  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.
