AD-compatible geometric distribution

Density and distribution function for the geometric distribution, written so that they can be taped by RTMB.

Usage

dgeom.ad(x, prob, log = FALSE)

pgeom.ad(q, prob, lower.tail = TRUE, log.p = FALSE)

Arguments

x, q: integer vector of counts
prob: probability of success in each trial, in (0,1]
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 $P[X > x]$.

Value

dgeom.ad gives the probability mass function and pgeom.ad gives the distribution function.

Details

stats already provides the geometric distribution, but its versions cannot be differentiated. These are AD-compatible replacements, reached automatically whenever an argument is an AD variable, so stats::dgeom and stats::pgeom are left untouched for ordinary use.

The parameterisation is the same as in stats: $X$ is the number of failures before the first success, so $$P(X = x;\,\pi) = \pi\,(1 - \pi)^{x}, \quad x = 0, 1, 2, \ldots$$ The density is obtained from the negative binomial with size = 1, for which RTMB provides an AD method; the distribution function $1 - (1-\pi)^{x+1}$ is elementary.

Examples

dgeom.ad(0:5, prob = 0.3)
#> [1] 0.300000 0.210000 0.147000 0.102900 0.072030 0.050421
pgeom.ad(0:5, prob = 0.3)
#> [1] 0.300000 0.510000 0.657000 0.759900 0.831930 0.882351