Related papers: The Probability Conflation: A Reply
Probability forecasts are intended to account for the uncertainties inherent in forecasting. It is suggested that from an end-user's point of view probability is not necessarily sufficient to reflect uncertainties that are not simply the…
Tail risk measures are fully determined by the distribution of the underlying loss beyond its quantile at a certain level, with Value-at-Risk, Expected Shortfall and Range Value-at-Risk being prime examples. They are induced by law-based…
We study tail probabilities via some Gaussian approximations. Our results make refinements to large deviation theory. The proof builds on classical results by Bahadur and Rao. Binomial distributions and their tail probabilities are…
We derive exponential bounds for tail of distribution for natural, i.e. under ordinary logarithm, normalized sums of arrays of random variables, not necessarily independent.
In public discussions of the quality of forecasts, attention typically focuses on the predictive performance in cases of extreme events. However, the restriction of conventional forecast evaluation methods to subsets of extreme observations…
The size that an epidemic can reach, measured in terms of the number of fatalities, is an extremely relevant quantity. It has been recently claimed [Cirillo & Taleb, Nature Physics 2020] that the size distribution of major epidemics in…
In this paper, according to a certain criterion, we divide the exponential distribution class into three subclasses. One of them is closely related to the regular-variation-tailed distribution class, so it is called the…
In this paper, we investigate risk measures such as value at risk (VaR) and the conditional tail expectation (CTE) of the extreme (maximum and minimum) and the aggregate (total) of two dependent risks. In finance, insurance and the other…
Returns distributions are heavy-tailed across asset classes. In this note, I examine the implications of this well-known stylized fact for the joint statistics of performance (absolute return) and Sharpe ratio (risk-adjusted return). Using…
Verifying probabilistic forecasts for extreme events is a highly active research area because popular media and public opinions are naturally focused on extreme events, and biased conclusions are readily made. In this context, classical…
Empirical likelihood is a well-known nonparametric method in statistics and has been widely applied in statistical inference. The method has been employed by Lu and Peng (2002) to constructing confidence intervals for the tail index of a…
One of the main goal of extreme value analysis is to estimate the probability of rare events given a sample from an unknown distribution. The upper tail behavior of this distribution is described by the extreme value index. We present a new…
We introduce a new actuarial tail-shape index, the $\theta$-index, based on a probability equal level relationship between Value at Risk and Expected Shortfall. The index is defined at each tail probability level as the parameter value for…
A weighted Gaussian approximation to tail product-limit process for Pareto-like distributions of randomly right-truncated data is provided and a new consistent and asymptotically normal estimator of the extreme value index is derived. A…
Individual probabilities refer to the probabilities of outcomes that are realized only once: the probability that it will rain tomorrow, the probability that Alice will die within the next 12 months, the probability that Bob will be…
In this note we prove bounds on the upper and lower probability tails of sums of independent geometric or exponentially distributed random variables. We also prove negative results showing that our established tail bounds are asymptotically…
We develop an econometric framework integrating heavy-tailed Student's $t$ distributions with behavioral probability weighting while preserving infinite divisibility. Using 432{,}752 observations across 86 assets (2004--2024), we…
Extreme quantile treatment effects (eQTEs) measure the causal impact of a treatment on the tails of an outcome distribution and are central for studying rare, high-impact events. Standard QTE methods often fail in extreme regimes due to…
We derive the tail inequalities between two random variables starting from inequalities between its moment, or more generally between its Lebesgue-Riesz norms, which holds true on certain sets of parameters. We consider some applications…
This paper introduces a novel measure to quantify the directional dependence of extreme events between two variables. The proposed approach is designed to capture asymmetric tail dependence by studying conditional tail expectations of…