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Preprints, Reports, Technical Reports

• G.N. Milstein, V. Spokoiny, Martingale approach in pricing European options under regime-switching, Preprint no. 1645, WIAS, Berlin, 2011.
  Abstract, Postscript (800 kByte), PDF (246 kByte)

  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, and the pricing function satisfies the Cauchy problem for a system of linear parabolic equations. It is shown that any European contingent claim is attainable using a generalized self-financing replicating strategy. For such a strategy, apart from the initial endowment, some additional funds are required both step-wise at the jump moments of the Markov chain and continuously between the jump moments. It is proved that the additional funds (the additional investments and consumptions) are present in the proposed strategy in the risk-neutral manner, hence the generalized self-financing strategy is self-financing in mean. A payment for the considered option should consist of two parts: the initial endowment and a fair insurance premium in order to compensate for contributions and consumptions arising in future.

Articles in Refereed Journals

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

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

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

• E. Giacomini, W. Härdle, V. Spokoiny, Inhomogeneous dependency modelling with time varying copulae, J. Bus. Econom. Statist., 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, Electron. J. Stat., 3 (2009) pp. 712--746.

• D. Belomestny, G.N. Milstein, V. Spokoiny, Regression methods in pricing American and Bermudan options using consumption processes, Quant. 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, Ann. Statist., 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, Ann. Statist., 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, J. Mach. Learn. Res., 9 (2008) pp. 1647--1678.
  PDF (290 kByte)

• I.G. Grama, V. Spokoiny, Statistics of extremes by oracle estimation, Ann. Statist., 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 Trans. Signal Process., 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, J. Comput. Appl. Math., 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 Process. Appl., 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, Ann. Statist., 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, J. Mach. Learn. Res., 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 Trans. Inform. Theory, 52 (2006) pp. 1322--1334.
  PDF (906 kByte)

• J. Polzehl, V. Spokoiny, Propagation-separation approach for local likelihood estimation, Probab. Theory 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, J. Amer. Statist. Assoc., 100 (2005) pp. 429--445.
  PDF (454 kByte)

• M. Giurcanu, V. Spokoiny, Confidence estimation of the covariance function of stationary and locally stationary processes, Statist. Decisions, 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, Probab. Theory 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, 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, Ann. Statist., 32 (2004) pp. 577--602.
  PDF (438 kByte)

• M.-Y. Cheng, J. Fan, V. Spokoiny, Dynamic nonparametric filtering with application to volatility estimation, , (2003) pp. 315-333.

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

• J. Polzehl, V. Spokoiny, Image denoising: Pointwise adaptive approach, Ann. Statist., 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, J. Amer. Statist. Assoc., 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 Process. Related Fields, 8 (2002) pp. 611-636.
  PDF (348 kByte), Postscript (343 kByte)

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

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

• W. Härdle, S. Sperlich, V. Spokoiny, Structural tests for additive regression, J. Amer. Statist. Assoc., 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, 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, Ann. Statist., 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, Ann. Statist., 29 (2001) pp. 595--623.
  PDF (247 kByte)

• J. Polzehl, V. Spokoiny, Functional and dynamic Magnetic Resonance Imaging using vector adaptive weights smoothing, J. Roy. Statist. Soc. Ser. C, 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, Math. Methods Statist., 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, Statist. Probab. Lett., 46 (2000) pp. 347--357.
  PDF (269 kByte)

• R. Liptser, V. Spokoiny, On estimating a dynamic function of a stochastic system with averaging, Stat. Inference Stoch. Process., 3 (2000) pp. 225--249.
  PDF (331 kByte), Postscript (344 kByte)

• J. Polzehl, V. Spokoiny, Adaptive Weights Smoothing with applications to image restoration, J. R. Stat. Soc. Ser. B Stat. Methodol., 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, Ann. Statist., 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)