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I want a formula of Gini's concentration ratio and the Lorenz concentration ratio involving three parameters about gamma Weibull distribution.

I have formula about gamma distribution that contains only one parameter. but I want it for three parameters. For the gamma distribution,the Gini concentration ratio is $ G= (Γ(α+1/2))/(√π Γ(α+1)) $.

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1 Answer 1

The Gini coefficient of any distribution $F$ with mean $\alpha$ is

$$\frac{1}{\alpha}\int_{-\infty}^\infty F(x)\left(1-F(x)\right) dx.$$

The three-parameter Gamma family is the location-scale extension of the Gamma distribution. That is, the location parameter $\mu$ and the scale parameter $\sigma\gt 0$ induce a new distribution

$$F_{\mu,\sigma}(z) = F\left(\frac{z-\mu}{\sigma}\right)$$

whose standard deviation has been multiplied by $\sigma$ and whose mean is $\sigma\alpha+\mu$.

Substitute $x = \frac{z-\mu}{\sigma}$ in the integral. Noting that $dz = \sigma\, d\left(\frac{z-\mu}{\sigma}\right) = \sigma dx,$ it is immediate that the Gini coefficient of $F_{\mu,\sigma}$ is

$$\frac{1}{\sigma\alpha+\mu}\int_{-\infty}^\infty F_{\mu,\sigma}(z)\left(1-F_{\mu,\sigma}(z)\right) dz = \frac{\sigma \alpha}{\sigma\alpha+\mu}\left(\frac{1}{\alpha}\int_{-\infty}^\infty F(x)\left(1-F(x)\right) dx\right).$$

The parenthesized expression on the right hand side is the Gini coefficient of $F$. This general result shows how the Gini coefficient responds to changes of location and scale.

For the Gamma distribution with shape parameter $\alpha$ the mean is $\alpha$ and its Gini coefficient is $1/(\alpha B(\alpha, 1/2))$ (equivalent to the expression in the question). Therefore the Gini coefficient for the three-parameter version equals $$\frac{\sigma\alpha}{\sigma\alpha+\mu} \frac{1}{\alpha B(\alpha, 1/2)}$$

provided $\mu \ne -\alpha$.

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I suppose Zahida need to find the Gini concentration ratio of three parameter Gamma Weibull distribution instead of three parameter Gamma distribution. –  Azeem Jun 23 at 13:41
    
@Azeem Thank you; I see why you would conjecture that this is what was requested. I addressed the Gamma distribution because (1) it, and not the Gamma-Weibull, has the "only one parameter" mentioned and (2) the question provides the correct Gini coefficient for that distribution, so I assumed that was the one that needed to be generalized to three parameters. Let's wait for Zahida to clarify which one is intended. –  whuber Jun 23 at 14:13

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