lino {VGAM} | R Documentation |
Maximum likelihood estimation of the 3-parameter generalized beta distribution as proposed by Libby and Novick (1982).
lino(lshape1="loge", lshape2="loge", llambda="loge", eshape1=list(), eshape2=list(), elambda=list(), ishape1=NULL, ishape2=NULL, ilambda=1, zero=NULL)
lshape1, lshape2 |
Parameter link functions applied to the two
(positive) shape parameters a and b.
See Links for more choices.
|
llambda |
Parameter link function applied to the
parameter lambda.
See Links for more choices.
|
eshape1, eshape2, elambda |
List. Extra argument for each of the links.
See earg in Links for general information.
|
ishape1, ishape2, ilambda |
Initial values for the parameters. A NULL value means
one is computed internally. The argument ilambda must
be numeric, and the default corresponds to a standard beta distribution.
|
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, b, lambda, respectively.
|
Proposed by Libby and Novick (1982), this distribution has density
f(y;a,b,lambda) = lambda^a y^(a-1) (1-y)^(b-1) / [B(a,b) (1 - (1-lambda)*y)^(a+b)]
for a > 0, b > 0, lambda > 0, 0 < y < 1.
Here B is the beta function (see beta
).
The mean is a complicated function involving the Gauss hypergeometric
function.
If X has a lino
distribution with parameters
shape1
, shape2
, lambda
, then
Y = λ*X / (1 - (1-λ)*X)
has a standard beta distribution with parameters shape1
,
shape2
.
Since log(lambda)=0 corresponds to the standard
beta distribution, a summary
of the fitted model performs a
t-test for whether the data belongs to a standard beta distribution
(provided the loge
link for lambda is used;
this is the default).
An object of class "vglmff"
(see vglmff-class
).
The object is used by modelling functions such as vglm
,
and vgam
.
The fitted values, which is usually the mean, have not been implemented
yet and consequently are NA
s.
Although Fisher scoring is used, the working weight matrices are positive-definite only in a certain region of the parameter space. Problems with this indicate poor initial values or an ill-conditioned model or insufficient data etc.
This model is can be difficult to fit. A reasonably good value of
ilambda
seems to be needed so if the self-starting initial
values fail, try experimenting with the initial value arguments.
Experience suggests ilambda
is better a little larger, rather
than smaller, compared to the true value.
T. W. Yee
Libby, D. L. and Novick, M. R. (1982) Multivariate generalized beta distributions with applications to utility assessment. Journal of Educational Statistics, 7, 271–294.
Gupta, A. K. and Nadarajah, S. (2004) Handbook of Beta Distribution and Its Applications, NY: Marcel Dekker, Inc.
y = rbeta(n=1000, exp(0.5), exp(1)) # Standard beta distribution fit = vglm(y ~ 1, lino, trace=TRUE) coef(fit, mat=TRUE) Coef(fit) fitted(fit)[1:4] summary(fit) # Nonstandard beta distribution y = rlino(n=1000, shape1=2, shape2=3, lambda=exp(1)) fit = vglm(y ~ 1, lino(lshape1=identity, lshape2=identity, ilambda=10)) coef(fit, mat=TRUE)