A Quantile Mixing Approach for the Combination of Experts' Models

A Quantile Mixing Approach for the Combination of Experts' Models

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In this article, we provide theoretical justifications for the use of convex combinations of quantile functions in order to combine experts' (or softwares) models. We assume that the experts' estimates are given in the form of two probability measures μ and ν. The use of quantile functions corresponds to choosing the coupling between the two experts models that solves a general optimal transport problem. We interpret the transport problem as a minimal disagreement cost between the experts. This transport interpretation allows us to combine models in a multi-dimensional setting. Moreover, we prove that the convex combination of quantile functions is the supremum, with respect to the second order stochastic dominance, of all convex combinations of random variables with respective distributions μ and ν. In particular, this has important consequences in insurance and reinsurance pricing. Finally, we indicate a procedure to optimally choose the weights of the convex combination which is adapted to small sample situations, as the reinsurance of natural catastrophes. We explicitly compute the asymptotic behavior of the optimal weights, thanks to a new result on the joint asymptotic behavior of order statistics of vectors with a given copula function. 

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