gbeta | R Documentation |
Calculates the Gini index for the Beta distribution with shape parameters a
(shape1
) and b
(shape2
).
gbeta(shape1, shape2)
shape1 |
A positive real number specifying the shape1 parameter |
shape2 |
A positive real number specifying the shape2 parameter |
The Beta distribution with shape parameters a
(argument shape1
) and b
(argument shape2
) and denoted as Beta(a,b)
, where a>0
and b>0
, has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Yee, 2022)
f(y) = \displaystyle \frac{1}{B(a,b)}y^{a-1}(1-y)^{b-1},
and a cumulative distribution function given by
F(y)= \displaystyle \frac{B(y;a,b)}{B(a,b)}
where 0 \leq y \leq 1
,
B(a,b) = \displaystyle \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}
is the beta function,
\Gamma(\alpha) = \int_{0}^{\infty}t^{\alpha-1}e^{-t}dt
is the gamma 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 can be computed as
G = \displaystyle \frac{2}{a}\frac{B(a+b,a+b)}{B(a,a)B(b,b)}.
A numeric value with the Gini index. A NA
is returned when a shape parameter is non-numeric or non-positive.
Juan F Munoz jfmunoz@ugr.es
Jose M Pavia pavia@uv.es
Encarnacion Alvarez encarniav@ugr.es
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.
Yee, T. W. (2022). VGAM: Vector Generalized Linear and Additive Models. R package version 1.1-7, https://CRAN.R-project.org/package=VGAM.
gf
, gunif
, gweibull
, ggamma
, gchisq
# Gini index for the Beta distribution with shape parameters 'a = 2' and 'b = 1'.
gbeta(shape1 = 2, shape2 = 1)
# Gini index for the Beta distribution with shape parameters 'a = 1' and 'b = 2'.
gbeta(shape1 = 1, shape2 = 2)
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