Calculates the Gini index for the F distribution with degrees of freedom \(\nu_1\) (df1) and \(\nu_2\) (df2).
gf(df1, df2)A numeric value with the Gini index. A NA is returned when degrees of freedom are non-numeric or \(df1 \leq 0\) or \(df2 < 2\) .
A positive real number specifying the degrees of freedom \(\nu_1\) of the F distribution.
A positive real number higher or equal than two specifying the degrees of freedom \(\nu_2\) of the F distribution.
Juan F Munoz jfmunoz@ugr.es
Jose M Pavia pavia@uv.es
Encarnacion Alvarez encarniav@ugr.es
The F distribution with \(\nu_1\) (argument df1) and \(\nu_2\) (argument df2) degrees of freedom and denoted as \(F_{\nu_1,\nu_2}\), where \(\nu_1>0\) and \(\nu_2 > 0\), has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995)
$$f(y) = \displaystyle \frac{\Gamma\left(\frac{\nu_{1}}{2} + \frac{\nu_{2}}{2}\right)}{\Gamma\left(\frac{\nu_{1}}{2}\right)\Gamma\left(\frac{\nu_{2}}{2}\right)}\left( \frac{\nu_{1}}{\nu_{2}}\right)^{\nu_{1}/2}y^{\nu_{1}/2-1}\left(1 + \frac{\nu_{1}y}{\nu_{2}}\right)^{-(\nu_{1}+\nu_{2})/2},$$
and a cumulative distribution function given by
$$F(y)= \displaystyle I_{\nu_{1}y/(\nu_{1}y + \nu_{2})}\left( \frac{\nu_{1}}{2}, \frac{\nu_{2}}{2} \right),$$
where \(y \geq 0\),
$$\Gamma(\alpha) = \int_{0}^{\infty}t^{\alpha-1}e^{-t}dt$$
is the gamma function,
$$I_{y}(a,b)=\displaystyle \frac{B(y;a,b)}{B(a,b)}$$
is the regularized incomplete beta function,
$$B(a,b) = \displaystyle \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$$
is the beta function,
and
$$B(y;a,b) = \displaystyle \int_{0}^{y}t^{a-1}(1-t)^{b-1}dt$$ is the incomplete beta function.
The Gini index, for \(\nu_2 \geq 2\), can be computed as $$G = 2\left(0.5 - \displaystyle \frac{\nu_{2} - 2}{ \nu_{2}}\int_{0}^{1}\int_{0}^{Q(y)}yf(y)dy\right),$$ where \(Q(y)\) is the quantile function of the F distribution.
Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1, chapter 14. Wiley, New York.
gchisq, ggamma, ggompertz, glnorm
# Gini index for the F distribution with 'df1 = 10' and 'df2 = 20' degrees of freedom.
gf(df1 = 10, df2 = 20)
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