Akshay Balsubramani
A pervasive issue in statistical hypothesis testing is that the reported $p$-values are biased downward by data "peeking" -- the practice of reporting only progressively extreme values of the test statistic as more data samples are collected. We develop principled mechanisms to estimate such running extrema of test statistics, which directly address the effect of peeking in some general scenarios.
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Estimation of extreme value copulas is often required in situations where available data are sparse. Parametric methods may then be the preferred approach. A possible way of defining parametric families that are simple and, at the same time, cover a large variety of multivariate extremal dependence structures is to build models based on spectral measures. This approach is considered here. Parametric families of spectral measures are defined as convex hulls of suitable basis elements, and parameters are estimated by projecting an initial nonparametric estimator on these finite-dimensional space…
The multivariate extremal index function relates the asymptotic distribution of the vector of pointwise maxima of a multivariate stationary sequence to that of the independent sequence from the same stationary distribution. It also measures the degree of clustering of extremes in the multivariate process. In this paper, we construct nonparametric estimators of this function and prove their asymptotic normality under long-range dependence and moment conditions. The results are illustrated by means of a simulation study.
Estimating the limiting shape of bivariate scaled sample clouds: with additional benefits of self-consistent inference for existing extremal dependence properties
本文针对二元缩放样本云的极限形状开发了首个统计推断方法，构建了联合尾部统一推断框架，并为多种极值依赖度量提供了首套自洽估计量，避免了现有方法可能产生的矛盾结论。
A new approach for evaluating time-trends in extreme values accounting also for spatial dependence is proposed. Based on exceedances over a space-time threshold, estimators for a trend function and for extreme value parameters are given, leading to a homogenization procedure for then applying stationary extreme value processes. Extremal dependence over space is further evaluated through variogram analysis including anisotropy. We detect significant inhomogeneities and trends in the extremal behaviour of daily precipitation data over a time period of 84 years and from 68 observational weather s…
Estimating Changepoints in Extremal Dependence, Applied to Aviation Stock Prices During COVID-19 Pandemic
本文利用二元Hüsler-Reiss分布建模航空股日收益率的极端依赖性，提出基于LRT和MIC的变点检测方法，并成功识别COVID-19期间印度航空股依赖性变点，为投资组合优化提供支持。
For stationary sequences, under general local and asymptotic dependence restrictions, any limiting point process for time normalized upcrossings of high levels is a compound Poisson process, i.e., there is a clustering of high upcrossings, where the underlying Poisson points represent cluster positions, and the multiplicities correspond to cluster sizes. For such classes of stationary sequences there exists the upcrossings index $η,$ $0\leq η\leq 1,$ which is directly related to the extremal index $θ,$ $0\leq θ\leq 1,$ for suitable high levels. In this paper we consider the problem of estimati…
The difficulties of estimating and representing the distributions of functional data mean that principal component methods play a substantially greater role in functional data analysis than in more conventional finite-dimensional settings. Local maxima and minima in principal component functions are of direct importance; they indicate places in the domain of a random function where influence on the function value tends to be relatively strong but of opposite sign. We explore statistical properties of the relationship between extrema of empirical principal component functions, and their counter…
We present a novel distribution-free approach, the data-driven threshold machine (DTM), for a fundamental problem at the core of many learning tasks: choose a threshold for a given pre-specified level that bounds the tail probability of the maximum of a (possibly dependent but stationary) random sequence. We do not assume data distribution, but rather relying on the asymptotic distribution of extremal values, and reduce the problem to estimate three parameters of the extreme value distributions and the extremal index. We specially take care of data dependence via estimating extremal index sinc…
Max-stable processes have been expanded to quantify extremal dependence in spatio-temporal data. Due to the interaction between space and time, spatio-temporal data are often complex to analyze. So, characterizing these dependencies is one of the crucial challenges in this field of statistics. This paper suggests a semiparametric inference methodology based on the spatio-temporal F-madogram for estimating the parameters of a space-time max-stable process using gridded data. The performance of the method is investigated through various simulation studies. Finally, we apply our inferential proce…
We study distributional robustness in the context of Extreme Value Theory (EVT). We provide a data-driven method for estimating extreme quantiles in a manner that is robust against incorrect model assumptions underlying the application of the standard Extremal Types Theorem. Typical studies in distributional robustness involve computing worst case estimates over a model uncertainty region expressed in terms of the Kullback-Leibler discrepancy. We go beyond standard distributional robustness in that we investigate different forms of discrepancies, and prove rigorous results which are helpful fo…
Improved Distribution Estimation in $\ell_\infty$
本文在 $\ell_\infty$ 范数下改进了离散概率分布的估计，给出了期望极小极大界和高概率尾界，解决了 Kontorovich 和 Painsky (JMLR, 2025) 提出的开放问题，包括最紧风险界的完全经验版本和最坏情况极值分布的形式，并报告了鼓励性的实证结果。
Max-stable processes are a common choice for modelling spatial extreme data as they arise naturally as the infinite-dimensional generalisation of multivariate extreme value theory. Statistical inference for such models is complicated by the intractability of the multivariate density function. Nonparametric, composite likelihood-based, and Bayesian approaches have been proposed to address this difficulty. More recently, a simulation-based approach using approximate Bayesian computation (ABC) has been employed for estimating parameters of max-stable models. ABC algorithms rely on the evaluation …
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