VGAM (version 1.1-6)

lomax: Lomax Distribution Family Function


Maximum likelihood estimation of the 2-parameter Lomax distribution.


lomax(lscale = "loglink", lshape3.q = "loglink", iscale = NULL,
      ishape3.q = NULL, imethod = 1, gscale = exp(-5:5),
      gshape3.q = seq(0.75, 4, by = 0.25),
      probs.y = c(0.25, 0.5, 0.75), zero = "shape")


lscale, lshape3.q

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

iscale, ishape3.q, imethod

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

gscale, gshape3.q, zero, probs.y


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


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 \(b > 0\), \(q > 0\), \(y \geq 0\). 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. This family function handles multiple responses.


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, inv.lomax, paralogistic, inv.paralogistic, simulate.vlm.


Run this code
ldata <- data.frame(y = rlomax(n = 1000, scale =  exp(1), exp(2)))
fit <- vglm(y ~ 1, lomax, data = ldata, trace = TRUE)
coef(fit, matrix = TRUE)
# }

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