VGAM (version 0.9-0)

paralogistic: Paralogistic Distribution Family Function

Description

Maximum likelihood estimation of the 2-parameter paralogistic distribution.

Usage

paralogistic(lshape1.a = "loge", lscale = "loge",
             ishape1.a = 2, iscale = NULL, zero = NULL)

Arguments

lshape1.a, lscale

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

ishape1.a, iscale

Optional initial values for a and scale.

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} which correspond to a, scale, respectively.

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

The 2-parameter paralogistic has density $$f(y) = a^2 y^{a-1} / [b^a {1 + (y/b)^a}^{1+a}]$$ for $a > 0$, $b > 0$, $y > 0$. Here, $b$ is the scale parameter scale, and $a$ is the shape parameter. The mean is $$E(Y) = b \, \Gamma(1 + 1/a) \, \Gamma(a - 1/a) / \Gamma(a)$$ provided $a > 1$; 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

Paralogistic, genbetaII, betaII, dagum, fisk, invlomax, lomax, invparalogistic.

Examples

pdata <- data.frame(y = rparalogistic(n = 3000, exp(1), exp(2)))
fit <- vglm(y ~ 1, paralogistic, pdata, trace = TRUE)
fit <- vglm(y ~ 1, paralogistic(ishape1.a = 2.3, iscale = 7),
            pdata, trace = TRUE, epsilon = 1e-8)
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)