Beta Distribution of the Second Kind
Maximum likelihood estimation of the 3-parameter beta II distribution.
betaII(lscale = "loglink", lshape2.p = "loglink", lshape3.q = "loglink",
iscale = NULL, ishape2.p = NULL, ishape3.q = NULL, imethod = 1,
gscale = exp(-5:5), gshape2.p = exp(-5:5),
gshape3.q = seq(0.75, 4, by = 0.25),
probs.y = c(0.25, 0.5, 0.75), zero = "shape")lscale, lshape2.p, lshape3.q |
Parameter link functions applied to the
(positive) parameters |
iscale, ishape2.p, ishape3.q, imethod, zero |
See |
gscale, gshape2.p, gshape3.q |
See |
probs.y |
See |
The 3-parameter beta II is the 4-parameter generalized beta II distribution with shape parameter a=1. It is also known as the Pearson VI distribution. Other distributions which are special cases of the 3-parameter beta II include the Lomax (p=1) and inverse Lomax (q=1). More details can be found in Kleiber and Kotz (2003).
The beta II distribution has density
f(y) = y^(p-1) / [b^p B(p,q) (1 + y/b)^(p+q)]
for b > 0, p > 0, q > 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) gamma(q - 1) / ( gamma(p) gamma(q))
provided q > 1; 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.
See the notes in genbetaII.
T. W. Yee
Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.
bdata <- data.frame(y = rsinmad(2000, shape1.a = 1, shape3.q = exp(2),
scale = exp(1))) # Not genuine data!
fit <- vglm(y ~ 1, betaII, data = bdata, trace = TRUE)
fit <- vglm(y ~ 1, betaII(ishape2.p = 0.7, ishape3.q = 0.7),
data = bdata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.