simplex {VGAM}

Simplex Distribution Family Function

Description

The two parameters of the univariate standard simplex distribution are estimated by full maximum likelihood estimation.

Usage

simplex(lmu = "logitlink", lsigma = "loglink", imu = NULL, isigma = NULL,
        imethod = 1, ishrinkage = 0.95, zero = "sigma")

Arguments

lmu, lsigma

Link function for mu and sigma. See Links for more choices.

imu, isigma

Optional initial values for mu and sigma. A NULL means a value is obtained internally.

imethod, ishrinkage, zero

See CommonVGAMffArguments for information.

Details

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.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Note

This distribution is potentially useful for dispersion modelling. Numerical problems may occur when mu is very close to 0 or 1.

Author(s)

T. W. Yee

References

Jorgensen, B. (1997). The Theory of Dispersion Models. London: Chapman & Hall

Song, P. X.-K. (2007). Correlated Data Analysis: Modeling, Analytics, and Applications. Springer.

See Also

dsimplex, dirichlet, rig, binomialff.

Examples

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)