ZeroModifiedGeometric {actuar} | R Documentation |

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

and arbitrary probability at zero `p0`

.

dzmgeom(x, prob, p0, log = FALSE) pzmgeom(q, prob, p0, lower.tail = TRUE, log.p = FALSE) qzmgeom(p, prob, p0, lower.tail = TRUE, log.p = FALSE) rzmgeom(n, prob, p0)

`x` |
vector of (strictly positive integer) quantiles. |

`q` |
vector of quantiles. |

`p` |
vector of probabilities. |

`n` |
number of observations. If |

`prob` |
parameter. |

`p0` |
probability mass at zero. |

`log, log.p` |
logical; if |

`lower.tail` |
logical; if |

The zero-modified geometric distribution with `prob`

*= p*
and `p0`

*= p0* is a discrete mixture between a
degenerate distribution at zero and a (standard) geometric. The
probability mass function is *p(0) = p0* and

*
p(x) = (1-p0)/(1-p) f(x)*

for *x = 1, 2, …*, *0 < p < 1* and *0 ≤ p0 ≤ 1*, where *f(x)* is the probability mass
function of the geometric.
The cumulative distribution function is

*
P(x) = p0 + (1 - p0) [F(x) - F(0)]/[1 - F(0)].*

The mean is *(1-p0)m* and the variance is
*(1-p0)v + p0(1-p0)m^2*,
where *m* and *v* are the mean and variance of
the zero-truncated geometric.

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

The special case `p0 == 0`

is the zero-truncated geometric.

If an element of `x`

is not integer, the result of
`dzmgeom`

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.

`dzmgeom`

gives the (log) probability mass function,
`pzmgeom`

gives the (log) distribution function,
`qzmgeom`

gives the quantile function, and
`rzmgeom`

generates random deviates.

Invalid `prob`

or `p0`

will result in return value
`NaN`

, with a warning.

The length of the result is determined by `n`

for
`rzmgeom`

, and is the maximum of the lengths of the
numerical arguments for the other functions.

Functions `{d,p,q}zmgeom`

use `{d,p,q}geom`

for all but
the trivial input values and *p(0)*.

Vincent Goulet vincent.goulet@act.ulaval.ca

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

`dgeom`

for the geometric distribution.

`dztgeom`

for the zero-truncated geometric distribution.

`dzmnbinom`

for the zero-modified negative binomial, of
which the zero-modified geometric is a special case.

p <- 1/(1 + 0.5) dzmgeom(1:5, prob = p, p0 = 0.6) (1-0.6) * dgeom(1:5, p)/pgeom(0, p, lower = FALSE) # same ## simple relation between survival functions pzmgeom(0:5, p, p0 = 0.2, lower = FALSE) (1-0.2) * pgeom(0:5, p, lower = FALSE)/pgeom(0, p, lower = FALSE) # same qzmgeom(pzmgeom(0:10, 0.3, p0 = 0.6), 0.3, p0 = 0.6)

[Package *actuar* version 3.1-4 Index]