Expected Shortfall Semi-Scale T-Distribution M-Estimator

The influence function of parametric t-distribution expected shortfall (ES) estimators has an approximately symmetric shape, for which large positive returns indicate large losses. We avoid this risk estimator’s unacceptable feature by introducing an ES semi-scale M-estimator for t-distributions, for which the usual t-distribution scale parameter is replaced by a semi-scale parameter. We derive the influence function of the ES semi-scale M-estimator, and show that its influence function has large values only for large negative returns as one expects, and only very small typically negative values for positive returns. The computation of an ES semi-scale M-estimator is shown to be a simple modification of a parametric t-distribution ES maximum-likelihood estimator (MLE), in which the scale MLE is replaced by a semi-scale estimator. We also derive the asymptotic variance expression for the ES semi-scale M-estimator, and show that its standard error is not very much larger than that of the t-distribution ES maximum-likelihood estimator.