gf: Gini index for the F distribution with user-defined degrees...

In giniVarCI: Gini Indices, Variances and Confidence Intervals for Finite and Infinite Populations

Gini index for the F distribution with user-defined degrees of freedom

Description

Calculates the Gini index for the F distribution with degrees of freedom \nu_1 (df1) and \nu_2 (df2).

Usage

gf(df1, df2)

Arguments

df1	A positive real number specifying the degrees of freedom \nu_1 of the F distribution.
df2	A positive real number higher or equal than two specifying the degrees of freedom \nu_2 of the F distribution.

Details

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.

Value

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 .

Author(s)

Juan F Munoz [e-mail]

Jose M Pavia [e-mail]

Encarnacion Alvarez [e-mail]

References

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.

See Also

gchisq, ggamma, ggompertz, glnorm

Examples

# Gini index for the F distribution with 'df1 = 10' and 'df2 = 20' degrees of freedom.
gf(df1 = 10, df2 = 20)

giniVarCI documentation built on May 29, 2024, 3:36 a.m.