Conference Agenda

Testing for independence can be done either time-point-wise or, more thoroughly, over periods of time up to the whole observed time horizon. Hence, we extended the testing procedure for locally stationary processes proposed in Beering (2021), which uses a characteristic function-based weighted distance inspired by the distance covariance defined by Székely et al. (2007) and its use by Jentsch et al. (2020). The refined testing procedure allows for time spans to be taken into consideration. As this test is supported by a bootstrap procedure, we present the theoretical results of both the real and the bootstrap world. Additionally, we show the practical performance of the testing procedure via a simulation study aiming to detect independence as well as dependence and a real-data example stemming from the field of structural health monitoring of infrastructural buildings.

References:

Beering, C. (2021). A functional central limit theorem and its bootstrap analogue for locally stationary processes with application to independence testing. Dissertation. Technische Universität Braunschweig.

Jentsch, C., Leucht, A., Meyer, M. and Beering, C. (2020). Empirical characteristic functions-based estimation and distance correlation for locally stationary processes. Journal of Time Series Analysis 41, 110-133.

Székely, G.J., Rizzo, M.L. and Bakirov, N.K. (2007). Measuring and testing dependence by correlation of distances. The Annals of Statistics 35, 2769-2794.

When we observe a stationary time series with observations missing at periodic time points, we can still estimate its marginal distribution well, but the dependence structure of the time series may not be recoverable at all, or the usual estimators may have much larger variance than in the fully observed case. We show how nonparametric estimators can often be improved by adding unbiased estimators. We consider a simple setting, first-order Markov chains on a finite state space, and an observation pattern in which a fixed number of consecutive observations is followed by an observation gap of fixed length, say workdays and weekends.

In this talk I will focus on the simplest reasonable scenario, namely when every third observation is missing. The new estimators perform astonishingly well, as illustrated with simulations for this scenario.

This talk is based on joint work with Anton Schick and Wolfgang Wefelmeyer.

For stationary $\mathbb{R}^d$-valued lattice processes on $\mathbb{Z}^g$, for $d,g\in \mathbb{N}$, we consider the estimation of the whole covariance function of the data. In this sense, we generalize the estimation of large covariance matrices via tapering as proposed by McMurry and Politis (2010) and Jentsch and Politis (2015) for univariate and multivariate time series, respectively, to more general lattice processes. Considering the vectorization $vec(\mathbf{X})$ of (multivariate) lattice data $\mathbf{X}$, we construct suitable (tapering) estimators for the covariance matrix $Cov(vec(\mathbf{X}))$ of the whole data set. Note that the dimension of $Cov(vec(\mathbf{X}))$ is growing with increasing sample size. We prove estimation consistency with respect to the spectral norm and discuss computational challenges caused by the high dimensionality of this task. To achieve efficiency gains, we discuss various forms of separability imposed on the covariance function and examine lattice processes in both non-separable and separable covariance setups. For this purpose, we propose an alternative tapered estimator tailored for separable covariance functions and establish its consistency under separability. To assess their performance, we conduct simulation studies exploring how these estimators behave with different separability and spatial dependence scenarios.