Title: Assessing differences across heavy tailed samples – With applications to environmental and financial data
Abstract:
Assessing significant differences between two independent samples of data is challenging when the focus is on extreme events and the distribution of data is heavy-tailed. While analysis of variance (ANOVA) is useful for assessing differences in the means, it is unsuitable for differences in tail behavior, especially when means do not exist or empirical estimation of means or higher moments is not consistent. Here, we propose an ANOVA-like decomposition to analyze tail variability, allowing for flexible representation of heavy tails through a set of user-defined extreme quantiles, possibly located outside the range of observations.
Assuming regular variation (i.e., power-laws), we introduce a test statistic to check for significant tail differences across multiple independent samples and derive its asymptotic distribution. This statistic can also be used to consistently identify a changepoint in a heavy-tailed data sequence. The changepoint is estimated as the position of the maximum of the statistic when checking for differences between the data subsets to the left and right of all changepoint candidates. The new methods are compared to competitor approaches, and we apply them to analyze tail behavior in various applications (precipitation, stock market indices and Bitcoin data.
Girard, S., Opitz, T., & Usseglio-Carleve, A. (2024). ANOVEX: ANalysis Of Variability for heavy-tailed EXtremes. Electronic Journal of Statistics, 18(2), 5258-5303.
Girard, S., Opitz, T., Usseglio-Carleve, A., & Yan, C. (2025+). Changepoint identification in heavy-tailed distributions. HAL preprint https://hal.science/hal-05044135/
Title: Maximum gap in complex Ginibre matrices
Abstract: We study the eigenvalue spacings of the Ginibre unitary ensemble, i.e. the spacings between the eigenvalues of an $n \times n$-matrix with iid complex Gaussian entries. Our goal is to understand the behavior of the maximum gap between eigenvalues in the bulk of the spectrum. To this end, we introduce an auxiliary point process obtained by thinning the eigenvalue set to retain only those points whose nearest-neighbor spacing exceeds a fixed threshold. We show that this thinned process, restricted to the open unit disk, converges to a Poisson point process as $n$ tends to infinity. Standard arguments then yield the asymptotic behavior of the maximum nearest-neighbor spacing in the bulk of the spectrum. This talk is based on joint work with Patrick Lopatto.