Generalized Beta Distribution of the Second Kind
Maximum likelihood estimation of the 4-parameter generalized beta II distribution.
genbetaII(lscale = "loglink", lshape1.a = "loglink", lshape2.p = "loglink",
lshape3.q = "loglink", iscale = NULL, ishape1.a = NULL,
ishape2.p = NULL, ishape3.q = NULL, lss = TRUE,
gscale = exp(-5:5), gshape1.a = exp(-5:5),
gshape2.p = exp(-5:5), gshape3.q = exp(-5:5),
zero = "shape")lss |
See |
lshape1.a, lscale, lshape2.p, lshape3.q |
Parameter link functions applied to the
shape parameter |
iscale, ishape1.a, ishape2.p, ishape3.q |
Optional initial values for the parameters.
A |
gscale, gshape1.a, gshape2.p, gshape3.q |
See |
zero |
The default is to set all the shape parameters to be
intercept-only.
See |
This distribution is most useful for unifying a substantial number of size distributions. For example, the Singh-Maddala, Dagum, Fisk (log-logistic), Lomax (Pareto type II), inverse Lomax, beta distribution of the second kind distributions are all special cases. Full details can be found in Kleiber and Kotz (2003), and Brazauskas (2002). The argument names given here are used by other families that are special cases of this family. Fisher scoring is used here and for the special cases too.
The 4-parameter generalized beta II distribution has density
f(y) = a y^(ap-1) / [b^(ap) B(p,q) (1 + (y/b)^a)^(p+q)]
for a > 0, b > 0, p > 0, q > 0, y >= 0.
Here B is the beta function, and
b is the scale parameter scale,
while the others are shape parameters.
The mean is
E(Y) = b gamma(p + 1/a) gamma(q - 1/a) / ( gamma(p) gamma(q))
provided -ap < 1 < aq; these are returned as the fitted values.
This family function handles multiple responses.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
This distribution is very flexible and it is not generally
recommended to use this family function when the sample size is
small—numerical problems easily occur with small samples.
Probably several hundred observations at least are needed in order
to estimate the parameters with any level of confidence.
Neither is the inclusion of covariates recommended at all—not
unless there are several thousand observations.
The mean is finite only when -ap < 1 < aq, and this can be
easily violated by the parameter estimates for small sample sizes.
Try fitting some of the special cases of this distribution
(e.g., sinmad, fisk, etc.) first, and
then possibly use those models for initial values for this
distribution.
The default is to use a grid search with respect to all
four parameters; this is quite costly and is time consuming.
If the self-starting initial values fail, try experimenting
with the initial value arguments.
Also, the constraint -ap < 1 < aq
may be violated as the iterations progress so it pays
to monitor convergence, e.g., set trace = TRUE.
Successful convergence depends on having very good initial
values. This is rather difficult for this distribution so that
a grid search is conducted by default.
One suggestion for increasing the estimation reliability
is to set stepsize = 0.5 and maxit = 100;
see vglm.control.
T. W. Yee, with help from Victor Miranda.
Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.
Brazauskas, V. (2002). Fisher information matrix for the Feller-Pareto distribution. Statistics & Probability Letters, 59, 159–167.
## Not run:
gdata <- data.frame(y = rsinmad(3000, shape1 = exp(1), scale = exp(2),
shape3 = exp(1))) # A special case!
fit <- vglm(y ~ 1, genbetaII(lss = FALSE), data = gdata, trace = TRUE)
fit <- vglm(y ~ 1, data = gdata, trace = TRUE,
genbetaII(ishape1.a = 3, iscale = 7, ishape3.q = 2.3))
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)
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