Characterizations of continuous distributions through inequalities involving the expected values of selected functions

Abstract

Characterizations of continuous distributions through inequalities involving the expected values of selected functions

Analysis and Applications of the Residual Varentropy of Random Lifetimes

“…Hence, by taking X t as reference, with g(x) = − log f (x) and integrating by parts, similarly as Eq. (3.9) of Goodarzi et al [18], we obtain (34).…”

“…Furthermore, a sharp uniform bound on varentropy for log-concave distributions is found in the work of Madiman [28]. An alternative way to calculate a bound for varentropy is discussed in Goodarzi et al [18] where the authors use some concepts of reliability theory. The generalization from log-concave to convex measures has been studied in the work of Li et al [26] where a bound on the varentropy for convex measures is discussed.…”

“…Hence, recalling (18), from ( 44) and (45) after some calculations, we obtain the residual varentropy of X (a) , for t > 0:…”

On lower bounds for the variance of functions of random variables

“…They had shown that Var[− ln f (X)] 2 9 , whereas by using (2.20), we can obtain lower bound 0.103985 for varentropy. The new lower bound is not sharper than the lower bound [12], however its calculation is very straightforward.…”

“…Let X have beta distribution with parameters a = 2 and b = 1. Goodarzi et al [12] obtained a lower bound for the varentropy of random variable X. They had shown that Var[− ln f (X)] 2 9 , whereas by using (2.20), we can obtain lower bound 0.103985 for varentropy.…”