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Preprints, Reports, Technical Reports

• F. Besold, V. Spokoiny, Adaptive weights community detection, Preprint no. 2951, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2951 .
  Abstract, PDF (628 kByte)
  Due to the technological progress of the last decades, Community Detection has become a major topic in machine learning. However, there is still a huge gap between practical and theoretical results, as theoretically optimal procedures often lack a feasible implementation and vice versa. This paper aims to close this gap and presents a novel algorithm that is both numerically and statistically efficient. Our procedure uses a test of homogeneity to compute adaptive weights describing local communities. The approach was inspired by the Adaptive Weights Community Detection (AWCD) algorithm by [2]. This algorithm delivered some promising results on artificial and real-life data, but our theoretical analysis reveals its performance to be suboptimal on a stochastic block model. In particular, the involved estimators are biased and the procedure does not work for sparse graphs. We propose significant modifications, addressing both shortcomings and achieving a nearly optimal rate of strong consistency on the stochastic block model. Our theoretical results are illustrated and validated by numerical experiments.

Articles in Refereed Journals

• A. Vasin, A. Gasnikov, P. Dvurechensky, V. Spokoiny, Accelerated gradient methods with absolute and relative noise in the gradient, Optimization Methods & Software, published online in June 2023, DOI 10.1080/10556788.2023.2212503 .

• V. Spokoiny, Dimension free non-asymptotic bounds on the accuracy of high dimensional Laplace approximation, SIAM/ASA Journal on Uncertainty Quantification, 11 (2023), pp. 1044--1068, DOI 10.1137/22M1495688 .
  Abstract
  This note attempts to revisit the classical results on Laplace approximation in a modern non-asymptotic and dimension free form. Such an extension is motivated by applications to high dimensional statistical and optimization problems. The established results provide explicit non-asymptotic bounds on the quality of a Gaussian approximation of the posterior distribution in total variation distance in terms of the so called empheffective dimension ( dimL ). This value is defined as interplay between information contained in the data and in the prior distribution. In the contrary to prominent Bernstein - von Mises results, the impact of the prior is not negligible and it allows to keep the effective dimension small or moderate even if the true parameter dimension is huge or infinite. We also address the issue of using a Gaussian approximation with inexact parameters with the focus on replacing the Maximum a Posteriori (MAP) value by the posterior mean and design the algorithm of Bayesian optimization based on Laplace iterations. The results are specified to the case of nonlinear regression.

• N. Puchkin, V. Spokoiny, Structure-adaptive manifold estimation, Journal of Machine Learning Research (JMLR). MIT Press, Cambridge, MA. English, English abstracts., 23 (2022), pp. 1--62.
  Abstract
  We consider a problem of manifold estimation from noisy observations. Many manifold learning procedures locally approximate a manifold by a weighted average over a small neighborhood. However, in the presence of large noise, the assigned weights become so corrupted that the averaged estimate shows very poor performance. We suggest a novel computationally efficient structure-adaptive procedure, which simultaneously reconstructs a smooth manifold and estimates projections of the point cloud onto this manifold. The proposed approach iteratively refines the weights on each step, using the structural information obtained at previous steps. After several iterations, we obtain nearly öracle" weights, so that the final estimates are nearly efficient even in the presence of relatively large noise. In our theoretical study we establish tight lower and upper bounds proving asymptotic optimality of the method for manifold estimation under the Hausdorff loss. Our finite sample study confirms a very reasonable performance of the procedure in comparison with the other methods of manifold estimation.

• CH. Bayer, D. Belomestny, P. Hager, P. Pigato, J.G.M. Schoenmakers, V. Spokoiny, Reinforced optimal control, Communications in Mathematical Sciences, 20 (2022), pp. 1951--1978, DOI 10.4310/CMS.2022.v20.n7.a7 .
  Abstract
  Least squares Monte Carlo methods are a popular numerical approximation method for solving stochastic control problems. Based on dynamic programming, their key feature is the approximation of the conditional expectation of future rewards by linear least squares regression. Hence, the choice of basis functions is crucial for the accuracy of the method. Earlier work by some of us [Belomestny, Schoenmakers, Spokoiny, Zharkynbay, Commun. Math. Sci., 18(1):109?121, 2020] proposes to reinforce the basis functions in the case of optimal stopping problems by already computed value functions for later times, thereby considerably improving the accuracy with limited additional computational cost. We extend the reinforced regression method to a general class of stochastic control problems, while considerably improving the method?s efficiency, as demonstrated by substantial numerical examples as well as theoretical analysis.

• A. Kroshnin, V. Spokoiny, A. Suvorikova, Statistical inference for Bures--Wasserstein barycenters, The Annals of Applied Probability, 31 (2021), pp. 1264--1298, DOI 10.1214/20-AAP1618 .

• L.-Ch. Lin, Y. Chen, G. Pan, V. Spokoiny, Efficient and positive semidefinite pre-averaging realized covariance estimator, Statistica Sinica, 31 (2021), pp. 1441--1462, DOI 10.5705/ss.202017.0489 .

• F. Bachoc, A. Suvorikova , D. Ginsbourger, J.-M. Loubes, V. Spokoiny, Gaussian processes with multidimensional distribution inputs via optimal transport and Hilbertian embedding, Electronic Journal of Statistics, 14 (2020), pp. 2742--2772, DOI 10.1214/20-EJS1725 .
  Abstract
  In this work, we propose a way to construct Gaussian processes indexed by multidimensional distributions. More precisely, we tackle the problem of defining positive definite kernels between multivariate distributions via notions of optimal transport and appealing to Hilbert space embeddings. Besides presenting a characterization of radial positive definite and strictly positive definite kernels on general Hilbert spaces, we investigate the statistical properties of our theoretical and empirical kernels, focusing in particular on consistency as well as the special case of Gaussian distributions. A wide set of applications is presented, both using simulations and implementation with real data.

