Generalised beta distribution of the second kind and its special cases

Density, distribution function, quantile function, and random generation for the generalised beta distribution of the second kind (GB2) and three of its special cases: the Dagum, Singh-Maddala (Burr XII) and Lomax (Pareto II) distributions.

Usage

dgb2(x, scale = 1, shape1, shape2, shape3, log = FALSE)

pgb2(q, scale = 1, shape1, shape2, shape3, lower.tail = TRUE, log.p = FALSE)

qgb2(p, scale = 1, shape1, shape2, shape3, lower.tail = TRUE, log.p = FALSE)

rgb2(n, scale = 1, shape1, shape2, shape3)

ddagum(x, scale = 1, shape1, shape2, log = FALSE)

pdagum(q, scale = 1, shape1, shape2, lower.tail = TRUE, log.p = FALSE)

qdagum(p, scale = 1, shape1, shape2, lower.tail = TRUE, log.p = FALSE)

rdagum(n, scale = 1, shape1, shape2)

dsinmad(x, scale = 1, shape1, shape3, log = FALSE)

psinmad(q, scale = 1, shape1, shape3, lower.tail = TRUE, log.p = FALSE)

qsinmad(p, scale = 1, shape1, shape3, lower.tail = TRUE, log.p = FALSE)

rsinmad(n, scale = 1, shape1, shape3)

dlomax(x, scale = 1, shape3, log = FALSE)

plomax(q, scale = 1, shape3, lower.tail = TRUE, log.p = FALSE)

qlomax(p, scale = 1, shape3, lower.tail = TRUE, log.p = FALSE)

rlomax(n, scale = 1, shape3)

Arguments

x, q

vector of quantiles.

scale

scale parameter \(b\), must be positive.

shape1

shape parameter \(a\), must be positive.

shape2

shape parameter \(p\), must be positive.

shape3

shape parameter \(q\), must be positive.

log, log.p

logical; if TRUE, probabilities/ densities \(p\) are returned as \(\log(p)\).

lower.tail

logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise \(P[X > x]\).

p

vector of probabilities.

n

number of random values to return.

Value

d... gives the density, p... gives the distribution function, q... gives the quantile function, and r... generates random deviates.

Details

This implementation allows for automatic differentiation with RTMB.

The GB2 distribution with scale \(b\) and shape parameters \(a\), \(p\) and \(q\) has density $$f(x) = \frac{a\, x^{ap - 1}}{b^{ap} B(p, q) \left(1 + (x/b)^a\right)^{p + q}}, \quad x > 0,$$ and its distribution function is a regularised incomplete beta function, $$F(x) = I_z(p, q), \qquad z = \frac{(x/b)^a}{1 + (x/b)^a}.$$ The special cases fix some of the shapes, and their arguments keep the names of the shapes that remain:

• Dagum: \(q = 1\), ddagum(x, scale, shape1, shape2).

• Singh-Maddala, or Burr type XII: \(p = 1\), dsinmad(x, scale, shape1, shape3).

• Lomax, or Pareto type II: \(a = p = 1\), dlomax(x, scale, shape3).

The log-logistic distribution dllogis is the GB2 with \(p = q = 1\), and the beta prime distribution dbetaprime the GB2 with \(a = b = 1\). The \(k\)-th moment exists for \(-ap < k < aq\).

The distribution functions are differentiable, so one-step-ahead (OSA) residuals via RTMB::oneStepPredict with method = "cdf" are supported.

References

McDonald, J. B. (1984) Some generalized functions for the size distribution of income. Econometrica, 52, 647-663.

Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences. Wiley.

See also

llogis, betaprime, pareto

Examples

set.seed(1)
x <- rgb2(5, scale = 2, shape1 = 3, shape2 = 1.5, shape3 = 0.8)
d <- dgb2(x, 2, 3, 1.5, 0.8)
p <- pgb2(x, 2, 3, 1.5, 0.8)
q <- qgb2(p, 2, 3, 1.5, 0.8)

d <- ddagum(x, 2, 3, 1.5)
d <- dsinmad(x, 2, 3, 0.8)
d <- dlomax(x, 2, 0.8)