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Non‐parametric Estimation of Extreme Risk Measures from Conditional Heavy‐tailed Distributions
Author(s) -
Methni Jonathan El,
Gardes Laurent,
Girard Stéphane
Publication year - 2014
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/sjos.12078
Subject(s) - mathematics , estimator , statistics , conditional probability distribution , quantile , conditional variance , covariate , moment (physics) , extreme value theory , kernel density estimation , conditional expectation , econometrics , parametric statistics , heavy tailed distribution , generalized extreme value distribution , kernel (algebra) , probability distribution , autoregressive conditional heteroskedasticity , volatility (finance) , physics , classical mechanics , combinatorics
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.