Statistics for stochastic processes

Estimation and inference of the discrete and continuous stochastic processes have always been considered as a challenging problem in statistics, since one has to take time dependence into account. Here various methods can be used especially parametric methods for time series, nonparametric procedures for diffusion processes and computational algorithms like MCMC. The application areas are very broad, ranging from financial modelling and engineering to building weather forecasts and energy economics.

Researchers

Dr.ir. G.N.J.C. (Joris) Bierkens

Dr. J. (Jakob) Söhl

Prof.dr. A.W. (Aad) van der Vaart

Key Publications

2024 - Fabian Mies, Mark Podolskij - The Annals of Statistics

Estimation of mixed fractional stable processes using high-frequency data

The linear fractional stable motion generalizes two prominent classes of stochastic processes, namely stable Lévy processes, and fractional Brownian motion. For this reason it may be regarded as a basic building block for continuous time models. We study a stylized model consisting of a superposition of independent linear fractional stable motions and our focus is on parameter estimation of the model. Applying an estimating equations approach, we construct estimators for the whole set of parameters and derive their asymptotic normality in a high-frequency regime. The conditions for consistency turn out to be sharp for two prominent special cases: (i) for Lévy processes, i.e. for the estimation of the successive Blumenthal-Getoor indices, and (ii) for the mixed fractional Brownian motion introduced by Cheridito. In the remaining cases, our results reveal a delicate interplay between the Hurst parameters and the indices of stability. Our asymptotic theory is based on new limit theorems for multiscale moving average processes.

2021 - Joris Bierkens, Sebastiano Grazzi, Frank van der Meulen, Moritz Schauer - Statistics and Computing

A piecewise deterministic Monte Carlo method for diffusion bridges

We introduce the use of the Zig-Zag sampler to the problem of sampling conditional diffusion processes (diffusion bridges). The Zig-Zag sampler is a rejection-free sampling scheme based on a non-reversible continuous piecewise deterministic Markov process. Similar to the Lévy–Ciesielski construction of a Brownian motion, we expand the diffusion path in a truncated Faber–Schauder basis. The coefficients within the basis are sampled using a Zig-Zag sampler. A key innovation is the use of the fully local algorithm for the Zig-Zag sampler that allows to exploit the sparsity structure implied by the dependency graph of the coefficients and by the subsampling technique to reduce the complexity of the algorithm. We illustrate the performance of the proposed methods in a number of examples.

2021 - Fabian Mies - Journal of the American Statistical Association

Functional Estimation and Change Detection for Nonstationary Time Series

Tests for structural breaks in time series should ideally be sensitive to breaks in the parameter of interest, while being robust to nuisance changes. Statistical analysis thus needs to allow for some form of nonstationarity under the null hypothesis of no change. In this article, estimators for integrated parameters of locally stationary time series are constructed and a corresponding functional central limit theorem is established, enabling change-point inference for a broad class of parameters under mild assumptions. The proposed framework covers all parameters which may be expressed as nonlinear functions of moments, for example kurtosis, autocorrelation, and coefficients in a linear regression model. To perform feasible inference based on the derived limit distribution, a bootstrap variant is proposed and its consistency is established. The methodology is illustrated by means of a simulation study and by an application to high-frequency asset prices.

2017 - Richard Nickl, Jakob Söhl - The Annals of Statistics

Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions

We consider nonparametric Bayesian inference in a reflected diffusion model \(dX_{t}=b(X_{t})\,dt+\sigma(X_{t})\,dW_{t}\), with discretely sampled observations \(X_{0},X_{\Delta},\ldots,X_{n\Delta}\). We analyse the nonlinear inverse problem corresponding to the “low frequency sampling” regime where Δ>0 is fixed and n→∞. A general theorem is proved that gives conditions for prior distributions Π on the diffusion coefficient σ and the drift function b that ensure minimax optimal contraction rates of the posterior distribution over Hölder–Sobolev smoothness classes. These conditions are verified for natural examples of nonparametric random wavelet series priors. For the proofs, we derive new concentration inequalities for empirical processes arising from discretely observed diffusions that are of independent interest.

2017 - Fengnan Gao, Aad van der Vaart - Stochastic Processes and their Applications

On the asymptotic normality of estimating the affine preferential attachment network models with random initial degrees

We consider the estimation of the affine parameter and power-law exponent in the preferential attachment model with random initial degrees. We derive the likelihood, and show that the maximum likelihood estimator (MLE) is asymptotically normal and efficient. We also propose a quasi-maximum-likelihood estimator (QMLE) to overcome the MLE’s dependence on the history of the initial degrees. To demonstrate the power of our idea, we present numerical simulations.

2015 - Richard Nickl, Markus Reiß, Jakob Söhl, Mathias Trabs - Probability Theory and Related Fields

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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