VGAM (version 0.8-7)

lomax: Lomax Distribution Family Function

Description

Maximum likelihood estimation of the 2-parameter Lomax distribution.

Usage

lomax(lscale = "loge", lshape3.q = "loge", escale = list(),
      eshape3.q = list(), iscale = NULL, ishape3.q = 2, zero = NULL)

Arguments

lscale, lshape3.q

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

escale, eshape3.q

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

iscale, ishape3.q

Optional initial values for scale 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} which correspond to scale, q, 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 Lomax distribution is the 4-parameter generalized beta II distribution with shape parameters $a=p=1$. It is probably more widely known as the Pareto (II) distribution. It is also the 3-parameter Singh-Maddala distribution with shape parameter $a=1$, as well as the beta distribution of the second kind with $p=1$. More details can be found in Kleiber and Kotz (2003).

The Lomax distribution has density $$f(y) = q / [b {1 + y/b}^{1+q}]$$ for $b > 0$, $q > 0$, $y > 0$. Here, $b$ is the scale parameter scale, and q is a shape parameter. The cumulative distribution function is $$F(y) = 1 - [1 + (y/b)]^{-q}.$$ The mean is $$E(Y) = b/(q-1)$$ provided $q > 1$; 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

Lomax, genbetaII, betaII, dagum, sinmad, fisk, invlomax, paralogistic, invparalogistic.

Examples

ldata = data.frame(y = rlomax(n = 1000, exp(1), exp(2)))
fit = vglm(y ~ 1, lomax, ldata, trace = TRUE)
fit = vglm(y ~ 1, lomax(iscale = exp(1), ishape3.q = exp(2)), ldata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)