Estimation of Spectral Risk Measure for Left Truncated and Right Censored Data

Left truncated and right censored data are encountered frequently in insurance loss data due to deductibles and policy limits. Risk estimation is an important task in insurance as it is a necessary step for determining premiums under various policy terms. Spectral risk measures are inherently coherent and have the benefit of connecting the risk measure to the user's risk aversion. In this paper we study the estimation of spectral risk measure based on left truncated and right censored data. We propose a non parametric estimator of spectral risk measure using the product limit estimator and establish the asymptotic normality for our proposed estimator. We also develop an Edgeworth expansion of our proposed estimator. The bootstrap is employed to approximate the distribution of our proposed estimator and shown to be second order accurate. Monte Carlo studies are conducted to compare the proposed spectral risk measure estimator with the existing parametric and nonparametric estimators for left truncated and right censored data. Our observation reveal that the proposed estimator outperforms all the estimators for small values of $k$ (coefficient of absolute risk aversion) and for small sample sizes for i.i.d. case. In the dependent case, it demonstrates superior performance for small $k$ across all sample sizes. Finally, we estimate the exponential spectral risk measure for two data sets viz; the Norwegian fire claims and the French marine losses.