VGAM (version 0.9-3)

genbetaII: Generalized Beta Distribution of the Second Kind

Description

Maximum likelihood estimation of the 4-parameter generalized beta II distribution.

Usage

genbetaII(lshape1.a = "loge", lscale = "loge", lshape2.p = "loge", lshape3.q = "loge",
          ishape1.a = NULL, iscale = NULL, ishape2.p = 1, ishape3.q = 1,
          zero = NULL)

Arguments

lshape1.a, lscale, lshape2.p, lshape3.q

Parameter link functions applied to the shape parameter a, scale parameter scale, shape parameter p, and shape parameter q. All four parameters are positive. See

ishape1.a, iscale

Optional initial values for a and scale. A NULL means a value is computed internally.

ishape2.p, ishape3.q

Optional initial values for 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 {1,2,3,4} which correspond to a, scale, p, q, respectivel

Value

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

Details

This distribution is most useful for unifying a substantial number of size distributions. For example, the Singh-Maddala, Dagum, Fisk (log-logistic), Lomax (Pareto type II), inverse Lomax, beta distribution of the second kind distributions are all special cases. Full details can be found in Kleiber and Kotz (2003), and Brazauskas (2002). The argument names given here are used by other families that are special cases of this family. Fisher scoring is used here and for the special cases too.

The 4-parameter generalized beta II distribution has density $$f(y) = a y^{ap-1} / [b^{ap} B(p,q) {1 + (y/b)^a}^{p+q}]$$ for $a > 0$, $b > 0$, $p > 0$, $q > 0$, $y \geq 0$. Here $B$ is the beta function, and $b$ is the scale parameter scale, while the others are shape parameters. The mean is $$E(Y) = b \, \Gamma(p + 1/a) \, \Gamma(q - 1/a) / (\Gamma(p) \, \Gamma(q))$$ provided $-ap < 1 < aq$; 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.

Brazauskas, V. (2002) Fisher information matrix for the Feller-Pareto distribution. Statistics & Probability Letters, 59, 159--167.

See Also

betaff, betaII, dagum, sinmad, fisk, lomax, invlomax, paralogistic, invparalogistic, lino.

Examples

gdata <- data.frame(y = rsinmad(3000, exp(2), exp(2), exp(1)))  # A special case!
fit <- vglm(y ~ 1, genbetaII, gdata, trace = TRUE)
fit <- vglm(y ~ 1, data = gdata, trace = TRUE,
            genbetaII(ishape1.a = 4, ishape2.p = 2.2, iscale = 7, ishape3.q = 2.3))
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)