View source: R/family.univariate.R
simplex | R Documentation |
The two parameters of the univariate standard simplex distribution are estimated by full maximum likelihood estimation.
simplex(lmu = "logitlink", lsigma = "loglink", imu = NULL, isigma = NULL,
imethod = 1, ishrinkage = 0.95, zero = "sigma")
lmu , lsigma |
Link function for |
imu , isigma |
Optional initial values for |
imethod , ishrinkage , zero |
See |
The probability density function can be written
f(y; \mu, \sigma) = [2 \pi \sigma^2 (y (1-y))^3]^{-0.5}
\exp[-0.5 (y-\mu)^2 / (\sigma^2 y (1-y) \mu^2 (1-\mu)^2)]
for 0 < y < 1
,
0 < \mu < 1
,
and \sigma > 0
.
The mean of Y
is \mu
(called mu
, and
returned as the fitted values).
The second parameter, sigma
, of this standard simplex
distribution is known as the dispersion parameter.
The unit variance function is
V(\mu) = \mu^3 (1-\mu)^3
.
Fisher scoring is applied to both parameters.
An object of class "vglmff"
(see vglmff-class
).
The object is used by modelling functions such as vglm
,
and vgam
.
This distribution is potentially useful for dispersion modelling.
Numerical problems may occur when mu
is very close to 0 or 1.
T. W. Yee
Jorgensen, B. (1997). The Theory of Dispersion Models. London: Chapman & Hall
Song, P. X.-K. (2007). Correlated Data Analysis: Modeling, Analytics, and Applications. Springer.
dsimplex
,
dirichlet
,
rig
,
binomialff
.
sdata <- data.frame(x2 = runif(nn <- 1000))
sdata <- transform(sdata, eta1 = 1 + 2 * x2,
eta2 = 1 - 2 * x2)
sdata <- transform(sdata, y = rsimplex(nn, mu = logitlink(eta1, inverse = TRUE),
dispersion = exp(eta2)))
(fit <- vglm(y ~ x2, simplex(zero = NULL), data = sdata, trace = TRUE))
coef(fit, matrix = TRUE)
summary(fit)
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