• D. Belomestny, J.G.M. Schoenmakers, V. Spokoiny, B. Zharkynbay, Optimal stopping via reinforced regression, Communications in Mathematical Sciences, 18 (2020), pp. 109--121, DOI 10.4310/CMS.2020.v18.n1.a5 .
  Abstract
  In this note we propose a new approach towards solving numerically optimal stopping problems via boosted regression based Monte Carlo algorithms. The main idea of the method is to boost standard linear regression algorithms in each backward induction step by adding new basis functions based on previously estimated continuation values. The proposed methodology is illustrated by several numerical examples from finance.

• L.-Ch. Lin, Y. Chen, G. Pan, V. Spokoiny, Efficient and positive semidefinite pre-averaging realized covariance estimator, Statistica Sinica, 31 (2021), pp. 1--22 (published online on 23.11.2020), DOI 10.5705/ss.202017.0489 .

• N. Puchkin, V. Spokoiny, An adaptive multiclass nearest neighbor classifier, ESAIM. Probability and Statistics, 24 (2020), pp. 69--99, DOI 10.1051/ps/2019021 .

• K. Efimov, L. Adamyan, V. Spokoiny, Adaptive nonparametric clustering, IEEE Transactions on Information Theory, 65 (2019), pp. 4875--4892, DOI 10.1109/TIT.2019.2903113 .
  Abstract
  This paper presents a new approach to non-parametric cluster analysis called adaptive weights? clustering. The method is fully adaptive and does not require to specify the number of clusters or their structure. The clustering results are not sensitive to noise and outliers, and the procedure is able to recover different clusters with sharp edges or manifold structure. The method is also scalable and computationally feasible. Our intensive numerical study shows a state-of-the-art performance of the method in various artificial examples and applications to text data. The idea of the method is to identify the clustering structure by checking at different points and for different scales on departure from local homogeneity. The proposed procedure describes the clustering structure in terms of weights $w_ij$ , and each of them measures the degree of local inhomogeneity for two neighbor local clusters using statistical tests of ?no gap? between them. The procedure starts from very local scale, and then, the parameter of locality grows by some factor at each step. We also provide a rigorous theoretical study of the procedure and state its optimal sensitivity to deviations from local homogeneity.

• F. Götze, A. Naumov, V. Spokoiny, V. Ulyanov, Large ball probabilities, Gaussian comparison and anti-concentration, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 25 (2019), pp. 2538--2563, DOI 10.3150/18-BEJ1062 .
  Abstract
  We derive tight non-asymptotic bounds for the Kolmogorov distance between the probabilities of two Gaussian elements to hit a ball in a Hilbert space. The key property of these bounds is that they are dimension-free and depend on the nuclear (Schatten-one) norm of the difference between the covariance operators of the elements and on the norm of the mean shift. The obtained bounds significantly improve the bound based on Pinsker?s inequality via the Kullback?Leibler divergence. We also establish an anti-concentration bound for a squared norm of a non-centered Gaussian element in Hilbert space. The paper presents a number of examples motivating our results and applications of the obtained bounds to statistical inference and to high-dimensional CLT.

• A. Naumov, V. Spokoiny, V. Ulyanov, Bootstrap confidence sets for spectral projectors of sample covariance, Probability Theory and Related Fields, 174 (2019), pp. 1091--1132, DOI 10.1007/s00440-018-0877-2 .

• V. Spokoiny, N. Willrich, Bootstrap tuning in Gaussian ordered model selection, The Annals of Statistics, 47 (2019), pp. 1351--1380, DOI 10.1214/18-AOS1717 .
  Abstract
  In the problem of model selection for a given family of linear estimators, ordered by their variance, we offer a new “smallest accepted” approach motivated by Lepski's device and the multiple testing idea. The procedure selects the smallest model which satisfies the acceptance rule based on comparison with all larger models. The method is completely data-driven and does not use any prior information about the variance structure of the noise: its parameters are adjusted to the underlying possibly heterogeneous noise by the so called “propagation condition” using bootstrap multiplier method. The validity of the bootstrap calibration is proved for finite samples with an explicit error bound. We provide a comprehensive theoretical study of the method and describe in details the set of possible values of the selector ( hatm ). We also establish some precise oracle error bounds for the corresponding estimator ( hattheta = tildetheta_hatm ) which equally applies to estimation of the whole parameter vectors, its subvector or linear mapping, as well as estimation of a linear functional.

• A. Naumov, V. Spokoiny, Y. Tavyrikov, V. Ulyanov, Non-asymptotic estimates of the closeness of Gaussian measures on the balls, Doklady Mathematics. Maik Nauka/Interperiodica Publishing, Moscow. English. Translation of the Mathematics Section of: Doklady Akademii Nauk. (Formerly: Russian Academy of Sciences. Doklady. Mathematics)., 98 (2018), pp. 490--493.

