arXiv subjects

Markus Reiß

Publications and source records attributed to Markus Reiß.

34 records · Page 2Linked to original sources

arXiv2016

Unbiased estimation of the volume of a convex body

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. No hypotheses on the boundary of the convex set are imposed. In a numerical study, we show that the estimator outperforms earlier estimators for the volume. In addition, an improved set estimator for the convex body itself is proposed.

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arXiv2016

Adaptive quantile estimation in deconvolution with unknown error distribution

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. This closes an important gap in the literature. Optimal adaptive estimation is obtained by a data-driven bandwidth choice. As a side result, we obtain optimal rates for the plug-in estimation of distribution functions with unknown error distributions. The method is applied to a real data example.

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arXiv2015

Volatility estimation under one-sided errors with applications to limit order books

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

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arXiv2015

Efficient estimation of functionals in nonparametric boundary models

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 Hölder 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. In that approach also non-asymptotic efficiency is obtained (UMVU: uniformly minimum variance among all unbiased estimators).The proofs rely essentially on martingale stopping arguments for counting processes and the point process geometry. The estimators are easily computable and a small simulation study confirms their applicability.

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arXiv2015

Nonparametric test for a constant beta between \Ito semi-martingales based on high-frequency data

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. Rate optimality of the test over smoothness classes is derived.

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arXiv2014

High-frequency Donsker theorems for Lévy measures

Donsker-type functional limit theorems are proved for empirical processes arising from discretely sampled increments of a univariate Lévy 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 Lévy measure. The results are applied to derive the asymptotic distribution of natural estimators for the distribution function of the Lévy jump measure. As an application we deduce Kolmogorov-Smirnov type tests and confidence bands.

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arXiv2014

Adaptive function estimation in nonparametric regression with one-sided errors

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

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arXiv2014

Estimating the quadratic covariation matrix from noisy observations: Local method of moments and efficiency

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 Cramér-Rao sense. Main findings are that nonsynchronicity of observation times has no impact on the asymptotics and that major efficiency gains are possible under correlation. Simulations illustrate the finite-sample behaviour.

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arXiv2013

Model selection in a sparse heterogeneous framework

We consider a Gaussian sequence space model $X_λ=f_λ + ξ_λ,$ where $ξ$ has a diagonal covariance matrix $Σ=\diag(σ_λ^2)$. We consider the situation where the parameter vector $(f_λ)$ is sparse. Our goal is to estimate the unknown parameter by a model selection approach. The heterogenous case is much more involved than the direct model. Indeed, there is no more symmetry inside the stochastic process that one needs to control since each empirical coefficient has its own variance. The problem and the penalty do not only depend on the number of coefficients that one selects, but also on their position. This appears also in the minimax bounds where the worst coefficients will go to the larger variances. However, with a careful and explicit choice of the penalty we are able to select the correct coefficients and get a sharp non-asymptotic control of the risk of our procedure. Some simulation results are provided.

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arXiv2013

Spectral covolatility estimation from noisy observations using local weights

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. The new estimator outperforms earlier nonparametric approaches in the considered model. We investigate its finite sample size characteristics in simulations and draw a comparison between the various proposed methods.

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arXiv2013

Testing the characteristics of a Lévy process

For $n$ equidistant observations of a Lévy process at time distance $Δ_n$ 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 $Δ_n=\frac1n$ and the low-frequency regime $Δ_n=1$ as well as intermediate cases. The approach via empirical characteristic function unifies existing theory and sheds new light on diverse results. Particular emphasis is given to asymptotic separation rates which reveal the complexity of these basic, but surprisingly non-standard inference questions.

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arXiv2012

A Donsker Theorem for Lévy Measures

Given $n$ equidistant realisations of a Lévy process $(L_t,\,t\ge 0)$, a natural estimator $\hat N_n$ for the distribution function $N$ of the Lévy measure is constructed. Under a polynomial decay restriction on the characteristic function $ϕ$, 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 ${\cal F}^{-1}[1/ϕ(-\cdot)]$. The class of Lévy processes covered includes several relevant examples such as compound Poisson, Gamma and self-decomposable processes. Main ideas in the proof include establishing pseudo-locality of the Fourier-integral operator and recent techniques from smoothed empirical processes.

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arXiv2011

Asymptotic equivalence for inference on the volatility from noisy observations

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 $σ$ and a nonstandard noise level. As an application, new rate-optimal estimators of the volatility function and simple efficient estimators of the integrated volatility are constructed.

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arXiv2011

Asymptotic Equivalence for Nonparametric Regression with Non-Regular Errors

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. The statistical model significantly differs from regression problems with Gaussian or regular errors, which are known to be asymptotically equivalent to Gaussian white noise models.

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arXiv2010

Asymptotic equivalence and sufficiency for volatility estimation under microstructure noise

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 $σ$. As an application, simple rate-optimal estimators of the volatility and efficient estimators of the integrated volatility are constructed.

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arXiv2006

Asymptotic equivalence for nonparametric regression with multivariate and random design

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.

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