Before revision: The estimation of risk measured in terms of a risk measure is typically done in two steps: in the first step, the distribution is estimated by statistical methods, either parametric or non-parametric. In the second step, the estimated distribution is considered as true distribution and the targeted risk-measure is computed. In the parametric case this is achieved by using the formula for the risk-measure in the model and inserting the estimated parameters. It is well-known that this procedure is not efficient because the highly nonlinear mapping from model parameters to the risk-measure introduces an additional biases. Statistical experiments show that this bias leads to a systematic underestimation of risk.
After revision: The estimation of risk measures recently gained a lot of attention, partly because of the backtesting issues of expected shortfall related to elicitability. In this work we shed a new and fundamental light on optimal estimation procedures in terms of bias. We show that once the parameters of a model need to be estimated, one has to take additional care when estimating risks. The typical plug-in approach, for example, introduces a bias which leads to a systematic underestimation of risk.