• A. Naumov, V. Spokoiny, V. Ulyanov, Bootstrap confidence sets for spectral projectors of sample covariance, Probability Theory and Related Fields, pp. published online on 26.10.2018, urlhttps://doi.org/10.1007/s00440-018-0877-2, DOI 10.1007/s00440-018-0877-2 .
  Abstract
  Let X1,?,Xn be i.i.d. sample in ?p with zero mean and the covariance matrix ?. The problem of recovering the projector onto an eigenspace of ? from these observations naturally arises in many applications. Recent technique from [Koltchinskii, Lounici, 2015] helps to study the asymptotic distribution of the distance in the Frobenius norm ?Pr?P?r?2 between the true projector Pr on the subspace of the r-th eigenvalue and its empirical counterpart P?r in terms of the effective rank of ?. This paper offers a bootstrap procedure for building sharp confidence sets for the true projector Pr from the given data. This procedure does not rely on the asymptotic distribution of ?Pr?P?r?2 and its moments. It could be applied for small or moderate sample size n and large dimension p. The main result states the validity of the proposed procedure for finite samples with an explicit error bound for the error of bootstrap approximation. This bound involves some new sharp results on Gaussian comparison and Gaussian anti-concentration in high-dimensional spaces. Numeric results confirm a good performance of the method in realistic examples.

• A. Naumov, V. Spokoiny, V. Ulyanov, Confidence sets for spectral projectors of covariance matrices, Doklady Mathematics. Maik Nauka/Interperiodica Publishing, Moscow. English. Translation of the Mathematics Section of: Doklady Akademii Nauk. (Formerly: Russian Academy of Sciences. Doklady. Mathematics)., 98 (2018), pp. 511--514.

• I. Silin, V. Spokoiny, Bayesian inference for spectral projectors of covariance matrix, Electronic Journal of Statistics, 12 (2018), pp. 1948--1987, DOI 10.1214/18-EJS1451 .

• A. Andresen, V. Spokoiny, Convergence of an alternating maximization procedure, Journal of Machine Learning Research (JMLR). MIT Press, Cambridge, MA. English, English abstracts., 53 (2017), pp. 389--429, DOI 10.1214/15-AIHP720 .

• Y. Nesterov, V. Spokoiny, Random gradient-free minimization of convex functions, Foundations of Computational Mathematics. The Journal of the Society for the Foundations of Computational Mathematics, 17 (2017), pp. 527--566.
  Abstract
  Summary: In this paper, we prove new complexity bounds for methods of convex optimization based only on computation of the function value. The search directions of our schemes are normally distributed random Gaussian vectors. It appears that such methods usually need at most nn times more iterations than the standard gradient methods, where nn is the dimension of the space of variables. This conclusion is true for both nonsmooth and smooth problems. For the latter class, we present also an accelerated scheme with the expected rate of convergence O(n2k2)O(n2k2), where kk is the iteration counter. For stochastic optimization, we propose a zero-order scheme and justify its expected rate of convergence O(nk1/2)O(nk1/2). We give also some bounds for the rate of convergence of the random gradient-free methods to stationary points of nonconvex functions, for both smooth and nonsmooth cases. Our theoretical results are supported by preliminary computational experiments.

• V. Spokoiny, Penalized maximum likelihood estimation and effective dimension, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 53 (2017), pp. 389--429, DOI 10.1214/15-AIHP720 .

• A. Kalinina, A. Suvorikova, V. Spokoiny, M. Gelfand, Detection of homologous recombination in closely related strains, Journal of Bioinformatics and Computational Biology, 14 (2016), pp. 1641001/1--1641001/12.

• A. Andresen, V. Spokoiny, Convergence for an alternation maximization procedure, Journal of Machine Learning Research (JMLR). MIT Press, Cambridge, MA. English, English abstracts., 17 (2016), pp. 1--53.

• A. Gasnikov, P. Dvurechensky, V. Spokoiny, P. Stetsyuk, A. Suvorikova, Superposition of the balancing algorithm and the universal gradient method for search of the regularized Wasserstein barycenter and equilibria in multistage transport models, Proceedings of Moscow Institute of Physics and Technology, 8 (2016), pp. 5--24.

• Y. Chen, V. Spokoiny, Modeling nonstationary and leptokurtic financial time series, Econometric Theory, (2015), pp. 703--728.

• A. Gasnikov, Y. Nesterov, V. Spokoiny, On the efficiency of a randomized mirror descent algorithm in online optimization problems, Computational Mathematics and Mathematical Physics, 55 (2015), pp. 580--596.

• A. Gasnikov, P. Dvurechensky, D. Kamzolov, Y. Nesterov, V. Spokoiny, P. Stetsyuk, A. Suvorikova, A. Chernov, Searching for equilibriums in multistage transport models (in Russian), Proceedings of Moscow Institute of Physics and Technology, 7 (2015), pp. 143--155.

• V.G. Gitis, A.B. Derendyaev, S.A. Pirogov, V. Spokoiny, E.F. Yurkov, Adaptive estimation of seismic parameter fields from earthquakes catalogs, Journal of Communications Technology and Electronics, 60 (2015), pp. 1459--1465.

• M. Panov, V. Spokoiny, Finite sample Bernstein--von Mises theorem for semiparametric problems, Bayesian Analysis, 10 (2015), pp. 665--710.

• P. Dvurechensky, Y. Nesterov, V. Spokoiny, Primal-dual methods for solving infinite-dimensional games, Journal of Optimization Theory and Applications, 166 (2015), pp. 23--51.

