VGAM (version 0.8-7)

betaII: Beta Distribution of the Second Kind

Description

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

Usage

betaII(lscale = "loge", lshape2.p = "loge", lshape3.q = "loge",
       escale = list(), eshape2.p = list(), eshape3.q = list(),
       iscale = NULL, ishape2.p = 2, ishape3.q = 2, zero = NULL)

Arguments

lscale, lshape2.p, lshape3.q

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

escale, eshape2.p, eshape3.q

List. Extra argument for each of the links. See earg in Links for general information.

iscale, ishape2.p, ishape3.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 $q > 1$; these are returned as the fitted values.

References

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

See Also

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

Examples

bdata = data.frame(y = rsinmad(2000, shape1.a = 1, 6, 2)) # Not genuine data!
fit = vglm(y ~ 1, betaII, bdata, trace = TRUE)
fit = vglm(y ~ 1, betaII(ishape2.p = 0.7, ishape3.q = 0.7),
           bdata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)