VGAM (version 0.9-4)

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",
       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.

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 $b > 0$, $p > 0$, $q > 0$, $y \geq 0$. 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, exp(2), exp(1)))  # Not genuine data!
fit <- vglm(y ~ 1, betaII, data = bdata, trace = TRUE)
fit <- vglm(y ~ 1, betaII(ishape2.p = 0.7, ishape3.q = 0.7),
            data = bdata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)