• V. Spokoiny, M. Zhilova, Bootstrap confidence sets under a model misspecification, The Annals of Statistics, 43 (2015), pp. 2653--2675.
  Abstract
  A multiplier bootstrap procedure for construction of likelihood-based confidence sets is considered for finite samples and possible model misspecification. Theoretical results justify the bootstrap consistency for small or moderate sample size and allow to control the impact of the parameter dimension: the bootstrap approximation works if the ratio of cube of the parameter dimension to the sample size is small. The main result about bootstrap consistency continues to apply even if the underlying parametric model is misspecified under the so called Small Modeling Bias condition. In the case when the true model deviates significantly from the considered parametric family, the bootstrap procedure is still applicable but it becomes a bit conservative: the size of the constructed confidence sets is increased by the modeling bias. We illustrate the results with numerical examples of misspecified constant and logistic regressions.

• A. Andresen, V. Spokoiny, Critical dimension in profile semiparametric estimation, Electronic Journal of Statistics, 8 (2014), pp. 3077--3125.

• N. Baldin, V. Spokoiny, Bayesian model selection and the concentration of the posterior of hyperparameters, Journal of Mathematical Sciences (New York), 203 (2014), pp. 761--776.

• D. Belomestny, V. Spokoiny, Concentration inequalities for smooth random fields, Theory of Probability and its Applications, 58 (2014), pp. 314--323.
  Abstract
  In this note we derive a sharp concentration inequality for the supremum of a smooth random field over a finite dimensional set. It is shown that this supremum can be bounded with high probability by the value of the field at some deterministic point plus an intrinsic dimension of the optimisation problem. As an application we prove the exponential inequality for a function of the maximal eigenvalue of a random matrix is proved.

• G. Milshteyn, V. Spokoiny, Construction of mean-self-financing strategies for European options under regime-switching, SIAM Journal on Financial Mathematics, ISSN 1945-497X, 5 (2014), pp. 532--556.
  Abstract
  The paper focuses on the problem of pricing and hedging a European contingent claim for an incomplete market model, in which evolution of price processes for a saving account and stocks depends on an observable Markov chain. The pricing function is evaluated using the martingale approach. The equivalent martingale measure is introduced in a way that the Markov chain remains the historical one. Due to the Markovian structure of the considered model, the pricing function satisfies the Cauchy problem for a system of linear parabolic partial differential equations. It is shown that any European contingent claim is attainable using a generalized replicating strategy which is self-financing in mean. For such a strategy, apart from the initial endowment and trading, some additional funds are required both step-wise at the jump moments of the Markov chain and continuously between the jump moments. The connection of the considered pricing and hedging problems with partial differential equations is very useful for computations.

• A. Zaytsev, E. Burnaev, V. Spokoiny, Properties of the Bayesian parameter estimation of a regression based on Gaussian processes, Journal of Mathematical Sciences (New York), 203 (2014), pp. 789--798.

• M. Zhilova, V. Spokoiny, Uniform properties of the local maximum likelihood estimate, Automation and Remote Control, 74 (2013), pp. 1656--1669.

• D. Belomestny, V. Spokoiny, Concentration inequalities for smooth random fields, Theory of Probability and its Applications, (2013), pp. 401--410.

• E. Burnaev, A. Zaytsev, V. Spokoiny, Non-asymptotic properties for Gaussian field regression, Automation and Remote Control, 74 (2013), pp. 1645--1655.

• E. Burnaev, A. Zaitsev, V. Spokoiny, Bernstein--von Mises theorem for regression based on Gaussian processes, Russian Mathematical Surveys, 68 (2013), pp. 954--956.

• E. Diederichs, A. Juditsky, A. Nemirovski, V. Spokoiny, Sparse non Gaussian component analysis by semidefinite programming, Journal of Machine Learning Research (JMLR). MIT Press, Cambridge, MA. English, English abstracts., (2013), pp. 1--28.

• F. Gach, R. Nickl, V. Spokoiny, Spatially adaptive density estimation by localised Haar projections, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 49 (2013), pp. 900--914.

• A. Zaitsev, E. Burnaev, V. Spokoiny, Properties of the posterior distribution of a regression model based on Gaussian random fields, Automation and Remote Control, 74 (2013), pp. 1645--1655.

• V. Spokoiny, M. Zhilova, Sharp deviation bounds for quadratic forms, Mathematical Methods of Statistics, 22 (2013), pp. 100--113.

• V. Spokoiny, W. Wang, W. Härdle, Local quantile regression (with rejoinder), Journal of Statistical Planning and Inference, 143 (2013), pp. 1109--1129.

• V. Spokoiny, Parametric estimation. Finite sample theory, The Annals of Statistics, 40 (2012), pp. 2877--2909.

• Y. Chen, W. Härdle, V. Spokoiny, GHICA --- Risk analysis with GH distributions and independent components, Journal of Empirical Finance, 17 (2010), pp. 255--269.

• E. Diederichs, A. Juditsky, V. Spokoiny, Ch. Schütte, Sparse non-Gaussian component analysis, IEEE Transactions on Information Theory, 56 (2010), pp. 3033--3047.

• P. Čížek, W. Härdle, V. Spokoiny, Adaptive pointwise estimation in time-inhomogeneous conditional heteroscedasticity models, The Econometrics Journal, 12 (2009), pp. 248--271.

