Abstract. Among the many possible ways to study the right tail of a real-valued random variable, a particularly general one is given by considering the family of its Wang distortion risk measures. This class of risk measures encompasses various interesting indicators, such as the widely used Value-at-Risk and Tail Value-at-Risk, which are especially popular in actuarial science, for instance. In this paper, we first build simple extreme analogues of Wang distortion risk measures and we show how this makes it possible to consider many standard measures of extreme risk, including the usual extreme Value-at-Risk or Tail-Value-at-Risk, as well as the recently introduced extreme Conditional Tail Moment, in a unified framework. We then introduce adapted estimators when the random variable of interest has a heavy-tailed distribution and we prove their asymptotic normality. The finite sample performance of our estimators is assessed on a simulation study and we showcase our techniques on two sets of real data.
In this paper, we introduce a new risk measure, the so‐called conditional tail moment. It is defined as the moment of order a ≥ 0 of the loss distribution above the upper α‐quantile where α ∈ (0,1). Estimating the conditional tail moment permits us to estimate all risk measures based on conditional moments such as conditional tail expectation, conditional value at risk or conditional tail variance. Here, we focus on the estimation of these risk measures in case of extreme losses (where α ↓0 is no longer fixed). It is moreover assumed that the loss distribution is heavy tailed and depends on a covariate. The estimation method thus combines non‐parametric kernel methods with extreme‐value statistics. The asymptotic distribution of the estimators is established, and their finite‐sample behaviour is illustrated both on simulated data and on a real data set of daily rainfalls.
BACKGROUND AND PURPOSE:Magnetic Resonance Imaging is the modality of choice to detect spinal cord lesions in patients with Multiple Sclerosis (MS). However, this imaging is challenging. New sequences such as phase-sensitive inversion recovery have been developed to improve detection. Our aim was to compare a 3D phase-sensitive inversion recovery and a conventional imaging dataset including postcontrast T2WI and T1WI to detect MS spinal cord lesions. MATERIALS AND METHODS:This retrospective single-center study included 100 consecutive patients with MS (mean age, 41 years) from January 2015 to June 2016. One senior neuroradiologist and 1 junior radiologist blinded to clinical data checked for new spinal cord lesions, individually analyzing conventional and 3D phase-sensitive inversion recovery datasets separately, placing a 3-week delay between the 2 readings. A consensus reading was done with a third senior neuroradiologist. A Wilcoxon test was used to compare the 2 imaging datasets. Intra-and interobserver agreement was assessed by the coefficient. RESULTS:3D phase-sensitive inversion recovery detected significantly more lesions than conventional imaging (480 versus 168, P Ͻ .001). Eleven patients had no detected lesions on T2WI, whereas 3D phase-sensitive inversion recovery detected at least 1 lesion. All postcontrast T1WI enhancing lesions were also visible on 3D phase-sensitive inversion recovery. The signal-to-noise ratio was significantly higher using 3D phase-sensitive inversion recovery (0.63 versus 0.46, P ϭ .03). Mean reading confidence was significantly higher using 3D phase-sensitive inversion recovery. Inter-and intraobserver agreement was good for both datasets. CONCLUSIONS:Our study showed that 3D phase-sensitive inversion recovery significantly improved detection of cervical spinal cord lesions, including both enhancing and nonenhancing lesions in patients with MS. ABBREVIATIONS:CNR ϭ contrast-to-noise ratio; PSIR ϭ phase-sensitive inversion recovery; MAGNIMS ϭ Magnetic Resonance Imaging in Multiple Sclerosis; MERGE ϭ Multi-Echo Recombined Gradient Echo; PST1-IR ϭ phase-sensitive T1-weighted inversion-recovery MR imaging
A general way to study the extremes of a random variable is to consider the family of its Wang distortion risk measures. This class of risk measures encompasses several indicators such as the classical quantile/Value-at-Risk, the Tail-Valueat-Risk and Conditional Tail Moments. Trimmed and winsorised versions of the empirical counterparts of extreme analogues of Wang distortion risk measures are considered. Their asymptotic properties are analysed, and it is shown that it is possible to construct corrected versions of trimmed or winsorised estimators of extreme Wang distortion risk measures who appear to perform overall better than their standard empirical counterparts in difficult finite-sample situations when the underlying distribution has a very heavy right tail. This technique is showcased on a set of real fire insurance data.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.