gb2 | R Documentation |
Density, distribution function, quantile function and random generation for the Generalized beta distribution of the second kind with parameters a
, b
, p
and q
.
dgb2(x, shape1, scale, shape2, shape3) pgb2(x, shape1, scale, shape2, shape3) qgb2(prob, shape1, scale, shape2, shape3) rgb2(n, shape1, scale, shape2, shape3)
x |
numeric; vector of quantiles. |
shape1 |
numeric; positive parameter. |
scale |
numeric; positive parameter. |
shape2, shape3 |
numeric; positive parameters of the Beta distribution. |
prob |
numeric; vector of probabilities. |
n |
numeric; number of observations. If |
The Generalized Beta distribution of the second kind with parameters shape1
= a, scale
= b, shape2
= p and shape3
= q has density
a(x/b)^(ap-1)/bB(p,q)(1+(x/b)^(a))^(p+q), x ≥ 0
for a > 0, b > 0, p > 0 and q > 0, where B(p,q) is the Beta function (beta
). If Z
follows a Beta distribution with parameters p and q and
y = \frac{z}{1-z},
then
x = b * y^{1/a}
follows the GB2 distribution.
dgb2
gives the density, pgb2
the distribution
function, qgb2
the quantile function, and rgb2
generates random deviates.
Monique Graf
Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences, chapter 6. Wiley, Ney York.
McDonald, J. B. (1984) Some generalized functions for the size distribution of income. Econometrica, 52, 647–663.
beta
for the Beta function and dbeta
for the Beta distribution.
a <- 3.9 b <- 18873 p <- 0.97 q <- 1.03 x <- qgb2(0.6, a, b, p, q) y <- dgb2(x, a, b, p, q)
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