VGAMextra (version 0.0-6)

genbetaIIDist: The Generalized Beta Distribution of the Second King

Description

Density, distribution function, inverse distribution (quantile function) and random generation for the Generalized Beta of the Second Kind (GB2).

Usage

dgen.betaII(x, scale = 1.0, shape1.a = 1.0, shape2.p = 1.0, shape3.q = 1.0, 
             log = FALSE)
  pgen.betaII(q, scale = 1.0, shape1.a = 1.0, shape2.p = 1.0, shape3.q = 1.0, 
             lower.tail = TRUE, log.p = FALSE)
  qgen.betaII(p, scale = 1.0, shape1.a = 1.0, shape2.p = 1.0, shape3.q = 1.0, 
             lower.tail = TRUE, log.p = FALSE)
  rgen.betaII(n, scale = 1.0, shape1.a = 1.0, shape2.p = 1.0, shape3.q = 1.0)

Value

dgen.betaII() returns the density (p.d.f), pgen.betaII() gives the distribution function (p.d.f), qgen.betaII() gives the quantile function (Inverse Distribution function), and rgen.betaII() generates random numbers from the GB2 distribution.

Arguments

x, q

Vector of quantiles.

p

Vector of probabilities.

n

Number of observations. If length(n) > 1, its length is taken to be the numbre required.

scale, shape1.a, shape2.p, shape3.q

Strictly positive scale and shape parameters.

log, log.p, lower.tail

Same meaning as in Beta

Author

V. Miranda and T. W. Yee

Details

The GB2 Distribution is defined by the probability density (pdf)

$$f(y) = \frac{a x^{ap - 1}}{b^{ap} B(p, q) [1 + (y/b)^{a}]^{p + q},}$$ for \(y > 0\), and \(b, a, p, q > 0\). Here, \(B(p, q)\) is the beta function as in beta.

The GB2 Distribution and the Beta Distribution (see Beta) are linked, as follows: Let \(X\) be a random variable with the Beta density and parameters \(p = shape_{1}\) and \(q = shape_{2}\). Then, introducing additional \(b = scale\) and \(a = shape\) parameters, the variable $$Y = \frac{(x/b)^{a}}{1 + (x/b)^{a}}$$ has the GB2 Distribution, with parameters \(b, a, p, q\).

The GB2 \(k^{th}\) moment exists for \(-ap < k < aq\) and is given by

$$E(Y^{k}) = \frac{b^{k} B(p + k/a, q - k/a)}{B(p, q)}$$

or, equivalently, $$E(Y^{k}) = \frac{b^{k} \Gamma(p + k/a) \Gamma(q - k/a)} {\Gamma(p) \Gamma(q)}).$$ Here, \(\Gamma(\cdot)\) is the gamma function as in gamma.

References

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, ch.6, p.255. Dover, New York, USA.

Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences. Wiley Series in Probability and Statistics. Hoboken, New Jersey, USA.

McDonald, J. B. and Xu, Y. J. (1995) A generalization of the beta distribution with applications. Journal of Econometrics, 66, p.133--152.

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

See Also

Beta, beta.

Examples

  # Setting parameters to both examples below.
  b  <- exp(0.4)      # Scale parameter.
  a  <- exp(0.5)      # Shape1.a
  p  <- exp(0.3)      # Shape2.p
  q  <- exp(1.4)      # Shape3.q
  
  # (1) ______________  
  probs.y <- seq(0.0, 1.0, by = 0.01)
  data.1  <- qgen.betaII(p = probs.y, scale = b, shape1.a = a, 
                        shape2.p = p, shape3.q = q)
  max(abs(pgen.betaII(q = data.1, scale = b, shape1.a = a, 
                     shape2.p = p, shape3.q = q)) - probs.y) # Should be 0.
 # \donttest{
  # (2)_________________
  xx    <- seq(0, 10.0, length = 200)
  yy    <- dgen.betaII(xx, scale = b, shape1.a = a, shape2.p = p, shape3.q = q)
  qtl   <- seq(0.1, 0.9, by = 0.1)
  d.qtl <- qgen.betaII(qtl, scale = b, shape1.a = a, shape2.p = p, shape3.q = q)
  plot(xx, yy, type = "l", col = "red", 
       main = "Red is the GB2 density, blue is the GB2 Distribution Function",
       sub  = "Brown dashed lines represent the 10th, ..., 90th percentiles",
       las = 1, xlab = "x", ylab = "", xlim = c(0, 3), ylim = c(0,1))
  abline(h = 0, col = "navy", lty = 2)
  abline(h = 1, col = "navy", lty = 2)
  lines(xx, pgen.betaII(xx, scale = b, shape1.a = a, 
                       shape2.p = b, shape3.q = q), col= "blue")
  lines(d.qtl, dgen.betaII(d.qtl, scale = b, shape1.a = a, 
                        shape2.p = p, shape3.q = q), 
                        type ="h", col = "brown", lty = 3)
 # }