On the generalized cumulative residual entropy with applications in actuarial science

Recently, Psarrakos and Navarro (2013) proposed a new measure of uncertainty which extends the cumulative residual entropy (CRE), called the generalized cumulative residual entropy (GCRE). In the present paper, new properties and applications in actuarial risk measures of the GCRE are explored. Bounds, stochastic order properties and characterization results of the new entropy are also discussed. It is shown that the GCRE is invariant under changes in location, and scale directly with scale of a random variable; the same properties also hold for the standard deviation. It is also proved that the GCRE of the first order statistics can uniquely determine the parent distribution. The Weibull distribution, which is commonly used in several fields of applied probability, is characterized by using the mentioned generalized measure. The GCRE is studied as a risk measure and is compared to the standard deviation and the right-tail risk measure, where the latter measure was introduced by Wang (1998). Several examples are also given to illustrate the new results.