Geodesic
On an inequality and the related characterization of the gamma distribution
Applications of Mathematics, Tome 38 (1993) no. 1, pp. 11-18

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In this paper we derive conditions upon the nonnegative random variable \xi under which the inequality $Dg(\xi)\leq cE\left[g'\left(\xi\right)\right]^2\xi$ holds for a fixed nonnegative constant $c$ and for any absolutely continuous function $g$. Taking into account the characterization of a Gamma distribution we consider the functional $U_\xi = \sup_g \frac{Dg\left(\xi\right)}{E\left[g'\left(\xi\right)\right]^2\xi}$ and establishing some of its properties we show that $U_\xi \geq 1$ and that $U_\xi =1$ iff the random variable $\xi$ has a Gamma distribution.
In this paper we derive conditions upon the nonnegative random variable \xi under which the inequality $Dg(\xi)\leq cE\left[g'\left(\xi\right)\right]^2\xi$ holds for a fixed nonnegative constant $c$ and for any absolutely continuous function $g$. Taking into account the characterization of a Gamma distribution we consider the functional $U_\xi = \sup_g \frac{Dg\left(\xi\right)}{E\left[g'\left(\xi\right)\right]^2\xi}$ and establishing some of its properties we show that $U_\xi \geq 1$ and that $U_\xi =1$ iff the random variable $\xi$ has a Gamma distribution.
DOI : 10.21136/AM.1993.104530
Classification : 60E15, 60E99, 62E10
Keywords: characterizations; Gamma distribution 
Koicheva, Maia. On an inequality and the related characterization of the gamma distribution. Applications of Mathematics, Tome 38 (1993) no. 1, pp. 11-18. doi: 10.21136/AM.1993.104530
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