Nonparametric Inference for VaR, CTE, and Expectile with High-order Precision

Value-at-Risk (VaR) and Conditional Tail Expectation (CTE) are the two most frequently applied risk measures in quantitative risk management. Recently, expectile has also attracted much attention as a risk measure due to its elicitability property. This paper establishes empirical likelihood based estimation with high-order precision for these three risk measures. The superiority of the estimation is justified both in theory and via simulation studies. Extensive simulation studies confirm that our method signifificantly improves the coverage probabilities for interval estimation of the three risk measures, compared to three competing methods available in the literature