# The Generalized Beta Distribution of the Second King Density, distribution function, inverse distribution (quantile function) and random generation for the Generalized Beta of the Second Kind (GB2). ```r 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) ``` ## 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` ## Details The GB2 Distribution is defined by the probability density (pdf) $$ f(y) = \frac{a x^{ap - 1}}{b^{ap} B(p, q) [1 + (y/b)^{a}]^{p + q},}%f(y) = a * x^(ap - 1) / b^(ap) * B(p, q) * [1 + (x/b)^a]^(p + q), $$ 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 $p = shape1$ and $q = shape2$. Then, introducing additional $b = scale$ and $a = shape$ parameters, the variable $$ Y = \frac{(x/b)^{a}}{1 + (x/b)^{a}}%Y = (x/b)^a / [1 + (x/b)^a] $$ has the GB2 Distribution, with parameters $b, a, p, q$. The GB2 $k^th$ moment exists for $-ap < k < aq$ and is given by $$ E(Y^{k}) = \frac{b^{k} B(p + k/a, q - k/a)}{B(p, q)}%E(Y^k) = b^k * B(p+ k/a, q - k/a) / B(p, q) $$ or, equivalently, $$ E(Y^{k}) = \frac{b^{k} \Gamma(p + k/a) \Gamma(q - k/a)}{\Gamma(p) \Gamma(q)}).%E(Y^k) = \Gamma(p + k/a) * \Gamma(q - k/a) / \Gamma(p) * \Gamma(q). $$ Here, $\Gamma(.)$ is the gamma function as in `gamma`. ## Returns `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. ## 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. ## Author(s) V. Miranda and T. W. Yee ## Note Values of the `shape2.p` parameter moderately close to zero may imply obtaning numerical values too close to zero or values represented as zero in computer arithmetic from the function `rgen.betaII()`. Additionally, for specific values of the arguments `x, q, p` such as `Inf, -Inf, NaN` and `NA`, the functions `qgen.betaII()`, `pgen.betaII()` and `qgen.betaII()` will return the limit when the argument tends to such value. In particular, the quantile `qgen.betaII()` retunrs zero for negative values and $Inf$ for missed probabilities greater than 1.0. ## See Also `Beta`, `beta`. ## Examples ```r # 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. # (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) ```

genbetaIIUC function