VGAM (version 1.1-6)

# simplex: 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.

## Value

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

## 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.

## 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.

dsimplex, dirichlet, rig, binomialff.

## Examples

Run this code
# NOT RUN {
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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