VGAM (version 1.1-14)

inv.paralogistic: Inverse Paralogistic Distribution Family Function

Description

Maximum likelihood estimation of the 2-parameter inverse paralogistic distribution.

Usage

inv.paralogistic(lscale = "loglink", lshape1.a = "loglink",
    iscale = NULL, ishape1.a = NULL, imethod = 1,
    lss = TRUE, gscale = exp(-5:5),
    gshape1.a = seq(0.75, 4, by = 0.25), probs.y = c(0.25, 0.5,
    0.75), zero = "shape")

Value

An object of class "vglmff"

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

Details

The 2-parameter inverse paralogistic distribution is the 4-parameter generalized beta II distribution with shape parameter \(q=1\) and \(a=p\). It is the 3-parameter Dagum distribution with \(a=p\). More details can be found in Kleiber and Kotz (2003).

The inverse paralogistic distribution has density $$f(y) = a^2 y^{a^2-1} / [b^{a^2} \{1 + (y/b)^a\}^{a+1}]$$ for \(a > 0\), \(b > 0\), \(y \geq 0\). Here, \(b\) is the scale parameter scale, and \(a\) is the shape parameter. The mean is $$E(Y) = b \, \Gamma(a + 1/a) \, \Gamma(1 - 1/a) / \Gamma(a)$$ provided \(a > 1\); these are returned as the fitted values. This family function handles multiple responses.

References

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

See Also

Inv.paralogistic, genbetaII, betaII, dagum, sinmad, fisk, inv.lomax, lomax, paralogistic, simulate.vlm.

Examples

if (FALSE) {
idata <- data.frame(y = rinv.paralogistic(3000, exp(1), sc = exp(2)))
fit <- vglm(y ~ 1, inv.paralogistic(lss = FALSE), idata, trace = TRUE)
fit <- vglm(y ~ 1, inv.paralogistic(imethod = 2, ishape1.a = 4),
            data = idata, trace = TRUE, crit = "coef")
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)   }