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 non parametric estimators for left truncated and right censored data. Based on our simulation study we estimate the exponential spectral risk measure for three data sets viz; Norwegian fire claims data set, Spain automobile insurance claims and French marine losses.

翻译：在保险损失数据中，由于免赔额和保单限额，左截断右删失数据频繁出现。风险估计是保险领域的重要任务，因为它是确定不同保单条款下保费的必要步骤。谱风险测度本质上是相合的，具有将风险测度与用户风险厌恶程度相联系的优点。本文研究基于左截断右删失数据的谱风险测度估计问题。我们利用乘积限估计量提出一种谱风险测度的非参数估计量，并建立了所提估计量的渐近正态性。我们还发展了所提估计量的埃奇沃斯展开。采用自助法逼近所提估计量的分布，并证明其具有二阶"准确性"。通过蒙特卡洛研究，将所提谱风险测度估计量与现有的左截断右删失数据参数及非参数估计量进行比较。基于模拟研究，我们估计了三个数据集（挪威火灾索赔数据集、西班牙汽车保险索赔数据集和法国海上损失数据集）的指数型谱风险测度。