dagum {VGAM} | R Documentation |
Maximum likelihood estimation of the 3-parameter Dagum distribution.
dagum(link.a = "loge", link.scale = "loge", link.p = "loge", earg.a=list(), earg.scale=list(), earg.p=list(), init.a = NULL, init.scale = NULL, init.p = 1, zero = NULL)
link.a, link.scale, link.p |
Parameter link functions applied to the
(positive) parameters a , scale , and p .
See Links for more choices.
|
earg.a, earg.scale, earg.p |
List. Extra argument for each of the links.
See earg in Links for general information.
|
init.a, init.scale, init.p |
Optional initial values for a , scale , and p .
|
zero |
An integer-valued vector specifying which
linear/additive predictors are modelled as intercepts only.
Here, the values must be from the set {1,2,3} which correspond to
a , scale , p , respectively.
|
The 3-parameter Dagum distribution is the 4-parameter generalized beta II distribution with shape parameter q=1. It is known under various other names, such as the Burr III, inverse Burr, beta-K, and 3-parameter kappa distribution. It can be considered a generalized log-logistic distribution. Some distributions which are special cases of the 3-parameter Dagum are the inverse Lomax (a=1), Fisk (p=1), and the inverse paralogistic (a=p). More details can be found in Kleiber and Kotz (2003).
The Dagum distribution has a cumulative distribution function
F(y) = [1 + (y/b)^(-a)]^(-p)
which leads to a probability density function
f(y) = ap y^(ap-1) / [b^(ap) (1 + (y/b)^a)^(p+1)]
for a > 0, b > 0, p > 0, y > 0.
Here, b is the scale parameter scale
,
and the others are shape parameters.
The mean is
E(Y) = b gamma(p + 1/a) gamma(1 - 1/a) / gamma(p)
provided -ap < 1 < a.
An object of class "vglmff"
(see vglmff-class
).
The object is used by modelling functions such as vglm
,
and vgam
.
If the self-starting initial values fail, try experimenting
with the initial value arguments, especially those whose
default value is not NULL
.
From Kleiber and Kotz (2003), the MLE is rather sensitive to isolated observations located sufficiently far from the majority of the data. Reliable estimation of the scale parameter require n>7000, while estimates for a and p can be considered unbiased for n>2000 or 3000.
T. W. Yee
Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ: Wiley-Interscience.
Dagum
,
genbetaII
,
betaII
,
sinmad
,
fisk
,
invlomax
,
lomax
,
paralogistic
,
invparalogistic
.
y = rdagum(n=3000, 4, 6, 2) fit = vglm(y ~ 1, dagum, trace=TRUE) fit = vglm(y ~ 1, dagum(init.a=2.1), trace=TRUE, crit="c") coef(fit, mat=TRUE) Coef(fit) summary(fit)