VGAM (version 0.8-7)

paralogistic: Paralogistic Distribution Family Function

Description

Maximum likelihood estimation of the 2-parameter paralogistic distribution.

Usage

paralogistic(lshape1.a = "loge", lscale = "loge", eshape1.a = list(),
             escale = list(), 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.

eshape1.a, escale

List. Extra argument for each of the links. See earg in Links for general information.

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)