• E. Giacomini, W. Härdle, V. Spokoiny, Inhomogeneous dependency modelling with time varying copulae, Journal of Business & Economic Statistics, 27 (2009), pp. 224--234.
  Abstract
  Measuring dependence in a multivariate time series is tantamount to modelling its dynamic structure in space and time. In the context of a multivariate normally distributed time series, the evolution of the covariance (or correlation) matrix over time describes this dynamic. A wide variety of applications, though, requires a modelling framework different from the multivariate normal. In risk management the non-normal behaviour of most financial time series calls for nonlinear dependency. The correct modelling of non-gaussian dependencies is therefore a key issue in the analysis of multivariate time series. In this paper we use copulae functions with adaptively estimated time varying parameters for modelling the distribution of returns, free from the usual normality assumptions. Further, we apply copulae to estimation of Value-at-Risk (VaR) of a portfolio and show its better performance over the RiskMetrics approach, a widely used methodology for VaR estimation

• Y. Golubev, V. Spokoiny, Exponential bounds for minimum contrast estimators, Electronic Journal of Statistics, 3 (2009), pp. 712--746.

• D. Belomestny, G.N. Milstein, V. Spokoiny, Regression methods in pricing American and Bermudan options using consumption processes, Quantitative Finance, 9 (2009), pp. 315--327.
  Abstract
  Here we develop methods for efficient pricing multidimensional discrete-time American and Bermudan options by using regression based algorithms together with a new approach towards constructing upper bounds for the price of the option. Applying sample space with payoffs at the optimal stopping times, we propose sequential estimates for continuation values, values of the consumption process, and stopping times on the sample paths. The approach admits constructing both low and upper bounds for the price by Monte Carlo simulations. The methods are illustrated by pricing Bermudan swaptions and snowballs in the Libor market model.

• V. Spokoiny, C. Vial, Parameter tuning in pointwise adaptation using a propagation approach, The Annals of Statistics, 37 (2009), pp. 2783--2807.
  Abstract, PDF (550 kByte)
  This paper discusses the problem of adaptive estimating a univariate object like the value of a regression function at a given point or a linear functional in a linear inverse problem. We consider an adaptive procedure originated from Lepski (1990) which selects in a data-driven way one estimate out of a given class of estimates ordered by their variability. A serious problem with using this and similar procedures is the choice of some tuning parameters like thresholds. Numerical results show that the theoretically recommended proposals appear to be too conservative and lead to a strong oversmoothing effects. A careful choice of the parameters of the procedure is extremely important for getting the reasonable quality of estimation. The main contribution of this paper is the new approach for choosing the parameters of the procedure by providing the prescribed behavior of the resulting estimate in the simple parametric situation. We establish a non-asymptotical “oracle” bound which shows that the estimation risk is, up to a logarithmic multiplier, equal to the risk of the “oracle” estimate which is optimally selected from the given family. A numerical study demonstrates the nice performance of the resulting procedure in a number of simulated examples.

• V. Spokoiny, Multiscale local change point detection with applications to Value-at-Risk, The Annals of Statistics, 37 (2009), pp. 1405--1436.
  Abstract, PDF (640 kByte)
  This paper offers a new procedure for nonparametric estimation and forecasting of time series with applications to volatility modeling for financial data. The approach is based on the assumption of local homogeneity: for every time point there exists a historical emphinterval of homogeneity, in which the volatility parameter can be well approximated by a constant. The procedure recovers this interval from the data using the local change point (LCP) analysis. Afterwards the estimate of the volatility can be simply obtained by local averaging. The approach carefully addresses the question of choosing the tuning parameters of the procedure using the so called “propagation” condition. The main result claims a new “oracle” inequality in terms of the modeling bias which measures the quality of the local constant approximation. This result yields the optimal rate of estimation for smooth and piecewise constant volatility functions. Then the new procedure is applied to some data sets and a comparison with a standard GARCH model is also provided. Finally we discuss applications of the new method to the Value at Risk problem. The numerical results demonstrate a very reasonable performance of the new method.

• K. Tabelow, J. Polzehl, V. Spokoiny, H.U. Voss, Diffusion tensor imaging: Structural adaptive smoothing, NeuroImage, 39 (2008), pp. 1763--1773.
  Abstract
  Diffusion Tensor Imaging (DTI) data is characterized by a high noise level. Thus, estimation errors of quantities like anisotropy indices or the main diffusion direction used for fiber tracking are relatively large and may significantly confound the accuracy of DTI in clinical or neuroscience applications. Besides pulse sequence optimization, noise reduction by smoothing the data can be pursued as a complementary approach to increase the accuracy of DTI. Here, we suggest an anisotropic structural adaptive smoothing procedure, which is based on the Propagation-Separation method and preserves the structures seen in DTI and their different sizes and shapes. It is applied to artificial phantom data and a brain scan. We show that this method significantly improves the quality of the estimate of the diffusion tensor and hence enables one either to reduce the number of scans or to enhance the input for subsequent analysis such as fiber tracking.

• A. Dalalyan, A. Juditsky, V. Spokoiny, A new algorithm for estimating the effective dimension-reduction subspace, Journal of Machine Learning Research (JMLR). MIT Press, Cambridge, MA. English, English abstracts., 9 (2008), pp. 1647--1678.
  PDF (290 kByte)

• I.G. Grama, V. Spokoiny, Statistics of extremes by oracle estimation, The Annals of Statistics, 36 (2008), pp. 1619--1648.
  Abstract, PDF (3712 kByte)
  We use the fitted Pareto law to construct an accompanying approximation of the excess distribution function. A selection rule of the location of the excess distribution function is proposed based on a stagewise lack-of-fit testing procedure. Our main result is an oracle type inequality for the Kullback-Leibler loss of the obtained adaptive estimator.

