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)