lomax: Lomax Distribution Family Function

Description Usage Arguments Details Value Note Author(s) References See Also Examples

View source: R/family.actuary.R

Description

Maximum likelihood estimation of the 2-parameter Lomax distribution.

Usage

1
2
3
4
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 >= 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.

Note

See the notes in genbetaII.

Author(s)

T. W. Yee

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

Examples

1
2
3
4
5
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)
Coef(fit)
summary(fit)

Example output

Loading required package: stats4
Loading required package: splines
VGLM    linear loop  1 :  loglikelihood = -159.2663
VGLM    linear loop  2 :  loglikelihood = -158.01733
VGLM    linear loop  3 :  loglikelihood = -157.96819
VGLM    linear loop  4 :  loglikelihood = -157.96804
VGLM    linear loop  5 :  loglikelihood = -157.96804
            loge(scale) loge(shape3.q)
(Intercept)    1.005489       1.984914
   scale shape3.q 
2.733245 7.278421 

Call:
vglm(formula = y ~ 1, family = lomax, data = ldata, trace = TRUE)


Pearson residuals:
                   Min      1Q   Median     3Q    Max
loge(scale)     -9.668 -0.8674 -0.07777 0.8655 1.5803
loge(shape3.q) -17.451  0.1186  0.24752 0.3377 0.3753

Coefficients: 
              Estimate Std. Error z value Pr(>|z|)    
(Intercept):1   1.0055     0.2956   3.402 0.000669 ***
(Intercept):2   1.9849     0.2618   7.582  3.4e-14 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Number of linear predictors:  2 

Names of linear predictors: loge(scale), loge(shape3.q)

Log-likelihood: -157.968 on 1998 degrees of freedom

Number of iterations: 5 

VGAM documentation built on Jan. 16, 2021, 5:21 p.m.