• V. Katkovnik, V. Spokoiny, Spatially adaptive estimation via fitted local likelihood techniques, IEEE Transactions on Signal Processing, 56 (2008), pp. 873--886.
  Abstract, PDF (1230 kByte)
  This paper offers a new technique for spatially adaptive estimation. The local likelihood is exploited for nonparametric modelling of observations and estimated signals. The approach is based on the assumption of a local homogeneity of the signal: for every point there exists a neighborhood in which the signal can be well approximated by a constant. The fitted local likelihood statistics is used for selection of an adaptive size of this neighborhood. The algorithm is developed for quite a general class of observations subject to the exponential distribution. The estimated signal can be uni- and multivariable. We demonstrate a good performance of the new algorithm for Poissonian image denoising and compare of the new method versus the intersection of confidence interval (ICI) technique that also exploits a selection of an adaptive neighborhood for estimation.

• Y. Chen, W. Härdle, V. Spokoiny, Portfolio value at risk based on independent components analysis, Journal of Computational and Applied Mathematics, 205 (2007), pp. 594--607.
  Abstract, PDF (446 kByte)
  Risk management technology applied to high-dimensional portfolios needs simple and fast methods for calculation of value at risk (VaR). The multivariate normal framework provides a simple off-the-shelf methodology but lacks the heavy-tailed distributional properties that are observed in data. A principle component-based method (tied closely to the elliptical structure of the distribution) is therefore expected to be unsatisfactory. Here, we propose and analyze a technology that is based on independent component analysis (ICA). We study the proposed ICVaR methodology in an extensive simulation study and apply it to a high-dimensional portfolio situation. Our analysis yields very accurate VaRs.

• G.N. Milstein, J.G.M. Schoenmakers, V. Spokoiny, Forward and reverse representations for Markov chains, Stochastic Processes and their Applications, 117 (2007), pp. 1052--1075.
  Abstract, PDF (274 kByte)
  In this paper we carry over the concept of reverse probabilistic representations developed in Milstein, Schoenmakers, Spokoiny (2004) for diffusion processes, to discrete time Markov chains. We outline the construction of reverse chains in several situations and apply this to processes which are connected with jump-diffusion models and finite state Markov chains. By combining forward and reverse representations we then construct transition density estimators for chains which have root-N accuracy in any dimension and consider some applications.

• D. Belomestny, V. Spokoiny, Spatial aggregation of local likelihood estimates with applications to classification, The Annals of Statistics, 35 (2007), pp. 2287--2311.
  Abstract, PDF (373 kByte)
  This paper presents a new method for spatially adaptive local (constant) likelihood estimation which applies to a broad class of nonparametric models, including the Gaussian, Poisson and binary response models. The main idea of the method is given a sequence of local likelihood estimates (”weak” estimates), to construct a new aggregated estimate whose pointwise risk is of order of the smallest risk among all “weak” estimates. We also propose a new approach towards selecting the parameters of the procedure by providing the prescribed behavior of the resulting estimate in the simple parametric situation. We establish a number of important theoretical results concerning the optimality of the aggregated estimate. In particular, our “oracle” results claims that its risk is up to some logarithmic multiplier equal to the smallest risk for the given family of estimates. The performance of the procedure is illustrated by application to the classification problem. A numerical study demonstrates its nice performance in simulated and real life examples.

• K. Tabelow, J. Polzehl, H.U. Voss, V. Spokoiny, Analyzing fMRI experiments with structural adaptive smoothing procedures, NeuroImage, 33 (2006), pp. 55--62.
  Abstract, PDF (281 kByte)
  Data from functional magnetic resonance imaging (fMRI) consists of time series of brain images which are characterized by a low signal-to-noise ratio. In order to reduce noise and to improve signal detection the fMRI data is spatially smoothed. However, the common application of a Gaussian filter does this at the cost of loss of information on spatial extent and shape of the activation area. We suggest to use the propagation-separation procedures introduced by Polzehl and Spokoiny (2006) instead. We show that this significantly improves the information on the spatial extent and shape of the activation region with similar results for the noise reduction. To complete the statistical analysis, signal detection is based on thresholds defined by random field theory. Effects of ad aptive and non-adaptive smoothing are illustrated by artificial examples and an analysis of experimental data.

• G. Blanchard, M. Kawanabe, M. Sugiyama, V. Spokoiny, K.-R. Müller, In search of non-Gaussian components of a high-dimensional distribution, Journal of Machine Learning Research (JMLR). MIT Press, Cambridge, MA. English, English abstracts., 7 (2006), pp. 247--282.
  Abstract, PDF (1502 kByte)
  Finding non-Gaussian components of high-dimensional data is an important preprocessing step for efficient information processing. This article proposes a new em linear method to identify the “non-Gaussian subspace” within a very general semi-parametric framework. Our proposed method, called NGCA (Non-Gaussian Component Analysis), is essentially based on the fact that we can construct a linear operator which, to any arbitrary nonlinear (smooth) function, associates a vector which belongs to the low dimensional non-Gaussian target subspace up to an estimation error. By applying this operator to a family of different nonlinear functions, one obtains a family of different vectors lying in a vicinity of the target space. As a final step, the target space itself is estimated by applying PCA to this family of vectors. We show that this procedure is consistent in the sense that the estimaton error tends to zero at a parametric rate, uniformly over the family. Numerical examples demonstrate the usefulness of our method.

