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Kuiper's $V_n$ statistic, a measure for comparing the difference of ideal distribution and empirical distribution, is of great significance in the goodness-of-fit test. However, Kuiper's formulae for computing the cumulative distribution…

The q-Gaussians are a class of stable distributions which are present in many scientific fields, and that behave as heavy tailed distributions for an especific range of q values. The identification of these values, which are used in the…

Consider $n$ i.i.d. random vectors on $\mathbb{R}^2$, with unknown, common distribution function $F$. Under a sharpening of the extreme value condition on $F$, we derive a weighted approximation of the corresponding tail copula process.…

Handling multiplicity without losing much power has been a persistent challenge in various fields that often face the necessity of managing numerous statistical tests simultaneously. Recently, $p$-value combination methods based on…

The rate of uniform convergence in extreme value statistics is non-universal and can be arbitrarily slow. Further, the relative error can be unbounded in the tail of the approximation, leading to difficulty in extrapolating the extreme…

The study of loss function distributions is critical to characterize a model's behaviour on a given machine learning problem. For example, while the quality of a model is commonly determined by the average loss assessed on a testing set,…

We study the problem of estimating the mean of a distribution in high dimensions when either the samples are adversarially corrupted or the distribution is heavy-tailed. Recent developments in robust statistics have established efficient…

This paper presents two results concerning uniform confidence intervals for the tail index and the extreme quantile. First, we show that it is impossible to construct a length-optimal confidence interval satisfying the correct uniform…

Suppose that $X$ is a bounded-degree polynomial with nonnegative coefficients on the $p$-biased discrete hypercube. Our main result gives sharp estimates on the logarithmic upper tail probability of $X$ whenever an associated extremal…

Outlier detection algorithms typically assign an outlier score to each observation in a dataset, indicating the degree to which an observation is an outlier. However, these scores are often not comparable across algorithms and can be…

It is argued that there is a need for fat-tailed distributions that become thin in the extreme tail. A 3-parameter distribution is introduced that visually resembles the t-distribution and interpolates between the normal distribution and…

We study the almost surely finite random variable $S$ defined by the distributional fixed-point equation \[ S \stackrel{d}{=} 1 + \max\{US', (1-U)S''\}, \qquad U \sim \mathrm{Unif}(0,1), \] where $S'$ and $S''$ are independent copies of…

In risk management, tail risks are of crucial importance. The assessment of risks should be carried out in accordance with the regulatory authority's requirement at high quantiles. In general, the underlying distribution function is…

For multivariate distributions in the domain of attraction of a max-stable distribution, the tail copula and the stable tail dependence function are equivalent ways to capture the dependence in the upper tail. The empirical versions of…

The Peaks-Over Threshold is a fundamental method in the estimation of rare events such as small exceedance probabilities, extreme quantiles and return periods. The main problem with the Peaks-Over Threshold method relates to the selection…

A fundamental problem in statistics and learning theory is to test properties of distributions. We show that quantum computers can solve such problems with significant speed-ups. In particular, we give fast quantum algorithms for testing…

Inference over tails is usually performed by fitting an appropriate limiting distribution over observations that exceed a fixed threshold. However, the choice of such threshold is critical and can affect the inferential results. Extreme…

The performance of quantum key distribution (QKD) heavily depends on statistical inference. For a broad class of protocols, the central statistical task is a random sampling problem, customarily addressed using a hypergeometric tail bound…

Insurance data can be asymmetric with heavy tails, causing inadequate adjustments of the usually applied models. To deal with this issue, hierarchical models for collective risk with heavy-tails of the claims distributions that take also…

Causal questions are omnipresent in many scientific problems. While much progress has been made in the analysis of causal relationships between random variables, these methods are not well suited if the causal mechanisms only manifest…