PhD Defense Christian Holberg
Title: Exploring Irregular Dynamics: Beyond Stationarity and Continuity
Abstract:
Contrary to the standard setting of independent and identically distributed (i.i.d.) data, stochastic processes may exhibit complex serial dependence structures and nonstationary. Both of these properties complicate the statistical analysis of such processes. In this thesis, we study processes that are either non-stationary or solutions to differential equations with event discontinuities. Cointegration assumes that the observed p-dimensional process is a linear mixing of k latent stationary components and p − k random walks. Inference is often predicted on knowing the exact number of stationary components, i.e., the cointegration rank. Crucially, this number is unknown in practice. In the first part of this thesis, we study the asymptotic distribution of different estimators under rank uncertainty. In particular, we establish central limit theorems for reduced rank estimators in the cointegrated vector autoregressive model under misspecified rank and present a new class of weighted reduced rank estimators that are arguably more robust to rank uncertainty. We then turn to the problem of uniform inference in cointegrated vector autoregressive processes. That is, we develop asymptotic approximations for two crucial covariance statistics that are valid uniformly across a parameter space including arbitrary cointegration ranks.
In the second part of the thesis, we establish a nonlinear generalization of cointegration. We derive identification results under varying assumptions on the class of admissible mixing transformations and the non-stationary component. Then, we develop a method for estimating that stationary component based on a single discretely sub-sampled trajectory of the observable process, xt, and show consistency under certain conditions.
Finally, in the last part of the paper, we consider the problem of deriving path-wise gradients of solutions to rough differential equations with endogenously defined discontinuities. Such discontinuities are termed event discontinuities. A canonical example is the spiking neuron model where an event is triggered every time the membrane potential of the neuron crosses a certain threshold upon which the potential is reset and the spike propagated to neighboring neurons. Thus, our results enable us to train spiking neuron models, where the inter-spike dynamics are governed by an SDE, using gradient-based optimization methods.
Supervisor: Professor, Susanne Ditlevsen, University of Copenhagen
Assessment Committee:
Professor Anders Rahbek (chair),University of Copenhagen
Professor Anastasios Magdalinos, University of Southampton
Associate Professor Blanka N. Horvath, Oxford University