gburr | R Documentation |
Calculates the Gini index for the Burr Type XII (Singh-Maddala) distribution with scale
parameter b
and shape parameters g
(shape.g
) and s
(shape.s
).
gburr(
scale = 1,
shape.g = 1,
shape.s = 1
)
scale |
A positive real number specifying the scale parameter |
shape.g |
A positive real number specifying the shape parameter |
shape.s |
A positive real number specifying the shape parameter |
The Burr Type XII (Singh-Maddala) distribution with scale
parameter b
, shape parameters g
(argument shape.g
) and s
(argument shape.s
) and denoted as BurrXII(b,g,s)
, where b>0
, g>0
and s>0
, has a probability density function given by (Kleiber and Kotz, 2003; Johnson et al., 1995; Rodriguez, 1977; Yee, 2022)
f(y) = \displaystyle \frac{gs}{b}\left(\frac{y}{b}\right)^{g-1}\left[1 + \left(\frac{y}{b}\right)^{g}\right]^{-(s+1)},
and a cumulative distribution function given by
F(y)=1-\left[1 + \displaystyle \left( \frac{y}{b}\right)^{g} \right]^{-s},
where y>0
.
The Gini index can be computed as
G = 2\left(0.5 - \displaystyle \frac{1}{E[y]}\int_{0}^{1}\int_{0}^{Q(y)}yf(y)dy\right),
where
Q(y)
is the quantile function of the Burr Type XII (Singh-Maddala) distribution, and E[y]
is the expectation of the distribution. The Burr Type XII (Singh-Maddala) distribution is related to the Pareto (IV) distribution: BurrXII(b,g,s) = ParetoIV(0,b,1/g,s)
.
A numeric value with the Gini index. A NA
is returned when any of the 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.
Rodriguez, R. N. (1977). A guide to the Burr type XII distributions. Biometrika, 64(1), 129-134.
Yee, T. W. (2022). VGAM: Vector Generalized Linear and Additive Models. R package version 1.1-7, https://CRAN.R-project.org/package=VGAM.
gparetoIV
, gpareto
, gparetoI
, gparetoII
, gparetoIII
, gfisk
# Gini index for the Burr Type XII distribution with 'scale = 1', 'shape.g = 2', 'shape.s = 1'.
gburr(scale = 1, shape.g = 2, shape.s = 1)
# Gini index for the Burr Type XII distribution with 'scale = 1', 'shape.g = 5', 'shape.s = 3'.
gburr(scale = 1, shape.g = 5, shape.s = 3)
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