We propose a distributionally robust approach to risk-sensitive estimation of an unknown signal x from an observed signal y. The unknown signal and observation are modeled as random vectors whose joint probability distribution is unknown, but assumed to belong to a given type-2 Wasserstein ball of distributions, termed the ambiguity set. The performance of an estimator is measured according to the conditional value-at-risk (CVaR) of the squared estimation error. Within this framework, we study the problem of computing affine estimators that minimize the worst-case CVaR over all distributions in the given ambiguity set. As our main result, we show that, when the nominal distribution at the center of the Wasserstein ball is finitely supported, such estimators can be exactly computed by solving a tractable semidefinite program. We evaluate the proposed estimators on a wholesale electricity price forecasting task using real market data and show that they deliver lower out-of-sample CVaR of squared error compared to existing methods.

翻译：本文提出一种分布鲁棒方法用于未知信号x在观测信号y下的风险敏感估计。未知信号与观测信号被建模为联合概率分布未知但假设属于给定类型-2 Wasserstein球（称为模糊集）的随机向量。估计器的性能通过平方估计误差的条件风险价值（CVaR）进行衡量。在此框架下，我们研究如何计算使得给定模糊集中所有分布的最坏情况CVaR达到最小的仿射估计器。作为主要结果，我们证明：当Wasserstein球中心的名义分布为有限支撑时，此类估计器可通过求解可处理的半定规划精确计算。我们利用真实市场数据在批发电价预测任务中评估所提估计器，结果表明其平方误差的样本外CVaR低于现有方法。