✨ TL;DR
This paper develops a distributionally robust estimation method that minimizes worst-case conditional value-at-risk (CVaR) of estimation error when the true distribution is uncertain but lies within a Wasserstein ball. The method can be computed via tractable semidefinite programming and outperforms existing approaches on electricity price forecasting.
Traditional estimation methods assume the joint probability distribution of signals and observations is known, which is unrealistic in practice. When the true distribution is uncertain or misspecified, standard estimators can perform poorly, especially in terms of tail risk. Existing robust estimation approaches often focus on worst-case expected error rather than tail risk measures, which are critical for risk-sensitive applications where extreme errors have disproportionate consequences. There is a need for estimation methods that are both robust to distributional uncertainty and explicitly account for tail risk through appropriate risk measures.
The authors formulate a distributionally robust optimization problem where the estimator minimizes the worst-case conditional value-at-risk (CVaR) of squared estimation error over all distributions within a type-2 Wasserstein ball centered at a nominal distribution. They restrict attention to affine estimators, which are linear functions of the observations. The key methodological contribution is reformulating this infinite-dimensional robust optimization problem as a tractable semidefinite program (SDP) when the nominal distribution has finite support. This reformulation exploits the structure of both the Wasserstein ambiguity set and the CVaR risk measure to convert the problem into a computationally solvable convex optimization problem.