A Euclidean Likelihood Estimator for Bivariate Tail Dependence

The spectral measure plays a key role in the statistical modeling of multivariate extremes. Estimation of the spectral measure is a complex issue, given the need to obey a certain moment condition. We propose a Euclidean likelihood-based estimator for the spectral measure which is simple and explicitly defined, with its expression being free of Lagrange multipliers. Our estimator is shown to have the same limit distribution as the maximum empirical likelihood estimator of J. H. J. Einmahl and J. Segers, Annals of Statistics 37(5B), 2953–2989 (2009). Numerical experiments suggest an overall good performance and identical behavior to the maximum empirical likelihood estimator. We illustrate the method in an extreme temperature data analysis.

Inference on the Angular Distribution of Extremes

Claudio Andri Semadeni

The spectral distribution plays a key role in the statistical modelling of multivariate extremes, as it defines the dependence structure of multivariate extreme-value distributions and characterizes the limiting distribution of the relative sizes of the components of large multivariate observations. No parametric family captures all possible types of multivariate dependence, and numerous parametric models have been proposed. Inference on the spectral distribution is typically based on the pseudo-angles of

large' observations under the assumption that they follow the spectral distribution. There has been little attention on studying the impact of this approximation on inference, and it turns out that it can yield significantly biased estimates. We provide a characterization of the angular distribution of excesses corresponding to the distribution of pseudo-angles of

large'observations that improves direct inference on the spectral distribution in the bivariate setting. Extremal dependence is at the heart of extreme value modelling and numerous measures to quantify it have been proposed in the literature. In many applications, datasets seem to exhibit asymmetry in the dependence between the variables. Many parametric multivariate extreme-value models can accommodate asymmetry in the sense that the spectral density can be asymmetric, resulting in a non-exchangeable dependence structure. There has been little attention paid to quantifying asymmetry at extreme levels, which can be useful for diagnosis and model checking. We propose a coefficient of extremal asymmetry that quantifies the asymmetry at extreme levels for pairs of variables. We also propose two non-parametric estimators of the coefficient of extremal asymmetry and compare their properties through numerical simulation. The two estimators have diametrically opposed bias-variance trade-offs. The estimator based on maximum empirical likelihood performs well and is nearly unbiased.

EPFL2020

Robust Multivariate and Nonlinear Time Series Models

Time series modeling and analysis is central to most financial and econometric data modeling. With increased globalization in trade, commerce and finance, national variables like gross domestic productivity (GDP) and unemployment rate, market variables like indices and stock prices and global variables like commodity prices are more tightly coupled than ever before. This translates to the use of multivariate or vector time series models and algorithms in analyzing and understanding the relationships that these variables share with each other. Autocorrelation is one of the fundamental aspects of time series modeling. However, traditional linear models, that arise from a strong observed autocorrelation in many financial and econometric time series data, are at times unable to capture the rather nonlinear relationship that characterizes many time series data. This necessitates the study of nonlinear models in analyzing such time series. The class of bilinear models is one of the simplest nonlinear models. These models are able to capture temporary erratic fluctuations that are common in many financial returns series and thus, are of tremendous interest in financial time series analysis. Another aspect of time series analysis is homoscedasticity versus heteroscedasticity. Many time series data, even after differencing, exhibit heteroscedasticity. Thus, it becomes important to incorporate this feature in the associated models. The class of conditional heteroscedastic autoregressive (ARCH) models and its variants form the primary backbone of conditional heteroscedastic time series models. Robustness is a highly underrated feature of most time series applications and models that are presently in use in the industry. With an ever increasing amount of information available for modeling, it is not uncommon for the data to have some aberrations within itself in terms of level shifts and the occasional large fluctuations. Conventional methods like the maximum likelihood and least squares are well known to be highly sensitive to such contaminations. Hence, it becomes important to use robust methods, especially in this age with high amounts of computing power readily available, to take into account such aberrations. While robustness and time series modeling have been vastly researched individually in the past, application of robust methods to estimate time series models is still quite open. The central goal of this thesis is the study of robust parameter estimation of some simple vector and nonlinear time series models. More precisely, we will briefly study some prominent linear and nonlinear models in the time series literature and apply the robust S-estimator in estimating parameters of some simple models like the vector autoregressive (VAR) model, the (0, 0, 1, 1) bilinear model and a simple conditional heteroscedastic bilinear model. In each case, we will look at the important aspect of stationarity of the model and analyze the asymptotic behavior of the S-estimator.

EPFL2010

Inference on the Angular Distribution of Extremes

Claudio Andri Semadeni

large' observations under the assumption that they follow the spectral distribution. There has been little attention on studying the impact of this approximation on inference, and it turns out that it can yield significantly biased estimates. We provide a characterization of the angular distribution of excesses corresponding to the distribution of pseudo-angles of

EPFL2020

Robust Multivariate and Nonlinear Time Series Models

EPFL2010