2,468 research outputs found
Quasi-Hadamard differentiability of general risk functionals and its application
We apply a suitable modification of the functional delta method to
statistical functionals that arise from law-invariant coherent risk measures.
To this end we establish differentiability of the statistical functional in a
relaxed Hadamard sense, namely with respect to a suitably chosen norm and in
the directions of a specifically chosen "tangent space". We show that this
notion of quasi-Hadamard differentiability yields both strong laws and limit
theorems for the asymptotic distribution of the plug-in estimators. Our results
can be regarded as a contribution to the statistics and numerics of risk
measurement and as a case study for possible refinements of the functional
delta method through fine-tuning the underlying notion of differentiabilit
Statistical Estimation of Composite Risk Functionals and Risk Optimization Problems
We address the statistical estimation of composite functionals which may be
nonlinear in the probability measure. Our study is motivated by the need to
estimate coherent measures of risk, which become increasingly popular in
finance, insurance, and other areas associated with optimization under
uncertainty and risk. We establish central limit formulae for composite risk
functionals. Furthermore, we discuss the asymptotic behavior of optimization
problems whose objectives are composite risk functionals and we establish a
central limit formula of their optimal values when an estimator of the risk
functional is used. While the mathematical structures accommodate commonly used
coherent measures of risk, they have more general character, which may be of
independent interest
Exponential functionals of Levy processes
This text surveys properties and applications of the exponential functional
of real-valued L\'evy processes .Comment: Published at http://dx.doi.org/10.1214/154957805100000122 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Estimating invariant laws of linear processes by U-statistics
Suppose we observe an invertible linear process with independent mean-zero
innovations and with coefficients depending on a finite-dimensional parameter,
and we want to estimate the expectation of some function under the stationary
distribution of the process. The usual estimator would be the empirical
estimator. It can be improved using the fact that the innovations are centered.
We construct an even better estimator using the representation of the
observations as infinite-order moving averages of the innovations. Then the
expectation of the function under the stationary distribution can be written as
the expectation under the distribution of an infinite series in terms of the
innovations, and it can be estimated by a U-statistic of increasing order
(also called an ``infinite-order U-statistic'') in terms of the estimated
innovations. The estimator can be further improved using the fact that the
innovations are centered. This improved estimator is optimal if the
coefficients of the linear process are estimated optimally
- …