GB2 (version 2.1.1)

gb2: The Generalized Beta Distribution of the Second Kind

Description

Density, distribution function, quantile function and random generation for the Generalized beta distribution of the second kind with parameters a, b, p and q.

Usage

dgb2(x, shape1, scale, shape2, shape3)
pgb2(x, shape1, scale, shape2, shape3)
qgb2(prob, shape1, scale, shape2, shape3)
rgb2(n, shape1, scale, shape2, shape3)

Arguments

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 length(n) > 1, the length is taken to be the number required.

Value

dgb2 gives the density, pgb2 the distribution function, qgb2 the quantile function, and rgb2 generates random deviates.

Details

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.

References

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.

See Also

beta for the Beta function and dbeta for the Beta distribution.

Examples

# NOT RUN {
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)
# }