Abstract
In this paper, we study the problem of defining statistical parameters when the uncertainty is expressed using a fuzzy measure. We extend the concept of monotone expectation in order to define a monotone variance and monotone moments. We also study parameters that allow the joint analysis of two functions defined over the same reference set. Finally, we propose some parameters over product spaces, considering the case in which a function over the product space is available and also the case in which such function is obtained by combining those in the marginal spaces.
Highlights
Fuzzy measures [1], known as capacities [2], non-additive measures or monotone measures [3], have shown to be a valuable tool for representing uncertainty, since they are able to cope with more general scenarios than probability measures do
Even though fuzzy measures have been successfully applied in a wide range of applications [4], no theory analogous to mathematical statistics has emerged around them in the general case, due to the difficulty of defining statistical parameters with a clear interpretation when additivity is replaced by monotonicity
We assume that the measurable space is endowed with a fuzzy measure, and we will study the definition of statistical parameters over the measurable function, in a similar way as statistical parameters over a random variable can be defined from a probability measure
Summary
Fuzzy measures [1], known as capacities [2], non-additive measures or monotone measures [3], have shown to be a valuable tool for representing uncertainty, since they are able to cope with more general scenarios than probability measures do. We assume that the measurable space is endowed with a fuzzy measure, and we will study the definition of statistical parameters over the measurable function, in a similar way as statistical parameters over a random variable can be defined from a probability measure. In this way, we attempt to handle more general scenarios than the ones covered by probability measures.
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