind forecast (ESWF) and the semiempirical Wang-Sheeley-Arge (WSA) model based on the in situ plasma measurements from the ACE spacecraft for the years 2011 to 2014. Read More

For linear inverse problems $Y=\mathsf{A}\mu+\xi$, it is classical to recover the unknown signal $\mu$ by iterative regularisation methods $(\hat \mu^{(m)}, m=0,1,\ldots)$ and halt at a data-dependent iteration $\tau$ using some stopping rule, typically based on a discrepancy principle, so that the weak (or prediction) error $\|\mathsf{A}(\hat \mu^{(\tau)}-\mu)\|^2$ is controlled. In the context of statistical estimation with stochastic noise $\xi$, we study oracle adaptation (that is, compared to the best possible stopping iteration) in strong squared-error $E[\|\hat \mu^{(\tau)}-\mu\|^2]$. We give sharp lower bounds for such stopping rules in the case of the spectral cutoff method, and thus delineate precisely when adaptation is possible. Read More

We demonstrate the use of machine learning algorithms in combination with segmentation techniques in order to distinguish coronal holes and filaments in SDO/AIA EUV images of the Sun. Based on two coronal hole detection techniques (intensity-based thresholding, SPoCA), we prepared data sets of manually labeled coronal hole and filament channel regions present on the Sun during the time range 2011 - 2013. By mapping the extracted regions from EUV observations onto HMI line-of-sight magnetograms we also include their magnetic characteristics. Read More

**Authors:**Christian Möstl, Tanja Rollett, Rudy A. Frahm, Ying D. Liu, David M. Long, Robin C. Colaninno, Martin A. Reiss, Manuela Temmer, Charles J. Farrugia, Arik Posner, Mateja Dumbović, Miho Janvier, Pascal Démoulin, Peter Boakes, Andy Devos, Emil Kraaikamp, Mona L. Mays, Bojan Vrsnak

**Category:**Solar and Stellar Astrophysics

The severe geomagnetic effects of solar storms or coronal mass ejections (CMEs) are to a large degree determined by their propagation direction with respect to Earth. There is a lack of understanding of the processes that determine their non-radial propagation. Here we present a synthesis of data from seven different space missions of a fast CME, which originated in an active region near the disk centre and, hence, a significant geomagnetic impact was forecasted. Read More

Based on observations of points uniformly distributed over a convex set in $\R^d$, a new estimator for the volume of the convex set is proposed. The estimator is minimax optimal and also efficient non-asymptotically: it is nearly unbiased with minimal variance among all unbiased oracle-type estimators. Our approach is based on a Poisson point process model and as an ingredient, we prove that the convex hull is a sufficient and complete statistic. Read More

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). Read More

In this study, we describe and evaluate shape measures for distinguishing between coronal holes and filament channels as observed in Extreme Ultraviolet (EUV) images of the Sun. For a set of well-observed coronal hole and filament channel regions extracted from SDO/AIA 193\r{A} images we analyze their intrinsic morphology during the period 2011 to 2013, by using well known shape measures from the literature and newly developed geometrical classification methods. The results suggest an asymmetry in the morphology of filament channels giving support for the sheared arcade or weakly twisted flux rope model for filaments. Read More

For nonparametric regression with one-sided errors and a boundary curve model for Poisson point processes we consider the problem of efficient estimation for linear functionals. The minimax optimal rate is obtained by an unbiased estimation method which nevertheless depends on a H\"older condition or monotonicity assumption for the underlying regression or boundary function. We first construct a simple blockwise estimator and then build up a nonparametric maximum-likelihood approach for exponential noise variables and the point process model. Read More

In the paper, we suggest three tests on the validity of a factor model which can be applied for both small dimensional and large dimensional data. Both the exact and asymptotic distributions of the resulting test statistics are derived under classical and high-dimensional asymptotic regimes. It is shown that the critical values of the proposed tests can be calibrated empirically by generating a sample from the inverse Wishart distribution with identity parameter matrix. Read More

