MATHICSE Technical Report : Drift Estimation of Multiscale Diffusions Based on Filtered Data

We study the problem of drift estimation for two-scale continuous time series. We set ourselves in the framework of overdamped Langevin equations, for which a single-scale surrogate homogenized equation exists. In this setting, estimating the drift coefficient of the homogenized equation requires pre-processing of the data, often in the form of subsampling; this is because the two-scale equation and the homogenized single-scale equation are incompatible at small scales, generating mutually singular measures on the path space. We avoid subsampling and work instead with filtered data, found by application of an appropriate kernel function, and compute maximum likelihood estimators based on the filtered process. We show that the estimators we propose are asymptotically unbiased and demonstrate numerically the advantages of our method with respect to subsampling. Finally, we show how our filtered data methodology can be combined with Bayesian techniques and provide a full uncertainty quantification of the inference procedure.

Drift Estimation of Multiscale Diffusions Based on Filtered Data

Assyr Abdulle, Giacomo Garegnani, Andrea Zanoni

2021

Statistical analysis of clusters of extreme events

Mária Süveges

The thesis is a contribution to extreme-value statistics, more precisely to the estimation of clustering characteristics of extreme values. One summary measure of the tendency to form groups is the inverse average cluster size. In extreme-value context, this parameter is called the extremal index, and apart from its relation with the size of groups, it appears as an important parameter measuring the effects of serial dependence on extreme levels in time series. Although several methods exist for its estimation in univariate sequences, these methods are only applicable for strictly stationary series satisfying a long-range asymptotic independence condition on extreme levels, cannot take covariates into consideration, and yield only crude estimates for the corresponding multivariate quantity. These are strong restrictions and great drawbacks. In climatic time series, both stationarity and asymptotic independence can be broken, due to climate change and possible long memory of the data, and not including information from simultaneously measured linked variables may lead to inefficient estimation. The thesis addresses these issues. First, we extend the theorem of Ferro and Segers (2003) concerning the distribution of inter-exceedance times: we introduce truncated inter-exceedance times, called K-gaps, and show that they follow the same exponential-point mass mixture distribution as the inter-exceedance times. The maximization of the likelihood built on this distribution yields a simple closed-form estimator for the extremal index. The method can admit covariates and can be applied with smoothing techniques, which allows its use in a nonstationary setting. Simulated and real data examples demonstrate the smooth estimation of the extremal index. The likelihood, based on an assumption of independence of the K-gaps, is misspecified whenever K is too small. This motivates another contribution of the thesis, the introduction into extreme-value statistics of misspecification tests based on the information matrix. For our likelihood, they are able to detect misspecification from any source, not only those due to a bad choice of the truncation parameter. They provide help also in threshold selection, and show whether the fundamental assumptions of stationarity or asymptotic independence are broken. Moreover, these diagnostic tests are of general use, and could be adapted to many kinds of extreme-value models, which are always approximate. Simulated examples demonstrate the performance of the misspecification tests in the context of extremal index estimation. Two data examples with complex behaviour, one univariate and the other bivariate, offer insight into their power in discovering situations where the fundamental assumptions of the likelihood model are not valid. In the multivariate case, the parameter corresponding to the univariate extremal index is the multivariate extremal index function. As in the univariate case, its appearance is linked to serial dependence in the observed processes. Univariate estimation methods can be applied, but are likely to give crude, unreasonably varying, estimates, and the constraints on the extremal index function implied by the characteristics of the stable tail dependence function are not automatically satisfied. The third contribution of the thesis is the development of methodology based on the M4 approximation of Smith and Weissman (1996), which can be used to estimate the multivariate extremal index, as well as other cluster characteristics. For this purpose, we give a preliminary cluster selection procedure, and approximate the noise on finite levels with a flexible semiparametric model, the Dirichlet mixtures used widely in Bayesian analysis. The model is fitted by the EM algorithm. Advantages and drawbacks of the method are discussed using the same univariate and bivariate examples as the likelihood methods.

EPFL2009

Inference and Computation for Sparsely Sampled Random Surfaces

Tomas Masák, Victor Panaretos, Tomas Rubin

Nonparametric inference for functional data over two-dimensional domains entails additional computational and statistical challenges, compared to the one-dimensional case. Separability of the covariance is commonly assumed to address these issues in the densely observed regime. Instead, we consider the sparse regime, where the latent surfaces are observed only at few irregular locations with additive measurement error, and propose an estimator of covariance based on local linear smoothers. Consequently, the assumption of separability reduces the intrinsically four-dimensional smoothing problem into several two-dimensional smoothers and allows the proposed estimator to retain the classical minimax-optimal convergence rate for two-dimensional smoothers. Even when separability fails to hold, imposing it can be still advantageous as a form of regularization. A simulation study reveals a favorable bias-variance tradeoff and massive speed-ups achieved by our approach. Finally, the proposed methodology is used for qualitative analysis of implied volatility surfaces corresponding to call options, and for prediction of the latent surfaces based on information from the entire dataset, allowing for uncertainty quantification. Our cross-validated out-of-sample quantitative results show that the proposed methodology outperforms the common approach of pre-smoothing every implied volatility surface separately. Supplementary materials for this article are available online.

TAYLOR & FRANCIS INC2022

Statistical analysis of clusters of extreme events

Mária Süveges

EPFL2009