• A. Goldenshluger, V. Spokoiny, Recovering convex edges of image from noisy tomographic data, IEEE Transactions on Information Theory, 52 (2006), pp. 1322--1334.
  PDF (906 kByte)

• J. Polzehl, V. Spokoiny, Propagation-separation approach for local likelihood estimation, Probability Theory and Related Fields, 135 (2006), pp. 335--362.
  Abstract, PDF (793 kByte)
  The paper presents a unified approach to local likelihood estimation for a broad class of nonparametric models, including, e.g., regression, density, Poisson and binary response models. The method extends the adaptive weights smoothing (AWS) procedure introduced by the authors [Adaptive weights smoothing with applications to image sequentation. J. R. Stat. Soc., Ser. B 62, 335-354 (2000)] in the context of image denoising. The main idea of the method is to describe a greatest possible local neighborhood of every design point in which the local parametric assumption is justified by the data. The method is especially powerful for model functions having large homogeneous regions and sharp discontinuities. The performance of the proposed procedure is illustrated by numerical examples for density estimation and classification. We also establish some remarkable theoretical non-asymptotic results on properties of the new algorithm. This includes the “propagation” property which particularly yields the root-$n$ consistency of the resulting estimate in the homogeneous case. We also state an “oracle” result which implies rate optimality of the estimate under usual smoothness conditions and a “separation” result which explains the sensitivity of the method to structural changes.

• A. Samarov, V. Spokoiny, C. Vial, Component identification and estimation in nonlinear high-dimensional regression models by structural adaptation, Journal of the American Statistical Association, 100 (2005), pp. 429--445.
  PDF (454 kByte)

• M. Giurcanu, V. Spokoiny, Confidence estimation of the covariance function of stationary and locally stationary processes, Statistics & Decisions. International Journal for Statistical Theory and Related Fields, 22 (2004), pp. 283--300.
  PDF (224 kByte)

• A. Goldenshluger, V. Spokoiny, On the shape-from-moments problem and recovering edges from noisy Radon data, Probability Theory and Related Fields, 128 (2004), pp. 123--140.
  PDF (148 kByte)

• G.N. Milstein, J.G.M. Schoenmakers, V. Spokoiny, Transition density estimation for stochastic differential equations via forward-reverse representations, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 10 (2004), pp. 281--312.
  Abstract, PDF (274 kByte)
  The general reverse diffusion equations are derived and applied to the problem of transition density estimation of diffusion processes between two fixed states. For this problem we propose density estimation based on forward?reverse representations and show that this method allows essentially better results to be achieved than the usual kernel or projection estimation based on forward representations only.

• V. Spokoiny, D. Mercurio, Statistical inference for time-inhomogeneous volatility models, The Annals of Statistics, 32 (2004), pp. 577--602.
  PDF (438 kByte)

• W. Härdle, H. Herwatz, V. Spokoiny, Time inhomogeneous multiple volatility modelling, Journal of Financial Econometrics, 1 (2003), pp. 55-95.

• J. Polzehl, V. Spokoiny, Image denoising: Pointwise adaptive approach, The Annals of Statistics, 31 (2003), pp. 30--57.
  Abstract, PDF (507 kByte)
  A new method of pointwise adaptation has been proposed and studied in Spokoiny (1998) in context of estimation of piecewise smooth univariate functions. The present paper extends that method to estimation of bivariate grey-scale images composed of large homogeneous regions with smooth edges and observed with noise on a gridded design. The proposed estimator $, hatf(x) ,$ at a point $, x ,$ is simply the average of observations over a window $, hatU(x) ,$ selected in a data-driven way. The theoretical properties of the procedure are studied for the case of piecewise constant images. We present a nonasymptotic bound for the accuracy of estimation at a specific grid point $, x ,$ as a function of the number of pixel $n$, of the distance from the point of estimation to the closest boundary and of smoothness properties and orientation of this boundary. It is also shown that the proposed method provides a near optimal rate of estimation near edges and inside homogeneous regions. We briefly discuss algorithmic aspects and the complexity of the procedure. The numerical examples demonstrate a reasonable performance of the method and they are in agreement with the theoretical issues. An example from satellite (SAR) imaging illustrates the applicability of the method.

• J.L. Horowitz, V. Spokoiny, An adaptive, rate-optimal test of linearity for median regression models, Journal of the American Statistical Association, 97 (2002), pp. 822--835.
  PDF (282 kByte), Postscript (694 kByte)

• R. Liptser, A.Y. Veretennikov, V. Spokoiny, Freidlin-Wentzell type moderate deviations for smooth processes, Markov Processes and Related Fields, 8 (2002), pp. 611-636.
  PDF (348 kByte), Postscript (343 kByte)

• V. Spokoiny, Variance estimation for high-dimensional regression models, Journal of Multivariate Analysis, 82 (2002), pp. 111--133.
  PDF (295 kByte), Postscript (278 kByte)

• L. Dümbgen, V. Spokoiny, Multiscale testing of qualitative hypotheses, The Annals of Statistics, 29 (2001), pp. 124--152.
  PDF (324 kByte)

• W. Härdle, S. Sperlich, V. Spokoiny, Structural tests for additive regression, Journal of the American Statistical Association, 96 (2001), pp. 1333--1347.
  PDF (382 kByte)

• J.L. Horowitz, V. Spokoiny, An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative, Econometrica. Journal of the Econometric Society, 69 (2001), pp. 599--631.
  PDF (422 kByte), Postscript (1830 kByte)

• M. Hristache, A. Juditsky, J. Polzehl, V. Spokoiny, Structure adaptive approach for dimension reduction, The Annals of Statistics, 29 (2001), pp. 1537--1566.
  Abstract, PDF (265 kByte)
  We propose a new method of effective dimension reduction for a multi-index model which is based on iterative improvement of the family of average derivative estimates. The procedure is computationally straightforward and does not require any prior information about the structure of the underlying model. We show that in the case when the effective dimension $m$ of the index space does not exceed $3$, this space can be estimated with the rate $n^-1/2$ under rather mild assumptions on the model.