We derive a nonparametric test for constant beta over a fixed time interval from high-frequency observations of a bivariate \Ito semimartingale. Beta is defined as the ratio of the spot continuous covariation between an asset and a risk factor and the spot continuous variation of the latter. The test is based on the asymptotic behavior of the covariation between the risk factor and an estimate of the residual component of the asset, that is orthogonal (in martingale sense) to the risk factor, over blocks with asymptotically shrinking time span. Read More

We consider a Gaussian sequence space model $X_{\lambda}=f_{\lambda} + \xi_{\lambda},$ where $\xi $ has a diagonal covariance matrix $\Sigma=\diag(\sigma_\lambda ^2)$. We consider the situation where the parameter vector $(f_{\lambda})$ is sparse. Our goal is to estimate the unknown parameter by a model selection approach. Read More

Donsker-type functional limit theorems are proved for empirical processes arising from discretely sampled increments of a univariate L\'evy process. In the asymptotic regime the sampling frequencies increase to infinity and the limiting object is a Gaussian process that can be obtained from the composition of a Brownian motion with a covariance operator determined by the L\'evy measure. The results are applied to derive the asymptotic distribution of natural estimators for the distribution function of the L\'evy jump measure. Read More

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. Read More

For $n$ equidistant observations of a L\'evy process at time distance $\Delta_n$ we consider the problem of testing hypotheses on the volatility, the jump measure and its Blumenthal-Getoor index in a non- or semiparametric manner. Asymptotically as $n\to\infty$ we allow for both, the high-frequency regime $\Delta_n=\frac1n$ and the low-frequency regime $\Delta_n=1$ as well as intermediate cases. The approach via empirical characteristic function unifies existing theory and sheds new light on diverse results. Read More

An efficient estimator is constructed for the quadratic covariation or integrated co-volatility matrix of a multivariate continuous martingale based on noisy and nonsynchronous observations under high-frequency asymptotics. Our approach relies on an asymptotically equivalent continuous-time observation model where a local generalised method of moments in the spectral domain turns out to be optimal. Asymptotic semi-parametric efficiency is established in the Cram\'{e}r-Rao sense. Read More

Quantile estimation in deconvolution problems is studied comprehensively. In particular, the more realistic setup of unknown error distributions is covered. Our plug-in method is based on a deconvolution density estimator and is minimax optimal under minimal and natural conditions. Read More

The optimal rate of convergence of estimators of the integrated volatility, for a discontinuous It\^{o} semimartingale sampled at regularly spaced times and over a fixed time interval, has been a long-standing problem, at least when the jumps are not summable. In this paper, we study this optimal rate, in the minimax sense and for appropriate "bounded" nonparametric classes of semimartingales. We show that, if the $r$th powers of the jumps are summable for some $r\in[0,2)$, the minimax rate is equal to $\min(\sqrt{n},(n\log n)^{(2-r)/2})$, where $n$ is the number of observations. Read More

Given $n$ equidistant realisations of a L\'evy process $(L_t,\,t\ge 0)$, a natural estimator $\hat N_n$ for the distribution function $N$ of the L\'evy measure is constructed. Under a polynomial decay restriction on the characteristic function $\phi$, a Donsker-type theorem is proved, that is, a functional central limit theorem for the process $\sqrt n (\hat N_n -N)$ in the space of bounded functions away from zero. The limit distribution is a generalised Brownian bridge process with bounded and continuous sample paths whose covariance structure depends on the Fourier-integral operator ${\cal F}^{-1}[1/\phi(-\cdot)]$. Read More

We propose localized spectral estimators for the quadratic covariation and the spot covolatility of diffusion processes which are observed discretely with additive observation noise. The eligibility of this approach to lead to an appropriate estimation for time-varying volatilities stems from an asymptotic equivalence of the underlying statistical model to a white noise model with correlation and volatility processes being constant over small intervals. The asymptotic equivalence of the continuous-time and the discrete-time experiments are proved by a construction with linear interpolation in one direction and local means for the other. Read More

We consider discrete-time observations of a continuous martingale under measurement error. This serves as a fundamental model for high-frequency data in finance, where an efficient price process is observed under microstructure noise. It is shown that this nonparametric model is in Le Cam's sense asymptotically equivalent to a Gaussian shift experiment in terms of the square root of the volatility function $\sigma$ and a nonstandard noise level. Read More

