Density, distribution function, quantile function and random generation for the generalized beta distribution.
dBE3(x, mu = 0.5, alpha = 1, beta = 1, tau = 0.5, log = FALSE)
pBE3(q, mu = 0.5, alpha = 1, beta = 1, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
qBE3(p, mu = 0.5, alpha = 1, beta = 1, tau = 0.5)
rBE3(n, mu = 0.5, alpha = 1, beta = 1, tau = 0.5)dBE3 gives the density, pBE3 gives the distribution function, qBE3 gives the quantile function, and rBE3 generates random deviates.
The length of the result is determined by n for rBE3, and is the maximum of the lengths of the numerical arguments for the other functions.
The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.
vector of quantiles.
vector of probabilities.
number of observations.
vector of \(\tau\)-quantiles of the distribution.
shape parameters of the distribution
corresponding quantile of the distribution (\(0<\tau<1\))
logical; if TRUE, probabilities \(p\) are given as \(\log p\).
logical; if TRUE (default), probabilities are \(P[X\leq x]\) otherwise, \(P[X>x]\).
Diego Gallardo and Marcelo Bourguignon
The probability density function for the generalized beta distribution is $$ f(y;\lambda,\alpha,\beta)=\frac{\lambda^\alpha y^{\alpha-1}(1-y)^{\beta-1}}{B(\alpha, \beta)[1-(1-\lambda)y]^{\alpha+\beta}}, \quad 0<y<1, $$ where \(\alpha, \beta>0\) and \(\lambda>0\). We consider the reparameterization in terms of the \(\tau\)-quantile of the distribution, say \(0<\mu<1\), taking $$ \lambda=\frac{(1-\mu)}{\mu}\frac{z_{\alpha,\beta}(\tau)}{[1-z_{\alpha,\beta}(\tau)]}, $$ with \(z_{\alpha,\beta}(\tau)\) denoting the \(\tau\)-quantile of the usual beta distribution with shape parameters \(\alpha\) and \(\beta\). The cumulative distribution function is given by $$ F(y;\lambda,\alpha,\beta)=I_{\lambda x/(1+\lambda x -x)}(\alpha, \beta), \quad 0<y<1, $$ where \(I_x(\alpha,\beta)=B_x(\alpha,\beta)/B(\alpha,\beta)\) is the incomplete beta funcion ratio, \(B_x(\alpha,\beta)=\int_0^x w^{\alpha-1}(1-w)^{\beta-1}dw\) is the incomplete beta function and \(B(\alpha,\beta)=\Gamma(\alpha)\Gamma(\beta)/\Gamma(\alpha+\beta)\) is the ordinary beta function. The quantile of the distribution can be represented as $$ q(\tau;\lambda,\alpha,\beta)=\frac{z_{\alpha,\beta}(\tau)}{\lambda[1-z_{\alpha,\beta}(\tau)]+z_{\alpha,\beta}(\tau)}, \quad 0<\tau<1. $$ Random generation can be performed using the stochastic representation of the model. If \(X_1 \sim \mbox{Gamma}(\alpha,\theta_1)\) and \(X_2 \sim \mbox{Gamma}(\beta,\theta_2)\), then $$ \frac{X_1}{X_1+X_2}\sim GB3(\alpha,\beta,\lambda), $$ where \(\lambda=\theta_1/\theta_2.\)
Libby, D. L. and Novick, M. R. (1982). Multivariate generalized beta-distributions with applications to utility assessment. Journal of Educational Statistics, 7.
rBE3(20, mu=0.5, alpha=2, beta=1)
dBE3(c(0.4,0.7), mu=0.5, alpha=2, beta=1)
pBE3(c(0.4,0.7), mu=0.5, alpha=2, beta=1)
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