• M. Hristache, A. Juditsky, V. Spokoiny, Direct estimation of the index coefficient in a single-index model, The Annals of Statistics, 29 (2001), pp. 595--623.
  PDF (247 kByte)

• J. Polzehl, V. Spokoiny, Functional and dynamic Magnetic Resonance Imaging using vector adaptive weights smoothing, Journal of the Royal Statistical Society. Series C. Applied Statistics, 50 (2001), pp. 485--501.
  Abstract, PDF (371 kByte)
  We consider the problem of statistical inference for functional and dynamic Magnetic Resonance Imaging (MRI). A new approach is proposed which extends the adaptive weights smoothing (AWS) procedure from Polzehl and Spokoiny (2000) originally designed for image denoising. We demonstrate how the AWS method can be applied for time series of images, which typically occur in functional and dynamic MRI. It is shown how signal detection in functional MRI and analysis of dynamic MRI can benefit from spatially adaptive smoothing. The performance of the procedure is illustrated using real and simulated data.

• V. Spokoiny, Data driven testing the fit of linear models, Mathematical Methods of Statistics, 10 (2001), pp. 465--497.
  PDF (373 kByte), Postscript (421 kByte)

• R. Liptser, V. Spokoiny, Deviation probability bound for martingales with applications to statistical estimation, Statistics & Probability Letters, 46 (2000), pp. 347--357.
  PDF (269 kByte)

• R. Liptser, V. Spokoiny, On estimating a dynamic function of a stochastic system with averaging, Statistical Inference for Stochastic Processes. An International Journal Devoted to Time Series Analysis and the Statistics of Continuous Time Processes and Dynamical Systems, 3 (2000), pp. 225--249.
  PDF (331 kByte), Postscript (344 kByte)

• J. Polzehl, V. Spokoiny, Adaptive Weights Smoothing with applications to image restoration, Journal of the Royal Statistical Society. Series B. Statistical Methodology, 62 (2000), pp. 335--354.
  Abstract, PDF (5980 kByte)
  We propose a new method of nonparametric estimation which is based on locally constant smoothing with an adaptive choice of weights for every pair of data-points. Some theoretical properties of the procedure are investigated. Then we demonstrate the performance of the method on some simulated univariate and bivariate examples and compare it with other nonparametric methods. Finally we discuss applications of this procedure to magnetic resonance and satellite imaging.

• V. Spokoiny, Adaptive drift estimation for nonparametric diffusion model, The Annals of Statistics, 28 (2000), pp. 815--836.
  PDF (160 kByte)

• R. Liptser, V. Spokoiny, Moderate deviations type evaluation for integral functionals of diffusion processes, Electron. J. Probab., 4(17) (1999) 25pp. (electronic)
  PDF(284 kByte)

• O. Lepski, V. Spokoiny, Minimax nonparametric hypothesis testing: the case of an inhomogeneous alternative, Bernoulli, 5 (1999) pp. 333--358.
  PDF(304 kByte)

• Y. Kutoyants, V. Spokoiny, Optimal choice of observation window for Poisson observations, Statist. Probab. Lett., 44 (1999) pp. 291--298.
  PDF(162 kByte)

• O. Lepski, A. Nemerovski, V. Spokoiny, On estimation of the $L_r$ norm of a regression function, Probab. Theory Related Fields, 113 (1999) pp. 245--273.
  PDF(327 kByte)

• A. Puhalskii, V. Spokoiny, On large-deviation efficiency in statistical inference, Bernoulli, 4 (1998) pp. 203--272.
  PDF(556 kByte)

• V. Spokoiny, Adaptive and spatially adaptive testing of a nonparametric hypothesis, Math. Methods Statist., 7 (1998) pp. 245--273.
  PDF(323 kByte)

• V. Spokoiny, Estimation of a function with discontinuities via local polynomial fit with an adaptive window choice, Ann. Statist., 26 (1998) pp. 1356--1378.
  PDF(176 kByte)

• O. Lepski, V. Spokoiny, Optimal pointwise adaptive methods in nonparametric estimation, Ann. Statist., 25 (1997) pp. 2512--2546.
  PDF(216 kByte)

• O. Lepski, E. Mammen, V. Spokoiny, Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors, Ann. Statist., 25 (1997) pp. 929--947.
  PDF(319 kByte)

• W. Härdle, V. Spokoiny, S. Sperlich, Semiparametric single index versus fixed link function modelling, Ann. Statist., 25 (1997) pp. 212--243.
  PDF(538 kByte)

• A. Shiryaev, V. Spokoiny, On sequential estimation of an autoregressive parameter, Stochastics Stochastics Rep., 60 (1997) pp. 219--240.
  PDF(422 kByte)

• A. Korostelev, V. Spokoiny, Exact asymptotics of minimax Bahdur risk in Lipschitz regression, Statistics, 28 (1996) pp. 13--24.
  PDF(180 kByte)

• V. Spokoiny, Adaptive hypothesis testing using wavelets, Ann. Statist., 24 (1996) pp. 2477--2498.
  PDF(360 kByte)