Asymptotic equivalence in Le Cam's sense for nonparametric regression experiments is extended to the case of non-regular error densities, which have jump discontinuities at their endpoints. We prove asymptotic equivalence of such regression models and the observation of two independent Poisson point processes which contain the target curve as the support boundary of its intensity function. The intensity of the point processes is of order of the sample size $n$ and involves the jump sizes as well as the design density. Read More

The basic model for high-frequency data in finance is considered, where an efficient price process is observed under microstructure noise. It is shown that this nonparametric model is in Le Cam's sense asymptotically equivalent to a Gaussian shift experiment in terms of the square root of the volatility function $\sigma$. As an application, simple rate-optimal estimators of the volatility and efficient estimators of the integrated volatility are constructed. Read More

A nonparametric procedure for robust regression estimation and for quantile regression is proposed which is completely data-driven and adapts locally to the regularity of the regression function. This is achieved by considering in each point M-estimators over different local neighbourhoods and by a local model selection procedure based on sequential testing. Non-asymptotic risk bounds are obtained, which yield rate-optimality for large sample asymptotics under weak conditions. Read More

We study two nonlinear methods for statistical linear inverse problems when the operator is not known. The two constructions combine Galerkin regularization and wavelet thresholding. Their performances depend on the underlying structure of the operator, quantified by an index of sparsity. Read More

The quasi-optimality criterion chooses the regularization parameter in inverse problems without taking into account the noise level. This rule works remarkably well in practice, although Bakushinskii has shown that there are always counterexamples with very poor performance. We propose an average case analysis of quasi-optimality for spectral cut-off estimators and we prove that the quasi-optimality criterion determines estimators which are rate-optimal {\em on average}. Read More

In this paper, we clarify the relations between the existing sets of regularity conditions for convergence rates of nonparametric indirect regression (NPIR) and nonparametric instrumental variables (NPIV) regression models. We establish minimax risk lower bounds in mean integrated squared error loss for the NPIR and the NPIV models under two basic regularity conditions that allow for both mildly ill-posed and severely ill-posed cases. We show that both a simple projection estimator for the NPIR model, and a sieve minimum distance estimator for the NPIV model, can achieve the minimax risk lower bounds, and are rate-optimal uniformly over a large class of structure functions, allowing for mildly ill-posed and severely ill-posed cases. Read More

We suppose that a L\'evy process is observed at discrete time points. A rather general construction of minimum-distance estimators is shown to give consistent estimators of the L\'evy-Khinchine characteristics as the number of observations tends to infinity, keeping the observation distance fixed. For a specific $C^2$-criterion this estimator is rate-optimal. Read More

We show that nonparametric regression is asymptotically equivalent in Le Cam's sense with a sequence of Gaussian white noise experiments as the number of observations tends to infinity. We propose a general constructive framework based on approximation spaces, which permits to achieve asymptotic equivalence even in the cases of multivariate and random design. Read More

As a main step in the numerical solution of control problems in continuous time, the controlled process is approximated by sequences of controlled Markov chains, thus discretising time and space. A new feature in this context is to allow for delay in the dynamics. The existence of an optimal strategy with respect to the cost functional can be guaranteed in the class of relaxed controls. Read More

We consider a stochastic delay differential equation driven by a general Levy process. Both, the drift and the noise term may depend on the past, but only the drift term is assumed to be linear. We show that the segment process is eventually Feller, but in general not eventually strong Feller on the Skorokhod space. Read More

**Affiliations:**

^{1}PMA,

^{2}WIAS

Asymptotic local equivalence in the sense of Le Cam is established for inference on the drift in multidimensional ergodic diffusions and an accompanying sequence of Gaussian shift experiments. The nonparametric local neighbourhoods can be attained for any dimension, provided the regularity of the drift is sufficiently large. In addition, a heteroskedastic Gaussian regression experiment is given, which is also locally asymptotically equivalent and which does not depend on the centre of localisation. Read More

We study the problem of estimating the coefficients of a diffusion (X_t,t\geq 0); the estimation is based on discrete data X_{n\Delta},n=0,1,... Read More