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How a probabilistic analogue of the mean value theorem yields stein-type covariance identities

Part of: Distribution theory Distribution theory - Probability

Published online by Cambridge University Press:  18 January 2022

Georgios Psarrakos
Georgios Psarrakos*
Affiliation:
University of Piraeus
*
*Postal address: Department of Statistics and Insurance Science, University of Piraeus, 18534 Piraeus, Greece. Email: gpsarr@unipi.gr
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Abstract

A method for the construction of Stein-type covariance identities for a nonnegative continuous random variable is proposed, using a probabilistic analogue of the mean value theorem and weighted distributions. A generalized covariance identity is obtained, and applications focused on actuarial and financial science are provided. Some characterization results for gamma and Pareto distributions are also given. Identities for risk measures which have a covariance representation are obtained; these measures are connected with the Bonferroni, De Vergottini, Gini, and Wang indices. Moreover, under some assumptions, an identity for the variance of a function of a random variable is derived, and its performance is discussed with respect to well-known upper and lower bounds.

Keywords

Stein-type covariance identityprobabilistic mean value theoremweighted distributionvariance identityrisk measureshazard rate

MSC classification

Primary: 60E05: Distributions
Secondary: 62E10: Characterization and structure theory
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 2 , June 2022 , pp. 350 - 365
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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