VGAM (version 0.7-1)

betaII: Beta Distribution of the Second Kind

Description

Maximum likelihood estimation of the 3-parameter beta II distribution.

Usage

betaII(link.scale = "loge", link.p = "loge", link.q = "loge",
       init.scale = NULL, init.p = 1, init.q = 1, zero = NULL)

Arguments

link.scale, link.p, link.q

Parameter link functions applied to the (positive) parameters scale, p and q. See Links for more choices.

init.scale, init.p, init.q

Optional initial values for scale, p and q.

zero

An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. Here, the values must be from the set {1,2,3} which correspond to scale, p, q, respectively.

Value

• An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Details

The 3-parameter beta II is the 4-parameter generalized beta II distribution with shape parameter $a=1$. It is also known as the Pearson VI distribution. Other distributions which are special cases of the 3-parameter beta II include the Lomax ($p=1$) and inverse Lomax ($q=1$). More details can be found in Kleiber and Kotz (2003).

The beta II distribution has density $$f(y) = y^{p-1} / [b^p B(p,q) {1 + y/b}^{p+q}]$$ for $b > 0$, $p > 0$, $q > 0$, $y > 0$. Here, $b$ is the scale parameter scale, and the others are shape parameters. The mean is $$E(Y) = b \, \Gamma(p + 1) \, \Gamma(q - 1) / (\Gamma(p) \, \Gamma(q))$$ provided $q > 1$.

References

Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ: Wiley-Interscience.

See Also

betaff, genbetaII, dagum, sinmad, fisk, invlomax, lomax, paralogistic, invparalogistic.

Examples

y = rsinmad(n=2000, a=1, 6, 2)  # Not genuine data!
fit = vglm(y ~ 1, betaII, trace=TRUE)
fit = vglm(y ~ 1, betaII(init.p=0.7, init.q=0.7), trace=TRUE, crit="c")
coef(fit, mat=TRUE)
Coef(fit)
summary(fit)