# The Simplex Distribution Functions Density, cumulative distribution function, quantile function and random variable generation for the simplex distribution with mean equal to `mu` and dispersion equal to `sig` ```r dsimplex(x, mu, sig) psimplex(q, mu, sig) qsimplex(p, mu, sig) rsimplex(n, mu, sig) psimplex.norm(q, mu, sig) qsimplex.norm(p, mu, sig) ``` ## Arguments - `x, q`: vector of quantiles - `p`: vector of probabilities - `n`: number of observations - `mu`: vector of means - `sig`: vector of square root of dispersion parameter of simplex distribution ## Details The simplex distribution has density $$ p(y) = (2\pi \sigma^2)^{-\frac{1}{2}} (y(1-y))^{-\frac{3}{2}} \exp(-\frac{1}{2\sigma^2} d(y;\mu)), y \in (0, 1)%p(y) = (2\pi \sigma^2)^{-1/2} (y(1-y))^{-3/2} e^(-1 / (2 \sigma^2) d(y;\mu)) $$ where $d(y;\mu)$ is a unit deviance function $$ d(y;\mu) = \frac{(y-\mu)^2}{y(1-y) \mu^2 (1-\mu)^2}%d(y;\mu) = (y - \mu)^2 / (y(1-y) \mu^2 (1-\mu)^2) $$ $\mu$ is the mean of simplex distribution and $\sigma^2$ the dispersion parameter. `qnorm` provides results up to about 6 digits. ## Returns `dsimplex` gives density function, `psimplex` gives the distribution function, `qsimplex` gives quantile function and `rsimplex` gives random number generated from the simplex distribution. `psim.norm` and `qsimplex.norm` gives the renormalized distribution and quantile function. ## References Barndorff-Nielsen, O.E. and Jorgensen, B. (1991) Some parametric models on the simplex. **Journal of Multivariate Analysis,** 39: 106--116 Jorgensen, B. (1997) **The Theory of Dispersion Models**. London: Chapman and Hall Song, P. and Qiu, Z. and Tan, M. (2004) Modelling Heterogeneous Dispersion in Marginal Models for Longitudinal Proportional Data. **Biometrical Journal,** 46: 540--553 ## Author(s) Peng Zhang and Zhenguo Qiu ## Examples ```r # simplex distribution function dsimplex(seq(0.01,0.99,0.01), 0.5, 1) psimplex(seq(0.01,0.99,0.01), 0.5, 1) qsimplex(seq(0.01,0.99,0.01), 0.5, 1) # random variable generation n <- 200 ga0 <- 1.5 ga1 <- 0.5 ga2 <- -0.5 sigma <- 4 T <- c(rep(0, n/2), rep(1, n/2)) S <- runif(n, 0, 5) eta <- ga0 + ga1 * T + ga2 * S mu <- exp(eta)/(1+exp(eta)) Y <- rep(0, n) for (i in 1:n){ Y[i] <- rsimplex(1, mu[i], sigma) } ```

simplex function