Inverse Paralogistic Distribution Family Function

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

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")

Arguments

lss

See CommonVGAMffArguments for important information.

lshape1.a, lscale

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

iscale, ishape1.a, imethod, zero

See CommonVGAMffArguments for information. For imethod = 2 a good initial value for ishape1.a is needed to obtain a good estimate for the other parameter.

gscale, gshape1.a

See CommonVGAMffArguments for information.

probs.y

See CommonVGAMffArguments for information.

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.

Value

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

References

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

Author

T. W. Yee

Note

See the notes in genbetaII.

See also

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

Examples

if (FALSE) { # \dontrun{
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)   } # }