In this paper we propose an optimal predictor of a random variable that has either an infinite mean or an infinite variance. The method consists of transforming the random variable such that the transformed variable has a finite mean and finite variance. The proposed predictor is a generalized arithmetic mean which is similar to the notion of certainty price in utility theory. Typically, the transformation consists of a parametric family of bijections, in which case the parameter might be chosen to minimize the prediction error in the transformed coordinates. The statistical properties of the estimator of the proposed predictor are studied, and confidence intervals are provided. The performance of the procedure is illustrated using simulated and real data.

翻译：本文提出了一种针对具有无限均值或无限方差的随机变量的最优预测方法。该方法通过对随机变量进行变换，使变换后的变量具有有限均值和有限方差。所提出的预测器是一种广义算术均值，类似于效用理论中的确定等价价格概念。通常情况下，变换由参数化的双射函数族构成，此时可通过最小化变换坐标下的预测误差来选择参数。本文研究了该预测器估计量的统计性质，并给出了置信区间。通过模拟数据和实际数据验证了该方法的有效性。