betaII {VGAM} | R Documentation |
Maximum likelihood estimation of the 3-parameter beta II distribution.
betaII(lscale = "loge", lshape2.p = "loge", lshape3.q = "loge", iscale = NULL, ishape2.p = NULL, ishape3.q = NULL, imethod = 1, gscale = exp(-5:5), gshape2.p = exp(-5:5), gshape3.q = exp(-5:5), 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.
betaff
,
genbetaII
,
dagum
,
sinmad
,
fisk
,
inv.lomax
,
lomax
,
paralogistic
,
inv.paralogistic
.
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)