Bounds for Gini’s mean difference based on first four moments, with some applications

Abstract

In this paper, we obtain lower and upper bounds for the Gini mean difference for the case of independent and identically distributed random variables based on the information about mean, variance, skewness, and kurtosis of the distribution. We also obtain some relationships between the three dispersion measures in the general case. The established results improve some well-known bounds and inequalities. These results are then used to sharpen some inequalities concerning Gini’s index, order statistics and premium principles. Examples demonstrate that the proposed bounds perform much better than the existing ones.

Appendix

For the convenience of readers and for making the paper self-contained, we list the following inequalities.

Inequality 105 in Hardy et al. (1952): If $\psi$ and $\chi$ are continuous, positive, and strictly monotonic, then $\psi ^{-1}\{\psi (\sum a)\}\ge \chi ^{-1}\{\chi (\sum a)\}$ whenever (1) $\psi$ and $\chi$ vary in opposite directions, or (2) $\psi$ and $\chi$ vary in the same direction and $\chi /\psi$ is monotonic.

Grüss inequality (Cerone and Dragomir (2007)): If g and h are integrable on [a, b] and $m\le g(x)\le M, n\le h(x)\le N$ for $a.e.\; x\in [a,b]$, then

$$\begin{aligned} \left| \frac{1}{b-a}\int _a^b g(x)h(x)dx-\frac{1}{b-a}\int _a^b g(x)dx\cdot \frac{1}{b-a}\int _a^b h(x)dx\right| \le \frac{1}{4}(M-m)(N-n). \end{aligned}$$

Sonin identity (Mitrinović et al. (1993), p. 246): If g and h are Lebesgue integrable on [a, b], $\gamma \in \mathbb {R}$, then

$$\begin{aligned}{} & {} \frac{1}{b-a}\int _a^b g(x)h(x)dx-\frac{1}{b-a}\int _a^b g(x)dx\cdot \frac{1}{b-a}\int _a^b h(x)dx \\{} & {} \quad = \frac{1}{b-a}\int _a^b \left( h(x)-\frac{1}{b-a} \int _a^b h(t)dt\right) (g(x)-\gamma )dx. \end{aligned}$$

Korkine identity (Mitrinović et al. (1993), p. 242): If g and h are Lebesgue integrable on [a, b], then

$$\begin{aligned}{} & {} \frac{1}{b-a}\int _a^b g(x)h(x)dx-\frac{1}{b-a}\int _a^b g(x)dx\cdot \frac{1}{b-a}\int _a^b h(x)dx \\{} & {} \quad = \frac{1}{2(b-a)^2}\int _a^b\int _a^b(h(x)-h(y))(g(x)-g(y))dxdy. \end{aligned}$$