VGAM (version 0.9-4)

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 = "logit", lsigma = "loge",
        imu = NULL, isigma = NULL,
        imethod = 1, shrinkage.init = 0.95, zero = 2)

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, shrinkage.init, zero

See CommonVGAMffArguments for more 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.

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 = logit(eta1, inverse = TRUE),
                                       dispersion = exp(eta2)))
(fit <- vglm(y ~ x2, simplex(zero = NULL), data = sdata, trace = TRUE))
coef(fit, matrix = TRUE)
summary(fit)