<abstract xmlns="http://www.w3.org/1999/xhtml">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 <italic>L</italic>2 distance between a point and a random variable. Similarly, the median is the same concept but when the distance is measured by the <italic>L</italic>1 norm.Also, a <italic>geodesic distance</italic> can be defined on the cone of strictly positive vectors in ℝ<italic>n</italic> in such a way that, the minimizer of the distance between a point and a collection of points is their geometric mean.That geodesic distance induces a distance on the class of strictly positive random variables, which in turn leads to an interesting notions of conditional expectation (or best predictors) and their estimators. It also leads to different versions of the Law of Large Numbers and the Central Limit Theorem. For example, the lognormal variables appear as the analogue of the Gaussian variables for version of the Central Limit Theorem in the 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 L2 distance between a point and a random variable. Similarly, the median is the same concept but when the distance is measured by the L1 norm.Also, a geodesic distance can be defined on the cone of strictly positive vectors in ℝn in such a way that, the minimizer of the distance between a point and a collection of points is their geometric mean.That geodesic distance induces a distance on the class of strictly positive random variables, which in turn leads to an interesting notions of conditional expectation (or best predictors) and their estimators. It also leads to different versions of the Law of Large Numbers and the Central Limit Theorem. For example, the lognormal variables appear as the analogue of the Gaussian variables for version of the Central Limit Theorem in the logarithmic distance.

Best predictors in logarithmic distance between positive random variables

Journal of Applied Mathematics, Statistics and Informatics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Best predictors in logarithmic distance between positive random variables"}

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...

Best predictors in logarithmic distance between positive random variables

Published Online: Dec 21, 2019

Page range: 15 - 28

DOI: https://doi.org/10.2478/jamsi-2019-0006

Keywords
Prediction in logarithmic distance, Law of large numbers in logarithmic distance, Central Limit Theorem in logarithmic distance, Logarithmic geometry for positive random variables

© 2019 Faculty of Natural Sciences, University of Saint Cyril and Metodius

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Best predictors in logarithmic distance between positive random variables

Published Online: Dec 21, 2019

Page range: 15 - 28

DOI: https://doi.org/10.2478/jamsi-2019-0006

KeywordsPrediction in logarithmic distance, Law of large numbers in logarithmic distance, Central Limit Theorem in logarithmic distance, Logarithmic geometry for positive random variables

© 2019 Faculty of Natural Sciences, University of Saint Cyril and Metodius

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Prediction in logarithmic distance, Law of large numbers in logarithmic distance, Central Limit Theorem in logarithmic distance, Logarithmic geometry for positive random variables