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)dgb2 gives the density, pgb2 the distribution
function, qgb2 the quantile function, and rgb2
generates random deviates.
numeric; vector of quantiles.
numeric; positive parameter.
numeric; positive parameter.
numeric; positive parameters of the Beta distribution.
numeric; vector of probabilities.
numeric; number of observations. If length(n) > 1, the length
is taken to be the number required.
Monique Graf
The Generalized Beta distribution of the second kind with parameters shape1 \(= a\), scale \(= b\), shape2 \(= p\) and shape3 \(= q\) has density
$$f(x)=\frac{a(x/b)^{ap-1}}{bB(p,q)(1+(x/b)^{a})^{p+q}}$$
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.
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.
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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