We aim to analyze the behaviour of a finitetime stochastic system, whose model is not available, in the context of more rare and harmful outcomes. Standard estimators are not effective in predicting the magnitude of such outcomes due to their rarity. Instead, we leverage Extreme Value Theory (EVT), the study of the long-term behaviour of normalized maxima of random variables. In this letter, we quantify risk using the upper-semideviation E(max{Y − E(Y), 0}), which is the expected exceedance of a random variable Y above its mean E(Y). To assess more rare and harmful outcomes, we propose an EVT-based estimator for this functional in a given fraction of the worst cases. We show that our estimator enjoys a closed-form representation in terms of a popular functional called the conditional value-at-risk. In experiments, we illustrate the extrapolation power of our estimator when a small number of i.i.d. samples is available (<50). Our approach is useful for estimating the risk of finite-time systems when models are inaccessible and data collection is expensive. The numerical complexity does not grow with the size of the state space. This letter initiates a broader pathway for estimating the risk of large-scale systems.

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