Prediction in logarithmic distance

Abstract

The metric properties of the set in which random variables take their values lead to relevant probabilistic concepts. For example, the mean of a random variable is a best predictor in that it minimizes the standard Euclidean distance or $L_2$ norm in an appropriate class of random variables. Similarly, the median is the same concept but when the distance is measured by the $L_1$ norm. These two predictors stem from the fact that the mean and the median, minimize the distance to a given set of points when distances in $\mathbb{R}$ or in $\mathbb{R}^n$ are measured in the aforementioned metrics.\\ It so happens that an interesting {\it logarithmic distance} can be defined on the cone of strictly positive vectors in $\mathbb{R}^n$ in such a way that the minimizer of the distance to a collection of points is their geometric mean.\\ This distance on the base space leads to an interesting distance on the class of strictly positive random variables, which in turn leads to an interesting class of best predictors predictors and their estimators, as well as a corresponding notion of conditional expectation. The appropriate version of the Law of Large Numbers and the Central Limit Theorem, can also be obtained. We shall see that, for example, the lognormal variables are the analogue of the Gaussian variables for the modified version of the Central Limit Theorem.