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")

Arguments

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

See CommonVGAMffArguments.

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 \(b > 0\), \(q > 0\), \(y \geq 0\). 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.

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

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

Examples

ldata <- data.frame(y = rlomax(n = 1000, scale =  exp(1), exp(2)))
fit <- vglm(y ~ 1, lomax, data = ldata, trace = TRUE)
#> Iteration 1: loglikelihood = -162.21432
#> Iteration 2: loglikelihood = -159.61744
#> Iteration 3: loglikelihood = -159.26851
#> Iteration 4: loglikelihood = -159.24586
#> Iteration 5: loglikelihood = -159.24529
#> Iteration 6: loglikelihood = -159.24528
#> Iteration 7: loglikelihood = -159.24528
coef(fit, matrix = TRUE)
#>             loglink(scale) loglink(shape3.q)
#> (Intercept)       1.479203          2.409792
Coef(fit)
#>     scale  shape3.q 
#>  4.389445 11.131641 
summary(fit)
#> 
#> Call:
#> vglm(formula = y ~ 1, family = lomax, data = ldata, trace = TRUE)
#> 
#> Coefficients: 
#>               Estimate Std. Error z value Pr(>|z|)    
#> (Intercept):1   1.4792     0.4166   3.551 0.000384 ***
#> (Intercept):2   2.4098     0.3836   6.283 3.33e-10 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Names of linear predictors: loglink(scale), loglink(shape3.q)
#> 
#> Log-likelihood: -159.2453 on 1998 degrees of freedom
#> 
#> Number of Fisher scoring iterations: 7 
#> 
#> Warning: Hauck-Donner effect detected in the following estimate(s):
#> '(Intercept):2'
#>