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Non-Parametric Probability Distributions Embedded Inside of a Linear Space Provided with a Quadratic Metric

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  • Pierpaolo Angelini

    (Department of Statistical Sciences, La Sapienza University, 00185 Roma, Italy)

  • Fabrizio Maturo

    (Department of Mathematics and Physics, University of Campania “Luigi Vanvitelli”, 81100 Caserta, Italy)

Abstract

There exist uncertain situations in which a random event is not a measurable set, but it is a point of a linear space inside of which it is possible to study different random quantities characterized by non-parametric probability distributions. We show that if an event is not a measurable set then it is contained in a closed structure which is not a σ -algebra but a linear space over R . We think of probability as being a mass. It is really a mass with respect to problems of statistical sampling. It is a mass with respect to problems of social sciences. In particular, it is a mass with regard to economic situations studied by means of the subjective notion of utility. We are able to decompose a random quantity meant as a geometric entity inside of a metric space. It is also possible to decompose its prevision and variance inside of it. We show a quadratic metric in order to obtain the variance of a random quantity. The origin of the notion of variability is not standardized within this context. It always depends on the state of information and knowledge of an individual. We study different intrinsic properties of non-parametric probability distributions as well as of probabilistic indices summarizing them. We define the notion of α -distance between two non-parametric probability distributions.

Suggested Citation

  • Pierpaolo Angelini & Fabrizio Maturo, 2020. "Non-Parametric Probability Distributions Embedded Inside of a Linear Space Provided with a Quadratic Metric," Mathematics, MDPI, vol. 8(11), pages 1-17, October.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:11:p:1901-:d:438112
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    References listed on IDEAS

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    Cited by:

    1. Angelini, Pierpaolo & Maturo, Fabrizio, 2022. "The price of risk based on multilinear measures," International Review of Economics & Finance, Elsevier, vol. 81(C), pages 39-57.
    2. Pierpaolo Angelini, 2023. "Probability Spaces Identifying Ordinal and Cardinal Utilities in Problems of an Economic Nature: New Issues and Perspectives," Mathematics, MDPI, vol. 11(20), pages 1-22, October.

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