R: The Zero-Truncated Geometric Distribution

ZeroTruncatedGeometric {actuar} | R Documentation

## The Zero-Truncated Geometric Distribution

### Description

Density function, distribution function, quantile function and random generation for the Zero-Truncated Geometric distribution with parameter `prob`.

### Usage

    dztgeom(x, prob, log = FALSE)
    pztgeom(q, prob, lower.tail = TRUE, log.p = FALSE)
    qztgeom(p, prob, lower.tail = TRUE, log.p = FALSE)
    rztgeom(n, prob)

### Arguments

`x` | vector of (strictly positive integer) quantiles.
`q` | vector of quantiles.
`p` | vector of probabilities.
`n` | number of observations. If `length(n) > 1`, the length is taken to be the number required.
`prob` | parameter. `0 < prob <= 1`.
`log, log.p` | logical; if `TRUE`, probabilities p are returned as log(p).
`lower.tail` | logical; if `TRUE` (default), probabilities are P[X ≤ x], otherwise, P[X > x].

### Details

The zero-truncated geometric distribution with `prob` = p has probability mass function

p(x) = p (1-p)^(x-1)

for x = 1, 2, … and 0 < p < 1, and p(1) = 1 when p = 1. The cumulative distribution function is

P(x) = [F(x) - F(0)]/[1 - F(0)],

where F(x) is the distribution function of the standard geometric.

The mean is 1/p and the variance is (1-p)/p^2.

In the terminology of Klugman et al. (2012), the zero-truncated geometric is a member of the (a, b, 1) class of distributions with a = 1-p and b = 0.

If an element of `x` is not integer, the result of `dztgeom` is zero, with a warning.

The quantile is defined as the smallest value x such that P(x) ≥ p, where P is the distribution function.

### Value

`dztgeom` gives the (log) probability mass function, `pztgeom` gives the (log) distribution function, `qztgeom` gives the quantile function, and `rztgeom` generates random deviates.

Invalid `prob` will result in return value `NaN`, with a warning.

The length of the result is determined by `n` for `rztgeom`, and is the maximum of the lengths of the numerical arguments for the other functions.

### Note

Functions `{d,p,q}ztgeom` use `{d,p,q}geom` for all but the trivial input values and p(0).

`rztgeom` uses the simple inversion algorithm suggested by Peter Dalgaard on the r-help mailing list on 1 May 2005 (https://stat.ethz.ch/pipermail/r-help/2005-May/070680.html).

### Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca

### References

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

### See Also

`dgeom` for the geometric distribution.

`dztnbinom` for the zero-truncated negative binomial, of which the zero-truncated geometric is a special case.

### Examples

    p <- 1/(1 + 0.5)
    dztgeom(c(1, 2, 3), prob = p)
    dgeom(c(1, 2, 3), p)/pgeom(0, p, lower = FALSE) # same
    dgeom(c(1, 2, 3) - 1, p)                        # same

    pztgeom(1, prob = 1)        # point mass at 1

    qztgeom(pztgeom(1:10, 0.3), 0.3)

[Package actuar version 3.1-2 Index]