On inequalities for moments and the covariance of monotone functions

Intuition based on the usual interpretation of the covariance of two random variables suggests that the inequality cov[f(X),g(X)]≥0 should hold for any random variable X and any two increasing functions f and g. The inequality holds indeed, but a proof is hard to find in the literature. In this pape...

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Bibliographic Details
Published in:Insurance, mathematics & economics mathematics & economics, 2014-03, Vol.55, p.91-95
Main Author: Schmidt, Klaus D.
Format: Article
Language:English
Subjects:
Actuaries
Collective model
Comonotonicity
Correlation
Covariance
Esscher premium
Inequality
Insurance
Mathematical analysis
Mathematical functions
Mathematics
Proof theory
Random sampling
Random variables
Reinsurance
Risk measures
Studies
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Summary:Intuition based on the usual interpretation of the covariance of two random variables suggests that the inequality cov[f(X),g(X)]≥0 should hold for any random variable X and any two increasing functions f and g. The inequality holds indeed, but a proof is hard to find in the literature. In this paper we provide an elementary proof of a more general inequality for moments and we present several applications in actuarial mathematics. •Any two increasing functions of a random variable have positive correlation.•Any two comonotone random variables have positive correlation.•Conditional tail expectation and tail value-at-risk are bounded below by the expectation.•The Esscher premium is bounded below by the net premium.•We give conditions for higher order moments of compound distributions to be finite.
ISSN:0167-6687
1873-5959
DOI:10.1016/j.insmatheco.2013.12.006