We study the model selection problem in a large class of causal time series models, which includes both the ARMA or AR($\infty$) processes, as well as the GARCH or ARCH($\infty$), APARCH, ARMA-GARCH and many others processes. To tackle this issue, we consider a penalized contrast based on the quasi-likelihood of the model. We provide sufficient conditions for the penalty term to ensure the consistency of the proposed procedure as well as the consistency and the asymptotic normality of the quasi-maximum likelihood estimator of the chosen model. We also propose a tool for diagnosing the goodness-of-fit of the chosen model based on a Portmanteau test. Monte-Carlo experiments and numerical applications on illustrative examples are performed to highlight the obtained asymptotic results. Moreover, using a data-driven choice of the penalty, they show the practical efficiency of this new model selection procedure and Portemanteau test.[-]

In this paper we study asymptotic properties of random forests within the framework of nonlinear time series modeling. While random forests have been successfully applied in various fields, the theoretical justification has not been considered for their use in a time series setting. Under mild conditions, we prove a uniform concentration inequality for regression trees built on nonlinear autoregressive processes and, subsequently, use this result to prove consistency for a large class of random forests. The results are supported by various simulations. (This is joint work with Mikkel Slot Nielsen.)[-]

Many effects of climate change seem to be reflected not in the mean temperatures, precipitation or other environmental variables, but rather in the frequency and severity of the extreme events in the distributional tails. The most serious climate-related disasters are caused by compound events that result from an unfortunate combination of several variables. Detecting changes in size or frequency of such compound events requires a statistical methodology that efficiently uses the largest observations in the sample.
We propose a simple, non-parametric test that decides whether two multivariate distributions exhibit the same tail behavior. The test is based on the entropy, namely Kullback–Leibler divergence, between exceedances over a high threshold of the two multivariate random vectors. We study the properties of the test and further explore its effectiveness for finite sample sizes.
Our main application is the analysis of daily heavy rainfall times series in France (1976 -2015). Our goal in this application is to detect if multivariate extremal dependence structure in heavy rainfall change according to seasons and regions.[-]

In time series analysis there is an apparent dichotomy between time and frequency domain methods. The aim of this paper is to draw connections between frequency and time domain methods. Our focus will be on reconciling the Gaussia likelihood and the Whittle likelihood. We derive an exact, interpretable, bound between the Gaussian and Whittle likelihood of a second order stationary time series. The derivation is based on obtaining the transformation which is biorthogonal to the discrete Fourier transform of the time series. Such a transformation yields a new decomposition for the inverse of a Toeplitz matrix and enables the representation of the Gaussian likelihood within the frequency domain. We show that the difference between the Gaussian and Whittle likelihood is due to the omission of the best linear predictions outside the domain of observation in the periodogram associated with the Whittle likelihood. Based on this result, we obtain an approximation for the difference between the Gaussian and Whittle likelihoods in terms of the best fitting, finite order autoregressive parameters. These approximations are used to define two new frequency domain quasi-likelihoods criteria. We show these new criteria yield a better approximation of the spectral divergence criterion, as compared to both the Gaussian and Whittle likelihoods. In simulations, we show that the proposed estimators have satisfactory finite sample properties.[-]

The class of integer-valued trawl processes has recently been introduced for modelling univariate and multivariate integer-valued time series with short or long memory.

In this talk, I will discuss recent developments with regards to model estimation, model selection and forecasting of such processes. The new methods will be illustrated in an empirical study of high-frequency financial data.

This is joint work with Mikkel Bennedsen (Aarhus University), Asger Lunde (Aarhus University) and Neil Shephard (Harvard University).[-]

It is generally admitted that financial time series have heavy tailed marginal distributions. When time series models are fitted on such data, the non-existence of appropriate moments may invalidate standard statistical tools used for inference. Moreover, the existence of moments can be crucial for risk management. This talk considers testing the existence of moments in the framework of standard and augmented GARCH models. In the case of standard GARCH, even-moment conditions involve moments of the independent innovation process. We propose tests for the existence of moments of the returns process that are based on the joint asymptotic distribution of the estimator of the volatility parameters and empirical moments of the residuals. To achieve efficiency gains we consider non Gaussian QML estimators founded on reparametrizations of the GARCH model, and we discuss optimality issues. We also consider augmented GARCH processes, for which moment conditions are less explicit. We establish the asymptotic distribution of the empirical moment Generating function (MGF) of the model, defined as the MGF of the random autoregressive coefficient in the volatility dynamics, from which a test is deduced. An alternative test is based on the estimation of the maximal exponent characterizing the existence of moments. Our results will be illustrated with Monte Carlo experiments and real financial data.[-]

Anomaly detection in random fields is an important problem in many applications including the detection of cancerous cells in medicine, obstacles in autonomous driving and cracks in the construction material of buildings. Scan statistics have the potential to detect local structure in such data sets by enhancing relevant features. Frequently, such anomalies are visible as areas with different expected values compared to the background noise where the geometric properties of these areas may depend on the type of anomaly. Such geometric properties can be taken into account by combinations and contrasts of sample means over differently-shaped local windows. For example, in 2D image data of concrete both cracks, which we aim to detect, as well as integral parts of the material (such as air bubbles or gravel) constitute areas with different expected values in the image. Nevertheless, due to their different geometric properties we can define scan statistics that enhance cracks and at the same time discard the integral parts of the given concrete. Cracks can then be detected using asuitable threshold for appropriate scan statistics. 9 In order to derive such thresholds, we prove weak convergence of the scan statistics towards a functional of a Gaussian process under the null hypothesis of no anomalies. The result allows for arbitrary (but fixed) dimension, makes relatively weak assumptions on the underlying noise, the shape of the local windows and the combination of finitely-many of such windows. These theoretical findings are accompanied by some simulations as well as applications to semi-artifical 2D-images of concrete.[-]

In this talk, I will present mixing properties for a broad class of Poisson count time series satisfying a certain contraction condition. Using specific coupling techniques, we obtain absolute regularity at a geometric rate not only for stationary Poisson-GARCH processes but also for models with an explosive trend. Easily verifiable sufficient conditions for absolute regularity can be deduced from our general results for a variety of models including classical (log-)linear models.[-]

Multivariate extreme value distributions are a common choice for modelling multivariate extremes. In high dimensions, however, the construction of flexible and parsimonious models is challenging. We propose to combine bivariate extreme value distributions into a Markov random field with respect to a tree. Although in general not an extreme value distribution itself, this Markov tree is attracted by a multivariate extreme value distribution. The latter serves as a tree-based approximation to an unknown extreme value distribution with the given bivariate distributions as margins. Given data, we learn an appropriate tree structure by Prim's algorithm with estimated pairwise upper tail dependence coefficients or Kendall's tau values as edge weights. The distributions of pairs of connected variables can be fitted in various ways. The resulting tree-structured extreme value distribution allows for inference on rare event probabilities, as illustrated on river discharge data from the upper Danube basin.[-]