| sinmad {VGAM} | R Documentation |
Maximum likelihood estimation of the 3-parameter Singh-Maddala distribution.
sinmad(link.a = "loge", link.scale = "loge", link.q = "loge",
earg.a=list(), earg.scale=list(), earg.q=list(),
init.a = NULL, init.scale = NULL, init.q = 1, zero = NULL)
link.a, link.scale, link.q |
Parameter link functions applied to the
(positive) parameters a, scale, and q.
See Links for more choices.
|
earg.a, earg.scale, earg.q |
List. Extra argument for each of the links.
See earg in Links for general information.
|
init.a, init.scale, init.q |
Optional initial values for a, 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 {1,2,3} which correspond to
a, scale, q, respectively.
|
The 3-parameter Singh-Maddala distribution is the 4-parameter generalized beta II distribution with shape parameter p=1. It is known under various other names, such as the Burr XII (or just the Burr distribution), Pareto IV, beta-P, and generalized log-logistic distribution. More details can be found in Kleiber and Kotz (2003).
Some distributions which are special cases of the 3-parameter Singh-Maddala are the Lomax (a=1), Fisk (q=1), and paralogistic (a=q).
The Singh-Maddala distribution has density
f(y) = aq y^(a-1) / [b^a (1 + (y/b)^a)^(1+q)]
for a > 0, b > 0, q > 0, y > 0.
Here, b is the scale parameter scale,
and the others are shape parameters.
The cumulative distribution function is
F(y) = 1 - [1 + (y/b)^a]^(-q).
The mean is
E(Y) = b gamma(1 + 1/a) gamma(q - 1/a) / gamma(q)
provided -a < 1 < aq.
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.
T. W. Yee
Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ: Wiley-Interscience.
Sinmad,
genbetaII,
betaII,
dagum,
fisk,
invlomax,
lomax,
paralogistic,
invparalogistic.
y = rsinmad(n=3000, 3, 5, 2) fit = vglm(y ~ 1, sinmad, trace=TRUE) fit = vglm(y ~ 1, sinmad, trace=TRUE, crit="c") coef(fit, mat=TRUE) Coef(fit) summary(fit)