The Efficiency Gap
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Parameter estimation via M- and Z-estimation is broadly considered to be equally powerful in semiparametric models for one-dimensional functionals. This is due to the fact that, under sufficient regularity conditions, there is a one-to-one relation between the corresponding objective functions - strictly consistent loss functions and oriented strict identification functions - via integration and differentiation. When dealing with multivariate functionals such as multiple moments, quantiles, or the pair (Value at Risk, Expected Shortfall), this one-to-one relation fails due to integrability conditions: Not every identification function possesses an antiderivative. The most important implication of this failure is an efficiency gap: The most efficient Z-estimator often outperforms the most efficient M-estimator, implying that the semiparametric efficiency bound cannot be attained by the M-estimator in these cases. We show that this phenomenon arises for pairs of quantiles at different levels and for the pair (Value at Risk, Expected Shortfall), where we illustrate the gap through extensive simulations.
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