Title: Asymptotics for L-statistics with dependent data and applications to risk measurement

Name: Hideatsu Tsukahara

Abstract:

For the class of distortion risk measures, a natural estimator has the form of
L-statistics.  We investigate the large sample properties of general L-statistics
based on weakly dependent data and apply them to our estimator. Under certain 
regularity conditions, which are somewhat weaker than the ones found in the literature, 
we prove that the estimator is strongly consistent and asymptotically normal. 
Furthermore we give a consistent estimator for its asymptotic variance using
spectral density estimators of a related stationary sequence. 
The behavior of the estimator is examined using simulation in a simple inverse-gamma 
autoregressive stochastic volatility model.  It is found both theoretically and by
simulation study that the estimator always suffers a negative bias.  
We will discuss bias correction methods in the i.i.d. case and the possibility of
their extension to the dependent case.  Also, we indicate how the asymtotics for our estimator 
can be extended to the estimator for law invariant risk measures, which is obtained by 
droppping comonotonic additivity requirement.  Some related statistical issues may be 
discussed if time allows.