Zero-inflated and reparameterised gamma distribution

Density, distribution function, and random generation for the zero-inflated gamma distribution reparameterised in terms of mean and standard deviation.

Usage

dzigamma2(x, mean = 1, sd = 1, zeroprob = 0, log = FALSE)

pzigamma2(q, mean = 1, sd = 1, zeroprob = 0, lower.tail = TRUE, log.p = FALSE)

rzigamma2(n, mean = 1, sd = 1, zeroprob = 0)

Arguments

x, q: vector of quantiles
mean: mean parameter, must be positive.
sd: standard deviation parameter, must be positive.
zeroprob: zero-inflation probability between 0 and 1.
log, log.p: logical; if TRUE, probabilities/ densities $p$ are returned as $\log(p)$.
lower.tail: logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise $P[X > x]$.
n: number of random values to return

Value

dzigamma2 gives the density, pzigamma2 gives the distribution function, and rzigamma2 generates random deviates.

Details

This implementation allows for automatic differentiation with RTMB.

Uses the same density as zigamma with $\text{shape} = \mu^2/s^2$ and $\text{scale} = s^2/\mu$: $$f(x;\,\mu,s,p_0) = p_0\,\mathbf{1}[x=0] + (1-p_0)\,f_{\mathrm{Gamma}}(x;\,\mu^2/s^2,\,s^2/\mu)\,\mathbf{1}[x>0].$$

Because the continuous part places no mass at zero, zeroprob is exactly $P(X = 0)$, and zero-inflation coincides with a hurdle model here. The two constructions only differ for discrete distributions, where the base distribution can generate zeros of its own. GAMLSS therefore calls distributions of this type zero-adjusted rather than zero-inflated, as in its ZAGA family.

Examples

x <- rzigamma2(1, 2, 1, 0.5)
d <- dzigamma2(x, 2, 1, 0.5)
p <- pzigamma2(x, 2, 1, 0.5)