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 $E (Y) = b Γ (p + 1) Γ (q - 1) / (Γ (p) Γ (q))$ 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.

Examples

Run this code

bdata <- data.frame(y = rsinmad(2000, shape1.a = 1, exp(2), exp(1))) # 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)

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