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Salamanca, J.J. (2020), Sets that maximize probability and a related variational problem. Can J Statistics. https://doi.org/10.1002/cjs.11578
In this work, the author considers random variables whose sample space has a certain complexity. To be more precise, when the sample space is modelled by a ‘differentiable’ metric space (i.e., the most general metric spaces where variational calculus can be performed). In fact, many random variables do not admit a vector space nor an affine space as their sample space. In this context, the manuscript deals with two problems: the first one studies how to capture the maximum probability under a volume restriction, that is, having fixed a volume, how to find the sets which attain the maximum probability. The second problem is the inverse of the former one: how to find the sets that have the minimum volume for a fixed probability. The author addresses the optimal sets for the second problem as a generalization of the classical notion of quantile.
In this paper, the solutions to both problems and a stability condition are given.
These problems have not been studied in the theory of probability before. Moreover, both problems have practical applications that are shown with several examples. Perhaps, the case of an electric power plant is the most illustrative example. Here, the author begins with a known distribution of the capturable energy on Earth. The two theoretical problems have their counterparts in this example as follows: on the one hand, the first problem reads: how to find the optimum configuration of the power plant when its area is fixed. On the other hand, for the second problem, how to find the configuration that minimizes the area of the power plant when the proportion of capturable energy that the power plant must recollect is fixed.